1. Understanding Quantum Correlations
Nicolas Gisin
Cyril Branciard, Nicolas Brunner
Group of Applied Physics
Geneva university
Switzerland
a + b= x.y
Alice
Bob x a b y -1 +1 -2
 0

2. Understanding Quantum Correlations
Intuition
decomposition into "simpler" correlations
simulation with "simpler“ correlations
resources provided by Q correlations
resources needed to simulate Q correlations
()=cos(/2) |00> + sin(/2) |11>
where , = ±1
no-signaling:
|11> 
()
|00>

3. 1. Bell locality By far the most natural assumption !
… refuted beyond (almost) any reasonable doubts.
Hence, quantum correlations happen, but the probabilities
of their occurrence are not determined by local variables.

4. Satigny – Geneva – Jussy
18.0 km δ
Geneva

5. How come the correlation ?
How can these two locations out there in space-time know about each other ?
There is no spooky action at a distance : there is not a first event that influences a second event.
Quantum correlation just happen, somehow from outside space-time : there is no story in space-time that can tell us how it happens.

6. 4. Leggett’s “locality”
Found.Phys. 10,1469,2003; Vienna, Nature 2006; A. Suarez, Found. Phys. 2008
Assume that locally everything is “normal”, i.e. that individual particles are always in pure states:
where
and
“Only” the correlations C are nonlocal.
They just happen, without any classical explanation.
They are only constraint by P  0

7. Leggett’s inequalities

8. Leggett’s inequalities

9. Leggett’s inequalities
Modern form
of Leggett’s
inequality
In strong contrast to
Bell’s inequalities, here
the bound depends on the
measurement settings

10. Experimental Setup
Traditional Type-II parametric down conversion source:
photon
HV: vis= 98.9±0.8%
±45°: vis=97.8±0.8%
max. coincidence rate: 630 s-1, accidentals: 0.3 s-1

11. Experimental refutation of
Leggett’s model
1.4
1.5
1.6
1.7
1.8
1.9 2
integration time: 4 x 15 sec / setting
maximal violation: L=1.925 ± (40.6 σ) at φ = -25° L=1.922 ± (38.1 σ) at φ = +25° QM L3
Leggett
-90°
-60°
-30°
30°
60°
90° 
PRL 99,210406,2007 PRL 99,210407,2007 Branciard et al. Quant-ph/
Nature Physics, in press, 2008
for 60 sec/setting:
L3(-30°)=5.7204± (83.7 σ)

12. 5. Simulation with a PR-box
Alice
Bob
x {0,1}
y{0,1}
a + b= x.y
a {0,1}
b {0,1}
a + b= x.y
A single bit of communication suffice to simulate
the PR-box (assuming shared randomness).
But the PR-box does not allow any communication.
Hence, the PR-box is a strictly weaker resource
than communication.
Prob(a=1|x,y) = ½, independent of y  no signaling
E(a,b|0,0) + E(a,b|0,1) + E(a,b|1,0) - E(a,b|1,1) = 4
Found.Phys. 24, 379, 1994

13. Simulating a singlet with a PR-box
where the are uniformly distributed on the sphere
and is defined by the PR-box as follows:
a + b= x.y
PRL 94,220403,2005
Quant-ph/

14. Does this help our understanding ?
After all, in a PR-box the correlation merely happen, without any explanation.
Yes, but this has to be the case!
Yes, but this is also the case in quantum physics (and in models à la Leggett)!
Moreover, a+b=x.y is really simply !
At least it helps me …

15. 6. Asymmetric detection loophole
Consider entanglement between an atom and a photon. In such a case the detection of the atom can be realised with quasi 100% efficiency.
Intuition predicts and computations confirm that the threshold photon-detection efficiency is lower in such an asymmetric situation compared to the symmetric case: CHSH: max entanglement  partial entanglemt 
A. Cabello and J.-A. Larsson, PRL 98, , N. Brunner et al., PRL 98, , 2007

16. Detection loophole in asymmetric entanglement with I3322
N. Brunner et al.
PRL 98, ,2007
1/2
2/3
Minimal detection efficiency < 0.5 !!
Connection to simulability with 1 bit of communication ?
product state
max entangl.

17. From asymmetric detection loophole to the impossibility of simulating with a PR-box
=x(,a)
=y(,b)
=(,a, a)
= (,b,b) x y
Assume some
correlation can
be simulated with
a PR-box: 
a+b=xy a b
Let xg and ag be 2 additional shared random bits
Asymmetric detection: Alice Bob
if xg = x(,a) = (, b, ag +xg.y(,b))
then =(,a,ag)
else “no output”

18. Impossibility of simulating very partially entangled states with a PR-box
The fact that it is possible to close the asymmetric detection loophole with a detector’s efficiency less than 50% and partially entangled states, implies the impossibility to simulate those states with a single PR-box.

19. Note on the role of marginals
We assumed a PR-box with trivial marginals and concluded that such a nonlocal resource can’t simulate quantum correlations with large marginals.
In Leggett’s model we imposed non-trivial marginals and concluded that this is incompatible with the quantum correlation corresponding to the singlets.
 it is especially hard to simulate simultaneously nonlocal correlation and non-trivial marginals.

