Two problems in Quantum Control andInformation

Thesis submitted in partial fulfillment

of the requirements for the degree of

MASTER OF SCIENCE

IN

COMPUTATIONAL NATURAL SCIENCES

by

Abhishek Deshpande200964003

Center for Computational Natural Sciences and Bioinformatics

International Institute of Information Technology

Hyderabad - 500 032, INDIA

August 2014

Copyright c© Abhishek Deshpande, 2014

All Rights Reserved

To my parents

Acknowledgments

It is my good fortune to have Prof. Harjinder Singh and Dr. Indranil Chakrabarty as my

advisors. Prof. Harjinder has been an unfailing source of inspiration for me. His insightful

comments have time and again opened up a whole new avenue of inquiry for me. Besides

supervising me in research, he has been a mentor to me. The discussions that I have had

with him have guided this thesis in a presentable form. Dr. Indranil has been like an elder

brother to me. He has always encouraged me to think independently. It is difficult for me

to imagine having better advisors.

I would like to thank all the faculty members at CCNSB for providing excellent research

environment in the center. In particular I would like to thank Dr. B. Prabhakar, Prof. A.

Mitra, Dr. Krishnan, Dr. Sau and our head of the department Dr. U. Deva Priyakumar

for all their guidance and support. Dr. Prabhakar has been at various times my teacher

and mentor. I am thankful for having received the benefit of his sagacity. Prof. A. Mitra

has been a constant source of support for me right from the time I joined college. Dr. U.

Deva Priyakumar has always motivated me to do research. His own research feats have

been cause for inspiration. I thank him for his constant support.

I would like to thank S.K Sazim and Prof. Pankaj Agarwal for useful discussions which

led to the work on different types of dissensions. I would also like to thank Prof. S.

v

Chaturvedi from the Dept.of Physics at the University of Hyderabad, and the faculty at the

Dept. of Mathematics at UoH for all the extensive courses that have built for me a strong

mathematical foundation.

Also, I would like to express my gratitude to my research group : N. Krishna Reddy,

Sourav Chatterjee and Maharshi Ray. The stimulating discussions that we had provided a

new perspective towards tackling problems in my research. I would also like to thank Prof.

Mahapatra from the University of Hyderabad whose discussions during the group meetings

gave a new dimension to my research. I also express my gratitude to Dr. J.C Tremblay

from FU Berlin and Prof. J. Manz from T.U Berlin whose brief visit at the University of

Hyderabad have opened new directions for research.

I would also like to thank S. Aravindan for the excellent management of the CCNSB

cluster and all the computational services which have helped me perform computer simu-

lations without any hassles. I would also like to thank Mr. Balsantosh and Mr. Girish for

providing administrative assistance.

Finally, I would like to thank my family for their love, support and encouragement.

Without their selfless support none of this would have been possible.

vi

Abstract

Control of quantum phenomenon using lasers has been a long standing problem since

their inception in 1960’s. There have been several approaches that have been developed

to control processes at a molecular level. Among them, optimal control theory has been

one of the most successful schemes developed to achieve the desired objectives. There were

other method like the well known Krotov method, the Zhu-Rabitz method and a recently

formulated two-point boundary-value quantum control paradigm has gone into developing

efficient schemes to deal with the iterative process of searching for optimized fields.

Our focus in this thesis has been on using OCT for electronic control. A recent exper-

iment by Iwakura, Yabushita and Kobayashi have demonstrated an alternative approach

i.e initiation of a pericyclic reaction in the electronic state by means of an ultrashort laser

pulse in the UV range. In accordance with this work, Manz et al [1] have used optimal

control theory to design a laser pulse to initiate a pericyclic reaction in the ground state.

In cognate with this approach, we use an excited state to mediate a reaction on a ground

state electronic potential surface and use optimal control theory in combination with time-

dependent Schrodinger equation to generate electric field to achieve the desired product.

The optimization of the field was carried out using the gradient descent technique. We have

managed to achieve an overlap of about 20%.

vii

Quantum control is intimately connected to quantum information. Researchers have

been trying to use optimal control for performing quantum information protocols e.g. the

use of optimal control theory to build quantum gates has been explored by [2]. We explore

some aspects of quantum information, in particular “state-merging”. In this protocol, mul-

tiple parties share a mixed quantum state. The objective here is to merge the state to one

of the parties. In the case of two parties A and B, the cost of state merging is given by

S(A|B). When we generalise the situation to three parties A,B and C, the cost of state-

merging is given by S(A|B)+S(C|A,B). The pertinent question we ask is whether the cost

of state merging can be reduced by performing measurement on the system. The answer is

negative in the case of two parties as shown by [3] i.e measurement can only increase the

cost of state merging. In fact they showed that this change in the cost of state-merging is

exactly equal to discord which is always positive. But the situation in case of three parties

is completely different. We make a deeper analysis of this situation in the thesis. We find

that the change in the cost of state merging is equal to D2 − D1 where D1 is dissension

and D2 is discord. Our results clarify that the change in cost of state merging is dependent

upon the competition between dissension and discord.

In addition we study the phenomenon of dissension in more detail. We define two tracks

for the definition of dissension. The first track is the usual way of defining dissension as the

difference between the quantum and classical versions of mutual information as mentioned

in [4].We have introduced a second track of defining dissension from the perspective of

considering all possible measurements. These approaches not only encompass measures of

correlation like discord but also give a unified view of the quantum correlation from the

view point of projective measurement done on the subsystem.

viii

Contents

Chapter Page

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Bond selective chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Single-parameter control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Pump Dump control . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Phase Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.3 STIRAP Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Multiple parameter control: Optimal Control Theory . . . . . . . . . . . . . 3

1.3.1 Closed loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Quantum Optimal Control Theory . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Derivation of control equations . . . . . . . . . . . . . . . . . . . . . 7

3 Electronic Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Pericyclic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Global optimization of product selectivity . . . . . . . . . . . . . . . 16

3.2.2 Time Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Quantum Information: An overview . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Gram Schmidt orthogonalisation . . . . . . . . . . . . . . . . . . . . 25

4.1.3 Operators and Spectral Decomposition . . . . . . . . . . . . . . . . . 25

4.1.4 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.5 Commutator and Anti-commutator . . . . . . . . . . . . . . . . . . . 26

4.1.6 Logical formalism of Quantum mechanics . . . . . . . . . . . . . . . 27

ix

4.1.7 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.8 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1.9 Schmidt decomposition and purification . . . . . . . . . . . . . . . . 294.1.10 Partial Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 Classical and Quantum Information . . . . . . . . . . . . . . . . . . . . . . 304.2.1 Shannon Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.2 Von-Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 Classical Relative entropy . . . . . . . . . . . . . . . . . . . . . . . . 314.2.4 Quantum relative entropy . . . . . . . . . . . . . . . . . . . . . . . . 314.2.5 Classical Conditional entropy . . . . . . . . . . . . . . . . . . . . . . 314.2.6 Quantum conditional entropy . . . . . . . . . . . . . . . . . . . . . . 324.2.7 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2.8 Quantum mutual information . . . . . . . . . . . . . . . . . . . . . . 324.2.9 Shannon’s noiseless coding theorem . . . . . . . . . . . . . . . . . . . 334.2.10 Schumacher’s quantum noiseless coding theorem . . . . . . . . . . . 33

4.3 Quantum Information Processing Protocols . . . . . . . . . . . . . . . . . . 344.3.1 Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.2 Superdense coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.3 Quantum Key Distribution . . . . . . . . . . . . . . . . . . . . . . . 374.3.4 Raw key exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.5 Key Sifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3.6 Key distillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Quantum Dissension and State merging . . . . . . . . . . . . . . . . . . . . . . . 405.0.7 Separability and Entanglement . . . . . . . . . . . . . . . . . . . . . 405.0.8 Quantum correlations: Going beyond entanglement . . . . . . . . . . 415.0.9 Quantum Discord . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.0.10 Quantum Dissension . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 Quantum Dissension: Track 1 and Track 2 . . . . . . . . . . . . . . . . . . . 465.1.1 Idea of correlation vectors . . . . . . . . . . . . . . . . . . . . . . . . 465.1.2 Track 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1.3 Track II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.1 Two qubit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.2 Three-qubit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.3 Four-qubit case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 State Merging: Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.1 One shot and multi-shot state merging . . . . . . . . . . . . . . . . . 565.3.2 Measurements and cost of state merging for three parties . . . . . . 575.3.3 One party measurement . . . . . . . . . . . . . . . . . . . . . . . . . 585.3.4 Two-party measurement . . . . . . . . . . . . . . . . . . . . . . . . . 59

x

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.1 Conferences and Posters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2 Journal not related to thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xi

List of Figures

Figure Page

1.1 Representation of quantum control schemes using single laser pulse parameteras a controller of the molecular dynamics to enhance a transition from agiven initial state to the desired target state. (a) The Tannor-Rice pump-dump approach. (b) The Brumer-Shapiro phase control approach. (c) TheSTIRAP control scheme. ‘A’ refers to the initial state; ‘B’ refers to the desiredtarget state; ‘i’ refers to the intermediate state; ’d’ refers to the decay states(unwanted channels) [T. Brixner and G. Gerber, Chem. Phys. Chem. 4, 418(2003)].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Closed-loop control [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Optimized laser pulses for 0 to 1 transition in 3-D HF molecule. (a2, b2, c2)Power spectra of the corresponding pulses.(a3, b3, c3) Population as a func-tion of time in each of the three states 0, 1 and 2.[6] . . . . . . . . . . . . . 11

3.1 Example of Pericyclic Reactions- Taken from http : //en.wikipedia.org/wiki/Pericyclicreaction 133.2 Cope rearrangement of semibullvalene, with pincer-motion-type direction-

ality of electronic fluxes indicated by curved arrows in Lewis structures(adapted from [7].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Flow chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Electric field as a function of time. . . . . . . . . . . . . . . . . . . . . . . . 203.5 Wave-packet as a function of reaction co-ordinate. . . . . . . . . . . . . . . 21

4.1 Bloch Sphere representation of a qubit. Here θ and φ are the polar andazimuthal angles respectively . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Diagrammatic representation of state merging as shown in [8] . . . . . . . . 57

xii

List of Tables

Table Page

5.1 Table Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

xiii

Chapter 1

Introduction

Control of molecular processes has been a dream that has only partly come true. Tradi-

tionally, chemical reactions were controlled by manipulating the thermodynamic parameters

like temperature, pressure, etc. But these controls were not completely efficient since the

experimenter did not have control over individual bonds that are being broken and formed

during the process. A chemical reaction, in addition to the desired products also produces a

number of unwanted products which one can minimize using these traditional methods with

limited success. In addition, when a system is provided energy, we have the phenomenon

of intramolecular energy transfer whereby the energy is not localised in a particular mode,

but rather distributed over several degrees of freedom.

Hence the need to control chemical reactions by directly altering the energy of the

molecules involved is warranted and this is achieved using lasers. Using lasers, we can

precisely control the energy distribution in various degrees of freedom. The electric field

of the laser field couples with the the dipole moment and provides a means for giving a

product with the desired interest. There have been various techniques developed for the

experimental realisation of laser control. The next few sections will briefly touch upon these

techniques.

1

1.1 Bond selective chemistry

Lasers have been viewed as molecular scale scissors which can selectively cut a bond

without disturbing the rest of the molecule. The pulses were engineered by tuning their

frequency to the characteristic frequency of the particular bond. This approach has been

advocated in some select molecules like : HOOH[9], C2HD[10] and HOD[11]. However

because of the phenomenon of intramolecular vibrational redistribution, this approach did

not work. This approach did not consider the complex structure of the rest of the molecule,

but rather concentrated on a particular bond.

1.2 Single-parameter control

Single parameter control schemes work by varying one single parameter e.g. phase,

frequency etc. in the control process. These schemes utilize various aspects of the common

mechanism of light controlled interferences. They are also known by the name of “coherent

control”.

1.2.1 Pump Dump control

Tannor, Rice and Kosloff [12][13] proposed this type of control for controlling molecular

processes using two laser pulses. A femtosecond pulse (pump pulse) was used to generate

a vibrational wave-packet on an excited electronic surface. After freely propagating the

packet on the excited electronic surface, another pulse(dump pulse) is used to bring back

the packet to the desired product channel on the ground electronic state.

1.2.2 Phase Control

This scheme was proposed by Brumer and Shapiro [14],[15] for controlling molecular

reactions where two monochromatic laser beams with commensurate phase differences were

2

used to create interference between two reaction pathways. By varying the phase difference

it is possible to control the ratio of the products formed from each reaction pathway. How-

ever, this method is not always very efficient due to difficulty in matching reactions rates,

extremely small absorption cross-sections and competing reaction pathways [16]. There

is a very significant problem of undesirable amplitude and phase locking between the two

incident laser fields when these experiments are conducted in optically dense media.

