Quantum Nonlocality:
How does Nature do it? And what can we do with it?
Nicolas Gisin GAP, University of Geneva, Switzerland
GAP Optique Geneva University
1. The scene:
what is nonlocality ?
2. How does Nature
perform the trick ?
3. What can nonlocal
correlations do for us ?
4. Deterministic nonlocal
variables
5. Hidden influences
1
x
Alice
GAP Optique Geneva University

a
P(a,b|x,y)
≠ P(a|x)·P(b|y)
The events at Alice and
Bob’s sides are not
independent !
It seems that somehow
the two sides are coordinated
or “interact” !?!
(but without signaling)
y
Bob

b
Spatially separated systems are not
logically separated.
 Quantum Physics is nonlocal
2
GAP Optique Geneva University
During my early carrier as a physicist...
when I was about 6 months old…
I learned the hard way that in order
to interact with an object I had either to
crawl to it or to throw something at it.
3
x
Alice
GAP Optique Geneva University

a
Assumptions:
y
1. locality:
P(a,b|x,y) = P(a|x)·P(b|y)
where =physical state of the
systems according to any
possible future theories.


2.a
Alice can freely choose her input
x and read the outcome a,
similarly for Bob.
2.b x & y are independent of 
I(x:)=I(y:)=0
Conclusion:
Bob

b
Bell inequalities
N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255,
Found. Phys. 42, 80 (2012)
NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009
4
x
y
Today, we don’t know :
P(a,b|x,y) = d P(a|x)·P(b|y)
 Bell inequality
Alice
GAP Optique Geneva University
Bell inequalities
Bob
1. Violation:



 all future theories have

to be nonlocal.
2. No violation:
 there is a separable quantum state and
a
measurements reproducing P(a,b|x,y). b
PRL 97, 120405, 2006; PRL 100, 210503, 2008.
N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255,
Found. Phys. 42, 80 (2012)
NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009
5
x
Alice
GAP Optique Geneva University

Don’t think of  as an old fashion
local hidden variable. Think of 
as the physical state of the
systems as described by any
possible future theory.

 Could be the state of the
entire universe, except that 
can’t determine x nor y.
a

Studying Bell’s inequality tells
us something about any possible
future theory compatible with
today’s experimental
observations.
y
Bob

b
N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255,
Found. Phys. 42, 80 (2012)
NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009
6
x
No signalling
Alice
GAP Optique Geneva University

y
Bob’s statistics are independent
of Alice’s input x.


Bob

Alice’s statistics are independent
of Bob’s input x.
a
b
 P (a, b | x, y)  P (b | y)
a
N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255,
Found. Phys. 42, 80 (2012)
NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009
7
x
Alice
GAP Optique Geneva University

a
Non realism ?
The only assumption in the
derivation of Bell inequality,
besides the locality assumption,
 is that x,y,a and b are 
classical variables. That is that
one directly access them, copy,
memories and broadcast them.
Non realism seems to me not an
alternative to nonlocality. And what
could “local non realism” mean ?!?
y
Bob

b
N. Gisin, Non-realism : deep thought or a soft option ? quant-ph/0901.4255,
Found. Phys. 42, 80 (2012)
NG, Quantum nonlocality : how does nature do it ?, Science, 326, 1357, 2009
8
The CHSH-Bell inequality
x,y,a,b = bits
E(x,y)=p(a=b|x,y)-p(a≠b|x,y)
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E(0,0)+E(0,1)+E(1,0)-E(1,1)
Alice
X=0
.
Bob
Y=0
.
.
x=1


