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Simulating Quantum Correlations
with Finite Communication
Ben Toner
(CWI, Amsterdam)
Oded Regev
(Tel Aviv University)
Proceedings of FOCS 2007,
quant-ph/0708.0827
QIP, 21 December, 2007
* Original
slides prepared by Oded.
Warm up problem: simulating EPR correlations
01001000011000
Alice
Bob
• Alice gets a unit vector aR3 and must outputs a bit .
• Bob gets a unit vector bR3 and must outputs a bit .
• Goal: the correlation E[] should satisfy
• Easy if Alice and Bob share |-i.
• Impossible with a shared random string and no communication
(Bell’s theorem)
Bell’s theorem
01001000011000
Alice
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•
Bob
Bell’s theorem: It is impossible to simulate these correlations
with a shared random string and no communication
• Werner & Wolf, quant-ph/0107093 (physics version);
• Arora & Barak, Complexity Theory: A Modern Approach (interactive proofs);
But can we quantify Bell’s theorem?
Yes! For example, allow some communication
between Alice and Bob after they receive questions.
How much communication is required?
More general problem
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•
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Bob
Alice
Fix some bipartite quantum state 
Alice gets a matrix A with 1 eigenvalues; outputs bit {-1,1}
Bob gets a matrix B with 1 eigenvalues; outputs bit {-1,1}
Goal: the correlation should be
Easy if Alice and Bob share .
Impossible with a shared random string and no communication
How much communication is required?
Simulating Quantum Correlations
(classical reformulation [Tsirelson87])
• Alice gets a unit vector aRn and
outputs a bit {-1,1}
• Bob gets a unit vector bRn and
outputs a bit {-1,1}
• Goal: the correlation E[] should satisfy
E[] = a,b
Previous Work
• Problem introduced by several authors
[Maudlin92,Steiner00,BrassardCleveTapp99]
• In the naïve protocol, Alice simply sends her vector to Bob;
this requires infinite communication
• For the case n=3 (EPR state), several protocols were
developed [BrassardCleveTapp99, Csirek00, CerfGisinMassar00]
with the best one requiring only one bit of communication
[TonerBacon03]
• For the general problem, best known protocol requires
n/2 bits [BaconToner0?]
Our Result:
The problem can be solved with
only 2 bits of communication
Outline
• The problem
• Getting strong enough correlations
• Getting the right correlations
Getting strong
enough correlations
A Naive Protocol with No Communication
• Alice and Bob share a random unit vector Rn
• Alice outputs sign(,a)
• Bob outputs sign(,b)
+1
-1
A Naive Protocol with No Communication
• Alice and Bob share a random
Rn
unit vector
• Alice outputs sign(,a)
• Bob outputs sign(,b)
• Analysis: if r=a,b then
therefore
+1
-1
-1
+1
a
b
Resulting Correlation Function
no correlation
The ‘Orthant’ Protocol
• Alice and Bob project their vectors on a random kdimensional subspace
• Alice tells Bob which of the 2k orthants her vector
lies in, and outputs +1
• Bob outputs +1 or -1 depending on whether his
vector lies in the half-space determined by the
orthant.
• This uses k bits of communication
(easy to improve to k-1).
Analysis of the ‘Orthant’ Protocol
• By using Gaussian random variables, we find out
that the correlation function is given by certain
areas on the sphere in k+1 dimensions
• For k=1 we get arcs on
k=1
the circle; area = angle
•For k=2 we get spherical
triangles:
area = 1+2+3-
•For k=3, we get spherical
tetrahedra…
k=2
Analysis of the ‘Orthant’ Protocol
• Problem: No closed formula is known for the
volume of a spherical tetrahedra.
• Solution: There is an expression for the derivative.
[Schlaefli1858]
Resulting Correlation Function
Strong enough!
Requires only 2 bits of
communication!!
Getting the right
correlations
Getting the Right Correlations
• Our goal is to have a protocol with correlations
h(r)=r
• However, all protocols we tried were either too
weak or too strong
• We show how to take any protocol with ‘strong
enough’ correlations, and transform it into a
protocol with the right correlation function h(r)=r
The Idea
[Krivine79]
• We define a transformation C from Rn to some other
Hilbert space with the property that for all a,bRn,
C(a),C(b)=f(a,b)
where f:[-1,1][-1,1] is some function with f(1)=1.
• Alice and Bob now run the original protocol on the vectors
C(a) and C(b)
• The resulting correlation function is
h(f(r))
where h is the original correlation function.
• If we take f=h-1, we obtain the right correlation function!
Idea - Continued
• Our goal is, therefore, to find a transformation C on
vectors such that for all a,bRn,
C(a),C(b)=h-1(a,b)
• Assume, for example, that h-1(x)=x3
• Then we can choose C to be the mapping
v  vvv
and then for any vectors a,b,
C(a),C(b)=aaa,bbb=a,b3=h-1(a,b)
as required.
Extending this Idea
• Now assume that h-1(x)=(x3+x)/2
• We can choose C to be the mapping
v  (vvv  v)/2
and this gives
C(a),C(b) = ½aaa  a , bbb  b
= ½a,b3 + ½a,b
= h-1(a,b)
as required.
Extending this Idea
• In general, we can find a mapping C as long as the
power series expansion of h-1 has only nonnegative
coefficients
• In order to apply this idea to the 2-bit ‘orthant’
protocol, we ‘simply’ have to analyze the power
series of the inverse of
• We omit the details…
Open Questions
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Is there any 1-bit protocol?
We conjecture that there is not.
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Extend to the more general problem of simulating
local measurements on quantum states.
Are there states for which there is no exact finite
bit simulation protocol?
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The CHSH game
Alice gets a bit a and outputs a bit 
Bob gets a bit b and outputs a bit 
Goal: =ab (i.e., output bits
should be equal unless a=b=1)
• No communication is allowed.
a1
b0
a0
b1
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Best strategy is to always output 0: they get 3 out
of the 4 possible questions right. True even with
randomness.
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If, however, they share an EPR state, they can get
success probability ~85% for each of the 4
questions.