1.2.3 STIRAP Control

In this scheme, population transfer is implemented by adiabatic passage techniques pro-

posed by Bergmann and is called stimulated Raman adiabatic passage (STIRAP)[[17][18][19][20]].

In this approach, two time-delayed nanosecond laser pulses are applied to a three-level sys-

tem to induce adiabatic population transfer between the two lower levels via an intermediate

upper level. The pumped pulse couples the initial and intermediate states, while the Stokes

pulse couples the intermediate and final states. By tuning the time delay between the

pulses, the transient population in the intermediate state is kept negligible which in turn

avoids the losses by radiative decay. This approach works well for small molecules, but for

large molecules successful adiabatic passage is difficult to achieve because of the increase in

density of levels [[20],[21],[22],[23]].

1.3 Multiple parameter control: Optimal Control Theory

Tannor and Rice [12] proposed a variational approach to control product channel using

a laser pulse. This concept of variational approach was expanded upon by Rabitz and co-

workers [24] to design pulses within the framework of OCT. Here too, we seek to maximise

a field dependent “cost functional”. The cost functional gives a measure of the objective to

which the desired task has been achieved and is constrained by the penalty term to min-

3

Figure 1.1 Representation of quantum control schemes using single laser pulse parameteras a controller of the molecular dynamics to enhance a transition from a given initial stateto the desired target state. (a) The Tannor-Rice pump-dump approach. (b) The Brumer-Shapiro phase control approach. (c) The STIRAP control scheme. ‘A’ refers to the initialstate; ‘B’ refers to the desired target state; ‘i’ refers to the intermediate state; ’d’ refers tothe decay states (unwanted channels) [T. Brixner and G. Gerber, Chem. Phys. Chem. 4,418 (2003)]..

imise the laser fluence. We use time-dependent Schrodinger equation in combination with

optimal control theory to achieve the desired pulse.

1.3.1 Closed loop control

A direct implementation of OCT is not always feasible in the laboratory because the

Hamiltonian that is taken for the calculations is approximate. Also inexact potential sur-

faces and interactions with the environment may limit the realisation of OCT.

4

Rabitz et al [25] have suggested the use of “closed loop” techniques whereby data from

experimental output are included in the optimization process to the shape of applied pulse

until the desired control objective has been successfully reached (Fig. 1.2).

Figure 1.2 Closed-loop control [5].

In this approach, an unshaped laser pulse is applied to the molecular system of inter-

est, and the resulting signal is detected. Then, these results are transferred to a computer

that runs a learning algorithm to get an improved pulse. This method is sometimes called

“adaptive feedback”. This takes into account any system-environment coupling that may

take place during the reaction.

A number of experiments have been conducted where AFC has been used to control

bound state multi-photon excitation in atoms [[26],[27],[28]], to manipulate the shape of

an atomic radial wave function (Rydberg wave packet)[29], to excite electronic transition

5

in molecules (laser dye IR125 in methanol solution)[30]. In their 1998 paper, Assion et

al.[31] studied the photo dissociation of the organo-metallic complex C5H5Fe(CO)2Cl. This

complex, due to the presence of particular kinds of iron-ligand bonds exhibits different

fragmentation channels upon excitation with the shaped pulse. Since then several groups

have demonstrated successful experimental applications of AFC [[32][33]].

6

Chapter 2

Optimal Control Theory

2.1 Quantum Optimal Control Theory

Modelling the process is one of the most important components of a quantum control

problem. Typically we wish to find the simplest description that reasonably models the

response of the physical system to all possible inputs. In this section, we give a flavour of

optimal control theory applied to quantum systems.

We consider a quantum system evolving according to the following equation:

ι~ ddt |ψ(t)〉 = H|ψ(t)〉.

H = H0 − µε(t)

H0 = T + V

2.1.1 Derivation of control equations

Let us consider the following quantum mechanical control problem: The goal is to find

a laser pulse ε(t) which drives a quantum system from it’s initial state ψ(0) to a state ψ(T )

in such a way that the expectation value of an operator O is maximized at the end of the

7

laser interaction: max J1 with J1[ψ] = |〈ψ(T )|O|ψ(T )〉|2.

The only constraint is that the operator O be Hermitian. In addition to the maximisation

of J1[ψ], we require that the fluence of the laser field is as small as possible which is cast in

the following mathematical form :

J2[ε] = −α∫ T

0 |ε(t)|2dt.

In addition to these terms, we have the constraint that the Schrodinger equation be satisfied

at all times during the propagation. Putting them together using Lagrange’s multipliers,

we have J [ε(t)] = 〈ψ(T )|O|ψ(T )〉 − α∫ T

0 |ε(t)|2dt − 2Re[

∫ T0 〈χ(T )| ddt + ι

~H|ψ(t)〉dt, where

ψ(T ) is the wave-packet at time t, O is the objective to be maximised, T is the final time,

ε(t) is the laser field at time t and χ(t) is a Lagrange multiplier.

To find the control field which maximises the “grand-functional”, we take first order deriva-

tives with respect to χ(t), ψ(t), ε(t) to yield the following set of equations:

ι~ ddt |ψ(t)〉 = H|ψ(t)〉. Here |ψ(0)〉 = ψinitial.

ι~ ddt |χ(t)〉 = H|χ(t)〉. Here |χ(T )〉 = Oψ(T ).

ε(t) = − 1~α〈χ(t)|µ|ψ(t)〉.

Attempting to solve these equations simultaneously lead to solutions which do not con-

verge. Hence methods such as “conjugate gradient” have been used by our group. In this

method, the grand functional is differentiated with respect to the electric field ε(t). The

wavefunction ψ(t) and the Langrange multiplier χ(t) are propagated in discrete time using

the split operator technique. A search along the Polak-Riebiere-Polyak direction is per-

formed [34].

Our group has performed the simulations using the Fourier Grid Hamiltonian method to

solve the time-independent Schrodinger equation in order to calculate bound state initial

8

wave functions and eigenvalues. Fourier transforms are used to diagonalise the Hamiltonian

matrix. The continuous range of co-ordinate range of co-ordinate values x is replaced by a

grid of discrete values xi. This gives us the bound states which are then propagated using

Split-operator technique. The potential energy part of the Hamiltonian is diagonal in the

coordinate representation, but the kinetic energy is diagonal in momentum representation.

Hence we need to convert the kinetic energy from the coordinate to the momentum rep-

resentation. The radial part of kinetic energy is calculated using Fast-fourier transform

[35], [13] and the angular part using Discrete Variable representation [36]. The details can

be found here in Chapter 2 of [6].

Using these techniques, our group has tried to generate population transfer between

the desired states in HF molecule. Please refer to the figure 2.1 for one-dimensional and

three-dimensional treatment of HF molecule. Next, our group has addressed the problem

of designing a laser pulse to accomplish controlled vibrational excitation in Fe−CO part of

the CO heme complex. Haemoglobin consists of a central heme group and a globin residue.

The Fe atom in the heme is bonded to the four nitrogen atoms of the porphyrin and one

nitrogen atom of the globin, leaving the sixth coordination position of the Fe atom free for

O2 to attach. This is how haemoglobin transports oxygen inside living organisms. However,

in case of CO poisoning, CO occupies the 6th coordination position in haemoglobin, thus

disabling oxygen carriage by the affected haemoglobin molecule. So the problem of breaking

the Fe− CO bond is of prime biological importance. Our group has designed laser pulses

for dissociating the Fe − CO bond in carboxy-haemoglobin and has been able to achieve

more than 90% population transfer at the end of the pulse duration.

9

Genetic algorithms have been developed by Sharma et al [6] to design pulses to achieve

a desired objective. The advantage of using genetic algorithms is that we have control over

the parameters as compared to the more conventional way of directly varying the electric

field within each small time interval. This way of pulse shaping allows us to select only a

small and limited number of parameters that are readily accessible to an experimentalist.

Such theoretically designed pulses may be more readily implemented in an experimental

situation. Genetic algorithm based approaches have been also used by other groups for the

design of quantum gates [37], [38], coherent control of rovibrational motion [39], control of

molecular orientation [40] etc. Our group has designed control pulses for several systems,

many of which had forbidden transitions. In each case, at most four distinct frequencies

were used and for each of the studied transitions, an overall excitation probability of greater

than 0.99 was achieved. Details of laser field parametrization and optimization using genetic

algorithms can be found in [6].

10

Figure 2.1 Optimized laser pulses for 0 to 1 transition in 3-D HF molecule. (a2, b2, c2)Power spectra of the corresponding pulses.(a3, b3, c3) Population as a function of time ineach of the three states 0, 1 and 2.[6]

11

Chapter 3

Electronic Control

3.1 Pericyclic Reactions

In organic chemistry, a pericyclic reaction [41] is a type of organic reaction wherein

the transition state of the molecule has a cyclic geometry, and the reaction progresses in

a concerted fashion. Pericyclic reactions are usually rearrangement reactions. They have

have the following properties :

1. They remain unaffected by solvent changes, the presence of radical initiators.

2. They proceed by a simultaneous series of bond breaking and bond making events in

a single kinetic step, often with high stereospecificity.

The four principle classes of reactions are as follows:

1. Cycloaddition reactions: It involves a concerted combination of two π-electron systems

to form a ring of atoms having two new σ bonds and two lesser π bonds.

2. Electrocyclic reactions: It is the concerted cyclization of a conjugated π-electron sys-

tem by converting one π-bond to a ring forming σ-bond.

12

3. Sigmatropic Rearrangements: Molecular rearrangements in which a σ-bonded atom,

grouped by one or more π-electron systems, shifts to a new location reorganization of

the π-bonds are called sigmatropic reactions.

4. Group transfer reaction: It is a pericyclic process where one or more groups of atoms

is transferred from a molecule to another.

5. Cheletropic reaction: It is a subclass of cycloaddition reactions, where one atom on

one of the reagents gets two new bonds, the only difference being that on one of the

reagents, both new bonds are being made to the same atom.

6. Dyotropic reaction: It is a type of pericyclic valence isomerization which involves

intramolecular migration of two sigma bonds simultaneously. It was first described

by Manfred T. Reetz in 1971.

Figure 3.1 Example of Pericyclic Reactions- Taken from http ://en.wikipedia.org/wiki/Pericyclicreaction

Laser control of pericyclic reactions in their electronic ground states is an arduous task

in chemical dynamics. Schwebel et al. in 1984 demonstrated initiation by means of high

overtone excitations of various mode selective vibrations, in the infrared (IR) frequency

domain. This approach is dependent on strong changes of the dipole moment. Recently,

13

Iwakura, Yabushita and Kobayashi could demonstrate an alternative approach; initiation of

a pericyclic reaction in the electronic ground state, by means of an ultrashort laser pulse in

the ultraviolet (UV ) frequency domain, within less than ten femtoseconds (fs). Accordingly,

we shall apply optimal control theory (OCT) [12] in order to design an ultrashort laser

pulse which initiates a pericyclic reaction : 3, 3-sigmatropic shift, or Cope rearrangement

of semibullvalene (C8H8, SBV ) in the electronic ground state 3.2. The reactant (R)

has a chain of six carbon atoms C6 = C7 − C8 − C2 − C3 = C4 (clockwise notation)

which is transformed into another chain C2 = C3 − C4 − C6 − C7 = C8 (same clockwise

notation, keeping the labels of the carbon atoms) for the products(P ); here R and P may be

distinguished by isotopic labelling. Apparently, the carbon-carbon bond C8−C2 (clockwise

notation) is broken during the pericyclic reaction, while the C4− C6 bond is formed. The

double bonds C3 = C4 and C6 = C7 are shifted to C2 = C3 and C7 = C8 respectively.

It would be rather interesting to explore the predictions of Reference [7] experimentally,

using e.g., modern techniques of femto- to attosecond spectroscopy[ [42]-[43]], see also Ref-

erences [44, 45, 46, 47] and the review [48]. Hence a deeper exploration of the efficiency

of the pump-dump mechanism for initiation of the specific pericyclic reaction is warranted,

and that is the agenda we pursue in the thesis.

3.2 Model

The initial (t = 0) state of the system representing R is the vibrational ground state (ν =

0) which is localized in the potential well of the electronic ground state (g) representing R,

ψg(t = 0) = ψg,ν=0, without any component in the excited electronic state (e),ψe(t = 0) = 0.

