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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
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Steve Jones, PIAF, 2 February ‘08
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
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)
Ψ = ci uii
∑ ψ i = d jj
∑ v j ϕ j
Steve Jones, PIAF, 2 February '08 5
Quantum Mechanics predicts, for certain entangled states, xA = xB and pA = - pB; by measuring
at A one can predict with certainty either xB or pB .
Therefore, elements of reality must exist for both xB and
pB , but QM doesn’t predict these
simultaneously.
• EPR conclude that Quantum Mechanics is incomplete.
The Einstein-Podolsky-Rosen paradox (1935)
Bob
XB, PBXA, PA
Alice Ψ
Steve Jones, PIAF, 2 February '08 6
• 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
ψ
Ψ
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 steering PRL 98, 140402 (2007)
Steve Jones, PIAF, 2 February '08 9
Mathematical definitions
P(A,B a ,b,c) = P(λ c)λ∑ P(A a,c,λ) PQ(B b,c,λ)
P(A,B a ,b,c) = P(λ c)λ∑ P(A a,c,λ) P(B b,c,λ)
P(A,B a ,b,c) = P(λ c)λ∑ PQ(A a,c,λ) PQ(B b,c,λ)
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
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.
–
–
Steve Jones, PIAF, 2 February '08 12
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
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
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
Hα ⊗ Hβ
S
Tr Sρ⎡⎣ ⎤⎦< 0
Tr Sσ⎡⎣ ⎤⎦≥0 σ
Tr Sρ⎡⎣ ⎤⎦
Steve Jones, PIAF, 2 February '08 15
Witnesses and experimental criteria
State Correlations
Entanglement
Entanglement witness
Separability criterion
SteeringSteerability
witness
EPR criterionSteering criterion
Bell-nonlocality
Bell-nonlocality
witnessBell inequality
• 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.P(A,B a ,b,c) = P(λ c)
λ∑ P(A a,c,λ) PQ(B b,c,λ)
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
• 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
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
1
nAi Bi
i=1
n
∑ < cn
• Assuming a LHS model for Bob, the following steering criteria must be satisfied:
• Consider the two-qubit Werner state
ρW = η ψ − ψ − + (1−η )I
4
• For n=2, this inequality is violated for
• For n=3, this drops to
η >1 / 2 ≈ 0.707
η >1 / 3 ≈ 0.577
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?