Entanglement in Field Theory

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Entanglement in Field Theory
School of Physics and Astronomy FACULTY OF
MATHEMATICAL AND PHYSICAL SCIENCES

• Vlatko Vedral
• University of Leeds, UK
• National University of Singapore

v.vedral_at_leeds.ac.uk
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In collaboration with

• Janet Anders (Singapore)
• Jacob Dunningham, Libby Heaney (Leeds)
• Also,
• Dago Kaszlikowski (Singapore), Jenny Hide,
  Cristhian Avila and Wonmin Son (Leeds).

See Many-body entanglement, Amico, Fazio,
Osterloh and Vedral, submitted to Rev. Mod. Phys.
(2007).
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Motivation

• Resource teleportation, cryptography,
  (measurement based) QC
• Fundamental Classical/Quantum transition,
  Non-locality, Failure of QM

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Motivation

• Resource teleportation, cryptography,
  (measurement based) QC
• Fundamental Classical/Quantum transition,
  Non-locality, Failure of QM
• Phase transitions new PTs, simulations,
  bio-coherence
• Other motivations

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"Physics is like sex. Sure, it may give some
practical results, but that's not why we do it."
- Feynman
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Why Fields?

• Many body physics field theory
• Relativistic incorporations
• Fields more fundamental than particles

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Particles

• ?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.

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System
First quantisation direct sum
Second quantised direct product
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QUESTION
When can the state be written as
This state is called disentangled (or separable).
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Fields
c.f. Yukawa, domain field theory, Prog. Theor.
Phys (1966)
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Defining Entanglement
Define x and p operators on L and R.
Reduce to two harmonic oscillators.
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Witnessing Entanglement
Test joint squeezing in X and P
This is the continuous variable equivalent of the
susceptibility argument presented for the
discrete spins.
See e.g. Brukner, Vedral, Zeilinger, PRA (2005).
Work with Heaney, Anders Kaszlikowski
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Surprises
Surprise I We can witness entanglement by using
macroscopic observables only (i.e. no need to do
difficult measurements). Surprise II No need
to know the eigenstates of the Hamiltonian! Surp
rise III Can find entanglement at any
temperature!
(Deutsch Any piece of matter will be a quantum
computer in the future).
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Controversies with fields
1. Single particle entanglement 2. Not real
entanglement (its all due to symmetrisation) 3.
There are superselection rules that prevent
it. 4. Choose your own objection
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Superposition Entanglement
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Only One Mystery
Feynman There is only one mystery in quantum
mechanics Therefore Superposition
Entanglement ( Non-locality?)
Key observation Single particle superposition in
first quantisation
becomes mode entanglement in second.
(See Terra Cunha, Dunningham Vedral, to appear
Proc. Roy Soc. M. R. Dowling, S. D. Bartlett,
T. Rudolph and R. W. Spekkens, Phys. Rev. A
(2006))
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Nonlocality

• John Bell (1964) - puts (non)locality on a firm
  footing
• Tan, Walls and Collett (1991) first suggested
  that a single particle could exhibit nonlocality
• Hardy (1994) tightens up some assumptions in TWC
• Greenberger, Horne, and Zeilinger (GHZ) still
  object not a real experiment/ multiparticle
  effect in disguise (1995)
• Other schemeslots of debateno clear consensus
  (1995 - 2007)
• What is needed is a feasible experiment to
  resolve the issue.

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Hardy Scheme
2
1
Reference L. Hardy, Phys. Rev. Lett. 73, 2279
(1994)
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Hardy Scheme
2
1
Reference L. Hardy, Phys. Rev. Lett. 73, 2279
(1994)
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Hardy Scheme
2
Bob
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Alice
Reference L. Hardy, Phys. Rev. Lett. 73, 2279
(1994)
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Hardy Scheme
2
Bob
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Experiment 1 Alice and Bob both measure the
number of photons on their path
Alice
They never both detect one
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Hardy Scheme
2
Bob
1
Experiment 2 Alice makes a homodyne detection
and Bob detects the number in path 2 If Bob
detects no particles the state at Alices
detectors is So, if Alice detects one, it must
be at c1 Conversely, if Alice detects one
particle at d1 and none at c1 then Bob cannot
detect none, i.e. he must detect one!
Alice
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Hardy Scheme
2
Bob
1
Experiment 3 The roles of Alice and Bob are
reversed. Alice measures the number of particles
on path 1 and Bob makes a homodyne detection. If
Bob detect one particle at d2 and nothing at c2
then Alice must detect one particle.
Alice
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Hardy Scheme
2
Bob
1
Experiment 4 Alice and Bob both make homodyne
detections
There is a finite probability that Alice detects
one particle at d1 and none at c1 AND Bob
detects one particle at d2 and none at c2
Alice
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Argument

• Experiment 1 Alice and Bob cannot both detect
  a particle in their path.

Experiment 2 If Alice detects one particle at
d1 and nothing at c1 it follows
that Bob must detect a particle on path 2.
Experiment 3 If Bob detects one particle at d2
and nothing at c2 it follows that Alice must
detect a particle on path 1.
Experiment 4 One possible outcome is that
Alice detects one particle at d1 and nothing at
c1 AND Bob detects one particle at d2 and nothing
at c2.
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Argument

• Experiment 1 Alice and Bob cannot both detect
  a particle in their path.

Experiment 2 If Alice detects one particle at
d1 and nothing at c1 it follows
that Bob must detect a particle on path 2.
Experiment 3 If Bob detects one particle at d2
and nothing at c2 it follows that Alice must
detect a particle on path 1.
CONTRADICTION
Experiment 4 One possible outcome is that
Alice detects one particle at d1 and nothing at
c1 AND Bob detects one particle at d2 and nothing
at c2.
NONLOCALITY
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Objections

• Greenberger, Horne, and Zeilinger (1995)
• Partly-cle states are unobservable - violate
  superselection rules
• Does not correspond to a real experiment

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Mixed States
Convenient to use coherent state inputs and
average over the phases at the end
Dunningham, Vedral, archive (2007).
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State truncation
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State truncation
x
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State truncation
x
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State truncation
x
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State truncation
Will become a mixed state when we average over
phases at the end
Not Entangled
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Results

• This gives all the same results as Hardys scheme
• Overall state just before Alice and Bobs beam
  splitters is

Each only makes a local operation Therefore
nonlocality is due to the single particle state
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Results

• This gives all the same results as Hardys scheme
• Overall state just before Alice and Bobs beam
  splitters is

Each only makes a local operation Therefore
nonlocality is due to the single particle state
Results are independent of ??- so we can average
over all phases and get the same result, I.e. a
mixed state input also works This takes care of
the GHZ objections
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Results

• This gives all the same results as Hardys scheme

Alice and Bob only make local operations Therefore
nonlocality is due to the single particle state
Results are independent of ??- so we can average
over all phases and get the same result, I.e. a
mixed state input also works This takes care of
the GHZ objections
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Conclusions

• Single particle superposition, entanglement and
  nonlocality the same.
• This is good news, since in quantum field theory
  particles are not
• fundamental. Particle is just an excitation of a
  mode of some field,
• therefore particles identical and there should be
  no fundamental
• difference between one, two and more particles.
• All entanglement therefore mode entanglement.
• Interesting question for mixed states is
  entanglement the same as
• nonlocality? What about mixedness and
  entanglement?

Dynamics Bose, Fuentes-Guridi, Knight, Vedral,
PRL (2001) Ferreira, Guerreiro, Vedral PRL
(2006).
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Funding