The wave functions satisfy the following time-dependent Schrodinger equation:

14

Figure 3.2 Cope rearrangement of semibullvalene, with pincer-motion-type directionalityof electronic fluxes indicated by curved arrows in Lewis structures (adapted from [7].)

ı~∂

∂t

ψe(t)

ψg(t)

=

[ T + Ve 0

0 T + Vg

− ε(t) dee deg

dge dgg

] ψe(t)

ψg(t)

This form of time dependent Schrodinger equation is applicable to the scenario of a

non-rotating molecule interacting with a linearly polarized laser pulse. The laser pulse

with electric field component E(t) should then be polarized parallel to the transition dipole

between the ground (g) and excited (e)electronic state. The relevant component is modelled

as deg = dge = 0.3934ea0 = 1 Debye in Condon approximation. Furthermore, we neglect

the dipole functions dgg = dee = 0. We suggest a duration τ of the laser pulse; we set τ

= 10 fs. Molecular rotations are negligible on this time scale. Further, we also want that

15

at the end (t = τ) of the laser pulse, the laser driven wavepacket ψg(ζ, t = τ) should agree

with the reactant wave-function, soon after the start of the Cope rearrangement this wave

function will be called the target wave function’ ψg,target(ζ). For convenience, we employ

ψg,target(ζ) = ψg,ν=0(ζ −4ζ) exp(ikζ).

3.2.1 Global optimization of product selectivity

Consider the target functional J = 〈ψ(tf )|P |ψ(tf )〉 where P is a projection operator

which selects the desired exit channel. We wish to optimize the cost functional. A con-

straint involving the total energy in the field is imposed on the optimization procedure.

This constraint takes the form

∫ tft0E(t)E∗(t)dt =

∫ tft0

[Re(E(t)2) + Im(E(t)2)] = ξ

where ξ is the energy of the pulse. The maximization of J , subject to the constraints

arising from the equation of motion of ψ and the total energy in the light field, can be

converted into an unconstrained optimization by using Lagrange multipliers χ and λ,

J = 〈ψ(tf )|P |ψ(tf )〉+ i∫ tft0

(〈χ|i ∂∂t −H|ψ〉) +λ(∫ tft0

[Re(E(t)2) + Im(E(t)2)]− ξ). The vari-

ation of J taken with respect to E(t). We find that :

1) ι~∂χ(t)∂t = Hχ(t) subject to the final condition |χ(tf )〉 = P |ψ(tf )〉

2) The electromagnetic field is related to the overlap function as :

E(t) = ιλ [〈χu(t)|µug|ψg(t)〉 − 〈ψu(t)|µug|χg(t)〉] = λ−1O(t)

16

3.2.2 Time Propagation

We have adopted an approach that is cognate with Schwender et al [49] to propagate

the wave function. We solve the Schrodinger equation with the split-operator technique.

The Schrodinger equation reads:

ı~∂

∂t

ψ0(R, t)

ψ1(R, t)

=

[ T 0

0 T

+

V0(R) W (R, t)

W (R, t) V1(t)

] ψ0(R, t)

ψ1(R, t)

where T is the kinetic energy operator and W (R, t) = E0E(t)µ(R)cos(ωt) is the laser

molecular interaction energy. The equations are solved subject to the initial conditions

ψ0(R, t = 0) = φv(R) and ψ1(R, t = 0) = 0. The initial trial wave-function is assumed to

be Gaussian. In order to solve this equation, we write the equation as follows:

ı~∂

∂t

T + V0 W

W T + V1

= exp(ı~

T 0

0 T

4t2

) exp(ı~

V0 W

W V1

4t) exp(ı~

T 0

0 T

4t2

)

Note that the first and the third terms are diagonal in this representation. So we can

easily evaluate them using the Fourier transform. Evaluating the potential energy matrix

requires a little bit more care. But we can diagonalize the matrix and then evaluate the

exponential term. This is what we do in our simulations which is depicted in the subsequent

section.

3.2.3 Numerical illustration

The iteration procedure we have used to determine the optimal pulse shape for enhanc-

ing production of a particular product of a reaction is as follows:

(a) Assume a form for the initial pulse shape E(t).

17

(b) Integrate the Schrodinger equation ι~(∂ψ(t)∂t ) = H(t)ψ(t)

forward to time tf , starting from the initial condition.

(c) Apply the projection operator that selects the exit channel to ψ(tf ) and obtain the value

of χ(tf ), regarded as an initial value for the backward propagation.

(d) Propagate χ(t) backwards in time using the Schrodinger equation:

ι~(∂χ(t)∂t ) = H(t)χ(t).

(e) During this propagation calculate the overlap function

O(t) = ι[〈χe(t)|ψg(t)〉]− 〈ψe(t)|µeg|χg(t)〉

(f ) When the backward propagation is complete renormalize the result to obtain the new

electromagnetic field pulse waveform

E(t) = O(t)( 1E

∫ tft0|O(t)|2dt)−1/2

(g ) Restart at step (b) with the result of step (f) for the waveform.

(h) Terminate the procedure when convergence has been achieved.

Refer to figure 3.3 for an illustration of the algorithm:

3.3 Results and Conclusions

We now depict the propagated wave-packet as a function of time. Note that the prop-

agated wave-packet has almost the same waveform as the target wave-packet 3.5. Also

note that the overlap is about 20% which is quite good considering the complex nature of

the processes involved. We are currently investigating the reasons for the relatively lower

overlap, but it nevertheless verifies the experimental observation of Reference [50]. The

approach we advocated is similar to that of Tannor-Rice pump dump scheme, but we have

focussed more on the initiation of a unimolecular reaction, completely in the electronic

18

Figure 3.3 Flow chart.

ground state, and with perfect product yield. Specifically for this purpose, we have used

OCT in the strong field limit. This is cognate with the paper of Manz el al [1], but we

have not used the MCTDH program as documented in [51, 52, 53]. The figure 3.4 depicts

the actual electric that needs to be supplied to obtain a result similar to the Tannor-Rice

pump dump scheme.

19

Figure 3.4 Electric field as a function of time.

Further, we wish to extend this work by replacing empirical potentials by quantum

potentials established by ab-initio calculations and 1-D model by a multi-dimensional one.

In addition, we would also like to suppress any wave-function that still exists in the excited

electronic state after the application of optimal pulse.

20

Figure 3.5 Wave-packet as a function of reaction co-ordinate.

21

Chapter 4

Quantum Information: An overview

4.1 Preliminaries

In this section, we will briefly review the basic notions of linear algebra before going into

mainstream quantum information theory.

4.1.1 Hilbert Spaces

A set V is a vector space over field F , if given two operations vector addition and scalar

multiplication, it satisfies the following axioms.

• Closure:

If X,Y ∈ V , then X + Y ∈ V .

αX ∈ V whenever X ∈ V for every α ∈ F .

• Associativity : u+ (v + w) = (u+ v) + w for all u, v, w ∈ V .

• Compatibility : a(bv) = (ab)v for all v ∈ V and a, b ∈ F .

• Commutativity : u+ v = v + u for all u, v ∈ V .

• Zero element : The set V contains an additive identity element, denoted by 0, such

that for any vector v ∈ V , 0 + v = v and v + 0 = v.

22

• Inverse : For each v ∈ V , an element v exists in V such that −v + v = 0. Here −v is

the additive inverse in F .

• Distributivity : a(v + w) = av + aw and (a + b)v = av + bv for all v, w ∈ V and

a, b ∈ F .

• Identity : 1.v = v for all v ∈ V . Here 1 is the multiplicative identity element in F .

We call a set B = {v1, v2, ..., vn} of vectors in V a basis of the vector space V if it is

a linearly independent set and it generates the whole space V i.e no vector in B can be

written as a linear combination of finitely many other vectors in B. A vector space with

basis {v1, v2, ..., vn} is said to have dimension = n i.e, the dimension of the vector space is

the number of elements in the basis of the vector space.

In order to add notions of length and distance to a vector space, we introduce a third

operation, called the inner product 〈u, v〉 : V × V → F . The inner product is defined as

:〈x, y〉 := x∗1y1 + ... + x∗nyn. For every inner product, we assign a length to a vector v via

the norm, ||v|| =√〈v, v〉. A vector space V on which a norm is defined is called a normed

vector space. For two vectors x, y of a normed space V , we can define a notion of distance

between two vectors x and y by using the norm:

d(x, y) = ||x− y||. (4.1)

If there exists such a distance function for a vector space V , we call V a metric space. A

metric space is complete if and only if every sequence x1, x2, .. of vectors in V for which

d(xn, xm) → 0, as both n and m independently tend toward infinity, converges in V . A

Hilbert space H is a real or complex inner product space which is also a complete metric

space with respect to the distance function induced by the inner product. Vectors associated

23

with the Hilbert space will be denoted in the bra-ket notation |ψ〉 etc, - these vectors are

called kets. We will define ket space as the space defined by all possible directions of the

ket vectors. For every ket, there is a corresponding bra associated with it. We will denote

it by 〈ψ|. The space generated by these bra vectors is called the bra space and is the dual

of the ket space.

A basis for the space C2 is given by {(0, 1), (1, 0)}, which in braket notation reads

{|0〉, |1〉}. This is known as the computational basis for the space C2. In general for the

space Cn, the computational basis will be written as {|1〉, ....|n〉}. Any vector |ψ〉 ∈ Cn can

then be expressed as:

|ψ〉 =

n∑i=1

αi|i〉.

We will now generalise our discussion to composite Hilbert spaces. Given two Hilbert

spaces, we can construct a larger Hilbert space by taking the tensor product A⊗B. Given

two orthogonal bases |v〉i and |w〉j , we can define the basis of A⊗B by the basis elements

{|v〉i ⊗ |w〉j}. The elements of the larger vector space obey the following properties.

• For any z ∈ C and arbitrary vectors |v〉A of A and |w〉B of B,

z(|v〉A ⊗ |w〉B) = (z|v〉A)⊗ |w〉B

• For arbitrary vectors |v1〉A and |v2〉A in A and |w〉B in B,

(|v1〉A + |v2〉A)⊗ |w〉B = |v1〉A ⊗ |w〉B + |v2〉A ⊗ |w〉B

• For arbitrary vectors |v〉 in A and |w1〉 and |w2〉 in B,

|v〉A ⊗ (|w1〉B + |w2〉B) = |v〉A ⊗ |w1〉B + |v〉A ⊗ |w2〉B

24

4.1.2 Gram Schmidt orthogonalisation

Given an arbitrary basis we can generate an orthonormal basis from it by using the

‘Gram-Schmidt Process’. The idea is to go through the vectors one by one and subtract off

that part of each vector that is not orthogonal to the previous ones.

Given a basisB = {|u1〉, |u2〉, ..., |un〉} of Rn, we generate a new basisB′ = {|v1〉, |v2〉, ..., |vn〉}

of Rn which is orthonormal.

• Set |w1〉 = |u1〉.

• For each i from 2 to n, set |wi〉 = |ui〉 −i−1∑j=1

〈ui|wj〉||wj ||2

|wj〉.

• For each i from 1 to n, set |vi〉 = |wi〉||wi|| .

4.1.3 Operators and Spectral Decomposition

We first define what is meant by a linear operator.

Definition 4.1.1. Linear operator: Consider a map F : V → W such that for every

{|vi〉; i = 1, 2, ..., n} and scalars {ai; i = 1, 2, 3..., n}, we have :

F (

n∑i=1

ai|vi〉) =

n∑i=1

aiF (|vi〉). Then F is said to be linear operator.

The trace of an operator is given by the sum of it’s diagonal elements when represented

in the matrix form.

Suppose A is any linear operator on a Hilbert space, V . Then there exists a unique

operator A† on V such that for all vectors |v〉, |w〉 ∈ V , (|v〉, A|w〉) = (A†|v〉, |w〉). This is

known as the adjoint of the operator A.

• Normal operator : An operator A is said to Normal iff AA† = A†A

• Unitary operator : An operator A is said to Unitary iff AA† = A†A = I

• Hermitian operator : An operator A is said to Hermitian iff A = A†

25

An important class of Hermitian operators is the projectors. Assume a m dimensional

vector subspace W of a n dimensional vector space V . Using Gram-Schmidt, it is possible

to construct an orthonormal basis |1〉, ......|k〉 for W such that :

P =

k∑i=1

|i〉〈i|. This is known as the projection operator.

Theorem 4.1.1. Spectral Decomposition Theorem : Let H be an normal operator acting

on a Hilbert space A. Then, there exists an orthonormal basis {|ei〉A} of A such that H is

diagonal with respect to this basis:

H =

dA∑i=1

λi|ei〉A〈ei|A, where all the eigenvalues λi of H are real numbers.

