Oded Regev

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Simulating Quantum Correlations with Finite
Communication
Oded Regev (Tel Aviv University) Ben
Toner (CWI, Amsterdam)
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Outline
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The Problem
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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 ab1)
• No communication is allowed
• Best strategy is to always output 0 they get 3
  out of the 4 possible questions right
• Moreover, even if they share a random string,
  their average success probability is at most 75
• However, if they share an EPR state, they can get
  success probability 85 for each of the 4
  questions

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Simulating Quantum Correlations

• Fix some bipartite quantum state ?
• Alice gets a matrix A with ?1 eigenvalues
  outputs a bit ??-1,1
• Bob gets a matrix B with ?1 eigenvalues outputs
  a bit ??-1,1
• Goal the correlation E?? should satisfy
• E?? Tr(A?B ? ?)
• If the parties share ?, this is easy
• Without shared entanglement, impossible
• However, what happens if we allow classical
  communication between Alice and Bob? How many
  bits do they need to exchange to simulate quantum
  correlations?

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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?

?a,b?1 ?a,b?0 ?a,b?-1
a
a
a
b
b
b
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Example CHSH

• Consider the special case in which Alice gets
  either
• a0(1,0) or a1(0,1)
• and Bob gets either
• b0(1,1)/?2 or b1(1,-1)/?2.
• Then ?ai,bj? -½ if ij1 and ½ otherwise
• Hence their goal is to output
• bits ?,? such that ??? with
• probability 85 if ij1, and
• ?? with probability 85
• otherwise

b0
a1
a0
b1
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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 n3 (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 TonerBacon06
• Another protocol achieves only logn/2 bits, but
  only on average (worst case communication is
  unbounded) DegorreLaplanteRoland07

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New Result
The problem can be solved with only 2 bits of
communication
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Getting strong enough correlations
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A Naïve Protocol with No Communication
1
-1
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A Naïve Protocol with No Communication
-1
1
1
a
-1
b
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Resulting Correlation Function
desired
result
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The Orthant Protocol

• Alice and Bob project their vectors on a random
  k-dimensional 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)

1
-1
a
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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 k1 dimensions
• For k1 we get arcs on
• the circle area angle

k1
k2

• For k2 we get spherical
• triangles
• area ?1?2?3-?
• For k3, we get spherical
• tetrahedra

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Resulting Correlation Function
k3
k2
k1
Strong enough! Requires only 2 bits of
communication!!
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Getting the Right Correlations
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The Idea
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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?3h-1(?a,b?)
• as required.

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Extending this Idea

• Now assume that h-1(x)(x3x)/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.

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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

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Open Questions

• Is there any 1-bit protocol?
• We conjecture that there isnt any
• Extend to the more general problem of simulating
  local measurements on quantum states