Istituto Nazionale di Fisica Nucleare Contents 1 Introduction 1 2 Prelude 3 2.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.2 Operators on Hilbert Spaces

UNIVERSITY OF CAMERINO

SCHOOL OF SCIENCE AND TECHNOLOGY

M. SC. DEGREE IN APPLIED MATHEMATICS

On The Estimation of Quantum Channelswith Environment Assistance

Master Thesisin

Quantum Information

Advisor

Prof. Stefano ManciniCandidate

Milajiguli Milajiguli

ACADEMIC YEAR 2014/2015

Contents

1 Introduction 1

2 Prelude 32.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . 42.3 Spectral Properties of Operators . . . . . . . . . . . . . . . . . . 62.4 Statistical Estimation . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4.1 Fisher estimator . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Bayes estimator . . . . . . . . . . . . . . . . . . . . . . . 10

3 Quantum Information Theory Tools 123.1 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . 123.2 Density operator and POVM . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Density operator . . . . . . . . . . . . . . . . . . . . . . 143.2.2 POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.3 The Pauli representation . . . . . . . . . . . . . . . . . . 16

3.3 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.1 Entanglement in pure states . . . . . . . . . . . . . . . . 183.3.2 Schmidt decomposition . . . . . . . . . . . . . . . . . . . 183.3.3 Entanglement in mixed states . . . . . . . . . . . . . . . 20

3.4 Quantum Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Entanglement monotones . . . . . . . . . . . . . . . . . . . . . . 23

4 Parameter Estimation of Quantum Channels 264.1 Cramer-Rao bound . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Optimal parameter estimation of depolarizing channel with SLD

Fisher information . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1 Optimal measurement operator . . . . . . . . . . . . . . . 29

4.3 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Optimal parameter estimation of depolarizing channel with quadratic

cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1

CONTENTS 2

5 The Environment Assistance Model 355.1 Parameterization of two qubits unitaries . . . . . . . . . . . . . . 355.2 Estimation of two qubit unitaries with environment assistance . . . 37

5.2.1 Fisher Estimation . . . . . . . . . . . . . . . . . . . . . . 395.2.2 Bayesian Estimation . . . . . . . . . . . . . . . . . . . . 43

6 Conclusion 47

7 Appendix 48

Bibliography 50

Chapter 1

Introduction

A quantum channel is a completely positive and trace preserving (CPTP) map onthe set of linear operators acting on a Hilbert space [1]. It generalizes the notionof classical channel as stochastic map on the set of probability distributions (overan alphabet). It is used as a model for information communication.

In a communication context it is necessary to have a priori knowledge of theproperty of a channel. Therefore, it is important to know how to estimate thechannel property in an efficient way, that is, as precisely as possible with minimumresources [2]. Quite generally a channel can be considered parametrized in such away that the estimation reduces to that of a parameter, say θ taking values in a setΘ.

There are two basic approaches in statistical estimation theory that can beborrowed in the quantum framework [3]

• Consider the parameter θ of a random variable X distributed according top(x;θ) and try to estimate it through a statistics, i.e. through a real-valuedfunction of a sample X1, ...,Xn (random variables independently and identi-cally distributed as X). Such a statement about θ can be viewed as a sum-mary of the information provided by the data and may be used as a guide toaction (we refer to this method as Fisher estimation).

• The parameter θ is itself a random variable (though inaccessible) with agiven probability distribution. This prior distribution (specified accordingto the problem) is modified in the light of the data to determine a posteriordistribution (the conditional distribution of θ given the data), which summa-rizes what can be said about θ on the basis of the assumptions made and thedata. The data come from the observation of another (accessible) randomvariable (we refer to this method as Bayesian estimation).

1

CHAPTER 1. INTRODUCTION 2

In estimating a quantum channel parameter, given a finite amount of input re-source, both the input probe state and the measurement of the output state need tobe optimized. This double maximization problem has been studied in the contextof estimation SU(d) unitary operation [4]. Estimating a noisy quantum channelhas been discussed in the literature [5, 6, 7]. There, the usefulness of entanglement(quantum correlations that cannot be explained by any classical and local theory)has been also pointed out.

In all these works the quantum channel can be regarded as a unitary acting onthe input system A and on the environment system E, with a fixed state for E, andthen tracing away E [8]. Recently however a more general communication schemehas been proposed [9] where the state of E is under control of a "helper", whoseaim is to facilitate the communication between sender and receiver. The role ofsuch a helper is passive in that she/he will simply tune the state of E without beinginvolved in the information encoding process.

Following this line, here we preliminarily address the issue of estimating aquantum channel with environment assistance, by analysing quantum channelsarising from unitaries acting on a 2 dimensional Hilbert space A and a 2 dimen-sional Hilbert space E.

The main and original results are that the Fisher estimation is hardly applica-ble in this context, while the Bayesian approach shows that entanglement is notenhancing the performance of estimation, but the state of the environment can doit. Hence an optimization over it is performed. In passing, we have also found theoptimal input for estimation of depolarizing channel (without environment assis-tance) via Fisher estimation, which concurs with the results [2].

The thesis is organized as follows.In Chapter 2 we revisit basic notions of linear algebra in the light of Dirac

(quantum) formalism as well as rudiments of estimation theory. Subsequently inChapter 3 we provide some basics tools of quantum information theory. Then, inChapter 4 we review the parameter estimation for the depolarizing quantum chan-nel by resorting to both Fisher and Bayes estimation. In chapter 5 we first presentthe model of quantum channel with environment assistance and then investigateits estimation by resorting to both Fisher and Bayes approaches. Finally, we drawour conclusions in Chapter 6.

Chapter 2

Prelude

In this preliminary chapter we revisit some basic concepts in linear algebra inthe light of Dirac formalism and then introduce elementary notions of statisticalestimation theory.

2.1 Hilbert SpacesDefinition 2.1.1. A Hilbert space H is a vector space over the complex field Csupplied with a scalar product such that it is complete in the norm induced by thescalar product[10].

There the vectors are n-tuples of complex numbers that can be simply repre-sented as column vectors. In the Dirac notation the ket |φ〉 indicates one of suchvector

|φ〉=

φ1φ2...

φn

, φ j ∈ C (2.1.1)

The scalar product in Cn between two vectors

|φ〉=

φ1φ2...

φn

, |ψ〉=

ψ1ψ2...

ψn

, (2.1.2)

is defined as ∑nj=1 φ∗j ψ j ∈ C. By Dirac notation it is denoted as 〈φ |ψ〉, being 〈φ |

the bra, i.e. the dual of |φ〉. Thus the bra 〈φ | can be interpreted as a row vector

3

CHAPTER 2. PRELUDE 4

which is conjugate transpose of |φ〉, so that 〈φ |ψ〉 becomes a row by columnproduct.

The norm induced by the scalar product is

‖|φ〉‖=√〈φ |ψ〉. (2.1.3)

Due to the finite dimensionality of Cn, its completeness with respect to the normis guaranteed. A set of vectors {|i〉}i for which 〈i| j〉= δi j is clearly orthonormal.

An orthonormal basis {|ei〉}ni=1 is a set of orthonormal vectors that span the

entire space H . The canonical basis is

|e1〉=

10...0

, |e2〉=

01...0

, ... |en〉=

00...1

(2.1.4)

The outer product in Cn between two vectors |φ〉, |ψ〉 is defined as:

|φ〉〈ψ|=

φ1ψ1 φ1ψ2 ... φ1ψnφ2ψ1 φ2ψ2 ... φ2ψn

......

...φnψ1 φnψ2 ... φnψn

(2.1.5)

2.2 Operators on Hilbert SpacesDefinition 2.2.1. A linear operator A is a map A : H →H such that

A(a |φ1〉+b |φ2〉) = aA |φ1〉+bA |φ2〉 ,

for all |φ1〉 , |φ2〉 ∈H and for all a,b ∈ C.Given two vectors |v〉 , |w〉 ∈H beside their inner product we can also con-

sider their outer product |v〉〈w| which results a linear operator. In fact

(|v〉〈w|) |φ〉= (〈w|φ〉) |v〉

Let {|ei〉}ni=1 be an orthonormal basis of H , so that given |v〉 ∈H we can

write |v〉= ∑i vi |ei〉 with vi = 〈ei|v〉 ∈ C, then it is(n

∑i=1|ei〉〈ei|

)|v〉=

n

∑i=1

vi |ei〉 , ∀|v〉 ∈H


which implies the completeness relation

n

∑i=1|ei〉〈ei|= I (2.2.1)

As a consequence, given A : H →H , it is

A =n

∑i=1|ei〉〈ei|A

n

∑j=1|e j〉〈e j|

=n

∑i, j=1〈ei|A |e j〉 |ei〉〈e j| (2.2.2)

with Ai j := 〈ei|A |e j〉 are the entries of an n×n matrix A in the basis {|ei〉}ni=1.

Definition 2.2.2. Given A : H →H , its adjoint operator A† : H →H is definedby means of

〈v|Aw〉= 〈A†v|w〉 , ∀|v〉 , |w〉 ∈H ,

where |Aw〉= A |w〉 and |A†v〉= A† |v〉

It results that

〈v|Aw〉=〈w|A†v〉∗

〈v|Aw〉∗ =〈w|A†v〉〈v|A |w〉∗ =〈w|A† |v〉

and using an orthonormal basis {|ei〉}ni=1(

〈ei|A |e j〉)∗

= 〈e j|A† |ei〉

i.e. A∗i j = Ai j means that the adjoint of A is represented by the transpose andconjugate matrix representing A. It is also |v〉† = 〈v|.

Definition 2.2.3. An operator U : H →H such that UU† =U†U = I is calledUnitary. An operator A : H →H such that A = A† is called self-adjoint (orHermitian). If AA† = A†A then it is called Normal.

Notice that unitary operators preserve the scalar product, hence the norm. Thatis, the scalar product of U |v〉 with U |w〉 equals to that of |v〉 with |w〉, ∀|v〉 , |w〉 ∈H .


2.3 Spectral Properties of OperatorsDefinition 2.3.1. An eigenvector (eigenstate) of a linear operator A : H →H isa (non-null) vector |i〉 such that A |i〉 = ai |i〉, where ai ∈ C is the correspondingeigenvalue.

Theorem 2.3.1. Given A : H →H normal, if {ai, |i〉}i is the set of its eigenvalues-eigenvectors, then {a∗i , |i〉}i is the set of eigenvalues-eigenvectors of A†.