4.1.4 Quantum Gates

In this subsection, we give definitions of some preliminary quantum gates that are used

frequently in quantum computing:

Xgate : −

0 1

1 0

(4.2)

Y gate : −

0 -i

i 0

(4.3)

Zgate : −

1 0

0 -1

(4.4)

4.1.5 Commutator and Anti-commutator

• The commutator between two operators is defined as:[A,B] = AB −BA.

• The anti-commutator between two operators is defined as: {A,B} = AB +BA

26

The most pertinent relationship of commutators is it’s relation to simultaneously diago-

nalisable matrices. We formally state it as a theorem below:

Theorem 4.1.2. Simultaneous diagonalisation theorem: Suppose A and B are two Hermi-

tian operators. Then [A,B] = 0 iff A and B are simultaneously diagonalisable.

4.1.6 Logical formalism of Quantum mechanics

The foundations of quantum mechanics have been established as a result of culmina-

tion of years of effort. Since then no known experiment has contradicted the predictions of

quantum mechanics. We now precisely state the four postulates of quantum mechanics.

Postulate1: There is an associated Hilbert space H of some dimension d for every

physical system, known as the state space of that system; and the system is completely

described by its state vector, a unit vector in the state space.

Postulate2: An Unitary transformation U describes the evolution of a closed quantum

system A. That is, if |ψA〉t1 and |ψA〉t2 are the wave functions of the system A at times

t1 and t2, they are related by a unitary operator UA which depends only on t1 and t2:

|ψA〉t2 = UA(t1, t2)|ψA〉t1.

Postulate3: An observable is a measurable quantity associated with an Hermitian oper-

ator O = O†. In measuring an observable O , the possible measurement outcomes are given

by the eigenvalues λj . These occur with probabilities equal to the square of the amplitude

for that outcome, and the system remains in an eigenstate of O.

Postulate4: If a composite system is composed of subsystems A and B which have

associated Hilbert spaces HA and HB, then the Hilbert space of the joint system is the

27

tensor product space HA ⊗HB.

4.1.7 Density Operator

Density operator is a linear operator that acts on the Hilbert space corresponding to the

state of the physical system. It obeys the following rules:

• ρ is positive

• Tr(ρ) = 1 where Tr is the trace of the density operator |ρ〉.

We also have a classification of states of the physical system

• Pure state: If the density operator satisfies ρ2 = ρ, then the states are called pure

states. There is a correspondence between the density operator and the vectors of the

Hilbert space H. ρ2 = ρ iff ρ = |ψ〉〈ψ|, where ψ ∈ H

• Mixed state: The complement of pure states are called mixed states. Hence all states

that are not pure are called mixed states. For every mixed state ρ2 < ρ. Any mixed

state can be written as a convex combination of pure states. ρ =∑i

|ψi〉〈ψi|.

A criterion to detect mixedness of a density operator: If ρ be any density operator then

Tr(ρ2) ≤ 1. This density operator will represent a pure state if and only if Tr(ρ2) = 1.

4.1.8 Bloch Sphere

There is a geometric way of looking at qubit i.e Bloch sphere. Consider the following

wave-function:

|ψ〉 = α|0〉+β|1〉, where α and β are the probability density amplitudes. Since |α|2+|β|2 = 1,

we may write the wave-function in this form :

28

|ψ〉 = expıγ(cos( θ2)|0〉+ expıφ sin( θ2)|1〉), where θ, γ, φ are real numbers and θ and φ are the

polar and azimuthal angles respectively. Please refer to the figure 4.1 for the details.

Figure 4.1 Bloch Sphere representation of a qubit. Here θ and φ are the polar and az-imuthal angles respectively

Many operations in quantum mechanics can be modelled as some transformations on the

Bloch sphere. However this intuition is limited since there is no elementary generalisation

of the Bloch sphere for multiple qubits.

4.1.9 Schmidt decomposition and purification

• Schmidt decomposition: For any pure state |ψAB〉 of a composite system AB, we

can always find orthonormal states |iA〉 for the system A, and orthonormal states

|iB〉 for the system B such that ψAB =∑i

λi|iA〉|iB〉, where λi are non-negative

real numbers satisfying∑i

λ2i = 1 known as Schmidt coefficients. We call the bases

|iA〉, |iB〉 Schmidt bases and the number of non-zero values λi is called the Schmidt

number for the state |ψ〉.

29

• Purification: Suppose we are given a state ρA of a quantum system A. It is possible

to introduce another system R and define a pure state |AR〉 for the joint system AR

such that ρA = trR(|AR〉〈AR|). This process is called purification.

4.1.10 Partial Trace

Consider a bipartite system in the state HA ⊗HB. Suppose the density operator repre-

senting the joint state is ρAB and {|ψiA〉, i = 1, 2...,m}, {|φiA〉, i = 1, 2..., n} be orthonormal

bases of the systems A and B respectively. Then the partial traces are given by:

ρA = TrB(ρAB) =∑j

〈φjB|ρAB|φjB〉 ρB = TrA(ρAB) =

∑i

〈φiB|ρAB|φiB〉

4.2 Classical and Quantum Information

In this section, we will explore some of the basic terminologies in quantum information

before stating our main contributions in the next chapter.

4.2.1 Shannon Entropy

Consider a random variable X over a set of events x such that event x occurs with

probability px, then the Shannon entropy of event x is −pxlog2(px). This quantity is always

non-negative, with H(X) = 0 if and only if the random variable X yields a definite outcome

x with px = 1. It takes a maximal value of log d for a random variable X generating d

possible outcomes with equal probabilities.

4.2.2 Von-Neumann entropy

In the quantum mechanical terms for a density operator ρ, the Von-neumann entropy is

defined as -Tr(ρlog(ρ)). The original motivation for the von Neumann entropy did not come

30

from an information-theoretical context, unlike the Shannon entropy. It came as an attempt

for extending a thermodynamical concept, the Gibbs entropy, to the quantum setting.

4.2.3 Classical Relative entropy

The relative entropy is a very handy measure that measures the distance between two

probability distributions i.e it gives an indication of how close two probability distributions

are. Suppose p(x) and q(x) are two probability distributions over the same index x. We

define relative entropy of p(x) to q(x) by : H(p(x)||q(x)) =∑x

p(x)logp(x)

q(x)= −H(X) +∑

x

p(x)log(q(x)).

It is not immediate how this measure is useful. The next theorem gives some motivation

as to why it is regarded as a distance measure.

Theorem 4.2.1. The relative entropy is non-negative. i.e H(p(x)||q(x)) ≥ 0, with equality

iff p(x) = q(x).

4.2.4 Quantum relative entropy

Cognate with the notion of classical relative entropy, we have a corresponding defini-

tion for quantum relative entropy. Suppose ρ and σ are density operators. S(ρ|||σ) =

Tr(ρlogρ) − Tr(ρlogσ). If λx are the eigen values of ρ, the Von-Neumann entropy can

be expressed as : -∑x

λxlogλx. As in the classical case quantum relative entropy is also

non-negative i.e, S(ρ||σ) ≥ 0 with equality iff ρ = σ. This is known as Klein’s inequality.

4.2.5 Classical Conditional entropy

Consider the joint entropy of the pair of random variables X and Y i.e H(X,Y ). The

joint entropy of X and Y is defined as follows:

31

H(X,Y ) = −∑x,y

p(x, y)log(p(x, y)). This measures the total uncertainty about the pair

(X,Y ). Suppose we know the value of Y , so we have acquired H(Y ) bits of information

about the pair (X,Y ). The remainder uncertainty is due to X given that we have already

know Y . The conditional entropy of X conditional on knowing on knowing Y is therefore

defined by: H(X|Y ) = H(X,Y )−H(Y )

4.2.6 Quantum conditional entropy

The classical entropy when generalised to the quantum domain gives us the quantum

conditional entropy. S(A|B) = S(A,B) − S(B). An important difference between the

classical and quantum case is that quantum conditional entropy can be negative e.g for an

entangled state.

4.2.7 Mutual Information

This term quantifies the amount of common information between two messages. From

Bayes probability distribution,

I(X : Y ) = H(X)−H(X|Y ) = H(X) +H(Y )−H(X,Y ) = H(Y )−H(Y |X)

Here H(X,Y ) represents the joint entropy of a pair of random variables X and Y . When

I(X : Y ) = 0, there is no classical correlation between X and Y .

4.2.8 Quantum mutual information

Quantum mutual information, or von Neumann mutual information, after John von

Neumann, is a measure of correlation between subsystems of quantum state. It is the

quantum mechanical analog of Shannon mutual information. I(X : Y ) = S(X) + S(Y ) −

S(X,Y ).

32

4.2.9 Shannon’s noiseless coding theorem

Shannon’s noiseless coding theorem gives us an estimate as to which we can compress

information that is produced by a classical source. Consider a sequence of random variables

X1, ...Xn, whose values represent the output of the source. Further the assumption is

that these variables are identical and independently distributed. The essential idea behind

Shannon’s theorem is that we divide possible sequences x1, x2, ....., xn into two types :

• Typical sequences: Sequences which occur with high probability.

• Atypical sequences: Sequences which occur very rarely

Now suppose that the output from the source is x1, ....., xn. We first check whether this

is a typical sequence. If not,we declare invalid. If the output is a typical sequence, we

record that fact. Since there are atmost 2nH(X) typical sequences, it only requires nH(X)

bits to uniquely identify a sequence. We choose a particular scheme for identification, and

compress the output from the source to the corresponding string of nH(X) bits describing

which typical sequence occured, which can later be decompressed. As n becomes large, the

probability of this protocol working reaches 1 i.e it is almost certain.

We now formally state Shannon’s noiseless coding theorem.

Theorem 4.2.2. Suppose X is an identically distributed independent distribution with en-

tropy rate H(X). Suppose R > H(X). Then there exists a reliable compression scheme of

rate R for the source. Conversely, if R < H(X), then any compression scheme will not be

reliable.

4.2.10 Schumacher’s quantum noiseless coding theorem

The quantum version of this is called Schumacher’s quantum noiseless coding theorem.

For description in the quantum setting, we assume that the identically distributed quantum

33

source will be described by a Hilbert space H and a density matrix ρ on that Hilbert space.

There exists a compression scheme for this source that consists of Cn and Dn. Cn is the

compression that takes states in H⊗n to a 2nR -dimensional state space which is the com-

pressed space. The operation D is the opposite of this operation. The combined operation is

CnDn. The key element for making the quantum noiseless channel coding theorem possible

is the quantum version of typical sequences. Let’s assume that the density operator has the

orthonormal decomposition

ρ =∑x

p(x)|x〉〈x| (4.5)

Note that here |x〉 are orthonormal basis and p(x) are the eigen values of ρ. As in the

classical case, we define a quantum analogue of typical sequences. We define a sequence

x1, x2, ..., xn to be ε-typical iff

2−n(H(x)+ε) ≤ p(x− 1, x2, ...xn) ≤ 2−n(H(x)−ε)

Also, note that the Shannon entropy here is replaced by the Von-neumann entropy. We

now formally state the Schumacher’s quantum noiseless coding theorem.

Theorem 4.2.3. Let {H, ρ} be an independent and identically distributed quantum source.

If R > S(ρ), then there exists a reliable compression scheme of rate R for the source {H, ρ}.

If R < S(ρ), then any compression scheme of rate R is not reliable.

4.3 Quantum Information Processing Protocols

This section will briefly review the quantum information protocols that have been de-

veloped over the past few decades. The protocols that we are primarily interested in are as

follows:

• Teleportation

34

• Superdense coding

• Quantum key distribution

4.3.1 Teleportation

The essential feature of entanglement is that one has to send information encoded in

terms of qubit from one place to another with the use of quantum channel. This has to

be done without actually physically sending the qubit through the quantum channel, but

rather using local operations and classical communication. The protocol was given by Ben-

nett. The fidelity of the protocol relies on the nature of the quantum channel. The key fact

about teleportation is that state is unknown to both the sender and the receiver.

Consider the following scenario. Alice and Bob share an EPR pair i.e they share

|0〉|0〉+|1〉|1〉√2

. Alice interacts the qubit that she has to send with her half of the EPR pair and

measures the two qubits in her possession, obtaining a possible of four classical outcomes :

{00, 01, 10, 11}. She sends this information to Bob and depending upon this classical mes-

sage Bob performs one of four operations on his EPR pair. Using this he can fully recover

the qubit that Alice wishes to send. Let’s illustrate this by an example. The state to be

teleported is ψ = α|0〉 + β|1〉, where α and β are unknown constants. After interaction

with the quantum channel, the new state is |ψ0〉 = 1√2[α|0〉(|00〉+ |11〉) + β|1〉(|00〉+ |11〉)].

