School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Introduction to entanglement Jacob Dunningham

School of somethingFACULTY OF OTHER

School of Physics and AstronomyFACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

Introduction to entanglement

Jacob Dunningham

Paraty, August 2007


www.quantuminfo.org

October 2004

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October 2010 (projected)

Overview

• Lecture1: Introduction to entanglement:

Bell’s theorem and nonlocality

Measures of entanglement

Entanglement witness

Tangled ideas in entanglement

Overview






• Lecture 2: Consequences of entanglement:

Classical from the quantum

Schrodinger cat states

Overview






• Lecture 2: Consequences of entanglement:

Classical from the quantum

Schrodinger cat states

• Lecture 3: Uses of entanglement:

Superdense coding

Quantum state teleportation

Precision measurements using entanglement

History

Both speakers yesterday referred to how

Schrödinger coined the term “entanglement” in 1935 (or earlier)

History

"When two systems, …… enter into temporary physical interaction due to known forces between them, and …… separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states]

have become entangled."

Schrödinger (Cambridge Philosophical Society)

Both speakers yesterday referred to how

Schrödinger coined the term “entanglement” in 1935 (or earlier)

Entanglement

Superpositions:

Superposed correlations:

Entanglement

(pure state)

Entanglement

Tensor Product:

Separable Entangled

Separability

Separable states (with respect to the subsystems

A, B, C, D, …)

Separability

Separable states (with respect to the subsystems

A, B, C, D, …)

Everything else is entangled

e.g.

The EPR ‘Paradox’

1935: Einstein, Podolsky, Rosen - QM is not complete

Either:

1. Measurements have nonlocal effects on distant parts of the system.

2. QM is incomplete - some element of physical reality cannot be accounted for by QM - ‘hidden variables’

An entangled pair of particles is sent to Alice and Bob. The spin in measured in the z, x (or any other) direction.

The measurement Alice makes instantaneously affects Bob’s….nonlocality? Hidden variables?


1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes.

CHSH:

S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2



CHSH:

S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2

ab

a’

b’

Alice’s axes: a and a’

Bob’s axes: b and b’



CHSH:

S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2

ab

a’

b’

0o (a)’ + + + + - - - -

45o (b)’ + + + - - - - +

90o (a’) + + - - - - + +

135o (b’) + - - - - + + +

S = +1 - (-1) +1 -1 = 2

S = +1 -(+1) +1 +1 = 2

Alice’s axes: a and a’

Bob’s axes: b and b’

Bell states

1100 +=+φ

€

φ− = 00 − 11

€

ψ + =i( 01 + 10 )

€

ψ− =i( 01 − 10 )


S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 ab

a’

b’

When =45o, we have S = > 2

i.e no local hidden variables

Without local hidden variables, e.g. for Bell states

E(a,b) = cos

E(a,b’) = cos = - sin

E(a’,b) = cos = sin

E(a’,b’) = cos

S = | 2 cos sin


Bipartite pure states:

Schmidt decomposition

Positive, real coefficients





Same coefficients

Measure of mixedness

Reduced density operators





Same coefficients

Measure of mixedness

Reduced density operators

Unique measure of entanglement (Entropy)

Example

Consider the Bell state:

Example


This can be written as:

Example


This can be written as:

Maximally entangled (S is maximised for two qubits)

“Monogamy of entanglement”


Bipartite mixed states:

• Average over pure state entanglement that makes up the mixture

• Problem: infinitely many decompositions and each leads to a different entanglement

• Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled)


Bipartite mixed states:

Entanglement of formation

von Neumann entropy

Minimum over all realisations of:

• Average over pure state entanglement that makes up the mixture

• Problem: infinitely many decompositions and each leads to a different entanglement

• Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled)

Entanglement witnesses

An entanglement witness is an observable that distinguishes entangled states from separable ones



Theorem: For every entangled state, there exists a Hermitian operator, A, such that Tr(A)<0 and Tr(A)>=0 for all separable states,

Corollary: A mixed state, , is separable if and only if:

Tr(A)>=0



Theorem: For every entangled state, there exists a Hermitian operator, A, such that Tr(A)<0 and Tr(A)>=0 for all separable states,

Corollary: A mixed state, , is separable if and only if:

Tr(A)>=0

Thermodynamic quantities provide convenient (unoptimised) EWs

Covalent bonding

Covalent bonding relies on entanglement of the electrons e.g. H2

Lowest energy (bound) configuration

Overall wave function is antisymmetric so the spin part is:

The energy of the bound state is lower than any separable state - witness

Covalent bonding is evidence of entanglement

Entangled

Covalent bonding

Covalent bonding relies on entanglement of the electrons e.g. H2

The energy of the bound state is lower than any separable state - witness

Covalent bonding is evidence of entanglement

NOTE: It is not at all clear that this entanglement could be used in quantum processing tasks.