Proof:

det(A†−a∗i I) = det(A−aiI)† = [det(A−aiI)]∗ = 0

where det denotes determinant of a matrix.

As a corollary we have that Hermitian operators have real eigenvalues and uni-tary operators have uni-modular eigenvalues.

Theorem 2.3.2. (Spectral Theorem) Given A : H → H normal, there existsan orthonormal basis given by its eigenvectors, in which its matrix represen-tation is diagonal. In other words, there exists an orthonormal basis given byits eigenvectors, in which its matrix representation is diagonal. In other words,there exists a unitary operator U : H →H such that A = UAdU†, with Ad =diag{a1,a2, ...,an} being a j, j = 1, ...,n the eigenvalues of A.

Proof: Let {ai, |i〉}i be the set of eigenvectors and eigenvalues of A, A can bewritten as:

A =n

∑i, j=1

(〈i|A | j〉) |i〉〈 j| (2.3.1)

construct U as

U =n

∑i, j=1|i〉〈 j| (2.3.2)

Then

U†AU =

(n

∑i, j=1| j〉〈 j|

)n

∑i, j=1

(〈i|A | j〉) |i〉〈 j|

(n

∑i, j=1|i〉〈 j|

)

=n

∑i= j=1

ai |i〉〈 j|= Ad

Using invertibility of UA =UAdU†.


A projection operator is an operator that can be formed by writing the outer prod-uct using a vector. That is, given |ϕ〉, the operator

P = |ϕ〉〈ϕ| (2.3.3)

is a projection operator. A projection operator is Hermitian. If the vector |ϕ〉 isnormalized, then a projection operator is equal to its own square:

P2 = P (2.3.4)

If P1 and P2 are projection operators that commute, meaning that P1P2 = P2P1,then their product is also a projection operator. Suppose that a given vector spacehas n dimensional and a basis given by |1〉 , |2〉 , ..., |n〉 and that n≥m. The operatorgiven by

P =m

∑i=1|i〉〈i| (2.3.5)

projects onto the subspace spanned by the set |1〉 , |2〉 , ..., |m〉.In summary, given A : H → H normal and {ai, |i〉}i are its eigenvalues-

eigenvectors, we can express A as (spectral decomposition)

A = ∑i

ai |i〉〈i|= ∑i

aiPi (2.3.6)

with Pi the projector onto the eigenvector |i〉’s subspace. They satisfy the relations

PiPj = Piδi j and P†i = Pi (2.3.7)

and completeness relation∑

iPi = I (2.3.8)

We denote by L (H ) the set of linear operators A : H →H . Given that a normaloperator A : H →H can be written as A=UAdU† with Ad = diag{a1,a2, ...,an},a function f : L (H )→ C acting on A must be intended as

f (A) =U f (Ad)U†,

with f (Ad) = diag{ f (a1) , f (a2) , ..., f (an)}

2.4 Statistical EstimationA standard problem in statistical estimation is to determine the parameters of a dis-tribution from a sample of data drawn from that distribution. Let {p(x,θ)},θ ∈Θ, denote an indexed family of densities for a random variable X , p(x,θ) ≥0,∫

p(x,θ)dx = 1 for all θ ∈Θ. Here Θ is called the parameter set. [11]


2.4.1 Fisher estimatorDefinition 2.4.1. An estimator for θ for the sample size n, (X1, ...,Xn), is a func-tion T : (X1, ...,Xn)→Θ, where (X1, ...,Xn) are identically and independently dis-tributed.

An estimator is meant to approximate the value of the parameter. Thereforeit is desirable to have some idea of the goodness of the approximation, hence theestimator. The difference T −θ is called error of the estimator. The error is itselfa random variable.

Definition 2.4.2. The bias of an estimator T for the parameter θ is the expectedvalue of the error of the estimator, i.e. Eθ (T −θ). The subscript θ means thatthe expectation is with respect to the density p(·;θ). The estimator is said to beunbiased if

Eθ (T ) = θ (2.4.1)

i.e., the bias is zero for all θ ∈Θ.

The bias is the expected value of the error, and the fact that it is zero does notguarantee that the error is low with high probability. We need to look at some lossfunction of the error. A good estimator should have a low expected squared errorand should have an error that approaches 0 as the sample size goes to infinity.

Definition 2.4.3. The Fisher information J(θ) is defined as:

J(θ) = Eθ

[∂

∂θln p(x;θ)

]2

(2.4.2)

The significance of the Fisher information is shown in the following theorem:

Theorem 2.4.1. (Cramer-Rao inequality) The mean-squared error of any unbi-ased estimator T (X) of the parameter θ is lower bounded by the reciprocal of theFisher information:

var(T )≥ 1J(θ)

. (2.4.3)

Proof: Consider data X with the density pθ (x). Let us denote V = ∂

∂θln pθ (x)=

∂

∂θpθ (x)

pθ (x)and let T (X) be an unbiased estimator for θ i.e.,

Eθ T (X) =∫

T (x)pθ (x)dx = θ ∀θ ∈Θ (2.4.4)


Differentiating (2.4.4) with respect to θ , we get

1 =∫

T (X)p′θ (x)dx =

∫T (x)

(∂

∂θln pθ (x)

)pθ (x)dx

=Eθ

(T (X)

∂

∂θln pθ (x)

)= Eθ (TV ) (2.4.5)

the mean value of V is:

EV =∫ ∂

∂θpθ (x)

pθ (x)pθ (x)dx

=∫

∂

∂θpθ (x)dx

=∂

∂θ1

=0 (2.4.6)

from (2.4.5) and (2.4.6) we have

1 = Eθ (TV ) = Eθ (V −EθV )(T −Eθ T ) (2.4.7)

By the Cauchy-Schwarz inequality

1 =(Eθ (V −EθV )(T −Eθ T ))2

≤Eθ (V −EθV )2 (T −Eθ T )2

=J(θ)var(T ) (2.4.8)

This completes the proof.Cramer-Rao inequality gives us a lower bound on the variance for all unbiased

estimators

Definition 2.4.4. An unbiased estimator T is said to be efficient if it meets theCramer-Rao bound with equality i.e., if var(T ) = 1

J(θ) .

Unbiasedness is a very desirable property of an estimator. Nevertheless, theapplicability of this condition is limited. There is an alternative approach whichis more generally applicable. Instead of seeking an unbiased estimator, one canmore modestly ask that the squared error be low only in some overall sense:∫

(T −θ)2w(θ)dθ (2.4.9)

minimize the average of the squared error for some weight function w(θ). Thisimplies that the average of bias is close to zero.

The estimator minimizing (2.4.9) formally coincides with the Bayes estimator,which we are going to introduce now, when θ is assumed to be a random variablewith probability density function w(θ)


2.4.2 Bayes estimatorSuppose that the unknown parameter θ is considered to be a random variablehaving a prior density function p(θ). Then p(x;θ) is now viewed as a conditionaldensity and written as p(x|θ), and we can express the joint density function of therandom sample (X1, ...,Xn) and θ as

p(x1, ...,xn,θ) = p(x1, ...,xn|θ) p(θ) (2.4.10)

and the marginal density function of the sample is given by

p(x1, ...,xn) =∫

Rθ

p(x1, ...,xn,θ)dθ (2.4.11)

where Rθ is the range of the possible value of θ . The other conditional densityfunction,

π(θ) = p(θ |x1, ...,xn) =p(x1, ...,xn,θ)

p(x1, ...,xn)=

p(x1, ...,xn|θ) p(θ)p(x1, ...,xn)

(2.4.12)

is referred to as the posterior density of θ . Thus the prior density p(θ) representsout information about θ prior to the observation of the outcomes of X1, ...,Xn, andthe posterior density function p(θ |x1, ...,xn) represents out information about θ

after having observed sample[3].Quite generally, we suppose that the consequences of estimating θ by a value

θ̂ are measured by cost function (loss function) C(θ , θ̂). Of the cost function, weshall assume that:

C(θ , θ̂)≥ 0 ∀θ , θ̂ (2.4.13)(2.4.14)

There are two commonly used cost functions, which are defined as:

• Quadratic cost function, C(θ , θ̂) = (θ − θ̂)2

• Absolute cost function, C(θ , θ̂) = |θ − θ̂ |

Definition 2.4.5. Given a cost function C(θ , θ̂), an estimator T minimizing∫C(θ , θ̂)π(θ)dθ (2.4.15)

is called a Bayes Estimator with respect to π(θ).


It is frequently reasonable to treat the parameter θ of a statistical problemas the realization of a random variable with known distribution rather than as anunknown constant.

To determine a Bayes estimator, first, consider the situation before any ob-servations are taken. Then, θ has prior distribution p(θ) and the Bayes estima-tor of θ is any number θ̂ minimizing average cost function E (C(θ ,T )). Oncethe data have been obtained and are given by the observed values of x of X , theprior distribution function is replaced by the posterior density π(θ) and the Bayesestimation is any number θ̂(x) minimizing the average posterior cost functionE (C(θ ,Tθ (x))|x) [12].

We are concerned here with the Bayes optimal strategy which minimizes theaverage cost.

Chapter 3

Quantum Information TheoryTools

In this Chapter we start with the postulates of quantum mechanics and discussrelated concepts. This provides a suitable background to deal with the problem ofquantum channels’ parameters estimation.

3.1 Postulates of Quantum MechanicsIn this work we will only consider finite dimensional Hilbert spaces which areisomorphic to Cn i.e. H w Cn.

Postulate 3.1.1 To an isolated quantum system is associated a space of statesH which is a Hilbert space. The state of the system is then represented by a unitnorm vector |φ〉 on such Hilbert space.

The simplest quantum system we can consider has associated Hilbert spaceC2. Hence a generic qubit state is described by a vector |φ〉 = α |0〉+β |1〉 nor-malized such that |α|2 + |β |2 = 1. The state of a qubit is a vector in a two dimen-sional complex vector space C2. Because |α|2 + |β |2 = 1, we can write a qubitas |φ〉 = eiγ(cos θ

2 |0〉+ eiϕ sin θ

2 |1〉), where θ ,ϕ,γ are real numbers. Since theuni-modular factor eiγ has no observable effects, we can parameterize a qubit as

|φ〉= cosθ

2|0〉+ eiϕ sin

θ

2|1〉 (3.1.1)

where θ ∈ [0,π] and ϕ ∈ [0,2π]. This means that the set of states {|φ(θ ,ϕ)〉}θ ,ϕ

corresponds to the surface of a 2-sphere S2 ⊂ R3 of radius one (known as theBloch sphere).

