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. Steve Jones, PIAF, 2 February '081 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?

3. Steve Jones, PIAF, 2 February '082 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

4. Steve Jones, PIAF, 2 February '083 EPR’s 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: The Einstein-Podolsky-Rosen paradox (1935)

5. Steve Jones, PIAF, 2 February '084 Quantum Mechanics predicts, for certain entangled states, x A = x B and p A = - p B ; by measuring at A one can predict with certainty either x B or p B. Therefore, elements of reality must exist for both x B and p B, but QM doesn’t predict these simultaneously. EPR conclude that Quantum Mechanics is incomplete. The Einstein-Podolsky-Rosen paradox (1935) Bob X B, P B X A, P A Alice

6. Steve Jones, PIAF, 2 February '085 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.” Schrodinger’s 1935 response to EPR

7. Steve Jones, PIAF, 2 February '086 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.

8. Steve Jones, PIAF, 2 February '087 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)

9. Steve Jones, PIAF, 2 February '088 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

10. Steve Jones, PIAF, 2 February '089 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?

11. Steve Jones, PIAF, 2 February '0810 Two types of problems 1. 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.

12. Steve Jones, PIAF, 2 February '0811 2. 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. Two types of problems

13. Steve Jones, PIAF, 2 February '0812 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).

14. Steve Jones, PIAF, 2 February '0813 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 Steerability Witnesses

15. Steve Jones, PIAF, 2 February '0814 Witnesses and experimental criteria StateCorrelations 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.

16. Steve Jones, PIAF, 2 February '0815 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.

17. Steve Jones, PIAF, 2 February '0816 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?

18. Steve Jones, PIAF, 2 February '0817 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). Loop-holes

19. Steve Jones, PIAF, 2 February '0818 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.

20. Steve Jones, PIAF, 2 February '0819 Steering criteria example Assuming a LHS model for Bob, the following steering criteria must be satisfied: Consider the two-qubit Werner state For n=2, this inequality is violated for For n=3, this drops to

21. Steve Jones, PIAF, 2 February '0820 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?