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School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES. Introduction to entanglement. Jacob Dunningham. Paraty, August 2007. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES. Vlatko pic. October 2004. 1. www.quantuminfo.org.

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Introduction to entanglement

School of Physics and Astronomy

FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

Introduction to entanglement

Jacob Dunningham

Paraty, August 2007


Introduction to entanglement

School of Physics and Astronomy

FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

Vlatko pic

October 2004

1

www.quantuminfo.org


Introduction to entanglement

School of Physics and Astronomy

FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

Vlatko pic

October 2005

October 2004

1

9

www.quantuminfo.org


Introduction to entanglement

School of Physics and Astronomy

FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

Vlatko pic

October 2005

October 2006

October 2004

1

9

~ 25

www.quantuminfo.org


Introduction to entanglement

School of Physics and Astronomy

FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES

October 2010 (projected)


Overview

Overview

  • Lecture1: Introduction to entanglement:

  • Bell’s theorem and nonlocality

  • Measures of entanglement

  • Entanglement witness

  • Tangled ideas in entanglement


Overview1

Overview

  • Lecture1: Introduction to entanglement:

  • Bell’s theorem and nonlocality

  • Measures of entanglement

  • Entanglement witness

  • Tangled ideas in entanglement

  • Lecture 2: Consequences of entanglement:

  • Classical from the quantum

  • Schrodinger cat states


Overview2

Overview

  • Lecture1: Introduction to entanglement:

  • Bell’s theorem and nonlocality

  • Measures of entanglement

  • Entanglement witness

  • Tangled ideas in entanglement

  • 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

History

Both speakers yesterday referred to how

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


History1

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

Entanglement

  • Superpositions:

  • Superposed correlations:

  • Entanglement

  • (pure state)


Entanglement1

Entanglement

  • Tensor Product:

Entangled

Separable


Separability

Separability

  • Separable states (with respect to the subsystems

  • A, B, C, D, …)


Separability1

Separability

  • Separable states (with respect to the subsystems

  • A, B, C, D, …)

  • Everything else is entangled

  • e.g.


The epr paradox

The EPR ‘Paradox’

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

  • Either:

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

  • 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?


Bell s theorem and nonlocality

Bell’s theorem and nonlocality

  • 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


Bell s theorem and nonlocality1

Bell’s theorem and nonlocality

  • 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

a

b

a’

b’

Alice’s axes: a and a’

Bob’s axes: b and b’


Bell s theorem and nonlocality2

Bell’s theorem and nonlocality

  • 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

a

0o (a)’ + + + + - - - -

45o (b)’ + + + - - - - +

90o (a’) + + - - - - + +

135o (b’) + - - - - + + +

b

a’

b’

Alice’s axes: a and a’

Bob’s axes: b and b’

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

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


Bell states

Bell states


Bell s theorem and nonlocality3

Bell’s theorem and nonlocality

a

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

b

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

a’

b’

E(a,b) = cos

E(a,b’) = cos = - sin

E(a’,b) = cos= sin 

E(a’,b’) = cos

S = | 2 cos  sin 

When =45o, we have S = > 2

i.e no local hidden variables


Measures of entanglement

Measures of entanglement

  • Bipartite pure states:

Schmidt decomposition

Positive, real coefficients


Measures of entanglement1

Measures of entanglement

  • Bipartite pure states:

Schmidt decomposition

Positive, real coefficients

Reduced density operators

Same coefficients

Measure of mixedness


Measures of entanglement2

Measures of entanglement

  • Bipartite pure states:

Schmidt decomposition

Positive, real coefficients

Reduced density operators

Same coefficients

Measure of mixedness

Unique measure of entanglement (Entropy)


Example

Example

  • Consider the Bell state:


Example1

Example

  • Consider the Bell state:

This can be written as:


Example2

Example

  • Consider the Bell state:

This can be written as:

Maximally entangled (S is maximised for two qubits)

“Monogamy of entanglement”


Measures of entanglement3

Measures 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)