12

CHAPTER 3. QUANTUM INFORMATION THEORY TOOLS 13

Postulate 3.1.2 The space of states of a composite physical system is the tensorproduct of the state spaces (Hilbert spaces) of its subsystems.

Suppose that H1 and H2 are two Hilbert spaces of dimension d1 and d2, wecan construct a larger Hilbert space H = H1⊗H2 which has dimension d1d2.Now question is how to represent state vectors in Hilbert space of the compositesystem. Let |φ〉 ∈H1 and |ψ〉 ∈H2 be two vectors that belong to the Hilbertspaces of the subsystems, then |ϕ〉= |φ〉⊗ |ψ〉 is a vector in H , which is a statevector of the composite physical system. Using linearity of tensor product, wecan expand this to an arbitrary vector from H1 and an arbitrary vector from H2 interms of the basis states in each of the respective Hilbert spaces. Denote the basisof H1 by |ui〉 and the basis of H2 by |v j〉, then

|φ〉= ∑i

αi |ui〉 , |ψ〉= ∑j

β j |v j〉

|ui〉⊗ |v j〉 forms a basis of the composite system H , then |ϕ〉 ∈H is simply:

|ϕ〉= ∑i, j

αiβ j |ui〉⊗ |v j〉

Postulate 3.1.3 The state change of an isolated physical system with associ-ated Hilbert space H is described by a unitary operator U : H →H

|φ〉 → |φ〉′ =U |φ〉

Postulate 3.1.4 To an observable physical quantity A is associated an Hermi-tian operator in the space of states (Hilbert space H ), i.e. A : H →H such thatA = A†. The possible measurement outcomes are the eigenvalues {a j} j (a j ∈R)of A. The probability that the outcome is a j, given that the system was in the state|φ〉, is

pφ (a j) = 〈φ | Pj |φ〉

with Pj =∣∣a j⟩⟨

a j∣∣ the projector onto the subspace of eigenvector

∣∣a j⟩

of A cor-responding to the eigenvalue a j .As consequence of a measurement outcome a j, the state of the system changes asfollows

|φ〉 → |φ〉′ =Pj |φ〉√pφ (a j)

. (3.1.2)


3.2 Density operator and POVM

3.2.1 Density operatorDefinition 3.2.1. Given a statistical mixture (ensemble) of quantum states {pi, |φi〉}i, where pi is probability mass function, on an Hilbert space H , the correspondingdensity (statistical) operator is defined as ρ := ∑i pi |φi〉〈φi|, where |φi〉 ∈H aregeneric states, not necessarily orthogonal.

Definition 3.2.2. If the rank of the density operator ρ ∈L (H ) is 1, the state ofthe system is said to be pure. If the rank of the density operator ρ ∈L (H ) isstrictly greater than 1, the state of the system is said to be mixed (maximally mixedif the rank equals to the dimension of H ).

If the state is pure, then it is known exactly. This is a state where single pi = 1and all others are zero. This means that the density operator for a pure state |φ〉can be written as

ρ = |φ〉〈φ | (3.2.1)

In this case the density operator satisfies ρ2 = ρ , which implies that it is a rank-1projection operator. A mixed state is a collection of different pure states, each ofthem occurring with a given probability.

Properties of the density operator

* The density operator is Hermitian, ρ = ρ†

* The trace of any density operator is 1

* For a pure state, since ρ2 = ρ,Tr(ρ2) = 1

* For a mixed state Tr(ρ2)< 1

* The eigenvalues of a density operator satisfy 0≤ λi ≤ 1

3.2.2 POVMAs it shown in Postulate 3.1.4, not like the classical mechanics where measure-ment generally does not have effect on the system, in quantum mechanical systemmeasurement has a profound impact, changing the state in an irreversible way.

Suppose we start with a physical system upon which we wish to perform mea-surements. The outcome of any given measurement depends upon the quantumstate of the system. According to the postulates in quantum mechanics, we as-sociate an Hilbert space H to the system. With each physical measurement that


can be performed upon the system, we associate an Hermitian operator M thatmaps H to itself. Hermitian operators have sets of orthonormal eigenvector andassociated real eigenvalues. An Hermitian operator can be written as follows:

M = ∑m j |m j〉〈m j| (3.2.2)

where m j is real, M |m j〉= m j |m j〉 for all j and 〈m j|mk〉= δk j . If a measurementM with associated Hermitian operator with eigenvectors |m j〉 and eigenvalues m jis performed on a system with mathematical state |x〉, the outcome of the mea-surement is a random variable. It must assume a value equal to an eigenvalue ofthe Hermitian operator M, and the probability that it assumes, the particular valuemk , is given by

Pr(mk|state = |x〉) = 〈mk|x〉〈x|mk〉 (3.2.3)

Suppose a measurement M with associated Hermitian operator M is performedupon a system whose state is known only statistically. That is, the state is knownto be a member of a set {|xk〉}. The probability that the state is |x j〉 is p j. We canthen use the postulates to write the probability that the outcome of measurementM is the particular eigenvalue m j:

Pr(mk|statistical state knowledge), Pr(mk)

= ∑ p j 〈mk|x j〉〈x j|mk〉 . (3.2.4)

Define the Hermitian operator ρ as follows:

ρ = ∑ p j |x j〉〈x j|= ∑ρ j |ρ j〉〈ρ j| (3.2.5)

where〈ρ j|ρk〉= δk j, ρ j ≥ 0, ∑ρ j = 1.

Note that |xk〉 are not necessarily orthogonal. On the right, we have written ρ inspectural decomposition. From (3.2.4) it is clear that

Pr(mk) = 〈mk|ρ|mk〉

Finally, it is straightforward to show that if f (t) is analytic (i.e., it has a powerseries expansion about t = 0), then the expected value of f (outcome of measure-ment M) is

E{ f (outcome)}= ∑ f (mk)Pr(mk) = Tr[ρ f (M)] (3.2.6)

where Tr(.) is the operation "trace" and f (M) is the power series expansion of fusing powers of M. This can be generalized by following notion–POVM.


Definition 3.2.3. Given a Hilbert space H , let us take a set of operators {Mm}m⊂L (H ) such that M†

mMm ≥ 0 and ∑m M†mMm = I, then the set {Mm}m is a proba-

bility operator value measure (POV M) [10].

Given the system on the state ρ , the possible measurement outcomes are la-belled by the index ’m’ and occurs with probability

Pr(m) = Tr(M†mρMm) (3.2.7)

As consequence of a measurement outcome the state of the system changes as

ρ → ρ′=

MmρM†m√

Pr(m). (3.2.8)

Now we review postulates in quantum mechanics in terms of density operator:

1. To an isolated quantum system is associated a Hilbert space H . Thestate of the system is then represented by a density operator on such Hilbertspace.

2. To a composite physical system is associated a Hilbert space that is the ten-sor product of the Hilbert spaces associated to its subsystems.

3. The state change of an isolated physical system with associated Hilbertspace H is described by a unitary operator U : H →H

ρ → ρ′=UρU†

4. A measurement process on H is described by means of POV M {Mm}m.

Notice that the projective measurement have been introduced in the Postulate 3.1.4are recovered for Mm becoming orthogonal projectors MmMm′ = δm,m′Mm. Insuch a case they represent the projectors onto the eigenvectors subspaces of somehermitian operator (observable) A whose expectation value reads EA = 〈A〉 =Tr(Aρ).

3.2.3 The Pauli representationPauli operators is a set of operators that is of fundamental importance in quantuminformation. They are defined as follows:

1. σ0 = I, the identity operator

σ0 |0〉= |0〉 , σ0 |1〉= |1〉 (3.2.9)


2. σx, it acts as follows:

σx |0〉= |1〉 , σx |1〉= |0〉 (3.2.10)

3. σy:σy |0〉=−i |1〉 , σy |1〉= i |0〉 (3.2.11)

4. σz, which acts as follows:

σz |0〉= |0〉 , σz |1〉=−|1〉 (3.2.12)

The Pauli operators constitute a basis for L (C2), the space of linear operators onC2. With respect to the canonical basis the matrix representation of Pauli operatorsσx,σy,σz are given by:

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

)(3.2.13)

The Pauli representation provides a means to write down a density operator ofa single qubit or two-qubit systems in terms of Pauli matrices. For a single qubitthe Pauli representation is given by

ρ =12 ∑

i=0,x,y,zciσi (3.2.14)

The coefficients in this expansion are given by

ci = Tr(ρσi)

The Pauli representation for a system of two qubits is given by

ρ =14 ∑

i, jci jσi⊗σ j (3.2.15)

where ci j = Tr(ρσi⊗σ j).

3.3 EntanglementEntanglement is one of the most unusual and fascinating aspects of quantum me-chanics. If two systems are entangled, this means that the values of certain proper-ties of one system are correlated with the values that those properties will assumefor another system. The properties can become correlated even when the twosystems are spatially separated [13].


3.3.1 Entanglement in pure statesFrom the postulates above it is clear that for any states |φ〉 ∈H1 and |ψ〉 ∈H2 thatbelong to the Hilbert spaces of the subsystems, the tensor product |ϕ〉= |φ〉⊗|ψ〉is a state vector of the composite physical system H . This naturally leads us tothe question: can any state of the composite system be written as a tensor productof the states of the subsystems?

Definition 3.3.1. The states |ΨAB〉 ∈HAB that can be written as |ΨAB〉= |ΨA〉⊗|ΨB〉 with |ΨA〉 ∈HA and |ΨB〉 ∈HB are called product states.

Those states |ΨAB〉 ∈HAB that are not product, i.e. |ΨAB〉 6= |ΨA〉 ⊗ |ΨB〉 forany |ΨA〉 ∈HA and |ΨB〉 ∈HB, are called entangled.

Notice that for product states it is assigned a precise state to each subsystemA and B (besides the state of the global system AB).This is no longer true for entangled states for which the state is precisely assignedonly to the global system AB.Hence, if the two systems are entangled, there is no way to characterize one with-out referring to the other.

3.3.2 Schmidt decompositionLet us now consider a generic state |ΨAB〉 ∈HAB = HA⊗HB with dimHA = dAand dimHB = dB, we will get

|ΨAB〉=dA

∑i=1

dB

∑j=1

ci j |iA jB〉 . (3.3.1)

The coefficients ci j ∈ C are called probability amplitudes, they are such that∑i, j |ci j|2 = 1 and can be rearranged in a matrix form as follows

C =

c11 c12 · · · c1dB

c21 c22 · · · c2dB...