.
Y=1
 SCHSH  E(0,0) + E(0,1) + E(1,0) - E(1,1)  2
But SCHSH(Quantum) = 22
9
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Nonlocality without Signaling
 It is very difficult to think of Nonlocality
without some sort of communication “behind
the scene”.
 Yet, quantum physics is nonlocal, but doesn’t
allow for “signaling”:
The statistics on Bob’s side do not depend on
Alice’s input:
p(b|x,y)ap(a,b|x,y) = p(b|y)
And similarly: p(a|x,y) = p(a|x).
10
No-cloning and no-signaling
NG, Phys. Lett. A 242, 1 (1998)
 Perfect cloning would allow one to exploit
nonlocal correlations for signaling:
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Alice
M
Source of
entangled
particules
Bob
}
*
clones
signaling
 Interestingly, optimal-but-imperfect quantum
cloning is precisely at the no-signaling limit !
Alice
M
Source of
entangled
particules
*
Bob
}
Laser amplifier:
stimulated &
NO signaling spontaneous emissions
Noisy
clones
11
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Nature is Nonlocal (but without signaling).
And so?
 Physics: Amazing ! How can these two locations
out there in space-time know about each other ?
 According to Quantum Physics Quantum
correlation just happen, somehow from outside
space-time :
there is no story in space-time
that can tell us how it happens.
 Computer Science: What can one do with
nonlocal non-signaling (quantum) correlations ?
12
PR-box: E(0,0) + E(0,1) + E(1,0) - E(1,1) = 4
x {0,1}
Alice
Bob
y{0,1}
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a + b= x.y
a {0,1}
b {0,1}
A single
bit of communication
Prob(a=1|x,y)
= ½, independentsuffice
of y  to
no simulate
signaling
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.
Found.Phys. 24, 379, 1994
13
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 The PR-box is the unique extremal nonlocal box
for binary inputs and outputs.
 Why is Quantum Physics not more nonlocal
than it is ?!?
 What could one do, had we PR-boxes ?
14
No-cloning from no-signaling
y
y
x
b
z
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a+b=x.y
a
 b+c = x.(y+z)
b
c
 If y+z=1, Bob and Charly can deduce Alice’s input
out of their outputs
 Signaling !!!
 In a non-signaling world, cloning of PR-boxes is impossible
A noisy PR box can be cloned iff it is noisy enough to be local.
15
From non-signaling nonlocality to
cryptographic keys
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 The quantum no-cloning theorem is the basis
for Quantum Key Distribution.
 Since no-cloning holds for all non-signaling
nonlocal correlations, QKD should be secure
against any non-signaling adversary, even postquantum (i.e. mastering technologies described
by any future theory, provided it is nonsignaling).
16
Alice
x=0 or 1
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0
a
1
Basics of QKD
Bob
y=0 or 1
0
1
b
After publicly announcing a fair sample of their data,
Alice and Bob’s information is entirely contained
in the conditional probability p(a,b|x,y)
If E(0,0)+E(0,1)+E(1,0)-E(1,1) is large enough,
then Alice and Bob can distil a secret key secure
against any possible individual attack by any
non-signaling adversary.
17
The no-signaling polytope
J.Barrett et al, PRL’05
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PR-box
Only one new vertex per CHSH facet of
the local polytope
a+b=xy
vertex:
local deterministic
correlation
QM
facet corresponding
to the CHSH-Bell :
E(0,0)+E(0,1)+
E(1,0)-E(1,1) 2
polytope of
local correlations
18
Studying
nonlocal
correlations
A bit more on the Theory side
from outside the Hilbert space
P(a,b|x,y) can be considered as a vector and
represented as a point in a vector space:
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No signalling
New question:
why are quantum
correlations not
more nonlocal ?
Bell inequality
Quantum
Local
19
20
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Deterministic nonlocal hidden
variables


a
b
Alice
(-)

a
GAP Optique Geneva University


=FAB( a,)

=SAB(b , a,)
e.g. ={,ra,rb}.
 1 if ra  1 / 2

FAB (a ,  )  
 1 if ra  1 / 2
Bob

t=2
t=1
t=0
 

1
1 a b
if
r

&
r


a
b
2
2 
 
 1
S AB (b , a ,  )  
1
1 a b
or
r

&
r

a
b

2
2
 1 else

21
Deterministic nonlocal hidden variables

b

a
Alice
(-)
GAP Optique Geneva University

a


=FAB( a,)

=SAB(b , a,)
Bob

t=2
t=1
t=0
t’=2
t’=1
t’=0

 =FBA(b ,)
 
 =SBA(a , b ,)
22

b

a
Alice
(-)
GAP Optique Geneva University


=FAB( a,)

=SAB(b , a,)
Bob

t=2
t=1
t=0
t’=2
t’=1
t’=0

 =FBA(b ,)
 
 =SBA(a , b ,)
Could there be , FAB, SAB, FBA and SBA s.t.
 