Now Alice sends the qubit through a CNOT gate obtaning |ψ1〉 = 1√2[α|0〉(|00〉 + |11〉) +

β|1〉(|01〉+|10〉)]. Now she sends the first qubit through the Hadamard gate obtaining : |ψ2〉

= 1√2[α(|0〉+ |1〉)(|00〉+ |11〉) +β(|0〉− |1〉)(|01〉+ |10〉)]. This can be rewritten by grouping

terms as: |ψ3〉 = 12 [|00〉(α|0〉+β|1〉)+ |01〉(α|1〉+β|0〉)+ |10〉(α|0〉−β|1〉)+ |11〉(α|1〉+β|0〉)].

The first two qubits are Alice’s qubits. Hence if she performs measurements on her qubits,

Bob’s systems can only be in one of these possible states. To know which state it is in,

35

Bob must be told of Alice’s measurement. Thus Alice conveys by classical means the result

of her measurement. Once Bob has been told classically by Alice about her result, Bob

can recover the state ψ by applying appropriate quantum gates. When the measurement

outcome for Alice is 00, Bob does not do anything. If the measurement outcome is 01, Bob

can fix up his state by applying the X gate. If the measurement outcome is 10, Bob can

apply the Z gate. If the measurement outcome is 11, Bob can first apply the X gate then

the Z gate.

00− > |ψ3(00)|〉 = [α|0〉+ β|1〉] (4.6)

01− > |ψ3(01)|〉 = [α|1〉+ β|0〉] (4.7)

10− > |ψ3(10)|〉 = [α|0〉 − β|1〉] (4.8)

11− > |ψ3(11)|〉 = [α|1〉 − β|0〉] (4.9)

4.3.2 Superdense coding

Superdense coding involves two parties Alice and Bob separated a long way from each

other. The final motive is to transmit classical information from Alice to Bob. Alice is

in possession of two classical bits of information which she wishes to send to Bob, but is

allowed to send a single qubit to Bob. The procedure uses very elementary ideas from

quantum computing to achieve this goal.

Suppose Alice and Bob share the state |0〉|0〉+|1〉|1〉√2

. Note that this state is a fixed state

which is prepared by some third party, sending one qubit to Alice and one qubit to Bob.

By sending the qubit in possession of her, Alice can communicate two bits of classical

information to Bob. We divide our analysis into cases :

• If Alice wishes to send the bit string ’00’ to Bob, she does nothing at all to her qubit.

36

• If Alice wishes to send the bit string ’01’ to Bob, she applies the phase flip Z to her

qubit.

• If Alice wishes to send the bit string ’10’ to Bob, she applies the phase flip X to her

qubit.

• If Alice wishes to send the bit string ’11’ to Bob, she applies the phase flip ıY to her

qubit.

00− > |ψ(00)|〉 =|00〉+ |11〉√

2(4.10)

01− > |ψ(01)|〉 =|00〉 − |11〉√

2(4.11)

10− > |ψ(10)|〉 =|10〉+ |01〉√

2(4.12)

11− > |ψ(11)|〉 =|01〉 − |10〉√

2(4.13)

As noted in the equations above, the measurements made by Alice produce the Bell basis.

These are orthonormal and can be distinguished by appropriate quantum measurement.

After Alice sends her qubit to Bob, Bob can do a measurement in the Bell basis and

determine which of the four strings Alice sent.

4.3.3 Quantum Key Distribution

The idea here is to prepare a secret communication channel between two parties so that a

third party does not eavesdrop a secret. The concept of entanglement is used very efficiently

here. A general key distribution protocol consists of three distinct stages :

• Raw key exchange

• Key Sifting

• Key distillation

37

4.3.4 Raw key exchange

This is the only quantum part of the quantum key distribution protocol. In general,

Alice and Bob exchange some quantum states i.e quantum information is passed along

a quantum channel irrespective of whether an eavesdropper is present or not. In all the

subsequent exchanges in the protocol, only classical channel will be used which is known as

classical post-processing.

4.3.5 Key Sifting

In general, both Alice and Bob mutually decide which measurements will be used for the

secret key. The decision making rules depend on which protocol is being used, and some

measurements will be discarded.

4.3.6 Key distillation

In practice channels may be lossy and there may be transmission errors, but the proto-

col still needs to work. Thus error correction and privacy amplification are required, which

are the first two steps in the key distillation phase of the classical post-processing of the

remaining secret key bits. The third process is authentication, which counteracts man-in-

the-middle attacks.

The most famous of quantum key distribution protocols is the BB84 protocol. Proposed

by Bennett in 1984, two parties Alice and Bob generate a secret key by sharing classical

public channel together with quantum communication channel which is insecure since there

is an eavesdropeer who can manipulate signals in some sense. The protocol is proposed

in such a way that the transmission of secret information is completely protected against

Eve′s success of knowing the secret data without it’s detection. Alice and Bob can detect

38

the eavesdropping by conducting random tests. They can then destroy the quantum channel

and start with a fresh channel to communicate secret information. The protocol, given by

Bennett and Brassard, is based on another invention of modern quantum mechanics, i.e.,

the existence of non-orthogonal states. After this, Ekert gave a protocol based on the use

of entangled states (i.e., Bell states).

39

Chapter 5

Quantum Dissension and State merging

In this chapter, we briefly start with the notions of entanglement and separability. Later

we move on to give a more nuanced form of quantum correlations. In particular, we focus

on discord and dissension and study dissension in two different tracks. The essential idea

of defining these tracks is to explore different types of correlations in the system. These

approaches not only encompass measures of correlation like discord but also give a unified

view of the quantum correlation from the view point of projective measurement done on

the subsystem. In addition we also probe the relation of discord and dissension with state

merging. We find that under certain special circumstances when dissension is greater than

discord, measurements can decrease the cost of state merging.

5.0.7 Separability and Entanglement

Assume a state is shared between multiple parties separated by a distance. The state

of the joint system is described by HA ⊗ HB ⊗ .. ⊗ HX , where the Hilbert space for each

system is represented by HA, HB, etc

• Separable state: If a general state can be represented as a convex combination of

product states as follows:

40

ρ =∑i

ωiρiA⊗ρiB...ρiX , 0 ≤ ωi ≤ 1,

∑i

ωi = 1, where ρiA, ρiB, ..., ρ

iX etc. are the desity

matrices of the respective Hilbert spaces.

Then this state is called a multipartite separable state.

• Entangled state: If a state cannot be expressed in the form of a separable state is

known as entangled state.

Note that if a separable state is shared between distant parties, it can be shared locally

by specific local operations between them through classical channels shared among the par-

ties. But it’s not true for entangled states. Besides, entanglement plays an important role in

quantum communication protocols like superdense coding[54], remote state preparation[55],

cryptography[56] and many more. Entanglement is indeed a real phenomenon ( Einstein

called it “spooky action at a distance” ), which has been demonstrated experimentally but

the mechanism however cannot be explained till date.

5.0.8 Quantum correlations: Going beyond entanglement

It is quite easy to quantify the amount of quantum correlation present in a pure bipartite

system. This can be attributed to certain standard measures like monotones quantifying

the amount of entanglement for both pure as well as mixed states. Two most important

measures of them are the negativity [57] and the concurrence [58, 59]. However, there are

certain issues in understanding the nature of correlations present in a mixed state. It is

not totally evident whether all the information-processing tasks that can be done more ef-

ficiently with quantum systems require entanglement as a resource. Several instances have

been reported where people have argued that various information processing tasks can be

carried out in the absence of entanglement. Then, the natural question arises if not en-

tanglement then what is responsible for such a behavior. It has been suggested that the

41

amount of correlation present in the composite system is not entirely captured by the en-

tanglement. The quantity which captures the quantum correlation and gives a meaningful

explanation of such behavior is the quantum discord [60, 61, 62, 4, 8]. The idea of quantum

discord [[60, 62]] is to capture all types of correlations including entanglement. It uses the

generalisation of two-variable mutual information to quantum domain. The difference of

two classically equivalent expressions when generalised to quantum domain gives discord.

It was also shown that discord reduces to Von-neumann entropy for a pure bipartite state.

Recently, another measure has been introduced which is called quantumness of correlation

[63], which has been defined for bipartite states by incorporating a specific measurement

scheme. In Ref [64], authors have given a unified view of the correlations in a given quan-

tum state by classifying it into entanglement, dissonance, and classical correlations. They

have used the concept of relative entropy as a distance measure of correlations. Entropic

methods have also been proposed as a measure of quantumness over classical correlations.

5.0.9 Quantum Discord

Information theoretic measures are those category of measures which are constructed

from the perspective of defining the notion of quantum correlation from the information

theoretic view point (entropic quantities). Unlike the entanglement monotones, these mea-

sures are not computable in a closed form. In spite of not being computable in a closed

form, they can be efficiently computed numerically for two-qubit system. The most impor-

tant of them is quantum discord which came into the limelight for understanding that the

quantum correlation in the mixed states goes beyond the notion of entanglement. Quantum

discord is defined as the difference between two quantum information-theoretic quantities,

whose classical counterparts are equivalent expressions for the classical mutual information.

Thus, the quantum discord is the difference between the total correlation and classical cor-

42

relation present in the bipartite quantum systems, thus quantifying the amount of quantum

correlation present in it. It is defined as [61, 60]

Q(ρAB) = I(ρAB)− J(ρAB). (5.1)

The total correlation, (ρAB), for a bipartite state ρAB is given by the expression

I(ρAB) = S(ρA) + S(ρB)− S(ρAB), (5.2)

where S(ρ) = −Tr(ρ log2 ρ) is the von Neumann entropy of the quantum state ρ, and ρA

and ρB are the reduced subsystems of the bipartite state ρAB. On the other hand, (ρAB)

can be thought of as the amount of classical correlation in ρAB, and is defined as [61]

J(ρAB) = S(ρA)− S(ρA|B). (5.3)

The basis used for measurement is given by :

|u1〉 = cos(t)|0〉+ sin(t)|1〉, |u2〉 = sin(t)|0〉 − cos(t)|1〉, here t ∈ [0, 2π]

S(ρA|B) = min{Bi}∑

i piS(ρA|i), is the average entropy of the entropies of the states ρA|i.

These entropies S(ρA|i) are the entropies of the subsystem A conditioned on a measurement

performed by B with a rank-one projection-valued measurement {Bi}.These states are given

by ρA|i = 1pi

TrB[(IA⊗Bi)ρ(IA⊗Bi)], with probability pi = TrAB[(IA⊗Bi)ρ(IA⊗Bi)].Here

I is the identity operator on the Hilbert space of A. It may be noted that the discord is a

positive quantity and vanishes on classical-classical and quantum-classical states.

5.0.10 Quantum Dissension

The notion of quantum discord has been generalised to three parties and has been given

the name “Quantum Dissension” by [4]. This quantity is obtained by considering three

43

different expressions of mutual information that same classically, but differ in the quan-

tum domain which are obtained by considering multiparticle-measurement. To obtain the

definition of dissension, we consider three variable mutual information:

I(X : Y : Z) = I(X : Y )− I(X,Y |Z). (5.4)

We have the following three forms of mutual information:

I(X : Y : Z) = H(X,Y )−H(Y |X) +H(X|Y )−H(X|Z)−H(Y |Z) +H(X,Y |Z) (5.5)

The given expression can be converted into the following form with one-conditional entropy:

J(X : Y : Z) = H(X)+H(Y )+H(Z)− [H(X,Y )+H(Y,Z)+H(X,Z)]+H(X,Y, Z) (5.6)

Also in terms of two conditional entropies, we have the following expression:

K(X : Y : Z) = [H(X) +H(Y ) +H(Z)]− [H(X,Y ) +H(X,Z)] +H(X|Y,Z). (5.7)

These three expressions for the three-variable mutual information are classically equiv-

alent, but not so in quantum domain. The difference of the three definitions can capture

various aspects of the quantum correlations and are known as ’dissension’.

Let us consider a three-qubit state ρXY Z . The quantum version of I(X : Y : Z),

obtained by replacing random variables with density matrices and Shannon entropy with

von Neumann entropy, reads,

I(X : Y : Z) = S(ρX) +S(ρY ) +S(ρZ)− [S(ρXY ) +S(ρY Z) +S(ρXZ)] +S(ρXY Z) (5.8)

The quantum version of J(X : Y : Z), obtained by defining conditional entropies, is given

by

J(X : Y : Z) = S(ρXY ) − S(ρY |ΠXj

) − S(ρX|ΠYj

) − S(ρX|ΠZj

) − S(ρY |ΠZj

) + S(ρX,Y |ΠZj

)

(5.9)

44

Here Πnj refers to a projective measurement on the subsystem ′n′. Quantum dissension

for single particle projective measurement is given by the difference of I(X : Y : Z) and

J(X : Y : Z), i.e.