... . . . ...cdA1 cdA2 · · · cdAdB

.

For a generic matrix with complex entries we can always write down the SingularValues Decomposition. If we apply this factorization to C, we obtain

C =UDV (3.3.2)

where


• U is a dA×dA unitary matrix,

• V is a dB×dB unitary matrix,

• D is a dA×dB diagonal matrix with non-negative real numbers on the prin-cipal diagonal. The diagonal entries,

√λa, of D are known as the singular

values of C.

The coefficients ci j read

ci j =min{dA,dB}

∑a=1

Uia√

λaVa j (3.3.3)

and we can use this result to rewrite the global state of the composite system |ΨAB〉as follows

|ΨAB〉=dA

∑i=1

dB

∑j=1

min{dA,dB}

∑a=1

Uia√

λaVa j |iA jB〉 .

Hence we have

|ΨAB〉=min{dA,dB}

∑a=1

√λa

(dA

∑i=1

Uia |iA〉)(

dB

∑j=1

Va j | jB〉)

where the vectors ∑dAi=1Uia |iA〉 = |ua〉A and ∑

dBj=1Va j | jB〉 = |va〉B form two new

basis, respectively for HA and HB.

Finally we can write the Schmidt decomposition of a pure bi-partite quantumstate:

|ΨAB〉=min{dA,dB}

∑a=1

√λa |ua〉A |va〉B (3.3.4)

where the coefficients√

λa, called the Schmidt coefficients, are such that ∑a λa = 1and they are also eigenvalues of C†C.

Definition 3.3.2. The Schmidt rank is equal to the number of non-zero Schmidtcoefficients (i.e. equals the rank of the matrix C ).

The Schmidt decomposition as it was defined for the state |ΨAB〉 of our bipar-tite system, give us a mathematical characterization of the entanglement:

• |ΨAB〉 is a separable state if and only if has Schmidt rank equal to one

• |ΨAB〉 is an entangled state if and only if it has Schmidt rank strictly greaterthan 1


It is worth noting that the maximum Schmidt rank equals d = min{dA,dB}.

Definition 3.3.3. (Maximally entangled state) States |ΨAB〉 having maximumSchmidt rank and Schmidt coefficients all equal to 1/d are called maximally en-tangled states:

|ΨAB〉=1√d

d

∑a=1|ua〉A |va〉B . (3.3.5)

Example. For bipartite systems the Hilbert space H =H1⊗H2 with dimH1 =dimH2 = 2 is spanned by the four Bell’s maximally entangled basis states

|Φ〉± =1√2(|00〉± |11〉), |Ψ〉± =

1√2(|01〉± |01〉) (3.3.6)

These states have remarkable properties namely the result of the measurementsfor both subsystems are perfectly correlated: we know nothing at all about thesubsystems, although we have maximal knowledge of the whole system becausethe state is pure.

3.3.3 Entanglement in mixed statesDefinition 3.3.4. States ρAB on HAB =HA⊗HB that can be written as the prod-uct of states ρA on HA and ρB on HB , i.e. ρAB = ρA⊗ ρB, are called productstates.

States ρAB on HAB = HA⊗HB that can be written as convex combination ofstates ρk

A on HA and ρkB on HB, i.e. ρAB = ∑k pkρk

A⊗ρkB, are called separable.

Those states ρAB on HAB =HA⊗HB that are not separable, i.e. ρAB 6=∑k pkρkA⊗

ρkB, are called entangled.

From the definition it is clear that product states are particular case of separablestates. Generally it is not easy to check if a mixed state is separable or entangled.

Theorem 3.3.1. . Given a density operator ρAB on HAB = HA⊗HB a necessarycondition for its separability is that ρTA ≥ 0 where TA denotes the transpositionwith only respect to the subsystem A (partial transposition).

Proof. By hypothesis ρAB is separable, hence it can be written as

ρAB = ∑k

pkρkA⊗ρ

kB

then it isρ

TAAB = ∑

kpk(ρ

kA)

T ⊗ρkB


where (ρkA)

T is the transpose of (ρkA). Being the latter positive, also its transposi-

tion is positive. Hence it is so (ρkA)

T .It is possible to prove that the criterion of positive partial transpose is nec-

essary and sufficient for a system of two qubits (Hilbert space isomorphic toC2⊗C2).

Given the Pauli representation of a density operator ρ (3.2.3), if it is separablestate, then

|c11|+ |c22|+ |c33| ≤ 1 (3.3.7)

3.4 Quantum ChannelDefinition 3.4.1. A quantum channel is a linear map

Φ : L (HA)→L (HB)

which is completely positive and trace preserving(CPTP), i.e. which satisfies fol-lowing properties:

i) linearity:Φ(aρ1 +bρ2) = aΦ(ρ1)+bΦ(ρ2)

for ρ1,ρ2 ∈L (HA) and a,b ∈ C.

ii) complete positivity,(Φ⊗ idR)(P)≥ 0,

for any P ∈L (HA)⊗L (HR) such that P≥ 0 and whatever is the extensionof the map Φ (i.e. whatever is the reference system R).

iii) preservation of traceTr(Φ(ρ)) = 1.

We have the following important result about the representation of quantumchannels.

Theorem 3.4.1. Kraus Representation Theorem A (linear) map Φ : L (HA)→L (HB) is CPTP if and only if

Φ(ρ) = ∑k

AkρA†k ,

for some set of operator {Ak}k which map HA into HB and such that ∑k A†kAk = I.


Proof: We first prove that any Φ written in the Kraus form is linear and CPTP.Linearity is evident. To demonstrate CP it suffices to show

〈ψ|(Φ⊗ id)(P) |ψ〉 ≥ 0,

for any |ψ〉 ∈HA⊗HR and for any positive operator P defined on HA⊗HR. ByKraus representation we have

∑k〈ψ|(Ak⊗ I)(P)(A†

k⊗ I) |ψ〉= ∑k〈φ |k P |φ〉k ,

where |φ〉k := (A†k⊗ I) |ψ〉 . Since P is positive the desired result follows immedi-

ately. Furthermore, ∑k A†kAk = I implies the preservation of trace.

Let us now prove that any (linear) CPTP map can be written in the Kraus form.Define

|ψ〉AR :=1√d

d

∑i=1|iA〉 |iR〉 ∈HA⊗HR, (3.4.1)

where d = dimHA = dimHB. Also define

|ψ̃AR〉 :=√

d |ψ〉AR =d

∑i=1|iA〉 |iR〉 , (3.4.2)

|ψA〉 :=∑i

ai |iA〉 ∈HA,ai ∈ C (3.4.3)

|ψ∗R〉 :=∑i

a∗i |iR〉 ∈HA,ai ∈ C (3.4.4)

Then we have the following "partial inner product"

〈ψ∗R|ψ̃AR〉= |ψA〉 . (3.4.5)

Hence, it is

Φ(|ψA〉〈ψA|) =Φ(〈ψ∗R|ψ̃AR〉〈ψ̃AR|ψ∗R〉)=〈ψ∗R|(Φ⊗ idR)(ρ̃AR)ψ

∗R〉

where ρ̃AR := |ψ̃AR〉〈ψ̃AR|. Since Φ is a CPTP map, the action of (Φ⊗ idR) onρ̃AR will give a density operator on L (HB⊗HR), say ρ̃

′BR. Then we can write

Φ(|ψA〉〈ψA|) = 〈ψ∗R|ρ̃′BR|ψ∗R〉

However the density operator ρ̃′BR can be associated to an ensemble of pure

states {λk, |α̃BRk 〉}k (which also provides its spectral decomposition), thus

Φ(|ψA〉〈ψA|) = ∑k

λk 〈ψ∗R|α̃BRk 〉〈α̃BR

k |ψ∗R〉 .


Now define a map Ak : HA→HB such that

Ak(|ψA〉) =√

λk 〈ψ∗R|α̃BRk 〉 .

It results a linear operator due to the linearity of the "partial inner product" of(3.4.5). As a consequence we have

Ak(|ψA〉) = ∑k

Ak |ψA〉〈ψA|A†k

Furthermore, being Φ trace preserving, we also have ∑k A†kAk = I.

It is worth remarking that since ρ̃′AR has a rank at most d2 (it has at most d2

non-zero eigenvalues), Φ possesses a Kraus representation with at most d2 Krausoperators.

Corollary Suppose {A1, ...,Am} and {B1, ...,Bn} are Kraus operators giving riseto the map Φ and Φ

′respectively. By appending zero operators to the shorter list

we may ensure that m = n. Then, Φ = Φ′if and only if there exists complex num-

bers ui j such that Ai = ∑ j ui jB j with ui j entries of a unitary transformation.

The Kraus representation can be thought as coming from a unitary (closed) evo-lution between the main system and an" environment" by disregarding the latterat the end, that is given a unitary U : HA⊗HE →HB⊗HF

Φ(ρ) = TrF [U(ρ⊗ρE)U†] (3.4.6)

Since ρE is fixed, the map Φ can be equivalently considered as coming froman isometry V : HA→HB⊗HF as

Φ(ρ) = TrF

(V ρV †

)(3.4.7)

Given Φ the isometry V is uniquely determined (up to unitaries from F to F). Thisis the content of the Stinespring’s Dilation Theorem [8].

3.5 Entanglement monotonesConsider a bipartite system with associated Hilbert space HA⊗HB (we considerHA 'HB and |HA|= |HB|= d).

Local operation and classical communication is:


* Perform local mechanical (local unitaries) operations on subsystems.

* Use classical communication to communicate measurement results.

A quantitative measure of entanglement E (also called entanglement mono-tone) must satisfy two fundamental properties [14]:

1. It must vanish for separable states, i.e. if ρAB is separable, then E (ρAB) = 0.

2. It cannot increase under Local Operation and Classical Communication(LOCC), i.e. if Λ : L (HA⊗HB)→ L (HA⊗HB) is a LOCC cptp act-ing as

Λ(ρAB) = ∑i(Ai⊗Bi)ρ

AB (Ai⊗Bi)† ,

thenE(

Λ(ρAB))≤ E (ρAB).

In the case of local unitary transformation we have the equality, i.e.

E((UA⊗UB)ρ

AB(UA⊗UB)†)= E (ρAB).

