Lecture Notes PH 411/511 ECE 598 A. La Rosa Portland State University

INTRODUCTION TO QUANTUM MECHANICS _____________________________________________________________________________________ Underlying subject of the PROJECT assignment

QUANTUM ENTANGLEMENT and TELEPORTATION

Fundamentals: Properties of QUANTUM SYSTEMS Implementation: Optical experiments using down-conversion

nonlinear processes Application: Quantum states teleportation References: 1. Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and

William K. Wootters. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993).

2. D. Bouwmeester, J. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger. Experimental quantum teleportation. Phil. Trans. R. Soc. Lond. A 356, 1733-1737 (1998).

3. Charles H. Bennett. Quantum Information and Computation. Physics Today 48, 24-30 (October 1995).

4. Mark Beck, Quantum Mechanics, Theory and Experiments, Oxford University Press (2012).

5. EPR Paradox timeline Yoon-Ho Kim, S. P. Kulik, and Y. Shih. Quantum Teleportation with a Complete Bell State

Measurement. Journal of Modern Optics 49, 221-236 (2002). C. K. Hong, Z. Y. Ou, and L. Mandel. Measurement of subpicosecond time intervals between

two photons by interference. Phys. Rev. Lett. 59, 2044 (1987). Hong-Ou-Mandel(HOM) interference [1] between independent photon sources (HOMI-IPS) is at the heart of quantum information processing involving the quantum interference of single photons.

Fig. 1 Principle of quantum teleportation proposed by Bennett [Ref 1 and 3].

Fundamentals. Entanglement between quantum systems is a pure quantum effect describing correlations between systems that are much stronger and richer than any classical correlations can be. Originally this property was introduced by Einstein, Podolsky and Rosen, and by Schrodinger and Bohr, in the discussion on the completeness of quantum mechanics and by von Neumann in his description of the measurement process. Since then entanglement has been seen as just one of the features which makes quantum mechanics so counterintuitive. Applications: However, recently the new field of quantum information theory has shown the tremendous importance of quantum entanglement also for the formulation of new methods of

information transfer and for algorithms exploiting the capability of quantum computers. (See Fig. 1). Optical implementation: While quantum computers need entanglement between a number of quantum systems, basic quantum communication schemes only rely on entanglement between the members of a pair of particles, directly pointing at a possible realization of such schemes by means of correlated photon pairs as produced by optical parametric down-conversion processes (see Fig. 2). The following are exerts from References 3 and 4, which outlines in better detail the quantum teleportation application. More specific papers, addressing the different aspects of the teleportation strategy shown in the figure above, are provided in Projects section of this course. http://www.pdx.edu/nanogroup/projects-1. Those papers are intended to serve the seed topics for your project assignment. You can choose the one that fits better your quantum curiosity and interest.

Fig. 2 Implementation of quantum teleportation via optical means [Ref 2].

Fundamental Properties of QUANTUM SYSTEMS

Superposition: A quantum computer can exist in an arbitrary complex linear combination of states, which evolve in parallel according to a unitary transformation.

Interference: parallel computation paths in the superposition (like paths of a particle through an interferometer) can reinforce or cancel one another, depending on their relative phase.

Nonclonability and uncertainty: An unknown quantum state cannot be accurately copied (cloned) nor can it be observed without being disturbed.

QUBITS1 Classical information can be represented as binary bits: 0’s and 1’s. All computer information is stored and processed as bits.

In quantum mechanics any two orthogonal states can be used to encode bits.

For example, the polarization state H could signify 0, while

polarization state V could signify 1.

A bit of information stored in this manner is known as qubit (i.e. quantum-bit)

At first, it may seem that there is little difference between a classical bit and a qbit.

But a classical bit, at any instant in time, can represent

either 0 or 1, but no both.

However, quantum systems can exist in superposition states.

For example, the linearly polarization state ( ) )/1(45 2 VH +=+= is a superposition of the states H and V .