D1(X : Y : Z) = J(X : Y : Z)− I(X : Y : Z) (5.10)

which can be re-written as

D1(X : Y : Z) = D(X,Y : Z) −D(X : Z) −D(Y : Z) −D(X : Y ) −D(Y : X). (5.11)

The above expression must be minimized above all one-particle measurement projects in

order for D1 to reveal the maximum possible quantum correlations. Mathematically, this

is δ1 = minπi

(D1(X : Y : Z)).

The generalization of K(X : Y : Z) in the quantum domain is as follows.

K(X : Y : Z) = [S(ρX) + S(ρY ) + S(ρZ)] − [S(ρXY ) + S(ρXZ)] + S(ρX|ΠjY Z) (5.12)

The two-particle projective measurement ought to be performed in the general basis:

|v1〉 = cosθ|00〉+eiφsinθ|11〉, |v2〉 = sinθ|00〉−eiφcosθ|11〉, |v3〉 = cosθ|01〉+eiφsinθ|10〉, |v4〉 =

sinθ|01〉 − eiφcosθ|10〉, where θ, φ ε [0,2π]. In this case, the quantum conditional en-

tropy is given as: S(ρX|Y Z) =∑

j pjS(ρX|ΠY Zj

) with pj = tr[(IX ⊗ ΠY Zj )ρ(IX ⊗ ΠY Z

j )] and

ρX|ΠY Zj

= 1pjtrY Z[(IX ⊗ΠY Z

j )ρ(IX ⊗ΠY Zj )].

To define quantum dissension for two-particle projective measurement, we take the dif-

ference, i.e.

D2(X : Y : Z) = K(X : Y : Z)− I(X : Y : Z) = S(ρX|ΠY Zj

) + S(ρY Z)− S(ρXY Z) (5.13)

which can be neatly reproduced as

D2(X : Y : Z) = D(X : Y, Z) (5.14)

45

i.e. D2 is just the quantum discord with a bipartite split of the system. One can minimizeD2

over all two-particle measurement projectors and obtain δ2=min(D2(X : Y : Z)). This is the

most generic expression since it includes all possible two-particle projective measurements.

Both δ1 and δ2 are not symmetric under the permutations of X, Y and Z. For a pure three-

qubit GHZ state, |ψ〉= 1√2(|000〉+|111〉), (δ1,δ2)=(-3.00,1.00) and for a pure three-qubit W

state, |ψ〉= 1√3(|001〉+|010〉+|100〉), (δ1,δ2)=(-1.75,0.92).

5.1 Quantum Dissension: Track 1 and Track 2

The idea of this section is to generalise the notion of quantum correlation for a n qubit

system in two distinct tracks. The first track describes the approach by which “Quantum

Discord” has been defined and then extend the concept for a multiparty system by consid-

ering equivalent expressions of n variable mutual information which are same classically but

different quantum mechanically. In the second track, we would redefine quantum correlation

from the perspective of all possible measurements. These approaches not only encompass

measures of correlation, but also give a unified view of quantum correlation from the view

of projective measurement done on the sub-system.

5.1.1 Idea of correlation vectors

Consider a state ρx1,x2,..,xn in Hilbert space H2⊗H2⊗ ...⊗H2. The mutual information

for this state without any conditional entropies is:

I0(ρx1,x2,..,xn) =n∑p=1

(−1)p−1n∑{lp}

H(xl1 , xl2 , .., xlp), (5.15)

Here {lp} in the sum denotes l1 < l2 < l3... < lp, and H(x), H(y) are the von-neumann

entropies of the qubits ρx and ρy. H(x, y) is the joint entropy of the state ρxy.

46

5.1.2 Track 1

Consider the most general situation where we have state ρx1,x2,....,xn with n number of

qubits. On basis of m-party joint measurement, we will have (n− 1) expressions for mutual

information {I1m(ρx1,x2,....,xn);m = 1, 2, ..., (n− 1)},

I1m =

m−1∑k=1

(−1)k−1n∑{lk}

H(xl1 , xl2 , .., xlk)

+ (−1)m−1∑

{km−1};k1=2

H(x1, xk1 , ..., xkm−1)

+

n∑p=m+1

(−1)p−1n∑{lp}

H(xl1 , .., xlp−m |xlp−m+1 , .., xlp)

(5.16)

where, H(xl1 , .., xlp−m |xlp−m+1 , .., xlp) denotes conditional entropy where joint measurement

have been done on parties xlp−m+1 , .., xlp . Hence we can define the dissension function,

δ1m = (−1)n(I0 − I1

m). (5.17)

The dissension function when minimized over all basis vectors gives us the expression

of dissension. The expressions of mutual information are asymmetric under exchange of

parties. For example, if we take m = 1, I11 can have n number of different expressions

which are very different from each others. In equation (5.16), if we put m = 1 we can have

one type of I11 , let name it I1

xn . Now exchanging xn with x1, x2, ....., xn−1 respectively one

can get other forms of mutual information. So we have n number of δ11 , let label them as

δ1xp = (−1)n(I0 − Ixp); p = 1, 2, ..., n. This leads us to define a correlation vectors

~δ11 = {δ1

xp ; p = 1, 2, ..., n}. (5.18)

47

5.1.3 Track II

In this track, we extend the definitions of mutual information in this track for all possible

m party conditionals. The idea is to capture all types of correlations present in the system.

Performing a measurement on all possible subsystems gives rise to maximum correlation

present in the system.

I2m =

m−1∑k=1

(−1)k−1n∑{lk}

H(xl1 , xl2 , .., xlk) + (−1)m−1

.

n−1∑{km−1};k1=2

{H(x1, xk1 , ..., xkm−1 , xkm−1+1)

− H(xkm−1+1|x1, xk1 , ..., xkm−1)}+ ........

+ H(x1, x2, xkn−m+2 , xkn−m+3 , ..., xn−1, xn)

− H(x2|x1, xkn−m+2 , ..., xn−1, xn)]

+n∑

p=m+1

(−1)p−1n∑{lp}

H(xl1 , .., xlp−m |xlp−m+1 , .., xlp)

(5.19)

The dissensions in this track are

δ2m = (−1)n(I0 − I2

m). (5.20)

As in Track 1, we have to minimize this dissension function over all basis vectors to give us

the expression of dissension. If we exchange parties, the mutual information in the equation

(5.19) will remain the same only for m = (n−1). If we consider m = 1 in the equation (5.19,

you will get one I21 , let define it as I2

xn . Now exchanging xn with x1, x2, ...., xn−1 respectively

you will get other expressions. If we label them as δ2xp = (−1)n(I0 − I2

xp); p = 1, 2, ..., n, we

can have the correlation vector

~δ21 = {δ2

xp ; p = 1, 2, ..., n}. (5.21)

48

5.2 Analysis

In this section, we will perform a detailed case analysis of the different tracks of dissen-

sions for two, three and four qubits.

5.2.1 Two qubit case

1. Classical or product states: We consider two-qubit states like ρ1 = | + +〉〈+ + |

and ρ2 = 12(| + +〉〈+ + | + | − −〉〈− − |). Here |±〉 = 1√

2(|0〉 + |1〉). The states mentioned

are purely classical and the values of correlation vectors are ~δ11(ρ1/2) = {0, 0}, δ2

1(ρ1/2) = 0

and ~δ11(ρ2) = {0, 0}, δ2

1(ρ2) = 0

2. Separable but not classical states: Here we consider a particular example of separable

state, ρsep = 12(|00〉〈00|+ |++〉〈++ |). The correlation vectors for this state are, ~δ1

1(ρsep) =

{0.15, 0.15}, δ21(ρsep) = 0.3. Non-zero values of vectors say, separable states have quantum

correlation in it. Now the question is, what is the upper bound of this type of correlation.

3. CQ/QC states: Consider the following states, one classical-quantum state ρCQ = 12(|+

+〉〈++|+|−0〉〈−0|) and other quantum-classical state ρQC = 12(|++〉〈++|+|0−〉〈0−|). The

values of correlation vectors are ~δ11(ρCQ) = {0, 0.2}, δ2

1(ρCQ) = 0.2 and ~δ11(ρQC) = {0.2, 0.0},

δ21(ρQC) = 0.2 respectively. Our correlation vectors gives information about which part of

the state is classical.

4. Maximally entangle Bell states: The values of correlation vectors for maximally

entangled Bell states are ~δ11 = {1, 1}, δ2

1 = 2.

5.2.2 Three-qubit case

The mutual information without conditional entropy is

I0(x, y, z) = H(x) +H(y) +H(z)− (H(x, y) +H(x, z)

+ H(y, z)) +H(x, y, z). (5.22)

49

Then the mutual information with one conditional entropy in track-1 are,

I1k(i, j, k) = H(i)− (H(i|j) +H(i|k) +H(j|k)) +H(ij|k), (5.23)

This expression is not symmetric in i, j. The dissensions in track-1 with one conditional

are

δ1k = I1

k − I0. (5.24)

Hence correlation vector in this setting is

~δ11 = {δ1

x, δ1y , δ

1z}. (5.25)

For conditionals in two variables we have the following mutual entropies

I12k = H(i) +H(j) +H(k)− (H(ik) +H(jk)) +H(k|ij). (5.26)

Here index 2 stands for conditional with two-variable.

δ12k = I1

2k − I0. (5.27)

Thus we have correlation vector

~δ12 = {δ1

2x, δ12y, δ

12z}. (5.28)

Let us consider conditionals in one variable:

I2k = H(ij)− (H(i|j) +H(j|i) +H(i|k) +H(j|k)) +H(ij|k). (5.29)

Dissensions for this case are :

δ2k = I2

k − I0. (5.30)

The corresponding correlation vector is

~δ21 = {δ2

x, δ2y , δ

2z}. (5.31)

50

Table 5.1 Table TitleState Track-1 Track-2

ρxpyz ~δ11 = {−1, 0, 0}; ~δ1

2 = {0, 1, 1} ~δ21 = {−2, 0, 0}; δ2

2 = 2

ρypxz ~δ11 = {0,−1, 0}; ~δ1

2 = {1, 0, 1} ~δ21 = {0,−2, 0}; δ2

2 = 2

ρzpxy ~δ11 = {0, 0,−1}; ~δ1

2 = {1, 1, 0} ~δ21 = {0, 0,−2}; δ2

2 = 2

In this track we can have another mutual information considering conditionals on two

variables

I2 = H(x) +H(y) +H(z) + (H(x|yz) +H(y|xz)

+ H(z|xy))− 2H(x, y, z). (5.32)

The corresponding dissension or correlation vector is

δ22 = I2 − I0. (5.33)

This quantity is nothing but symmetric discord in i : jk setting. Now, we consider three

biseparable three qubit pure states i.e., ρipjk = |0〉i〈0| ⊗ |Bell〉jk〈Bell| where i 6= j 6= k

and |Bell〉 is any Bell state. We have calculated correlation vectors for these states and are

given in the following table. The table above elucidates an important strategy of correlation

measures. Our correlation vectors not only can detect pure biseparable states easily but

also it can say about the position where state is biseparable.

Let us consider an example of a mixed state of the form

ρ123 =1

2(|0〉〈0| ⊗ |+〉〈+| ⊗ |−〉〈−|+ |1〉〈1| ⊗ |0〉〈0| ⊗ |+〉〈+|) (5.34)

Here, the states of parties 1 and 3 are both classical and of 2 is quantum. If we calculate

correlation vectors of this state (5.34) then ~δ11 = {0.8, 1.4, 0.8}, ~δ1

2 = {0, 0.6, 0} in track-1

and in track-2 ~δ21 = {0.2, 1.6, 0.2}, δ2

2 = 0.6. These values depicts that only mode 2 is

quantum.