Other possible (not compulsory) properties are [14]:

1. Normalization, i.e. for ρ = |Φ〉〈Φ| with

|Φ〉= 1√d

∑i|ai〉|bi〉,

{|ai〉}i and {|bi〉}i being orthonormal bases of HA and HB respectively, itis E (ρAB) = logd.

2. Asymptotic continuity, i.e.

‖ρABn −σ

ABn ‖1→ 0⇒ |E (ρAB

n )−E (σABn )|

2logdn→ 0,

for sequence of states ρABn , σAB

n acting on Hilbert spaces HA,n⊗HB,n ofdimension d2

n .

3. Convexity, i.e. for ρAB = ∑i piρABi it is

E (ρAB)≤∑i

piE (ρABi ).


Here we introduce some measures of entanglement:

1. Negativity which is defined as

N(ρ) :=‖ρTA‖1−1

2(3.5.1)

and corresponds to the absolute value of the sum of negative eigenvalues ofρTA . Where :

• ρTA is the partial transpose of ρ with respect to subsystem A.

• ‖X‖1 = Tr|X |= Tr√

X†X is the trace norm.

2. Concurrence [15] which is defined for a mixed state of two qubits as:

C(|Ψ〉) = | 〈Ψ|σy⊗σy |Ψ∗〉 | (3.5.2)

|Ψ∗〉 denotes complex conjugation of |Ψ〉.

Chapter 4

Parameter Estimation ofQuantum Channels

Given an unknown quantum channel Γ, how can we identify it? Suppose the chan-nel to be parametrized as {Γθ (Ψ)}θ∈Θ (for the mathematical simplicity, we re-strict ourselves on smooth patametric family). Once an input state Ψ for the chan-nel is fixed, we have a parametric family {Γθ (Ψ)}θ∈Θ} of output state. Thus theproblem is reduced to a parameter estimation problem for a family {Γθ (Ψ)}θ∈Θ}.It can be stated as finding an optimal input state for the channal and an optimalestimator for the output state[5].

Furthermore, to conceive the possibility of exploiting entanglement in the es-timation process, we will also consider an extension of the channel as

Γθ ⊗ id, (4.0.1)

where id is the identity channel acting on L (H ).

4.1 Cramer-Rao boundIn statistics an estimator is Efficient Estimator if it is unbiased and attains theCramer-Rao lower bound [section 2.4.1]. When it comes to quantum channels, isthere any quantity similar to this?

Suppose θ is a real parameter coupled some quantum system through a knownconditional density operator ρθ defined on the associated Hilbert space H . AndΘ̂ is a random variable that takes values θ . Let Θ̃ be an Hermitian operator asso-ciated with some measurement Θ̃. The outcome of the measurement is to be anestimate of parameter θ . The bias of the estimate is defined as:

E(Θ̃outcome−θ) = Trρθ (Θ̃−θ I) (4.1.1)

26

CHAPTER 4. PARAMETER ESTIMATION OF QUANTUM CHANNELS 27

Theorem 4.1.1. If we have following condition, which means that the estimate isunbiased for θ in some interval about the point s0,

E(Θ̃outcome−θ) = Trρθ (Θ̃−θ I) = 0,|θ −θ0|< ε, ε > 0 (4.1.2)

then the variance of the difference between the estimate and the true value of θ

must satisfy the inequality when θ = θ0.

Var(Θ̃outcome−θ0) = Tr[ρθ0(Θ̃−θ0I)2]≥ J−1 (4.1.3)

where

J = Tr[ρθ0(L(θ0))2] (4.1.4)

and L(θ) satisfies∂

∂θρθ =

12[L(θ)ρθ +ρθ L(θ)] (4.1.5)

this is called the symmetric logarithmic derivative (SLD) [16].The inequality (4.1.3) is called quantum Cramer-Rao inequality. (4.1.4) is

called quantum Fisher information with L(θ) SLD[17].The lower boundJ−1 , the inverse of SLD Fisher information, in the Cramer-

Rao inequality is achievable (at least locally). In other words J−1 gives the ulti-mate limit of estimation. Hence the larger the SLD Fisher information is, the moreaccurately we can estimate the parameter θ . Thus, from this prospective of view,our problem becomes finding an optimal input for the channel that maximizes theSLD Fisher information of the corresponding parametric family of output states.

4.2 Optimal parameter estimation of depolarizingchannel with SLD Fisher information

Let H2 = C2 and let the channel Lθ : L (H2)→ L (H2) be the depolarizingchannel

Lθ ρ = θρ +1−θ

2I (4.2.1)

which maps a density operator ρ to density operator which is a mixture of ρ .Toassure the complete positivity of the channel Lθ , the parameter θ must lies in theinterval [−1

3 ,1].As we discussed before, now our task is maximize the SLD Fisher information

of the parametric family {Lθ (Ψ)}θ∈Θ} with respect to the input state Ψ. Theconvexity of the SLD Fisher information assures that the maximum is attained by


a pure state [5]. So the input probe we are considering is Ψ = |Ψ〉〈Ψ|. Representit in the Schmidt decomposition:

|Ψ〉=√

x |0〉⊗ |e0〉+√

1− x |1〉⊗ |e1〉 (4.2.2)

where x is a real number between 0 and 1, {|0〉 , |1〉} and {|e0〉 , |e1〉} are orthonor-mal basis for the first and second probe particle, respectively . Consider the ex-tension of the channel (4.0.1), input one qubit of the pair into the channel keepingthe other untouched and have the output state:

Ψout(θ) = (Lθ ⊗ I) |Ψ〉〈Ψ| , (4.2.3)

Matrix representation of the output state in under the canonical basis is:

Ψout(θ) =12

x(θ +1) 2

√(1− x)xθ 0 0

2√

(1− x)xθ (1− x)(θ +1) 0 00 0 (1− x)(1−θ) 00 0 0 x(1−θ)

(4.2.4)

The symmetric logarithmic derivative (SLD) given by (4.1.5) is:

Lθ =

3θ 2−8xθ+6θ−13θ 3+θ 2−3θ−1

4√

(1−x)xθ(2−3θ)+1 0 0

4√

(1−x)xθ(2−3θ)+1

8xθ

3θ 3+θ 2−3θ−1 +1

θ+1 0 00 0 1

θ−1 00 0 0 1

θ−1

(4.2.5)

where

∂

∂θΨout(θ) =

x2

√(1− x)x 0 0√

(1− x)x 1−x2 0 0

0 0 x−12 0

0 0 0 − x2

(4.2.6)

and the SLD Fisher information (4.1.4) results

Jθ =3θ −8x2 +8x+1−3θ 3−θ 2 +3θ +1

(4.2.7)

For every θ , the SLD Fisher information attains its maximum values 3(1−θ)(1+3θ)

at x = 12 . This implies that the optimal input is the maximally entangled state

|Ψ〉= 1√2(|0〉⊗ |e0〉+ |1〉⊗ |e1〉) (4.2.8)


4.2.1 Optimal measurement operatorTo find the optimal measurement operator, first let us consider the single qubitcase ρθ = Γ(σ), this is equivalent to let x = 1 or x = 0 in (4.2.2). Notice that hereany input can be seen as optimal. Choose optimal input as

σ =

(1 00 0

)(4.2.9)

The corresponding output is

ρσ =12

(1+θ 0

0 1−θ

)(4.2.10)

and the Fisher information is Jθ = 11−θ 2 . Suppose the measurement operator (the

estimator) is of the form

T =

(a b+ ic

b− ic d

)a,b,c ∈ R (4.2.11)

Impose unbiasedness and minimum variance, i.e. the Cramer-Rao lower boundon the estimator T :

Tr(ρθ T ) =θ (4.2.12)

Tr(ρθ (T −θ)2) =1−θ2 (4.2.13)

we have

12(a+d)+

12(a−d)θ =θ

12(a2 +d2)+

12(a2−2a−d2−2d)θ +

12(2−2a+2d)θ 2 =1−θ

2

that is

d =−a (4.2.14)a =1 (4.2.15)

with b,c are arbitrary. Setting them to zero for simplicity we have

T =

(1 00 −1

)(4.2.16)

From this result we conclude that the optimal input state must be an eigenstate ofT , i.e. σ = |ϕ〉〈ϕ| and T |ϕ〉= |ϕ〉.


Now let us move on to the two qubit case ρθ = (Γθ⊗ id)(σ) where the optimalinput is the maximally entangled state

σ =12

1 0 0 10 0 0 00 0 0 01 0 0 1

(4.2.17)

The corresponding output reads

σ =12

1+θ

2 0 0 θ

0 1−θ

2 0 00 0 1−θ

2 0θ 0 0 1+θ

2

(4.2.18)

From what we learnt about the single qubit case, we can assume the followingform for the operator T:

T =

a 0 0 b0 c 0 00 0 c 0b 0 0 a

a,b,c ∈ R (4.2.19)

which has |Φ+〉= (|00〉+ |11〉)/√

2 as eigenvector (notice that σ = |Φ+〉〈Φ+|).Impose

Tr(ρθ T ) =θ (4.2.20)

Tr(ρθ (T −θ)2) =1+2θ −3θ 2

3(4.2.21)

from (4.2.20) we get

a+ c =0 (4.2.22)12(a+2b− c) =1 (4.2.23)

from (4.2.21) we have

12(a2 +b2 + c2) =

13

(4.2.24)

12(a2 +b2− c2 +4ab−2a−2c) =

23

(4.2.25)

(1−a−2b+ c) =−1 (4.2.26)


with the hermiticity of T we the have following solution:

a = 1, b = 2, c =−1, (4.2.27)

Hence the measurement operator reads:

T =13

1 0 0 20 −1 0 00 0 −1 02 0 0 1

(4.2.28)

4.3 Bayesian approachOur estimation problem is to make measurements upon a system whose state de-pends statistically upon a parameter θ and to use the outcome to estimate θ , whichis not known a priori. The parameter is an assumed value of a random variablewhose a priori density p(θ) is given [section 2.4.2]. Suppose we have a randomvariable Θ̂ that takes on values on Θ with probability density p(θ). This ran-dom variable is coupled to the state of a quantum system through the conditionaldensity operator ρθ . We wish to perform a measurement Π(θ) with associatedHermitian operator Π upon the quantum system, the outcome of which is an esti-mate of θ . We wish to minimize the average cost for the quadratic cost functionwhich is given by

C̄(x) =∫

Θ

∫Θ

(θ − θ̂)2 p(θ)Tr[Π(θ̂)Ψ(θ)]dθ̂dθ (4.3.1)

where p(θ) is the a priori probability distribution of θ , Π(θ̂) is a probabilityoperator measure (POVM) and

∫Θ

Π(θ̂)dθ̂ = I. It is assumed that we have not apriori knowledge about θ , that is, p(θ) uniformly distributed. We want to find theoptimal probe |Ψ〉 and the POVM Π(θ̂) minimizing the average cost C̄(x). Forconvenience we introduce the risk operator

W (θ) =∫

p(θ)(θ − θ̂)2Ψ(θ)dθ =W (2)−2θW (1)+θ

2W (0) (4.3.2)

whereW (k) =

∫p(θ)θ k

Ψ(θ)dθ (4.3.3)

The average cost is then

C̄(x) = TrΓ, Γ≡∫

Π(θ)W (θ) (4.3.4)

For a fixed probe state |Ψ〉 the optimal POVM Π(θ) is derived from the nec-essary and sufficient conditions to minimize the average cost [18, 19]


(1) Γ = Γ†, and [W (θ)−Γ]Π(θ) = 0 for all θ ,

(2) W (θ)−Γ≥ 0 for all θ .

