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. 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

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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


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

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


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

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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