FAB( a,) = SBA(a , b,) ?
Theorem: NO !
 Impossibility of covariant deterministic nonlocal
Proof: SBA would be independent of b hidden-variable extensions of quantum theory
 locality  Bell inequality. NG, PRA 83, 020102, 2011
Quantum correlations can’t be described with local
variables, nor can they be described with
deterministic nonlocal variables.
23
Finite-Speed Hidden-Influence
Explanations of Quantum
Correlations lead to Signalling
Nicolas Gisin
GAP Optique Geneva University
J.-D. Bancal, S. Pironio, A. Acin, Y.-C. Liang, V. Scarani
Group of Applied Physics, University of Geneva
 A principle of continuity
 Quantum nonlocality based on finite-speed
causal influences leads to superluminal
signalling, arXiv:1110.3795
 Nature is nonlocal in the sense of discontinuous
24
GAP Optique Geneva University
Newton’s Nonlocality
A stone moved on the moon
would immediately affect
the gravitational field
on earth.
How can these two locations out there in
Space-time know about each other ?
25
GAP Optique Geneva University
Let’s read Newton’s words:
That Gravity should be innate, inherent and essential to
Matter, so that one Body may act upon another at a
Distance thro’ a Vacuum, without the mediation of any
thing else, by and through which their Action and Force
may be conveyed from one to another, is to me so great
an Absurdity, that I believe no Man who has in philosophical
Matters a competent Faculty of thinking, can ever fall into it.
Gravity must be caused by an Agent acting constantly
according to certain Laws, but whether this Agent be
material or immaterial, I have left to the Consideration
of my Readers.
Isaac Newton
Papers & Letters on Natural Philosophy and related documents
Edited by Bernard Cohen, assisted by Robert E. Schofield
Harvard University Press, Cambridge, Massachusetts, 1958
26
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Signalling = non-physical communication
 To send information one has to encode it into a
physical support and send this support to the
receiver.
Landauer: Information is physical.
 Any other way of communicating would be
signalling; it would be non-physical
communication.
 Moreover, signalling would allow faster-than-light
communication. But no-signalling is even more
fundamental than relativity.
27
Principle of Continuity
GAP Optique Geneva University
 Everything (mass, energy and information) propagates
gradually and continuously through space as time
passes.
 Nothing jumps instantaneously from here to there
(no instantaneous teleportation).
 Correlations can have only two types of explanations.
Either common local cause or influences at finite speed
t
x
28
Principle of Continuity
Explanations of correlations
by local common causes
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Variables
(hidden)
Local
Bell’s theorem
Explanations of correlations
by an event influencing
another one
Influences
(hidden)
Finite speed
This talk
Contradiction with quantum
predictions
Falsified explanation
29
Satigny – Geneva – Jussy
N
W
E
GAP Optique Geneva University
S
Nature 454, 861, 2008
Cocciaro et al., PLA 375, 379, 2011
Experimental lower bound on the speed of hidden influence
30
And so ?
GAP Optique Geneva University
 The influence may merely propagate faster,
 or may not exist at all.
 2-party experiments will never be able to
exclude hidden influences, only set lower
bounds on its speed.
31
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 -causality
 Assume that a hidden influence propagates at speed
v < .
 v can be larger than c
(defined in a universal privileged frame).
 Whenever an event happens, the rest of the universe is informed at
speed v.
 Whenever the hidden influence arrives on time, future events are
correlated as predicted by QM.
 Whenever the hidden influence doesn’t arrive on time, events can
only be Bell-local correlated
(i.e. correlated by local variables).
  -causal predictions may differ from quantum theory
 Bohm
We shall see that hidden influence at finite speed leads
to signalling at the macroscopic level, i.e. hidden
influences at finite speed can’t remain hidden
32
tpriv.
-causality
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Light cone


v-cone
xprivileged
33
tpriv.
-causality leads to signalling
ABD is quantum
B
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
C

ACD is quantum
D

A

Alice
BC is local
even if
conditioned
on A and D
xprivileged
Bob
Charlie
Dave
34
Q predictions for v-causality
GAP Optique Geneva University
BC|AD local  p(a,b,c,d|x,y,z,w)=p(a,d|x,y,z,w)
//
 p(b|y, x,a,w,d, /
z)
 p(c|z, x,a,w,d, /
y)
V-causality
Q prediction:
p(a, b, c, d | x, y, z, w)  Tr( Aax  Ddw   )  Tr(Bby  adxw )  Tr(Ccz  adxw )
where
x
w
x
w
A