The optimal solution for a single parameter estimation with a quadratic cost iswell know [16, 20]. The optimal POVM constructed by finding the eigenstate of|θ〉 of the minimazing operator Θ̃ which is defined by

Θ̃W (0)+W (0)Θ̃ = 2W (1) (4.3.5)

That is, Π(θ) = |θ〉〈θ | so that Θ̃ |θ〉 = θ |θ〉. We then have Γ = W (2) −Θ̃W (0)Θ̃ from which the conditions (1) and (2) are easily verified.

For a discrete system, one can find the optimal POVM with finite elements.Let the spectral decomposition of W (0) for two qubit system be

W (0) =4

∑i=1

wi |wi〉〈wi| . (4.3.6)

Then the minimizing operator is

Θ̃ =4

∑i, j=1

2wi +w j

|wi〉〈wi|W (1) |w j〉〈w j| . (4.3.7)

Let the spectral decomposition of Θ̃ be

Θ̃ =4

∑i=1

θi |θi〉〈θi|

The optimal POVM is given by

Π(θ) =4

∑i=1

δ (θ −θi) |θi〉〈θi| (4.3.8)

This implies that the measurement has four outputs at most and we then estimatethe channel parameter as one of four θi’s.

4.4 Optimal parameter estimation of depolarizingchannel with quadratic cost function

Consider the state (4.2.2) as the input into the depolarizing channel (4.2.1) withextention (4.0.1). The output state Ψ(θ) reads as (4.2.4).


The average cost for the quadratic cost function (4.3.1) is given by

C̄(x) =∫ 1

− 13

∫ 1

− 13

(θ − θ̂)2 p(θ)Tr[Π(θ̂)Ψ(θ)]dθ̂dθ (4.4.1)

where p(θ) = 3/4. Furthermore, the elements of the risk operator (4.3.2) result:

W (0) =12

x(θ +1) 2

√(1− x)xθ 0 0

2√

(1− x)xθ (1− x)(θ +1) 0 00 0 (1− x)(1−θ) 00 0 0 x(1−θ)

(4.4.2)

W (1) =1

27

8x 7

27

√(1− x)x 0 0

7√

(1− x)x 8(1− x) 0 00 0 1− x 00 0 0 x

(4.4.3)

W (2) =1

27

6x 5

√(1− x)x 0 0

5√

(1− x)x 6(1− x) 0 00 0 1− x 00 0 0 x

(4.4.4)

The minimizing operator Θ̃ is:

Θ̃ =227

−4x+7(x−1)r+10 2

√(1− x)x(7−4r) 0 0

2√

(1− x)x(7−4r) −x(7r−4)−6 0 00 0 1

9 00 0 0 1

9

(4.4.5)

with r =√

3(x−1)x+1


After diagonalizing Θ̃ we have the following eigenstates and eigenvalues:

|θ1〉=

2x+√

1−12(x−1)x−1

4√

(1−x)x

100

θ1 =19

(3+√

1−12(x−1)x)

|θ2〉=

2x−√

1−12(x−1)x−1

4√

(1−x)x

100

θ2 =19

(3−√

1−12(x−1)x)

|θ3〉=

0010

θ3 =19

|θ4〉=

0001

θ4 =19

The average cost

C̄1(x) = Tr(W (2)− Θ̃W (0)Θ̃) =

881

(1+(x− 12)2) (4.4.6)

is minimized by the maximally entangled input state,

|Ψ〉= 1√2(|0〉⊗ |e0〉+ |1〉⊗ |e1〉)

i.e., when x = 12 , for which θ1 =

59 and θ2 = θ3 = θ4 =

19 . Therefore the optimal

measurement is actually constructed by the two projectors

Π1 = |Ψ〉〈Ψ| , Π2 = I−|Ψ〉〈Ψ| (4.4.7)

with the associated guess θ1 = 59 and θ2 = 1

9 respectively. And the minimumaverage cost reads C̄1(x) = 8

81

Chapter 5

The Environment AssistanceModel

We introduce a communication model where there is a third party, other than thesender and receiver, who has access to the environment input system. We theninvestigate the role of environment in channel estimation. This will be done byrestricting to qubit channels having dilations in one qubit environments. Hencewe start characterizing two-qubits unitaries.

5.1 Parameterization of two qubits unitariesEvery quantum channel can be viewed as arising from the unitary interaction ofa system with its environment [section 3.4]. The resulting entanglement betweensystem and environment is lost when the environment is ’traced out’ thereby,destroying the purity of the signal states and introducing noise into the system.Specifically, letting A denote the system and E its environment, the action of thechannel Ψ on A is obtained as

Φ(ψ) = Tr[UAE (ψ⊗η)U†

AE

](5.1.1)

where the unitary matrix UAE entangles the system and environment, and η is astate in E.

Below we will restrict to unitaries U acting on two qubits Hilbert space HAE =HA⊗HE = C2⊗C2 [21].

Lemma For any unitary operator UAE , there exist local unitary operators UA,UE ,VA,VEand a unitary operator Ud = e−iH such that

UAE =UA⊗UEUdVA⊗VE , (5.1.2)

35

CHAPTER 5. THE ENVIRONMENT ASSISTANCE MODEL 36

where Ud is diagonal matrix. Denote the diagonal elements of Ud by αx,αy,αz,and H = αxσx⊗σx +αyσy⊗σy +αzσz⊗σz, with σx,σy,σz the Pauli matrices[section3.2.3][21]. Constructive proof of this lemma is given in [22]

Let us define

|Φ1〉= |Φ〉+ , |Φ2〉=−i |Φ〉− , |Φ3〉= |Ψ〉− , |Φ4〉=−i |Ψ〉+ . (5.1.3)

which is called Magic Basis, defined in the same way as the Bell Basis(3.3.6)except for some global phases. Hence we have

H = ∑k={x,y,z}

αkσk⊗σk =4

∑j=1

λ j |Φ j〉〈Φ j| (5.1.4)

Thus operator Ud is diagonal in the magic basis and of the form

Ud =4

∑k=1

e−iλk |Φk〉〈Φk| (5.1.5)

where the eigenvalues (phases) λ j are

λ1 =αx−αy +αz, λ2 =−αx +αy +αz,

λ3 =−αx−αy−αz, λ4 = αx +αy−αz (5.1.6)

λ4 ≥ λ1 ≥ λ2 ≥ λ3. (5.1.7)

Notice that the real numbers αx,αy,αz are not unique. This is so for two rea-sons: Firstly operators of the type ±σ ⊗σ are local and commute with H so thatwe can always extract such a local operator from the local parts in this decom-position and include it in H. This alters the corresponding coefficient αx,αy,αzby ±π

2 . Secondly there are certain local transformations of H which conserve itsform but permute the coefficients α and change the sign of two of them. The localunitaries which cause such a transformation are of the types ±iσ⊗ I and±iI⊗σ .So we can restrict on

π

4≥ αx ≥ αy ≥ |αz| (5.1.8)

Furthermore, the maximal amount of entanglement created by Ud is symmetricaround π

4 in αx,αy,αz .To show this we define matrices d with diagonal elements{π

4 +αx,αy,αz} and d̃ with diagonal elements {π

4 −αx,αy,αz}, then we have

Ud =−iσAx U∗d̃ σ

Ex

where U∗d̃

denotes the complex conjugate of Ud̃ , in the canonical basis. Since localunitary operators do not change the entanglement

E (Ud |Ψ〉) = E (U∗d̃ σEx |Ψ〉)


Moreover, since all measures are determined by the Schmidt coefficients and theyare real, we have E (|Ψ〉) = E (|Ψ∗〉). Thus

E (Ud |Ψ〉) = E (Ud̃(σEx |Ψ〉)∗)

It is clear that the maximal amount of entanglement created by Ud is the sameas the one created byUd̃ , Using the same argument for other angles, proves thesymmetry [2].

Because any measure of entanglement is not changed by local unitary oper-ators, the maximal amount of entanglement that can be produced by applying ageneral unitary UAE is the same as the one created by Ud . This means that we haveto deal with unitaries that are determined by only three parameters (αx,αy,αz),which are used in order to describe a general unitary operator working on twoqubits[22].

Hence, when studying entanglement created by a two qubits unitaries, we canrestrict ourselves to the case

π

2≥ αx ≥ αy ≥ αz ≥ 0 (5.1.9)

This means that the noise of the channel can be entirely attributed to losinginformation into the environment, which raises the question how much better onecould communicate over the channel if one had access to the environment.

5.2 Estimation of two qubit unitaries with environ-ment assistance

R

η

A

E F

B

out

U

in

Figure 5.1: Quantum channel with environment assistance model. The quantumchannel maps between input system A and output system B arises from a unitaryU by tracing out F . The input system A can be entangled with a reference systemR. The environment system E can be controlled by the helper.


Consider quantum channel from system A (input) to system B (output) asshown in Fig.5.1.

A general two-qubit unitary interaction can be described by 15 real parame-ters. But as we showed in Sec.5.1., this can be reduced to 3. Hence the parameterspace given by

S = {(αx,αy,αz) :π

2≥ αx ≥ αy ≥ αz ≥ 0} (5.2.1)

describes all two-qubits unitaries up to local basis choice and complex conjuga-tion. The parameters {αx,αy,αz} are π

2 periodic and symmetric around π

4 . Theyform a tetrahedron with vertices (0,0,0),

(π

2 ,0,0),(

π

2 ,π

2 ,0)

and(

π

2 ,π

2 ,π

2

)shown

in Fig.5.2. Familiar two-qubits unitaries (gates) can easily be identified withinthis parameter space: for instance, (0,0,0) represents the identity I,

(π

2 ,0,0)

rep-resents the CNOT,

(π

2 ,π

2 ,0)

the DCNOT (double controlled not), and(

π

2 ,π

2 ,π

2

)the SWAP gate, respectively [9].

