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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. Interesting questions that I don’t plan to address….

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steering witnesses and criteria for the non existence of local hidden state lhs models

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

interesting questions that i don t plan to address
Interesting questions that I don’t plan to address…
  • Is steering an argument for the epistemic view of quantum states?
  • But isn’t that what Schrodinger meant…?
  • Do you consider contextuality for any of

this?

Steve Jones, PIAF, 2 February \'08

outline or what i actually will talk about
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

Steve Jones, PIAF, 2 February \'08

slide4

The Einstein-Podolsky-Rosen paradox (1935)

EPR’s assumptions:

  • Completeness:

“Every element of the physical reality must have a counterpart in the physical theory”.

  • Reality:

Accurate prediction ofa physical quantity → element of reality associated to it.

  • Local Causality:

No action at a distance

They considered a nonfactorizable state of the form:

Steve Jones, PIAF, 2 February \'08

the einstein podolsky rosen paradox 1935
The Einstein-Podolsky-Rosen paradox (1935)

Alice

Bob

XA, PA

XB, PB

Quantum Mechanics predicts, for certain entangled states, xA = xB and pA = - pB; by measuring at A one can predict with certainty either xBor pB .

Therefore, elements of reality must exist for both xBand pB , but QM doesn’t predict these simultaneously.

  • EPR conclude that Quantum Mechanics is incomplete.

Steve Jones, PIAF, 2 February \'08

schrodinger s 1935 response to epr
Schrodinger’s 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.”

Steve Jones, PIAF, 2 February \'08

slide7

Schrodinger’s 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 Bob’s state if she can prepare different ensembles of states for Bob by performing (at least 2) different measurements on her system.

Steve Jones, PIAF, 2 February \'08

what about mixed states
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 steeringPRL 98, 140402 (2007)

Steve Jones, PIAF, 2 February \'08

mathematical definitions
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

Steve Jones, PIAF, 2 February \'08

why experimental steering criteria
Why experimental steering criteria?
  • Foundational arguments aside for a moment.
  • Demonstration of the EPR effect: local causality is false or Bob’s system cannot be quantum (quantum mechanics is incomplete)
  • Easier to get around detection loophole than Bell’s
  • Hopefully applications in quantum information processing tasks?

Steve Jones, PIAF, 2 February \'08

two types of problems
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.

Steve Jones, PIAF, 2 February \'08

two types of problems1
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.

Steve Jones, PIAF, 2 February \'08

review linear entanglement witnesses
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).

Steve Jones, PIAF, 2 February \'08

steerability witnesses
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

Steve Jones, PIAF, 2 February \'08

witnesses and experimental criteria
Witnesses and experimental criteria
  • Witnesses: surfaces on the space of states;
  • Experimental criteria: surfaces on the space of correlations.

Steve Jones, PIAF, 2 February \'08

experimental steering criteria
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.

Steve Jones, PIAF, 2 February \'08

loop holes
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?

Steve Jones, PIAF, 2 February \'08

loop holes1
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 Bob’s detectors).

Steve Jones, PIAF, 2 February \'08

loop holes2
Loop-holes
  • Detection loop-hole:
    • Clearly this loop-hole will affect a demonstration of steering.
    • If Alice’s 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.

Steve Jones, PIAF, 2 February \'08

steering criteria example
Steering criteria example
  • Consider the two-qubit Werner state
  • Assuming a LHS model for Bob, the following steering criteria must be satisfied:
  • For n=2, this inequality is violated for
  • For n=3, this drops to

Steve Jones, PIAF, 2 February \'08

summary and open problems
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?

Steve Jones, PIAF, 2 February \'08

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