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?