You will often hear people distinguish “useful” entanglement from other sorts

Detecting Entanglement

• State tomography

•Bell’s inequalities

•Entanglement witnesses (EW)

Detecting Entanglement

• State tomography

•Bell’s inequalities

•Entanglement witnesses (EW)

Remarkable features of entanglement

• It can give rise to macroscopic effects

• It can occur at finite temperature (i.e. the system need not be in the ground state)

• We do not need to know the state to detect entanglement

• It can occur for a single particle

Remarkable features of entanglement

• It can give rise to macroscopic effects

• It can occur at finite temperature (i.e. the system need not be in the ground state)

• We do not need to know the state to detect entanglement

• It can occur for a single particle

Let’s consider an example that exhibits all these features….

Molecule of the Year

Molecule of the Year

Overall state: Atoms are not entangled

Use Entanglement Witnesses for free quantum fields

e.g. Bosons

Free quantum fields


e.g. Bosons

Free quantum fields

“Biblical” operators - more on these later…..


e.g. Bosons

Want to detect entanglement between regions of space

Free quantum fields

Energy

• Particle in a box of length L

• In each dimension:

where

Energy

• In each dimension:

where

• For N separable particles in a d-dimensional box of length L, the minimum energy is:

• Particle in a box of length L





Energy as an EW

• M spatial regions of length L/M

Energy as an EW

• M spatial regions of length L/M

Internal energy, temperature, and equation of state

Internal energy, temperature, and equation of state

Thermodynamics

Ketterle’s experiments

The critical temperature for BEC in an homogeneous trap is:

Comparing with the onset of entanglement across the system

These differ only by a numerical factor of about 2 !

Entanglement as a phase transition

Ketterle’s experiments

Typical numbers:

This gives:

In experiments, the temperature of the BEC is typically:

Entanglement in a BEC (even though it can be written as a product state of each particle)

Munich experiment

A reservoir of entanglement - changes the state of the BEC

Ref: I. Bloch et al., Nature 403, 166 (2000)

Entanglement & spatial correlations

The Munich experiment demonstrates long-range order (LRO)

It is tempting to think that LRO and entanglement are the same

Interference term Phase coherence

Entanglement & spatial correlations

The Munich experiment demonstrates long-range order (LRO)

They are, however, related Ongoing research

It is tempting to think that LRO and entanglement are the same

Interference term Phase coherence

A GHZ-type state is clearly entangled:

BUT


1. Entanglement does not depend on how we divide the system

2. A single particle cannot be ‘entangled’

3. Nonlocality and entanglement are the same thing

Entanglement and subsystems

Entanglement depends on what the subsystems are

Entanglement and subsystems

Entanglement depends on what the subsystems are

Entangled

Single particle entanglement?

“Superposition is the only mystery in quantum mechanics”

What about entanglement?

R. P. Feynman




R. P. Feynman

Instead of the superposition of a single particle, we can think of the entanglement of two different variables:




R. P. Feynman

Is this all just semantics?

Can we measure any real effect, e.g. violation of Bell’s inequalities?

Instead of the superposition of a single particle, we can think of the entanglement of two different variables:


Single photon incident on a 50:50 beam splitter:




Entanglement must be due to the single particle state

Entangled “Bell state”


“The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates. . . It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.”

A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942)pp. 30-31.

“The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates. . . It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.”

A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942)pp. 30-31.

We need a field theory treatment of entanglement

Nonlocality and entanglement

Nonlocality implies position distinguishability, which is not necessary for entanglement

Confusion arises because Alice and Bob are normally spatially separated

Nonlocality and entanglement

Nonlocality implies position distinguishability, which is not necessary for entanglement

Confusion arises because Alice and Bob are normally spatially separated

Example:

This state is local, but can be considered to have entanglement

PBS

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Summary

• What is entanglement

• Bell’s theorem and nonlocality

• Measures of entanglement

• Entanglement witness in a BEC

• Confusing concepts in entanglement

Documents

School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Introduction to entanglement Jacob Dunningham