Transcript Document

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
Bob
beamsplitter
Alice
I(a) Einstein’s objection to the
Copenhagen Interpretation: Nonlocality
Bob
beamsplitter
Alice
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)
The [Copenhagen] Interpretation
of ||2, I think, contradicts the
postulates of relativity. (1927)
Einstein’s (obvious) conclusion:
Quantum Mechanics is incomplete.
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)
Bob
“empty
wave”
• Alice sees the particle
because it really is in
beamher wave packet.
splitter
• Bob doesn’t because
it’s not in his.
Alice • There is no action at a
distance here.
I(b) Einstein, Podolsky & Rosen (1935)
•
•
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.
Bohr
Einstein
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.
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: Entanglement.
I also have a cute name for the
nonlocal EPR-effect: Steering.
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.
Schrödinger’s Car?
9
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 theory
can explain the predicted correlations
between Bob’s and Alice’s results.
Alain Aspect
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 !
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-steering
3. 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 Definitions
For 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 Inequality
From 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: A, A,B, B {1,1}
These can arise from measuring a Pauli operator (e.g. 
ˆ X ) on
a qubit (= a spin-1/2 particle).
Bell-nonlocality (CHSH, 1969)

AB  AB  AB  AB  2 
EPR-steering (CJWR, 2009)
ˆ XB  A
ˆ ZB  2
A

Non-separability
(entanglement witness, mid-90s)

ˆ XA
ˆ XB  
ˆ ZA
ˆ ZB 1

* The first EPR-Steering inequality was derived by Reid (1989), as a more literal generalization of EPR.
II(e) Three Classes of States
By considering all possible inequalities, we proved that
the three classes of states are distinct (WJD’07).
Bell-nonlocal states
steerable states
non-separable 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
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.
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?)
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
ˆ XB  A
ˆ ZB  2
Pretend that you recall the n=2 EPR-s inequality A
1 n
ˆ kB  Cn where C is a constant,
Here we generalize to  Ak
n
n k1
ˆ kB } is a set of n Pauli (spin) operators in directions {uk}.
and { 

The Punch-line
How to arrange the spin directions to demonstrate EPRsteering, 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).
n
1
B
This makes the Cn in
“easy” to calculate.
ˆ
A


k k  Cn
n k1
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.
Example state which demonstrates
Bell-nonlocality and EPR-steering.
C2
C3
C4
This is
measured
to ≈ the
purity p.
States with
purity below
here violate
no Bellinequalities.
Example state
which is “Belllocal” but which
demonstrates
EPR-steering.
separable states
C6
C∞
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 Bellnonlocality 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.
IV(b) Ongoing and Future Work
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