A qubit in this state signifies both 0 and 1; a property referred to as quantum parallelism. (Quantum parallelism can give quantum information processing an advantage over classical information processing)

Pair of Qubits (A two-photon system) Consider the spontaneous parametric down-conversion process, shown in Fig. 3.

In this process a single photon from a pump laser incident in a crystal is split into two photons, called the signal and the idler inside the crystal.

The signal and the idler emerge from the crystal at essentially the same time. For Type-I down-conversion the two down-converted photons have the same polarization, and orthogonal to that of the pump.

ωp

ω s

ω i Fig. 3 Type-I spontaneous parametric down-conversion. Polarizations of

the signal and idler photons are orthogonal to that of the pump

In order to describe the polarization state of the two-photons system, the polarization of each photon must be specified (when possible).

For the case shown in the Fig. 3 the polarization state is,

is HHHH, = (1)

If one places half-wave polarizers along the photons propagation direction, different two-photon polarization states can result,

is HH , is VH , is HV , is VV (2)

Entangled states Fig. 4 show a variation with respect to the one in Fig. 3. This time, there are two crystals

sandwiched together, with their orientation rotated by 90o with respect to each other. One crystal converts vertically (horizontally) polarized pump photons into horizontally (vertically) polarized signal and idler photons. If the pump is polarized at 45o, each of these processes is equally likely.

ωp

ω s

ω i

45ο polarized light

Fig. 4 Setting to produce entangle states, as the one in Eq. (3).

If the crystals are thin enough, observers detecting the signal and idler photons have no information about which crystal a given photon was produced in. (Actually photons with a given

polarization are emitted in a conical region of some thickness. It turn out the cones corresponding to each polarization intersect in some regions. Hence, photon contained in that intersected regions have a polarization that cannot be distinguished. ) If the photons are indistinguishable, their polarization state is a superposition of the two possible states generated by the down-conversion process. One possible state of the two photon system in Fig. 4 is,

) ( isissi VVHH +=2

1φ (3)

Here the two photons are in the is HH state and in the is VV state at the same time.

(Not as meaning that they are in one state is HH or the other is VV ).

Notice siφ cannot be expressed as the product of a (state of the signal photon) × (state of the idler photon)

States of the combined system that cannot be written as a single product the product of states of the individual particles are known as entangled states.

EBITS

Let’s call 0=H , and 1 =V Under certain experimental conditions (see paragraph below Fig. 4) Alice can have a qubit, and Bob can have a qubit in such a way that neither qubit by itself has a definite state of the type

A0 ,

A1 ,

A .

A B

Qubit is not in a individual Qubit is not in a individual

A0 ,

A1 ,

A , or else

B0 ,

B1 ,

B , or else

ωp

45ο polarized light

A

B

?

?

Fig. 5 What we mean is that that pair can exists in an entangled state

like for example

)( 11002

1 BABAAB

+= Entangled two-qubit

i.e. it is a state of the global system.

ωp

)( 11002

1 BABAAB

+= 45ο polarized

light

Α

Β

There are four two-qubit entangled states that we will find useful

)( 11002

1 BABAAB

±=±φ

(4) )( 0110

21

BABAAB±=±ψ

The no-cloning theorem

TELEPORTATION It is possible to transmit an unknown quantum state with perfect fidelity if the sender and the receiver have at their disposal two resources:

- The ability to send classical messages, and - Entanglement of qubits between the sender and the receiver

aψ

A

B aψ

Following Ref. 4

• For each of the three qubits we will have,

aa0=↔ AA

0=↔ BB0=↔

aa1 = AA

1 = BB1 =

• The diagram above indicates that Alice and Bob have a qubit, and that these

qubits are entangled.

