Platonic Love at a Distance:the EPR paradox revisited

arXiv:0909.0805

Howard Wiseman, Steve Jones, (Eric Cavalcanti), Dylan Saunders, and Geoff Pryde

Centre for Quantum Dynamics

Brisbane Australia

Outline of this talk

I. A cartoon history of quantum nonlocality (Einstein, Heisenberg, Schrödinger, Bell).

II. Three types of quantum nonlocality.

III. Experimental metaphysics.

IV. Conclusions.

I.A Cartoon History of Quantum Nonlocality

I(a) Einstein’s objection to the Copenhagen Interpretation: Nonlocality

beam-splitter

Bob

Alice

beam-splitter

Bob

Alice

I(a) Einstein’s objection to the Copenhagen Interpretation: Nonlocality

The [Copenhagen] Interpretation of ||2, I think, contradicts the

postulates of relativity. (1927)

The [measurement] at the position of the reflected

packet thus exerts a kind of action (reduction of the

wave packet) at the distant point occupied by the transmitted packet, and one sees that this

action is propagated with a velocity greater than

that of light. (1930)

Einstein’s (obvious) conclusion: Quantum Mechanics is incomplete.

• Alice sees the particle because it really is in her wave packet.

• Bob doesn’t because it’s not in his.

• There is no action at a distance here.

beam-splitter

Bob

Alice

“empty wave”

It seems to me that this difficulty cannot be overcome unless the description of the process in terms of the Schrödinger wave

is supplemented by some detailed specification of the localization of the particle during its propagation. (1927)

I(b) Einstein, Podolsky & Rosen (1935)

BohrEinstein

Consider two well-separated particles in a state such that pA=pB and qA=-qB.

Then without disturbing Bob’s particle, Alice can find out pB by

measuring pA, or qB by measuring qA. But no quantum state has definite

values for both pB and qB.

Therefore the quantum state is not a complete description for Bob’s

particle.

• Einstein’s 1927 argument for the incompleteness of the QM used a single-particle scenario. Its weakness is that it relies on the long-range coherence of the wavefunction. For experiments, this would require additional testing and/or assumptions.

• This “weakness” is overcome in the 1935 EPR argument.

Note that an assumption of locality is still central (implicitly) in the EPR argument.

I(c) Schrödinger: Steering (1935)

I have a cute name for these correlated pure states:

EntanglementEntanglement.

I also have a cute name for the nonlocal EPR-effect: SteeringSteering.And I’ve generalized the idea to

allow Alice not just two, but arbitrarily many different

measurements, so she can remotely prepare arbitrarily many different

sorts of pure states for Bob.

To me, the generality of steering suggests that this nonlocality is not due to the incompleteness of QM.

Rather, I think this nonlocality is an unavoidable consequence of QM.

Instead, I suspect QM will break down for distant entangled systems.

9

Schrödinger’s Car?

I(d) Bell-Nonlocality (1964, 1981)

Schrödinger was right about nonlocality being inherent to QM. By

considering a specific example of measurements on entangled states I

showed that QM can’t be “completed” as Einstein envisaged.

That is, no local hidden variable That is, no local hidden variable theory can explain the predicted theory can explain the predicted correlations between Bob’s and correlations between Bob’s and

Alice’s results. Alice’s results.

And, unfortunately for Schrödinger and Einstein, our experiments have

proven that QM is correct in its predictions for two distant entangled

systems.

The world is nonlocal !The world is nonlocal !

Alain Aspect

II. Three Types of

Quantum Nonlocality

II(a) Three Notions of Nonlocality

• 1935 EPR correlations = steering “…as a consequence of two different measurements performed upon the first system, the [distant] second system may be left in states with two different [types of] wavefunctions.”

• 1935 Schrödinger’s entanglement = nonseparability. “Maximal knowledge of a total system does not necessarily include total knowledge of all its parts, not even when these are fully separated from each other.”

• 1964 Bell nonlocality “In a theory in which parameters … determine the results of individual measurements, … there must be a mechanism whereby the setting of one measurement device can influence the reading of another instrument, however remote.”

II(b) Nonlocality for Mixed States

For all pure bipartite states, with perfect detection, the following are equivalent:1. The state is nonseparable (cannot be prepared locally)2. The state can be used to demonstrate EPR-steering3. The state can be used to demonstrate Bell-nonlocality.

For mixed bipartite states, Werner (1989) showed that (1)and (3) are not equivalent; (3) is strictly stronger than (1).

What about (2)? Is (1)(2)? Is (2)(3)? Or neither?To answer this we need a rigorous definition of steering.

II(c) Operational DefinitionsFor a given bipartite state, and given measurement strategies for Alice and Bob, we can define when the statistics of Alice’s and Bob’s measurement results demonstrate these forms of nonlocality. (HMW, Jones & Doherty, PRL’07)• They demonstrate Bell-nonlocality iff their results could not have arisen from correlations between a random local hidden variable (LHV) for Alice and the same for Bob. • They demonstrate EPR-steering by Alice iff their results could not have arisen from correlations between a random LHV for Alice and a random pure state which Bob measures.• They demonstrate non-separability iff their results could not have arisen from correlations between a random pure state which Alice measures and the same for Bob.

Thus Bell-nonlocality EPR-steering non-separability

II(d) Three Types of InequalityFrom these conditions, we can derive inequalities that must be violated to demonstrate the three types of nonlocality. (Cavalcanti, Jones, HMW and Reid, PRA 80, 032112 (2009) )*

Consider two pairs of binary measurements: These can arise from measuring a Pauli operator (e.g. ) on a qubit (= a spin-1/2 particle).