20. Leggett’s “locality” revisited
Assume that locally everything is “normal”, i.e. that each particle is always in a non maximally mixed state:
where
and
where 01.
 …  similar inequalities prove incompatibility with singlets
Branciard et al. Quant-ph/
Nature Physics, 2008;
Renner et al, PRL 2008

21. 7. Correlated local flips
Let’s try to make up the non-trivial marginal afterwards.
Let 1 1 f
and let the outcomes , pass through a Z channel:
1-f
Let the flip probabilities f and f be determined by a common
variable [0,1]: 1 f f
 no flip
 flip  but not 
where
 flip  and 
quant-ph/

22. Local flips for quantum correlations
Let
and look for the corresponding
unbiased correlation: 
and !
where is the original input moved back one step one the Hardy ladder :
Hence, we almost succeeded in simulating any 2 qubit state with a PR-box
… but we had to assume f  f, i.e. bz  az !
quant-ph/

23. 7. Correlated local flips
Lemma If
then there is
and local flip probabilities f and f such that
In words: all marginals can be realised via correlated local flips.
quant-ph/

24. 8. The M-box (Millionaire-box)
Alice
Bob
x [0,1]
y[0,1]
a + b= (xy)
a {0,1}
b {0,1}
M-box are non-signaling.
a M-box allows one to simulate a PR-box.
a M-box violates maximally all the Inn22 Bell inequalities.
quant-ph/

25. Simulating entangled qubits with 4 PR-boxes and 1 M-box
quant-ph/

26. Simulating entangled qubits with 4 PR-boxes and 1 M-box
a+b=xy
a+b=xy b1 a2 b2 a1 az bz
a+b=(xy) a b
quant-ph/

27. Simulating entangled qubits with 4 PR-boxes and 1 M-box
a+b=xy
a+b=xy b1 a2 b2 a1 az bz
a+b=(xy) a b
a+b=xy b a 1 1 2 2

28. Simulating entangled qubits with 4 PR-boxes and 1 M-box 
local flip fa
local flip fb  
quant-ph/

29. 9. decomposition into local+nonlocal
EPR2
Phys. Lett. A 162, 25, 1992
PQM = pl.PL + (1-pl).PNL
lemma: if PL(,|a,b) = PL(,|az,bz)
then pl  1-sin()
V. Scarani,
Quant-ph/
PRA 2008
proof:

30. 9. decomposition into local+nonlocal
PQM = pl.PL + (1-pl).PNL
pl = 1-sin()
V. Scarani,
Quant-ph/
PRA 2008
In the slice around the equator
the nonlocal part reduces
to a scalar product:
but … this slice tends to zero for the singlet !?!

31. Simulating PNL with nonlocal non-signaling resources
PNL can be simulated with 4 PR-boxes and one M-box, in a way very similar the presented one.
Consequently, partially entangled states can be simulated using nonlocal resources only in a fraction sin() of all cases: the less () is entangled, the less frequently one needs nonlocal resources.
However, the nonlocal resources (seldomly) needed to simulate partially entangled states are definitively larger than those (always) required to simulate maximally entangled states.

32. Conclusions
Q nonlocality is a mature topic. Lots of progress have been achieved, but many important and fascinating questions are still open.
Quantum correlations are very peculiar. They combine nonlocal correlations with non-trivial marginals in a way that is difficult to reproduce.
Bell-type inequalities can be derived for all kinds of hypothesis, not only Bell locality, and all sorts of nonlocal resources.
In counting resources required to simulate () one should distinguish the amount of resources and the frequency at which one has to use them.
There are connections to experiments: - moving masses to ensure space-like separation - east-west Bell tests with good synchronization - asymmetric atom-photon entanglement

33. Let’s test these hypothetical preferred reference frame
Alice and Bob,
east-west orientation,
perfect synchronization
with respect to earth
w.r.t any frame moving perpendicular to the
A-B axis
in 12 hours all hypothe-
tical privileged frames are scanned.
A B
Ph. Eberhard, private communication

34. D. Salard et al., Nature, 2008

35. c (°) Bound on VQI/c D. Salard et al., Nature, 2008
Bound assuming the Earth’s speed is  300 km/s
Bound assuming  = 90o

36. Satigny Jussy Geneva Franson interferometer = 60 s Piezo 1573.5 nmFG
APD FM
Satigny δ
Jussy
17.5 km
quantum channel
classical channels
10.7 km
8.2 km
18.0 km
TAC
Geneva
FBG C
PPLN L F
Laser
Piezo
= 60 s
1573.5 nm
1568.5 nm
A. Kent
arXiv:gr-qc/
Franson
interferometer
quant-ph/
PRL 2008

37. Piezo He-Ne Laser - BS + Mirror Mirror 100 nm 4V Single-photon
detector 4V
Photodiode
quant-ph/

38. Bell test with true space-like separation
source A B
time
space
A macroscopic mass has
significantly moved
 60 s
 18 km
 7 s
The photon enters
the interferometer
In usual Bell tests, detection events only trigger the motion of electrons
of insufficient mass to finish the measurement process.
quant-ph/
PRL 2008

39. Visibility > 90%  nonlocal correlations between
truly space-like separated events.
quant-ph/ ; PRL 2008