There are four two-qubit entangled states that are possible:

( ) 11 00 2

1BABAAB

+=+φ

( ) 11 00 2

1BABAAB

−=−φ

(1)

( ) 01 10 2

1BABAAB

+=+ψ

( ) 01 10 2

1BABAAB

−=−ψ

They are known as the Bell states.

The Bell states form a basis; i.e. any two-qubit state can be expressed as a linear combination of the four Bell states.

A pair of entangled qubits, shared by separate parties, is known as a ebit.

• At the beginning:

Alice has a qubit in the state,

aaa1 0 βαψ += The state Alice wants to teleport (2)

Alice and Bob have a shared ebit in the state AB

+φ ,

( ) 11 00 2

1BABAAB

+=+φ (3)

The total state of the system is then,

ABa

+⊗= φψψ (4)

( ) ( ) 11 00 2

1 1 0 BABAaa

+⊗+= βαψ

which can be written as,

( )

( )

( )

( ) 1 0 - 21

1 0 21

1 0 21

1 0 21

++⊗

++⊗

+−⊗

++⊗=

−

+

−

+

BBaA

BBaA

BBaA

BBaA

αβψ

αβψ

βαφ

βαφψ

(5)

Proof:

( ) ( ) 11 00 2

1 1 0 BABAaa

+⊗+= βαψ

ψ ( ) ( ) 11 2

1 1 0 00 2

1 1 0 BAaaBAaa

⊗++⊗+= βαβα

( ) ( ) 1 1 1 1 0 2

1 0 01 0 0 2

1 BAaAaBAaAa

⊗++⊗+= βαβα

( )

( )

( )

( ) 1 1 1 2

1

1 1 0 2

1

0 01 2

1

0 0 0 2

1

BAa

BAa

BAa

BAa

β

α

β

α

⊗+

⊗+

⊗+

⊗=

( )

( )

( )

( ) 1 1 1 2

1

1 1 0 2

1

0 01 2

1

0 0 0 2

1

BAa

BAa

BAa

BAa

β

α

β

α

⊗+

⊗+

⊗+

⊗=

( )

( )

( )

( ) 1 0 0 2

1

1 0 1 2

1

0 10 2

1

0 1 1 2

1

BAa

BAa

BAa

BAa

β

α

β

α

⊗±

⊗±

⊗±

⊗±

( )( )( )

( ) 1

1

0

0

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφψ

⊗−

⊗+

⊗−

⊗=

−

+

−

+

( )

( )

( )

( ) 1 0 0 2

1

1 0 1 2

1

0 10 2

1

0 1 1 2

1

BAa

BAa

BAa

BAa

β

α

β

α

⊗+

⊗−

⊗+

⊗−

(5b)

From ( ) 11 00 2

1AaAaa

+=+

Aφ , ( ) 11 00

21

AaAaaA−=−φ

one obtains

aA

+φ + aA

−φ = Aa

00 2 and aA

+φ - aA

−φ = Aa

11 2 (5c)

(5c) in (5b)

( )( )( )

( ) 1

1

0

0

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφψ

⊗−

⊗+

⊗−

⊗=

−

+

−

+

( )( )

( )

( ) 1 21

1 0 1 2

1

0 10 2

1

0 - 21

BaAaA

BAa

BAa

BaAaA

βφφ

α

β

αφφ

⊗++

⊗−

⊗+

⊗−

−+

−+

(5d)

From ( ) 01 10 2

1AaAaaA

+=+ψ , ( ) 01 10 2

1AaAaaA

−=−ψ

one obtains

aA

+ψ + aA

−ψ = Aa

10 2 and aA

+ψ - aA

−ψ = Aa

01 2 (5e)

(5e) in (5d)

( )( )( )