D



A

D
d
a
d
 adxw  a
Tr ( Aax  Ddw   )
 for the state and setting of our paper,
p(abc|x,y,z,w) is signalling iff a=1 & x=1 & z=0
p(bcd|x,y,z,w) is signalling iff z=1 & w=0
arXiv:1110.3795
35
Principle of Continuity
GAP Optique Geneva University
Explanation of correlations by
local common causes and influences
Variables and Influences
(hidden)
Local and Finite speed
Rest of this talk
36
-causality
leads to signalling
y
z
w
binary
inputs 0/1
x
A
B
Alice
Bob
a
b
GAP Optique Geneva University

binary
outcomes 1

C

Charlie
c
D

xpriv.
Dave
d
Theorem: If p(a,b,c,d|x,y,z,w) is non-signalling
and p(b,c|y,z, a,x,d,w) is local for all a,x,d,w,
then
J7
Where J = -3A1 - B0 - B1 - C0 - 3D0 - A1B0 - A1B1 + A0C0
+ 2A1C0 + A0D0 + B0D1 - B1D1 - C0D0 - 2C1D1
+ A0B0D0 + A0B0D1 + A0B1D0 - A0B1D1 - A1B0D0
- A1B1D0 + A0C0D0 + 2A1C0D0 + 2A0C1D1
37
tpriv.
J = -3A0 - B0 - B1 - C0 - 3D0 - A1B0 - A1B1 + A0C0
+ 2A1C0 + A0D0 + B0D1 - B1D1 - C0D0 - 2C1D1
+ A0B0D0 + A0B0D1 + A0B1D0 - A0B1D1 - A1B0D0
- A1B1D0 + A0C0D0 + 2A1C0D0 + 2A0C1D1
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B

C

D

A

Alice
xprivileged
Bob
Charlie
Dave
38
J = -3A0 - B0 - B1 - C0 - 3D0 - A1B0 - A1B1 + A0C0
+ 2A1C0 + A0D0 + B0D1 - B1D1 - C0D0 - 2C1D1
+ A0B0D0 + A0B0D1 + A0B1D0 - A0B1D1 - A1B0D0
- A1B1D0 + A0C0D0 + 2A1C0D0 + 2A0C1D1
tAny
priv. v-causal model predicts the same value for J as QM
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B
C


-causal predictions differ
from Q theory, but since J
doesn’t contain any term
involving B and C, the D
-causal prediction for J
is merely the Q value.
Moreover, in an experiment
B and C do not need to be
A
in the same run.
 measured
No B-CCharlie
timing issue
!
Alice Bob
Dave
xprivileged
39
-causality leads to signalling
GAP Optique Geneva University
Fact: there are quantum states and measurements predicting J>7
Theorem: If p(a,b,c,d|x,y,z,w) is non-signalling
and p(b,c|y,z, a,x,d,w) is local for all a,x,d,w,
then
J7
Consequence: Since any v-causal model predicts that
p(b,c|y,z, a,x,d,w) is local,
p(a,b,c,d|x,y,z,w) must be signalling.
arXiv:1110.3795
40
-causality leads to
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supraluminal communication
41
Principle of Continuity
Explications of correlations
by local common causes
GAP Optique Geneva University
Variables
Explications of correlations
by an event influencing
another
Influences
(hidden)
(hidden)
Local
Finite speed
Bell’s theorem
This talk
Contradiction with quantum
predictions
Signalling
Contradiction with no faster
than light communication
Falsified explanation
Falsified explanation
Nature doesn’t satisfy the principle of continuity
Nature is nonlocal
42
arXiv:1110.3795
GAP Optique Geneva University
Conclusion
 There is no spooky action at a distance: there is
not a first event that influences a second event.
 Quantum correlation just happen, without any
time-ordering, somehow from outside spacetime !
 There is no story in space-time that tells us
how nonlocal Q correlations happen.
Shall scientists give up the Grand Enterprise of
explaining how Nature does it?
Explaining = telling stories
 We need to enlarge our story-tool-box to
explain nonlocal correlations
43
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New tools for our story-tool-box
Instead of claiming with Einstein that God doesn’t
play dice, let’s ask why he plays dice.
The answer is that so Nature can be nonlocal
without signalling (without non-physical
communication).
 Let’s tell stories in which random events can
manifest themselves at several locations,
 i.e. stories that include nonlocal randomness.
44