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

1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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