( ) 1

1

0

0

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφψ

⊗−

⊗+

⊗−

⊗=

−

+

−

+

( )( )( )( ) 1

21

1 - 21

0 21

0 - 21

BaAaA

BaAaA

BaAaA

BaAaA

βφφ

αψψ

βψψ

αφφ

⊗++

⊗−

⊗++

⊗−

−+

−+

−+

−+

1 21

1 21

0 21

0 21

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφψ

⊗

−

⊗

+

⊗

−

⊗

=

−

+

−

+

( )( )( )( ) 1

21

1 21

0 21

0 21

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφ

⊗+

⊗+

⊗+

⊗+

+

−

+

−

1 21

1 21

0 21

0 21

BaA

BaA

BaA

BaA

βφ

αψ

βψ

αφψ

⊗

−

⊗

+

⊗

−

⊗

=

−

+

−

+

( )( )( )

( ) 0 21

0 21

1 21

1 21

BaA

BaA

BaA

BaA

αφ

βψ

αψ

βφ

⊗+

⊗+

⊗+

⊗+

−

+

−

+

( )

( )

( )

( ) 0 - 1 21

0 1 21

1 0 21

1 0 21

BBaA

BBaA

BBaA

BBaA

αβφ

βαψ

αβψ

βαφψ

⊗

−

+⊗

+

−⊗

−

+⊗

=

−

+

−

+

( )

( )

( )

( ) 1 0 21

1 0 21

1 0 21

1 0 21

BBaA

BBaA

BBaA

BBaA

αβψ

αβψ

βαφ

βαφψ

+−⊗

+⊗

+

−⊗

+⊗

=

−

+

−

+

( )

( )

( )

( ) 1 0 - 21

1 0 21

1 0 21

1 0 21

++⊗

++⊗

+−⊗

++⊗=

−

+

−

+

BBaA

BBaA

BBaA

BBaA

αβψ

αβψ

βαφ

βαφψ

(5)

• If Alice performs a measurement and let’s assume she finds her qubits are in

the state,

aA

+φ

then, according to (5), Bob’s qubit is projected into the state

( ) 1 0 BB

βα + , which is the desired teleported state.

Thus, by simply performing a Bell measurement, Alice successfully teleport the state to Bob, with no further action on Bob’s part.

• If Alice’s masurement projects her qubits into the state

aA

−φ ,

then, according to (5), Bob’s qubit is projected into the state

( ) 1 0 BB

βα − .

Subsequently, Bob can transform this state into the desired state by applying a π shift to the B

1 component of his qubit.

For that purpose, he uses a device (a polarizer) whose matrix representation is,

−

=10

01~T

For,

=

−

−

=

− β

αβ

αβ

α

1001

~T

Thus, upon receiving information from Alice (that her qubits have collapsed to state

aA

−φ ), Bob knows which transformation to apply to his entangled

qubit in order to obtained the desired state.

In short, is is possible to teleport an unknown quantum state aaa

1 0 βαψ += by making it to become part of a stationary state of a compound system

ABa+⊗= φψψ .

The quantum teleportation is actually done through the Alice-Bob entangled

system ( ) 11 00 2

1BABAAB

+=+φ . Amazingly, a measurement on

ψ made by Alice affects the entangled system own by Bob. Once Alice tell Bob what to do, then Bob can perform the appropriate measurement to attain (construct) ψ APPENDIX Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels Charles H. Bennett, Gilles Brassard, Claude Crepeau, Richard Jozsa, Asher Peres, and William K. Wootters Phys. Rev. Lett. 70, 1895 (1993).

An unknown quantum state ψ can be disassembled into, then later reconstructed from, purely classical information and purely nonclassical Einstein-Podolsky-Rosen (EPR) correlations. To do so the sender, "Alice," and the receiver, "Bob," must prearrange the sharing of an EPR-correlated pair of particles. Alice makes a joint measurement on her EPR particle and the unknown quantum system, and sends Bob the classical result of this measurement. Knowing this, Bob can convert the state of his EPR particle into an exact replica of the unknown state ψ which Alice destroyed.

1 Mark Beck, Quantum Mechanics, Theory and Experiments, Oxford University Press (2012).