51

5.2.3 Four-qubit case

{I1l , I

12l andI1

3l} respectively in track-1 and in track-2 {I2l , I

22l andI3} respectively. We

have consider four sets of (i, j, k, l) only for I1l i.e., first consider the set (i = x, j = y, k =

z, l = w) then get other three just by replacing w ↔ x, w ↔ y and w ↔ z. So, dissensions

in track-1 with one conditional are

δ1l = I1

l − I0. (5.35)

The correlation vector in this setting is

~δ11 = {δ1

x, δ1y , δ

1z , δ

1w}. (5.36)

Similarly, we can define I12l. The corresponding dissensions as

δ12l = I1

2l − I0. (5.37)

Hence we have the correlation vector

~δ12 = {δ1

2x, δ12y, δ

12z, δ

12w}. (5.38)

The other correlation vector will be

~δ13 = {δ1

3x, δ13y, δ

13z, δ

13w}. (5.39)

Here, δ13l = I1

3l − I0 and l is the index which occurs most on the left side of the bar. Now

we can have other dissensions considering all possible conditionals. The dissensions in this

case are

δ2l = I2

l − I0. (5.40)

The corresponding correlation vector is

~δ21 = {δ2

x, δ2y , δ

2z , δ

2w}. (5.41)

52

The dissensions with conditional in two indices are

δ22l = I2

2l − I0, (5.42)

The corresponding vector

~δ22 = {δ2

2x, δ22y, δ

22z, δ

22w}. (5.43)

The last of vectors is written below:

δ23 = I3 − I0, (5.44)

5.3 State Merging: Concept

Since a long time quantum entanglement was of only philosophical interest and the re-

searchers were focussing on addressing the questions that were related with the quantum

mechanical understanding of various fundamental notions like reality and locality. However,

for the last two decades world had seen that quantum entanglement is not just a philosoph-

ical dilemma but also a reality as far as the laboratory preparation of entangled qubits

are concerned. The research that was conducted during these period did not cater to the

existence, but rather it’s use as a resource to carry out information processing protocols

like quantum teleportation [65], cryptography [66], superdense coding [54], remote state

preparation [67]etc. It was subsequently evident that entanglement plays a pivotal role in

all these information processing protocols. One of such important information processing

tasks is state merging. By state merging we refer to a situation, where we consider two

parties Alice and Bob where Bob has some prior information Y and the other party Alice

has some missing information X (where X and Y are random variables). Now here one

can ask a very simple question that if Bob wants to learn about X, how much additional

information Alice needs to send him. It is a very well known problem in classical infor-

mation theory. Apparently it looks like Alice needs to send H(X) bits of information to

53

Bob, however it had been seen that only H(X|Y ) bits of information will suffice [68]. This

quantity is known as the conditional entropy of X given that Y has occurred. This partial

information that Alice needs to send to Bob is always a positive quantity.

However, when we pose the similar problem in the context of quantum information, we

have a different interpretational view associated with it. Here we consider Alice and Bob

possessing an unknown state with the total density operator ρAB. We can equivalently view

this as a source emitting a sequence of unknown states, from an ensemble ρAB known to

both Alice and Bob. However the probability distribution of the ensemble is unknown to

both the parties. The job is to find out the exact state shared between Alice and Bob.

In the quantum case, the amount of information that Alice needs to send to Bob is given

by S(A|B), the conditional Von-Neumann entropy of A conditioned on B. There is yet

another way of visualising this problem of state merging. We imagine this state ρAB being

part of a larger pure state ρABR = |ψ〉ABR〈ψ|ABR with a state reference system R. Faithful

state transfer means that the transferred state has high fidelity with the original state.More

formally, we define:

([8, Definition 1]) Consider a pure state |ψ〉ABR shared between two parties A, B and a

reference R. Let Alice and Bob have further registers A0, A1 and B0, B1, respectively. We

call a joint operation M : A,A0 ⊗ BB0 → A1 ⊗ B1B′B state merging of ψ with error ε, if

it is LOCC and, with ρA1B1B′BR = (M ⊗ idR)(|ψ〉ABR ⊗ (|φ〉K)A0B0),

F (ρA1B1B′BR, )|φ〉LA1B1⊗ |ψ〉B′BR) ≥ 1− ε

54

with maximally entangled states |φ〉K , |φ〉L on A0B0, A1B1 of Schmidt rank K,L, re-

spectively. Here, B’ is a local ancilla at Bobs end. The quantity logK-logL is called the

entanglement cost of the protocol. In the case of many copies of the same state, |ψ〉 = |ψ〉⊗n,

we call 1n(logK-logL) the entanglement rate of the protocol. R is called an achievable rate

if there exist, for n → ∞, merging protocols of rate approaching R and error approaching

0. The least achievable rate is the merging cost of |ψ〉.

A fundamental result in quantum information theory is given by the next theorem:

[8, Theorem 2 ]) For a state ρAB shared by Alice and Bob, the entanglement cost of merg-

ing is equal to the quantum conditional entropy S(A|B) = S(B)S(AB). When the S(A|B)

is positive, then merging is possible if and only if R > S(A|B) holds true. When S(A|B)

is negative, then merging is possible by local operations and classical communication, and

moreover R < S(A|B) maximally entangled states are obtained per input copy.

The theorems written above give terse definitions of the quantum state merging protocol.

In simple words, the amount of partial information that Alice needs to send Bob is given

by the quantum conditional entropy, which is exactly the same as in the classical case but

with he Shannon entropy changed to Von-Neumann entropy. But in the quantum world,

conditional entropy can be negative. Bob can secure the full state only by classical commu-

nication and additionally Alice and Bob have the potential to transfer additional quantum

information in the future at no additional cost. As in [8], we consider three instructive

examples to depict the various signs of the conditional entropy.

1. The state that Alice possesses can be represented by the maximally mixed density

matrix: ρA = 12(|0〉〈0|A+|1〉〈1|A) and Bob has the state |0〉B. In this case, S(A|B) = 1

and Alice should send one qubit through the quantum channel to transfer her state

to Bob.

55

2. We consider the classically correlated state ρAB = 12(|00〉〈00|AB + |11〉〈11|AB). Let

us imagine this state as being part of a more pure state with reference system R,

|ψ〉ABR〉 = 1√2(|0〉A|0〉B|0〉R + |1〉A|1〉B|1〉R). In this case, S(A|B) = 0, and hence no

quantum information has to be sent. Alice can measure her state in the |0〉|±〉|1〉 and

inform Bob of the result. Depending upon the outcome of the measurement, Bob and

R will share one of two states φ|±〉〉BR = 1√2(|0〉B|0〉R|±〉|1〉B|1〉R) and by a local oper-

ation, Bob can always transform the state to |ψA′BR = 1√2(|0〉A′ |0〉B|0〉R|±〉|1〉A′ |1〉B|1〉R

with A′ being ancilla at Bob’s site. Alice has thus managed to send her state to Bob

while fully preserving theie entanglement with R.

3. For the state |φ〉+ = 1√2(|0〉A0〉B+ |1〉|A1〉B), S(A|B) = 1, and Alice and Bob can keep

this shared EPR pair to allow future transmission of quantum information, since te

pure state is known and Bob can create the EPR pair locally.

Pictorially, we can depict state merging by the following figure 5.1:

5.3.1 One shot and multi-shot state merging

Consider the following scenario. If many copies of a state |ψ〉C1C2R are distributed parties

C1 and C2, then the sender C1 can first send his system to the receiver at a compression

rate arbitrarily close to S(C1)ψ. The second sender follows by merging his system with the

receiver at the rate approaching S(C2|C1)ψ. If the second sender gives his state first,the

rates S(C2) and S(C1|C2) are achieved. The compression protocols although optimal in

rates require the use of time-sharing for achieving rates which are not corner points of the

rate region. By “time-sharing”, we mean partitioning a large supply of states and applying

different protocols to each set. Hence this approach is not feasible if a single copy of state

is available. Dutil and Hayden[69] showed the existence of multi-party merging protocols

56

Figure 5.1 Diagrammatic representation of state merging as shown in [8]

which work even if there is only one copy of state available. In addition they also extended

he notion of state-merging in general to m-parties.

5.3.2 Measurements and cost of state merging for three parties

We show in this section that measurements can actually decrease the cost of state merg-

ing for three parties.

Let us consider the state merging scenario in the tripartite situation. Let’s say A and

C both want to merge their information to B.(Here we are considering multi-shot state

merging.)

We make our analysis by dividing into two cases:

57

• One-party measurement

• Multi-party measurement

5.3.3 One party measurement

In order to make the analysis for one party measurement, we need to simulate an ar-

bitrary quantum operation ε (including measurements) on B. We assume D an ancilla to

initially be in a pure state |0〉, and a unitary interaction U between B and D. The primes

will denote the state of the system after U has acted. From this scenario, we can infer

the following points. We have S(AC,B) = S(AC,BD) as since D starts out in a product

state with ABC. We also have I(AC : BD) = I(A′C ′ : B′D′) . A discarding quantum

information cannot decrease mutual information, we have I(A′C ′ : B′) ≤ I(A′C ′ : B′D′).

We also have I(AC : BD) = I(AC : B). Hence I(A′C ′ : B′) ≤ I(A′C ′ : B′D′) = I(AC :

BD) = I(AC : B) which implies I(A′C ′ : B′) ≤ I(AC : B). These statements imply

that S(A′C ′ : B′) ≥ S(AC|B). But we can never conclude that S(A′|B′) ≥ S(A|B) or

S(A|B) ≥ S(A′|B′).

The state ρABC after one-particle measurement of the subsystem B changes to ρ′ABC =∑j

pjρA|j ⊗ πj ⊗ ρC|j . Here {πj} are the projection operators. The unconditional post-

conditionals states of A,B and C are ρ′A =∑j

pjρA|j , ρ′B =

∑j

pjπj and ρ′C =∑j

pjρC|j .

After measurement, I1(A′ : B′ : C ′) = S(A′) + S(B′) + S(C ′)− S(A′, B′)− S(B′, C ′)−

S(A′, C ′) + S(A′, B′, C ′) = S(A′) + S(C ′) + H(p) − {H(p) +∑j

S(ρA|j)} − S(A′, C ′) −

S(B′, C ′)+S(A′, B′, C ′) = S(A′)+S(C ′)+∑j

S(ρA|j)−S(A′, C ′)−S(B′, C ′)+S(A′, B′, C ′).

58

Now, I0(A : B : C) = S(A)+S(B)+S(C)−S(A,B)−S(B,C)−S(A,C)+S(A,B,C). The

quantum dissension D1 is given by: minπi

(I1 − I0) = S(A|B)− S(A′|B′). The change in the

cost of state-merging due to one-particle measurement is given by δ1 = S(A′|B′)−S(A|B) =

−D1.

5.3.4 Two-party measurement

In this situation, we can bring in an ancilla D(|0〉) and apply a unitary interaction between

A,B and D. We have S(C,AB) = S(C,ABD). Also we have I(C : AB) = I(C ′ : A′B′D′).

Similar to the reasoning for the one-party measurement, since discarding quantum infor-

mation cannot decrease mutual information, we have I(C ′ : A′B′) ≤ I(C ′ : A′B′D′). Hence

I(C ′ : A′B′) ≤ I(C ′ : A′B′D′) = I(C : AB). Therefore, I(C ′ : A′B′) ≤ I(C : AB) and

hence S(C ′|A′B′) ≥ S(C|AB).

The state ρABC after two particle measurement of the subsystem AB changes to ρ′ABC =∑j

pjπj ⊗ ρC|j . Now ρAB =∑j

pjπj and ρC =∑j

pjρC|j .

I2(A′ : B′ : C ′) = S(A′)+S(B′)+S(C ′)−S(A′, B′)−S(B′, C ′)−S(A′, C ′)+S(A′, B′, C ′) =

S(A′) + S(B′) + S(C ′)−H(p)− S(A′, C ′)− S(B′, C ′) +H(p) +∑j

S(ρC|j)

The quantum dissension is given by: D2 = minπi{I2 − I0}. The change in cost of state

merging considering two party measurement is δ2 = S(C ′|A′B′) − S(C|AB) > 0 . Hence,

the total change in the cost of state merging before and after the measurement is δ2 + δ1 =

D2−D1. We have to thus infer about the sign of D2−D1. If D2 < D1, we can decrease the

cost of state merging i.e measurements can decrease the cost of state merging. In essence,

what this means is that in order to reduce the cost of state merging, we can first perform

measurements on the system, and then perform the state merging protocol.

59

Chapter 6

Conclusions

We have managed to establish a relation between the change in the cost of state-merging

for three party systems before and after measurement to the different types of dissensions.

We show that by making measurements, we can decrease the cost of state merging for three

parties if D1 > D2. This gives an operational interpretation for dissension in terms of

state-merging.

Further, we have tried to classify different types of dissensions based on the number

of measurements performed on the systems. The first track describes the approach by

which “Quantum Discord” has been defined and then extend the concept for a multiparty

system by considering equivalent expressions of n variable mutual information which are

same classically but different quantum mechanically. In the second track, we would redefine

quantum correlation from the perspective of all possible measurements. This track gives

the maximum amount of correlation present in the system. These tracks of dissensions can

elucidate an important strategy of correlation measures. As a specific example mentioned

in the chapter 5, our correlation vectors not only can detect pure biseparable states easily

but also it can say about the position where state is biseparable.

