Steering witnesses and criteria for the (non-)existence of local hidden state (LHS) models

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Steering witnesses and criteria for the
(non-)existence of local hidden state (LHS) models

• Eric Cavalcanti, Steve Jones, Howard Wiseman
• Centre for Quantum Dynamics, Griffith University

Steve Jones, PIAF, 2 February 08
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Interesting questions that I dont plan to
address

• Is steering an argument for the epistemic view of
  quantum states?

• But isnt that what Schrodinger meant?

• Do you consider contextuality for any of
• this?

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Outline (or what I actually will talk about)

• History and definitions
• Steering criteria vs Steerability witnesses
• (and Bell inequalities vs Bell-nonlocality
  witnesses)
• Loopholes
• Example
• Open problems

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The Einstein-Podolsky-Rosen paradox (1935)

• EPRs assumptions
• Completeness
• Every element of the physical reality must have
  a counterpart in the physical theory.
• Reality
• Accurate prediction of a physical quantity ?
  element of reality associated to it.
• Local Causality
• No action at a distance
• They considered a nonfactorizable state of the
  form

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The Einstein-Podolsky-Rosen paradox (1935)

• Quantum Mechanics predicts, for certain entangled
  states, xA xB and pA -
  pB by measuring at A one can predict with
  certainty either xB or pB .
• Therefore, elements of reality must exist for
  both xB and pB , but QM
  doesnt predict these simultaneously.
• EPR conclude that Quantum Mechanics is incomplete.

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Schrodingers 1935 response to EPR

• Schrodinger introduced the terms entangled and
  steering to describe the state and situation
  introduced by EPR.
• By the interaction the two representatives (or
  -functions) have become entangled.
• What constitutes the entanglement is that
  is not a product of a function for x and a
  function for y.

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Schrodingers 1935 response to EPR

• Schrodinger emphasized that in the EPR paradox,
  and steering in general, the choice of
  measurement at one side is important.

• Alice can steer Bobs state if she can prepare
  different ensembles of states for Bob by
  performing (at least 2) different measurements on
  her system.

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What about mixed states?

• Both EPR and Schrodinger considered pure states
  in their 1935 works.
• For pure states entangled steerable (Bell
  nonlocal)
• Even with improvements in modern experiments we
  must deal with states which are mixed.
• How does all this generalize?
• EPR paradox EPR-Reid criteria
• Schrodinger steering PRL 98, 140402
  (2007)

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Mathematical definitions
Separable A local hidden state (LHS) model for
both parties
Non-steerable A local hidden state (LHS) model
for one party
Bell local A local hidden variable (LHV) model
for both parties
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Why experimental steering criteria?

• Foundational arguments aside for a moment.
• Demonstration of the EPR effect local causality
  is false or Bobs system cannot be quantum
  (quantum mechanics is incomplete)
• Easier to get around detection loophole than
  Bells
• Hopefully applications in quantum information
  processing tasks?

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Two types of problems

• Experimental steering
• Given sets of measurements for Alice and Bob and
  a preparation procedure, can the experimental
  outcomes associated with this setup demonstrate
  steering?
• That is, do they violate the assumption of a
  local hidden state model for Bob?
  
• Definition
• Any sufficient criterion for experimental
  steering will be called a steering criterion.

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Two types of problems

• State steerability
• Given a quantum state, can it demonstrate
  steering with some measurements for Alice and
  Bob?
• Definition
• Any sufficient criterion for state
  steerability will be called a steerability
  witness.

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Review (linear) Entanglement witnesses

• Reasoning There exists a plane separating a
  convex set (separable states) and a point outside
  of it (the entangled state).
• The same is true for any convex set (e.g.
  non-steerable states).

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Steerability Witnesses

• Lemma A bipartite density matrix on
  is steerable if and only if there exists a
  Hermitian operator such that
• and for all
  non-steerable density matrices .
• However, the measurements required to determine
  do not necessarily violate a LHS
  model.
• Compare with Bell-nonlocality witnesses vs Bell
  inequalities

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Witnesses and experimental criteria
State Correlations
Entanglement Entanglement witness Separability criterion
Steering Steerability witness EPR criterion Steering criterion
Bell-nonlocality Bell-nonlocality witness Bell inequality

• Witnesses surfaces on the space of states
• Experimental criteria surfaces on the space of
  correlations.

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Experimental steering criteria

• Bell inequalities are experimental criteria
  derived from LHV models.
• Violation implies failure of LHV theories.
• Analogously, experimental steering criteria are
  derived from the LHS model (for Bob).
• Violation implies steering.

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Loop-holes

• All experimental tests of Bell inequalities have
  suffered from the detection and/or locality
  loop-hole.
• How do loop-holes affect the experimental
  demonstration of steering?

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Loop-holes

• Locality loop-hole
• Not obvious that this loop-hole would apply to a
  demonstration of steering.
• Although, to be rigorous, one must assume that
  once Bob obtains his system, Alice cannot affect
  it (or the outcomes reported by Bobs detectors).

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Loop-holes

• Detection loop-hole
• Clearly this loop-hole will affect a
  demonstration of steering.
• If Alices detectors are inefficient
• ? harder for her to steer to a given
  ensemble.
• As for Bell nonlocality, there will be a
  threshold detection efficiency that allows a
  loop-hole free demonstration.
• The threshold efficiency for steering will be
  lower than for Bell nonlocality.

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Steering criteria example

• Consider the two-qubit Werner state

• Assuming a LHS model for Bob, the following
  steering criteria must be satisfied

• For n2, this inequality is violated for

• For n3, this drops to

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Summary and open problems

• LHS model is the correct formalisation of the
  concept of steering introduced by Schrodinger as
  a generalisation of the EPR paradox
• Steerability witnesses and steering criteria
• Is there a general algorithm to generate all
  steering criteria?
• What is the set of steerable states?
• e.g., are there asymmetric steerable states?
• Can the concept of Bell-nonlocality witnesses
  help in studying the set of Bell-local states?
• Applications of steering to quantum information
  processing tasks?
• What features of toy models allow steering in
  general?