For the sake of simplicity, in our model, we only consider controlled unitaries(αx,0,0) which lie on a edge of the tetrahedron

x

y

z

0,0,0

0,0,

2

0,

2,2

2,2,2

Figure 5.2: Tetrahedron representing all two-qubit unitaries. We consider the edgelying along αx axis

Replacing αx by θ we deal with (controlled) unitary having spectrum {eiθ ,eiθ ,e−iθ ,e−iθ}and parameterization (θ ,0,0). The parameter θ lies in the interval 0≤ θ ≤ π

2 . Ex-plicitly such unitaries can be written as:

Uθ = e−iθ2 σx⊗σx = (cos

θ

2)I− isin(

θ

2)σx⊗σx (5.2.2)


Then the goal is to estimate the parameter θ from the channel arising fromsuch unitaries by using an input state possibly entangled across systems A and R|ψ〉=

√x |r0〉⊗|a0〉+

√1− x |r1〉⊗|a1〉 and a state of environment |η〉= α |0〉+

β |1〉. Hence the input state for the system RAE reads

|Ψ〉in =(√

x|r0a0〉+√

1− x|r1a1〉)(α|0〉+β |1〉) , (5.2.3)

where {|r0〉, |r1〉} , {|a0〉 , |a1〉} are canonical basis for the system R and system Arespectively. By the action of IR⊗UAE , |ψ〉in transforms into

|Ψ〉out =

[1+ eiθ

2

(α√

x|r0a0〉+α√

1− x|r1a1〉)+

1− eiθ

2

(β√

x|r0a1〉+β√

1− x|r1a0〉)]|0〉

+

[1+ eiθ

2

(β√

x|r0a0〉+β√

1− x|r1a1〉)+

1− eiθ

2

(α√

x|r0a1〉+α√

1− x|r1a0〉)]|1〉.

DefineΨ := TrF |ψ〉out〈ψ|. (5.2.4)

Its matrix representation Ψ(θ), under the basis {|r0a0〉, |r0a1〉, |r1a0〉, |r1a1〉} reads

14

∣∣1+ eiθ

∣∣2 x γξ x γξ√

x(1− x)∣∣1+ eiθ

∣∣2√x(1− x)−γξ x

∣∣1− eiθ∣∣2 x

∣∣1− eiθ∣∣2√x(1− x) −γξ

√x(1− x)

−γξ√

x(1− x)∣∣1− eiθ

∣∣2√x(1− x)∣∣1− eiθ

∣∣2 (1− x) −γξ (1− x)∣∣1+ eiθ∣∣2√x(1− x) γξ

√x(1− x) γξ (1− x)

∣∣1+ eiθ∣∣2 (1− x)

,

(5.2.5)where it has been set ξ := α∗β +αβ ∗, γ = eiθ − e−iθ .

5.2.1 Fisher EstimationFor the output state (5.2.5) we compute ∂

∂θΨ̂(θ) resulting in

12

−xsinθ ixξ cosθ iξ cosθ

√(1− x)x −sinθ

√(1− x)x

−ixξ cosθ xsinθ sinθ√

(1− x)x −iξ cosθ√(1− x)x

−iξ cosθ√(1− x)x sinθ

√(1− x)x sinθ(1− x) −iξ cosθ(1− x)

−sinθ√(1− x)x iξ cosθ

√(1− x)x iξ cosθ(1− x) sinθ(x−1)

(5.2.6)

Then the SLD, given by (4.1.5), becomes

Lθ =

(1−2x) tan( θ

2 )x 0 0

(x−1) tan( θ

2 )√(1−x)x

0(2x−1)cot( θ

2 )x

√1x −1cot

(θ

2

)0

0√

1x −1cot

(θ

2

)0 0

(x−1) tan( θ

2 )√(1−x)x

0 0 0

(5.2.7)


and the SLD Fisher information (4.1.4) is

Jθ = 1 (5.2.8)

The Cramer-Rao lower bound in our model thus results equal to 1, meaning thatthe minimum variance of the unbiased estimator would be 1. However, the exis-tence of an unbiased estimator is not guaranteed.

A general Hermitian operator (measurement operator) acting on two qubits isof the form:

a b1 +b2i c1 + c2i d1 +d2ib1−b2i j e1 + e2i f1 + f2ic1− c2i e1− e2i k h1 +h2id1−d2i f1− f2i h1−h2i g

(5.2.9)

For simplicity, we restrict our estimation problem in following cases:

I Factorable input between systems R and A and state |0〉 for E, correspondingto x = 1, ξ = 0:

|Ψ〉in = |r0a00〉 (5.2.10)

The resulting output state is14e−iθ (1+ eiθ)2 0 0 0

0 −14e−iθ (−1+ eiθ)2 0 0

0 0 0 00 0 0 0

(5.2.11)

with bias= 12((a− j)cos(θ)+a−2θ + j) as defined in Eq.(4.1.1). By equat-

ing the bias to 0 we can realize that there does not exist an unbiased estimator.We look for an estimator (measurement operator) that approximate θ withsmallest bias, in other words, let the bias close to zero as much as it possible.Since the bias is related to the unknown parameter θ , suppose we do not haveany knowlege about θ , θ is equally distributed in the interval [0, π

2 ], then wecan minimized the average error square∫ π

2

0

2π

bias2dθ (5.2.12)

Setting irrelevant elements to zero the measurement operator becomes:

T =

a 0 0 00 j 0 00 0 0 00 0 0 0

(5.2.13)


with eigenvalues {0,0,a, j}. Now our problem becomes find a, j such that themeasurement operator constructed by them minimizes the average squarederror (5.2.12) under following constraints:

0≤ a≤ π

20≤ j ≤ π

2

since the eigenvalues of the measurement operator are the estimation of θ ,and θ ∈ [0, π

2 ].

Optimal measurement operator is

T =

0.545268 0 0 0

0 1.5708 0 00 0 0 00 0 0 0

(5.2.14)

with average squared error 0.0933 and associated guesses

θ̂1 = 1.5708, θ̂2 = 0.545268, (5.2.15)

II Maximally entangled input between systems R and A and state |0〉 for E,corresponding to x = 1/2,ξ = 0:

|Ψ〉in =1√2(|r0a0〉+ |r1a1〉) |0〉 (5.2.16)

Out put reads:14(cos(θ)+1) 0 0 1

4(cos(θ)+1)0 1

4(1− cos(θ)) 14(1− cos(θ)) 0

0 14(1− cos(θ)) 1

4(1− cos(θ)) 014(cos(θ)+1) 0 0 1

4(cos(θ)+1)

(5.2.17)

optimal measurement operator is:

T =

a 0 0 d0 j e 00 e k 0d 0 0 g

(5.2.18)

with bias = 14(cosθ(a+2d−2e+g− j− k)+a+2d+2e+g+ j+ k−4θ)

as defined in Eq.(4.1.1). Also in this case there does not exist an unbiasedestimator. We thus proceed like in the case I.


The eigenvalues of T are:

12

(−√

a2−2ag+4d2 +g2 +a+g),

12

(√a2−2ag+4d2 +g2 +a+g

),

12

(−√

4e2 + j2−2 jk+ k2 + j+ k),

12

(√4e2 + j2−2 jk+ k2 + j+ k

)Optimal measurement operator is:

T =

0.795413 0 0 0.135942

0 0.851467 0.719329 00 0.719329 0.851467 0

0.135942 0 0 0.0232394

(5.2.19)

minimum average squared error 0.0933071, associated guesses

θ̂1 = 1.5708, θ̂2 = 0.818647, θ̂3 = 0.132138, θ̂4 = 0 (5.2.20)

The minimum average squared eror did not change, this implies that entan-glement can not improve the estimation.

III Factorable input between systems R and A and superposition of states for E,corresponding to x = 1,ξ =±1

|Ψ〉in =1√2|r0a0〉(|0〉± |1〉) (5.2.21)

The output state is:14e−iθ (1+ eiθ)2 1

4e−iθ (−1+ eiθ)(1+ eiθ) 0 0−1

4e−iθ (−1+ eiθ)(1+ eiθ) −14e−iθ (−1+ eiθ)2 0 0

0 0 0 00 0 0 0

(5.2.22)

Proceeding like in the previous cases I and II, the optimal measurement op-erator reads:

0.230038 0.55536i 0 0−0.55536i 1.34076 0 0

0 0 0 00 0 0 0

(5.2.23)

with average squared error 0.0142 and associated guesses

θ̂1 = 1.5708, θ̂2 = 0 (5.2.24)

We conclude that by changing the environment’s state we can get a better estima-tion in the sence of smaller bias.


5.2.2 Bayesian EstimationFollowing Sec.2.4.2 we now assume to have no prior knowledge about θ , that is,a prior probability distribution function of θ is p(θ) = 2

π.

Consider the output state (5.2.5) and risk operator (4.3.2) , the elements of riskoperator (4.3.3) read:

W (k) :=2π

∫π/2

0θ

kΨ(θ)dθ , (5.2.25)

we straightforwardly obtain

W (0) =

(2+π)x(2π) iξ x

πiξ√

x(1−x)π

(2+π)√

x(1−x)2π

−iξ xπ

(π−2)x2π

(π−2)√

x(1−x)2π

−iξ√

x(1−x)π

−iξ√

x(1−x)π

(π−2)√

x(1−x)2π

(π−2)(1−x)2π

−iξ (1−x)π

(2+π)√

x(1−x)2π

iξ√

x(1−x)π

iξ (1−x)π

(2+π)(1−x)2π

,

W (1)=

(4π+π2−8)x

8πiξ x

πiξ√

x(1−x)π

(4π+π2−8)√

x(1−x)8π

−iξ xπ

(8−4π+π2)x8π

(8−4π+π2)√

x(1−x)8π

−iξ√

x(1−x)π

−iξ√

x(1−x)π

(8−4π+π2)√

x(1−x)8π

(8−4π+π2)(1−x)8π

−iξ (1−x)π

(4π+π2−8)√

x(1−x)8π

iξ√

x(1−x)π

iξ (1−x)π

(4π+π2−8)(1−x)8π

,

and W (2) reads(π3+6π2−48)x

24πi (π−2)ξ x

πi (π−2)ξ

√x(1−x)

π

(π3+6π2−48)√

x(1−x)24π

i (2−π)ξ xπ

(48−6π2+π3)x24π

(48−6π2+π3)√

x(1−x)24π

i (2−π)ξ√

x(1−x)π

i (2−π)ξ√

x(1−x)π

(48−6π2+π3)√

x(1−x)24π

(48−6π2+π3)(1−x)24π

i (2−π)ξ (1−x)π

(π3+6π2−48)√

x(1−x)24π

i (π−2)ξ√

x(1−x)π

i (π−2)ξ (1−x)π

(π3+6π2−48)(1−x)24π

.