Measures of entanglement4

Measures 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)

Entanglement of formation

von Neumann entropy

Minimum over all realisations of:


Entanglement witnesses

Entanglement witnesses

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


Entanglement witnesses1

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


Entanglement witnesses2

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

Thermodynamic quantities provide convenient (unoptimised) EWs


Covalent bonding

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:

Entangled

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

Covalent bonding is evidence of entanglement


Covalent bonding1

Covalent bonding

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

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

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

Covalent bonding is evidence of entanglement


Detecting entanglement

Detecting Entanglement

  • State tomography

  • Bell’s inequalities

  • Entanglement witnesses (EW)


Detecting entanglement1

Detecting Entanglement

  • State tomography

  • Bell’s inequalities

  • Entanglement witnesses (EW)


Remarkable features of entanglement

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 entanglement1

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….


Introduction to entanglement

Molecule of the Year


Introduction to entanglement

Molecule of the Year

Overall state:

Atoms are not entangled


Free quantum fields

Free quantum fields

Use Entanglement Witnesses for free quantum fields

e.g. Bosons


Free quantum fields1

Free quantum fields

Use Entanglement Witnesses for free quantum fields

e.g. Bosons

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


Free quantum fields2

Free quantum fields

Use Entanglement Witnesses for free quantum fields

e.g. Bosons

Want to detect entanglement between regions of space


Energy

Energy

  • Particle in a box of length L

where

  • In each dimension:


Energy1

Energy

  • Particle in a box of length L

where

  • In each dimension:

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


Energy as an ew

Energy as an EW

  • M spatial regions of length L/M


Energy as an ew1

Energy as an EW

  • M spatial regions of length L/M


Thermodynamics

Thermodynamics

  • Internal energy, temperature, and equation of state

  • Internal energy, temperature, and equation of state


Ketterle s experiments

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 experiments1

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

Munich experiment

  • A reservoir of entanglement - changes the state of the BEC

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


Entanglement spatial correlations

Entanglement & spatial correlations

  • The Munich experiment demonstrates long-range order (LRO)

Interference term Phase coherence

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


Entanglement spatial correlations1

Entanglement & spatial correlations

  • The Munich experiment demonstrates long-range order (LRO)

Interference term Phase coherence

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

A GHZ-type state is clearly entangled:

BUT

  • They are, however, related Ongoing research


Tangled ideas in entanglement

Tangled ideas in entanglement

  • 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 and subsystems

  • Entanglement depends on what the subsystems are


Entanglement and subsystems1

Entanglement and subsystems

  • Entanglement depends on what the subsystems are

Entangled


Single particle entanglement

Single particle entanglement?

  • “Superposition is the only mystery in quantum mechanics”

R. P. Feynman

What about entanglement?


Single particle entanglement1

Single particle entanglement?

  • “Superposition is the only mystery in quantum mechanics”

R. P. Feynman

What about entanglement?

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


Single particle entanglement2

Single particle entanglement?

  • “Superposition is the only mystery in quantum mechanics”

R. P. Feynman

What about entanglement?

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

Is this all just semantics?

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


Single particle entanglement3

Single particle entanglement?

  • Single photon incident on a 50:50 beam splitter:


Single particle entanglement4

Single particle entanglement?

  • Single photon incident on a 50:50 beam splitter:


Single particle entanglement5

Single particle entanglement?

  • Single photon incident on a 50:50 beam splitter:

Entangled “Bell state”

Entanglement must be due to the single particle state


Introduction to entanglement

  • “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.


Introduction to entanglement

  • “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 and entanglement

  • Nonlocality implies position distinguishability, which is not necessary for entanglement

  • Confusion arises because Alice and Bob are normally spatially separated


Nonlocality and entanglement1

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

1

2

PBS


Summary

Summary

  • What is entanglement

  • Bell’s theorem and nonlocality

  • Measures of entanglement

  • Entanglement witness in a BEC

  • Confusing concepts in entanglement


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