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School of Physics
something
and Astronomy
FACULTY OF MATHEMATICAL
OTHER
AND PHYSICAL SCIENCES
Introduction to entanglement
Jacob Dunningham
Paraty, August 2007
School of Physics and Astronomy
FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES
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October 2004
1
www.quantuminfo.org
School of Physics and Astronomy
FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES
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Vlatko
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October 2004
October 2005
1
9
www.quantuminfo.org
School of Physics and Astronomy
FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES
QuickTime™ and a
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Vlatko
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October 2004
October 2005
October 2006
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9
~ 25
www.quantuminfo.org
School of Physics and Astronomy
FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES
October 2010 (projected)
Overview
• Lecture1: Introduction to entanglement:
Bell’s theorem and nonlocality
Measures of entanglement
Entanglement witness
Tangled ideas in entanglement
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
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
Both speakers yesterday referred to how
Schrödinger coined the term “entanglement” in 1935 (or earlier)
History
Both speakers yesterday referred to how
Schrödinger coined the term “entanglement” in 1935 (or earlier)
"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)
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?
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 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 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
0o
(a)’
+
+
+
+
-
-
-
-
45o (b)’
+
+
+
-
-
-
-
+
90o (a’)
+
+
-
-
-
-
+ +
135o (b’)
+
-
-
-
-
+
+ +
a

b
a’
b’
Alice’s axes: a and a’
S = +1 - (-1) +1 -1 = 2
Bob’s axes: b and b’
S = +1 -(+1) +1 +1 = 2
Bell states
   00  11
  00  11

   i( 01  10 )


  i( 01  10 )
Bell’s theorem and nonlocality
S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2
a

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 =
i.e no local hidden variables
>2
Measures of entanglement
Bipartite pure states:
Schmidt
decomposition
Positive, real coefficients
Measures of entanglement
Bipartite pure states:
Schmidt
decomposition
Positive, real coefficients
Reduced density operators
Same coefficients
Measure of
mixedness
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
Consider the Bell state:
Example
Consider the Bell state:
This can be written as:
Example
Consider the Bell state:
This can be written as:
Maximally entangled (S is maximised for two qubits)
“Monogamy of entanglement”
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 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
An entanglement witness is an observable that distinguishes
entangled states from separable ones
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 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 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 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
• 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
Free quantum fields
Use Entanglement Witnesses for free quantum fields
e.g. Bosons
Free quantum fields
Use Entanglement Witnesses for free quantum fields
e.g. Bosons
“Biblical” operators - more on these
later…..
Free quantum fields
Use Entanglement Witnesses for free quantum fields
e.g. Bosons
Want to detect entanglement between regions of space
Energy
• Particle in a box of length L
where
• In each dimension:
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:
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Energy as an EW
• M spatial regions of length L/M
Energy as an EW
• M spatial regions of length L/M
Thermodynamics
Internal energy, temperature, and equation of state
Internal energy, temperature, and equation of state
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)
Interference term
Phase coherence
It is tempting to think that LRO and entanglement are the same
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
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”
R. P. Feynman
What about entanglement?
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 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 entanglement?
Single photon incident on a 50:50 beam
splitter:
Single particle entanglement?
Single photon incident on a 50:50 beam
splitter:
Single particle entanglement?
Single photon incident on a 50:50 beam
splitter:
Entangled
“Bell state”
Entanglement must be due to the single particle 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
1
PBS
2
Summary
• What is entanglement
• Bell’s theorem and nonlocality
• Measures of entanglement
• Entanglement witness in a BEC
• Confusing concepts in entanglement