For a fixed probe state |Ψ〉 the optimal POVM is constructed by finding theeigenstate of |θ〉 of the minimizing operator Θ̃ which is defined by (4.3).

In our case equation(4.3) has solution, but not unique[23].


The general solution of the equation(4.3) is (see Appendix for the derivation):

Θ̃=

k1(1

x −1)+ (2x−1)δ

4xγk∗2(1

x −1)− i(2x−1)ξ τ

2xγ

(x−1)(

2k∗2+iξ τ

γ

)2√

(1−x)x

(x−1)(

4k1− δ

γ

)4√

(1−x)x

k2(1

x −1)+ i(2x−1)ξ τ

2xγk3(1

x −1)+ (2x−1)µ

4xγ

k3(x−1)√(1−x)x

+µ

√1x−1

4γ

(x−1)(

2k2− iξ τ

γ

)2√

(1−x)x

(x−1)(

2k2− iξ τ

γ

)2√

(1−x)xk3(x−1)√(1−x)x

+µ

√1x−1

4γk3 k2

(x−1)(

4k1− δ

γ

)4√

(1−x)x

(x−1)(

2k∗2+iξ τ

γ

)2√

(1−x)xk∗2 k1

(5.2.26)

where k1,k2,k3 are arbitrary constants in C and

δ = (π−2)(π

3 +4π2−8π−32ξ

2)γ = π

3−4πξ2−4π

µ = π4−2π

3 +16π−64ξ2

τ = π3−4π

2−8π +32

Average cost function with this general minimizing operator (5.2.26) is:

C̄(x) = Tr(W (2)− Θ̃W (0)Θ̃)

=π2 (π (96−16π +π3)−192

)−16

(−192+96π−6π3 +π4)ξ 2

48π2 (π2−4(ξ 2 +1))

which is not dependent on the variable x, and minimized whenξ =±1Since all those minimizing operators with arbitrary constants k1,k2,k3 give

the same minimum average cost, we construct the optimal measurement operatorΠ(θ) for the simplest case: k1 = k2 = k3 = 0 . In this case, the correspondingminimizing operator reads

Θ̃0 =

(2x−1)δ4xγ

− i(2x−1)ξ τ

2xγ

i(x−1)ξ τ

2γ√

(1−x)x(x−1)δ

4γ√

(1−x)x

i(2x−1)ξ τ

2xγ

(2x−1)µ4xγ

µ

√1x−1

4γ

−i(x−1)ξ τ

2γ√

(1−x)x

− i(x−1)ξ τ

2γ√

(1−x)x

µ

√1x−1

4γ0 0

− (x−1)δ4γ√

(1−x)xi(x−1)ξ τ

2γ√

(1−x)x0 0

(5.2.27)


The eigenvalues of Θ̃0 are:

θ01 =−16πξ 2 +2(π−4)r0 +π4−8π2 +16π

4π (π2−4(ξ 2 +1)), (5.2.28)

θ02 =−16πξ 2−2(π−4)r0 +π4−8π2 +16π

4π (π2−4(ξ 2 +1)), (5.2.29)

θ03 =

(8r0−2π

(8ξ 2 + r0−8

)+π4−8π2)(x−1)

4π (π2−4(ξ 2 +1))x, (5.2.30)

θ04 =

(−8r0 +2π

(−8ξ 2 + r0 +8

)+π4−8π2)(x−1)

4π (π2−4(ξ 2 +1))x(5.2.31)

wherer0 =

√64ξ 4 +(64−32π2 +π4)ξ 2 +π4

and the corresponding eigenvectors are:

|θ01〉=

√− x

x−1i(π2−8)ξ x

(−8ξ 2+r0+π2)√

(1−x)xi(8ξ 2+r0−π2)(π2−8)ξ

1

|θ02〉=

√− x

x−1

− i(π2−8)ξ x

(8ξ 2+r0−π2)√

(1−x)x

− i(−8ξ 2+r0+π2)(π2−8)ξ

1

|θ03〉=

x−1√(1−x)x

i(−8ξ 2+r0+π2)√

(1−x)x

(π2−8)ξ x

− i(−8ξ 2+r0+π2)(π2−8)ξ

1

|θ04〉=

x−1√(1−x)x

− i(8ξ 2+r0−π2)√

(1−x)x

(π2−8)ξ xi(8ξ 2+r0−π2)(π2−8)ξ

1

(5.2.32)

where |α|2 + |β |2 = 1, θ ∈ [0,π],φ ∈ [0,2π].the optimal state of environment which minimizes the average cost function

is: √2

2(|0〉± |1〉) (5.2.33)

The optimal input state is:

|Ψ〉= 1√2

(√x|r0a0〉+

√1− x|r1a1〉

)(|0〉± |1〉) , (5.2.34)


for which

θ̂01 =−2√

2π

+π

4+

1√2, θ̂02 =

14

(8√

2π−2√

2+π

),

θ̂03 =

(−2π√

2+8√

2+π2)(x−1)

4πx, θ̂04 =

(2π√

2−8√

2+π2)(x−1)

4πx(5.2.35)

Therefore the optimal measurement is

Πθ =4

∑k=1

δ (θ −θk)|θk〉〈θk|, (5.2.36)

with the associated guesses θ01,θ02,θ03,θ04.

The minimum average cost isπ(96−16π+π3)−192

48(π2−4)

Chapter 6

Conclusion

In conclusion, in this thesis we have addressed the problem of parameter esti-mation in quantum channels employing environment assistance. Specifically wehave considered quantum channels arising from unitaries acting on a 2 dimen-sional Hilbert space A and a 2 dimensional Hilbert space E after tracing awaythe latter. Such unitaries can be parametrized by three real parameters spanning atetrahedron on R3. Restricting our attention to one edge of the tetrahedron we endup with the estimation of a single parameter for which we exploited the freedomof the "helper" in tuning the state of the system E.

Using the Fisher estimation approach the Cramer-Rao bound has been de-rived. Then the optimal input state and measurement operator have been foundfor a small range of values of the parameter. Applicability of this method to theentire range seems hard due to the difficulties in finding an umbiased and efficientestimator. We have then opted for Bayesian estimation. With this approach theminimum average cost function turns out to be independent from the purity of theinput state on A, i.e. entanglement with a reference system R does not help in thiscase. However, the state of the environment plays a key role when minimizing theaverage cost function.

As future development we could foresee the extension of the present investiga-tion to all unitaries in the tetrahedron. Looking far afield a complementary schemecould be conceived in which we have feedback assistance in the estimation, that isonce fixed the unitary from systems A and E to systems B and F (or an isometryfrom system A to systems B and F), the helper can perform a measurement on Fand then communicate with the receiver B.

47

Chapter 7

Appendix

The vectorization of a matrix is a linear transformation which converts the matrixinto a column vector. Specifically, the vectorization of anm×n matrix A, denotedby VecA, is the mn× 1 column vector obtained by stacking the columns of thematrix A on top of one another:

VecA = [a1,1, . . . ,am,1,a1,2, . . . ,am,2, . . . ,a1,n, . . . ,am,n]T

We can write equation (4.3) in the form:

(I4⊗W (0)+W (0)T⊗ I4)VecΘ̃ = 2VecW (1) (7.0.1)

where I4 is 4×4 identity matrix. In this form, the equation can be seen as a linearsystem of dimension 42× 42. The general solution of this linear system can bewritten as:

Θ̃ = Θ0 + Θ̃0 (7.0.2)

where Θ0is a particular solution of (4.3), Θ̃ is the general solution of equation

Θ̃W (0)+W (0)Θ̃ = 0 (7.0.3)

48

CHAPTER 7. APPENDIX 49

The particular solution of (4.3) is:

(−2+π)(2x−1)(π(π(4+π)−8)−32ξ 2)4πx(π2−4(ξ 2+1))

− i(−4+π)(−8+π2)(2x−1)ξ2πx(π2−4(ξ 2+1))

−i(−4+π)(−8+π2)

√1x−1ξ

2π(π2−4(ξ 2+1))(−2+π)

√1x−1(π(π(4+π)−8)−32ξ 2)

4π(π2−4(ξ 2+1))i(−4+π)(−8+π2)(2x−1)ξ

2πx(π2−4(ξ 2+1))(2x−1)(−64ξ 2+π4−2π3+16π)

4πx(π2−4(ξ 2+1))√1x−1(−64ξ 2+π4−2π3+16π)

4π(π2−4(ξ 2+1))i(−4+π)(−8+π2)

√1x−1ξ

2π(π2−4(ξ 2+1))i(−4+π)(−8+π2)

√1x−1ξ

2π(π2−4(ξ 2+1))√1x−1(−64ξ 2+π4−2π3+16π)

4π(π2−4(ξ 2+1))00

(−2+π)√

1x−1(π(π(4+π)−8)−32ξ 2)

4π(π2−4(ξ 2+1))

−i(−4+π)(−8+π2)

√1x−1ξ

2π(π2−4(ξ 2+1))00

(7.0.4)

CHAPTER 7. APPENDIX 50

The null space of (7.0.3)

1x −1

00

x−1√(1−x)x

00000000

x−1√(1−x)x

001

,

01x −1x−1√(1−x)x

0000000000

x−1√(1−x)x

10

,

0000

1x −1

00

x−1√(1−x)xx−1√(1−x)x

0010000

,

00000

1x −1x−1√(1−x)x

00

x−1√(1−x)x

100000

(7.0.5)

Thus, we have the solution (5.2.26)

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Documents

Istituto Nazionale di Fisica Nucleare Contents 1 Introduction 1 2 Prelude 3 2.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.2 Operators on Hilbert Spaces