60

Related Publications

6.1 Conferences and Posters

• Abhishek Deshpande, S.K. Sazim, Indranil Chakrabarty, Pankaj Agarwal, Quantum

correlation vector: A unified approach for multiparty quantum correlations, ICQIQC

(2013).

6.2 Journal not related to thesis

• Abhishek Deshpande, Manoj Gopalkrishnan, Autocatalysis in reaction networks,

http://arxiv.org/pdf/1309.3957v3.pdf

61

Bibliography

[1] T Bredtmann and J Manz. Optimal control of the initiation of a pericyclic reaction

in the electronic ground state. J. Chem. Sci, 124(1):121–129, 2012.

[2] Regina de Vivie-Riedle Carmen Tesch. Phys. Rev. Lett., 89(157901), 2002.

[3] Animesh Datta Vaibhav Madhok. Phys. Rev. A., 83(032323), 2011.

[4] I Chakrabarty, P Agrawal, and A Pati. Quantum dissension: Generalizing quantum

discord for three-qubit states. Eur. Phys. J. D, 65(605).

[5] H Rabitz, R Vivie-Riedle, M Motzkus, and K Kompa. Science, 288(824), 2000.

[6] S Sharma. Design of optimal laser pulses for controlling molecular processes. Ph.D

thesis, International Institute of Information Technology, 2011.

[7] D Andrae, I Barth, T Bredtmann, H Hege, J Manz, F Marquardt, and B Paulus. J.

Phys. Chem. B., 115(5476), 2011.

[8] M Horodocki, J Oppenheim, and A Winter. Quantum state merging and negative

information. IEEE Transactions on Information Theory.

[9] T Ticich, M Likar, H Dubal, L Butler, and F Crim. Optimal control of the initiation

of a pericyclic reaction in the electronic ground state. J. Chem. Phys., 87(5820), 1987.

62

[10] T Arusi-Parpar, R Schmid, R Li, I Bar, and Rosenwaks. S. Optimal control of the

initiation of a pericyclic reaction in the electronic ground state. Chem. Phys. Lett,

268(163), 1997.

[11] F Crim. Optimal control of the initiation of a pericyclic reaction in the electronic

ground state. Science, 249(1387), 1990.

[12] D Tannor and H Rice. J. Chem. Phys, 83(5013), 1985.

[13] D Tannor, R Kosloff, and S Rice. J. Chem. Phys, 85(5085), 1986.

[14] P Brumer and M Shapiro. Chem. Phys. Lett., 126(541), 1986.

[15] P Brumer and M Shapiro. Faraday Discuss. Chem. Soc, 82(177), 1986.

[16] C Brif, R Chakrabarti, and H Rabitz. New J. Phys, 12(075008), 2010.

[17] U Gaubatz, P Rudecki, M Becker, S Schiemann, M Klz, and K Bergmann. Chem.

Phys. Lett., 149(463), 1988.

[18] U Gaubatz, P Rudecki, S Schiemann, M Klz, and K Bergmann. J. Chem. Phys.,

92(5363), 1990.

[19] G Coulston and K Bergmann. J. Chem. Phys., 96(3467), 1992.

[20] K Bergmann, H Theuer, and B Shore. Rev. Mod. Phys., 70(1003), 1998.

[21] T Halfmann, L Yatsenko, M Shapiro, B Shore, and K Bergmann. Phys. Rev. A, 58,

1998.

[22] T Halfmann and K Bergmann. J. Chem. Phys., 104(7068), 1996.

[23] N Vitanov, T Halfmann, B Shore, and K Bergmann. Ann. Rev. Phys. Chem., 52(763),

2001.

63

[24] A Pierce, M Dahleh, and H Rabitz. Phys. Rev. A, 37(4950), 1988.

[25] R Judson and H Rabitz. Phys. Rev. Lett., 68(1500), 1992.

[26] D Meshulach and Y Silberberg. Nature, 396(239), 1998.

[27] D Meshulach and Y Silberberg. Phys. Rev. A, 60(1287), 1999.

[28] T Hornung, R Meier, D Zeidler, K Kompa, D Proch, and M Motzkus. Appl. Phys. B,

71(277), 2000.

[29] T Weinacht, J Ahn, and P Buckbaum. Nature, 397(233), 1999.

[30] C Bardeen, V Yakovlev, K Wilson, S Carpenter, P Weber, and W Warren. Chem.

Phys. Lett, 280(151), 1997.

[31] A Assien and T Baumert. Science, 282(919), 1998.

[32] D Yelin, D Meshulach, and Y Silberberg. Opt. Lett., 22(1793), 1997.

[33] D Tannor, R Kosloff, and A Bartana. Faraday Discuss, 113(365), 1999.

[34] M Feit and J Fleck. J. Chem. Phys., 78(301), 1983.

[35] R Kosloff. J. Phys. Chem, 92(2087), 1988.

[36] J Light, I Hamilton, and V Lill. J. Chem. Phys., 82(1400), 1985.

[37] M Tsubouchi and T Momose. Phys. Rev. A., 77(052326), 2008.

[38] C Gollub and R Vivie-Riedle. Phys. Rev. A., 78(033424), 2008.

[39] M Tsubouchi, A Khramov, and T Momose. Phy. Rev. A., 77(023405), 2008.

[40] O Atebak, D Dion, and A Haj-Yedder. J. Phys. B: At. Mol. Opt. Phys., 36(4667),

2003.

64

[41] Jerry March. March’s advanced organic chemistry. John Wiley and sons, 5, 2001.

[42] B McFarland, J Farell, P Buckbaum, and M Guhr. Science, 322(1232), 2008.

[43] E Goulielmakis, Z Loh, A Wirth, R Santra, N Rohringer, V Yakovlev, S Zherebstov,

T Pfeifer, A Azzeer, M Kling, S Leone, and F Krausz. Nature, 466(739), 2010.

[44] H Niikura, D Villeneuve, and P Corkum. Phys. Rev. Lett., 94(083003), 2005.

[45] S Chelkowski, G Yudin, and A Bandrauk. J. Phys. B., 39, 2006.

[46] A Debnarova, S Techert, and S Schmatz. J. Chem. Phys., 125(224101), 2006.

[47] I Schweigert and S Mukamel. Phys. Rev. A., 76(012504), 2007.

[48] F Krausz and M Ivanov. Rev. Mod. Phys., 81(163), 2009.

[49] P Schwender, F Seyl, and R Schinke. Photodissociation of in strong laser fields.

Chemical Physics, 217:233–247, 1997.

[50] I Iwakura, A Yabushita, and T Kobayashi. Chem. Lett., 39(374), 2010.

[51] M Schroder, J L Carreon-Macedo, and A Brown. Phys. Chem. Chem. Phys., 10(850),

2008.

[52] J M Combes, A Grossman, and P Tchamitchian. Wavelets.

[53] C Chandre, S Wiggins, and T Uzer. Physica D, 181(171).

[54] C Bennett and S Wiesner. Phys. Rev. Lett., 69(2881), 1992.

[55] C Bennett and S Wiesner. Phys. Rev. Lett., 69(2881), 1992.

[56] C Bennett and S Wiesner. Phys. Rev. Lett., 69(2881), 1992.

65

[57] G Vidal and R Werner. A computable measure of entanglement. Phys. Rev. A.,

65(032314), 2002.

[58] H Scott and W Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett.,

78(5022).

[59] W Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys.

Rev. Lett, 80(2245), 1998.

[60] H Ollivier and W Zurek. Phys. Rev. Lett., 88(017901), 2002.

[61] L Henderson and V Vedral. J. Phys. A, 34(6899), 2001.

[62] S Luo. Phys. Rev. A., 77(042303), 2008.

[63] S.B Zheng. Phys. Rev. A., 74(054303), 2006.

[64] A Keet, B Fortescue, D Markham, and B Sanders. Phys. Rev. A., 82(062315), 2010.

[65] C H Bennett, G Brassard, C Crepeau, R Jozsa, A Peres, and A K Wooters. Teleporting

an unknown quantum state via dual classical and einstein-podolsky-rosen channels.

Phys. Rev. Lett., 70(13), 1993.

[66] N Gisin. Rev. Mod. Phys., 74(145), 2002.

[67] A Pati. Phys. Rev. A., 63(014320), 2001.

[68] D Slepian and J K Wolf. Noiseless coding of correlated information sources. IEEE

Trans. Inf. Theory, 19(461).

[69] N Dutil and P Hayden. One-shot multiparty state merging. IEEE Transactions on

Information Theory, 69(2881), 2010.

66

[70] S Schirmer, H Fu, and A Solomon. Complete controllability of quantum systems.

Phys.Rev. A, 63(063410), 2003.

[71] Balakrishnan N, Kalyanaram C, and Sathyamurthy N. Physics Reports, 280(79).

[72] E Heller. Acc. Chem. Res, 14(368), 1981.

[73] I Bar, Y Cohen, D David, S Rosenwaks, and J Valentini. Optimal control of the

initiation of a pericyclic reaction in the electronic ground state. J. Chem. Phys.,

93(2146), 1990.

[74] M Shapiro, J Hepburn, and P Brumer. Chem. Phys. Lett., 149(451), 1988.

[75] C Bisgaard, O Clarkin, G Wu, A Lee, O Gebner, C Hayden, and A Stolow. Science,

323(1463), 2009.

[76] H Worner, J Bertrand, D Kartashov, P Corkum, and D Villeneuve. Nature, 466(604),

2010.

[77] M Ben-Nun and T Martinez. Nature, 466(739), 2010.

[78] A Sugita, M Mashino, M Kawasaki, Y Mathumi, R J Gordon, and R Bersohn. J.

chem. Phys., 112(2164), 2000.

[79] B Kohler, J L Krause, F Raksi, K R Wilson, R M Whitnell, V V Yakovlev, and Yan

Y. J. Acc. Chem. Res, 28(133), 1995.

[80] C Chen and D S Elliott. Phys. Rev. A, 53(272), 1996.

[81] D Goswami and A S Sandhu. Adv. Multi-photon Processes Spectrosc, 13(132), 2001.

[82] S M Park, S P Lu, and R J Gordon. J. Chem. Phys., 94(8622), 1991.

[83] P Brumer and M Shapiro. Annu. Rev. Phys. Chem., 43(257), 1992.

67

[84] N P Moore, G M Menkir, A N Markewitch, and P G R J Levis.

[85] S A-Gaspard P Tersini S Tannor D J Kosloff R, Rice.

[86] Williams R M Papanikolas J M Leone S R Uberna R, Khalil M. J. Chem. Phys,

108(9259), 1998.

[87] A Shi S, Woody and H Rabitz. J. Chem. Phys., 6870(88), 1998.

[88] S Sharma, H Singh, and C G Balint-Kurti. J. Chem. Phys., 132(064108), 2010.

[89] S Sharma, H Singh, N J Harvey, and C G Balint-Kurti. Vir. J. Bio. Phys. Res,

133(174103), 2010.

[90] S Sharma and H Singh. Chem. Phys., 390(68), 2011.

[91] W Jakubetz, J Manz, and H J Schreiber. Chem. Phys. Lett, 165(100), 1990.

[92] Y Kurosaki, M Artamonov, T S Ho, and H Rabitz. J. Chem. Phys., 131(044306),

2009.

[93] M Artamonov, T S Ho, and Rabitz. Chem. Phys., 328(147), 2006.

[94] T Hornung, R Meier, and M Motzkus. Chem. Phys. Lett, 326(445), 2000.

[95] J Herek, W Wohllehen, R Cogdell, Zeidler, and M Motzkus. Nature, 417(533), 2002.

[96] J Savolainen, R Fanciulli, N Dijkhuizen, A Moore, J Hauer, T Buckup, and

M Motzkus. Proc. Natl. Acad. Sci. USA, 105(7641), 2008.

[97] V Leimann, S Arrivo, J Melinger, and E Heilweil. Chem. Phys., 233(207), 1998.

[98] D Meshulach and Y Silberberg. Nature, 396(239), 1998.

[99] T Hornung, R Meier, and M Motzkus. Chem. Phys. Lett, 326(445), 2000.

68

[100] R Bartels, S Backus, E Zeek, L Misoguti, G Vdovin, I Christov, M Murnane, and

H Kapteyn. Nature, 406(164), 2000.

[101] D Kosloff and R Kosloff. J. Comp. Phys., 52(35), 1983.

69