Bell-nonlocality (CHSH, 1969)

EPR-steering (CJWR, 2009)

Non-separability (entanglement witness, mid-90s)

€

A, ′ A ,B, ′ B ∈ {−1,1}

€

AB + ′ A B + A ′ B − ′ A ′ B ≤ 2

€

A ˆ σ XB + ′ A ˆ σ Z

B ≤ 2

€

ˆ σ X

€

ˆ σ XA ˆ σ X

B + ˆ σ ZA ˆ σ Z

B ≤1

* The first EPR-Steering inequality was derived by Reid (1989), as a more literal generalization of EPR.

II(e) Three Classes of StatesBy considering all possible inequalities, we proved that the three classes of states are distinct (WJD’07).

non-separable states

steerable states

Bell-nonlocal states

e.g. for Werner states of two qubits with purity 0<p<1,Bell-nonlocality exists only if p > 0.6595… [Acin+’06] EPR-Steering exists if and only if p > 1/2 [WJD’07]Non-separability exists if and only if p > 1/3

[Werner’89]

III. Experimental Metaphysics

III(a) Abner Shimony

EPR-Steering implies a still-weaker sort of connection, which I suggest could be called Platonic love at a distance.

(Entanglement might be sympathy at a distance?)

As Heisenberg said, with bipartite states, it can be that measurement at one point “exerts a kind of action … at the distant

point”.

But it is not the kind of action at a distance in Newtonian gravity, for

example --- even the violation of a Bell-inequality does not mean that Alice can

send a signal to Bob.

But it shows a connection between the two parties stronger than that of a common

cause.

In 1984 I called this passion at a distance.

III(b) Testing nonlocality:the power of n

Consider again mixed Werner states of two qubits with purity p.

Let n be the number of different measurement settings used by Alice and Bob.

• for n=2, Bell-nonlocality exists if p > 0.707 [CHSH’69]

• for n=465, Bell-nonlocality exists if p > 0. 7056 [Vertesi’08]

• for n=∞, Bell-nonlocality exists only if p > 0.6595 [Acin+’06]

How about for EPR-steering?

Traditionally (i.e. following EPR) one considers only n=2.

• for n=2, EPR-steering exists if p > 0.707 [CJWR’09]

• for n=∞, EPR-steering exists if and only if p > 0.5 [WJD’07]

New EPR-Steering Inequalities

Pretend that you recall the n=2 EPR-s inequality

Here we generalize to where Cn is a constant,

and { } is a set of n Pauli (spin) operators in directions {uk}. €

A ˆ σ XB + ′ A ˆ σ Z

B ≤ 2

€

1

nAk

ˆ σ kB

k=1

n

∑ ≤ Cn

€

ˆ σ kB

The Punch-lineHow to arrange the spin directions to demonstrate EPR-steering, a.k.a. Platonic love at a distance?

Bob’s measurement directions are the vertices with: n = 2 (square), 3 (octahedron), 4 (cube), 6 (icosahedron) and 10 (dodecahedron).

This makes the Cn in “easy” to calculate.

€

1

nAk

ˆ σ kB

k=1

n

∑ ≤ Cn

How Cn is calculated (and observed)Recall that Alice may be trying to cheat, by randomly choosing a pure state from some ensemble, sending it to Bob, and then reporting Ak based on that state to maximize the correlation Sn.

• The maximum Sn that Alice can achieve by trying to cheat is the bound Cn which must be exceeded to show steering by definition.

• Interestingly, the arrangement of pure states in Alices’ “optimal trying-to-cheat ensemble” varies with n.

• For n = 2, 3, 4, the optimal pure states are face-centres (as in quantum random access codes). For n = 6, 10, they are vertices.

III(c) Experiment

• The Controlled-Z gate creates a state ≈ a polarization singlet. • The azimuthal angle between the two Hanle wedge depolarizers

(DP) controls the amount of depolarization, creating Werner states of any desired purity p. [Puentes et al Opt. Lett. 31, 2057 (2006)]

Experimental Results: It works!

States with purity above here violate the CHSH (n=2) Bell-inequality.

States with purity below here violate no Bell-inequalities.

C2

C3 C4 C6

Example state which demonstrates Bell-nonlocality and EPR-steering.

Example state which is “Bell-local” but which demonstrates EPR-steering.

C∞This is measured to ≈ the purity p.

separable states

How close Alice got to Cn by trying to cheat, sending pure states to Bob.

IV. Conclusions

• Einstein, Podolsky, and Rosen developed their thought-experiment to demonstrate the nonlocality of the collapse in the Copenhagen interpretation.

• Schrödinger called it “steering”, but thought it could only be avoided if QM itself was wrong. (It’s not.)

• We have (finally, in 2007!) given a formal definition for EPR-steering, and proven that it is a form of nonlocality strictly intermediate between Bell- nonlocality and entanglement.

• Unlike the case of Bell-nonlocality, increasing the number of measurement settings n per side beyond two dramatically increases the robustness of the EPR-steering phenomenon to noise (i.e. mixture).

• We have demonstrated this experimentally using settings based on Platonic solids with n=2, 3, 4, & 6.

1. EPR-steering has been demonstrated experimentally before in a number of systems. But some of these experiments lack a rigorous theoretical treatment.

2. Nobody has done a loop-hole free Bell-nonlocality test, simultaneously closing the

– Detection efficiency loop-hole– Space-like-separation loop-hole.

A loop-hole free EPR-steering test should be easier.3. We showed (WJD’07) that EPR-steering can be

defined as a quantum information task with partial lack of trust between parties. We believe it can also be used to make quantum key distribution more secure, and perhaps in other QIP tasks.

IV(b) Ongoing and Future Work