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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

Eric Cavalcanti, Steve Jones, Howard Wiseman

Centre for Quantum Dynamics, Griffith University

Steve Jones, PIAF, 2 February ‘08

- 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

- 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

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

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 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

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

- 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

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

- 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

- 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.

- Given sets of measurements for Alice and Bob and a preparation procedure, can the experimental outcomes associated with this setup demonstrate steering?

Steve Jones, PIAF, 2 February '08

- 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

- 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

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: surfaces on the space of states;
- Experimental criteria: surfaces on the space of correlations.

Steve Jones, PIAF, 2 February '08

- 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

- 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

- 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

- 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

- 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

- 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