Grothendieck's inequality for most matrices · References Motivation Deﬁnition Values of the game Quantum (Hilbertian) value of the game We deﬁne the Quantum, or Hilbertian, value

XOR gamesGrothendieck’s inequality

The resultsThe techniques

References

Grothendieck’s inequality for most matrices

Carlos Hugo Jiménez 1 C. Palazuelos 2 I. Villanueva1

1Universidad Complutense de Madrid

2ICMAT

WFAV 2013

Carlos Hugo Jiménez , C. Palazuelos , I. Villanueva Grothendieck’s inequality for most matrices



References

MotivationDefinitionValues of the game

Matrices as games

Given a matrix G = (gij)ni,j=1 we normalize it so that∑

i,j

|gij | = 1

and we factor G as a the elementwise product of a sign matrix

A = εij

times a matrixP = (pij)

representing a probability distribution on the coordinates (i , j).




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XOR games

A XOR game G = (A,P) with n inputs on each side is definedby a sign matrix

A = (εij)

together with a probability distribution

P = (pij)

Alice and Bob receive as inputs 1 ≤ i , j ≤ n respectively withprobability pij and each of them must answer a numberαi , βj ∈ {−1,1}, so that εij = αiβj .




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Classical (scalar) value of the game

We define the classical, or scalar, value of the game as thesupremum of the numbers

supα,β

∑i,j

pijεijαiβj

where αi , βj = ±1.




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Quantum (Hilbertian) value of the game

We define the Quantum, or Hilbertian, value of the game as thesupremum of the numbers

supui ,vj

∑i,j

pijεij〈ui |vj〉

where ui , vj are norm one vectors in a Hilbert space.




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Grothendieck’s inequality

Grothendieck

Grothendieck’s inequality tells us that for every matrix/game theratio between both values verifies

1 ≤ ρA :=sup

∑i,j pijεij〈ui |vj〉

sup∑

i,j pijεijαiβj≤ KG

where we know

1.67696... ≤ KG ≤ 1.7822...




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What happens for a typical matrix?

In Quantum Information, the Hilbertian model describes thevalue you can achieve in this context with quantum resources,and the scalar value describes what you can do with classical(not quantum) resources.A natural question arises: do you typically get someimprovement? or the maximum possible improvement? or dowe need some kind of special structure in the game forimprovements/maximum improvements to happen?




References



In Quantum Information, the Hilbertian model describes thevalue you can achieve in this context with quantum resources,and the scalar value describes what you can do with classical(not quantum) resources.A natural question arises: do you typically get someimprovement? or the maximum possible improvement? or dowe need some kind of special structure in the game forimprovements/maximum improvements to happen?




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From the point of view of functional analysis the question alsoseems relevant. For instance, one could wonder whether onecan ε-approximate Grothendieck’s constant in a set of measure1, or not 0. Or whether for most matrices the ratio between thehilbertian and the scalar value stays close to 1




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The quantum valueBounds for the classical values

Uniform case

If pij =1n2 for every (i , j), then

ωQ =2± o(1)√

n

with probability 1− o(1), where o(1) stands for a sequence thatconverges to 0 as n→∞.




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Non uniform case

For G = (A,P) in the spheres of (Rn2, ‖ · ‖2) or (Rn2

, ‖ · ‖1),

ωQ =(2± o(1))√

nn

∑ij

p2ij

12

with probability 1− o(1).




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Uniform case

1.2789...√n

≤ ωC ≤1.6651...+ o(1)√

n

with probability 1− o(1)




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Uniform case

If Aij is a Gaussian symmetric matrix then

supαi=±1

∑aijαiαj = (1.527...+ o(1))n

32




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Non uniform case

1.2789...√n

n

∑ij

p2ij

12

≤ ωC ≤1.6651...+ o(1)√

nn

∑ij

p2ij

12




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Ratio

In particular we have that

1 < 1.2011... ≤sup

∑i,j εij〈ui |vj〉

sup∑

i,j εijαiβj≤ 1.563 < 1.6769... ≤ KG




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Distribution of eigenvalues in a random matrix

Classical upper value

The classical upper value follows from a Chernoff bound plusthe union bound




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Classical lower value

The classical lower value is algorithmic.




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The Marcenko-Pastur law

Given a random matrix An, we can define s1(An), . . . , sn(An) tobe the singular values of 1√

n An, and we define their distributionfunction

FAn(x) =1n

n∑i=1

χ(si(An) ≤ x),

We let now g(x),G(x) be the density and the distributionfunction of the quarter-circle law:

g(x) =1π

√4− x2χ(0 ≤ x ≤ 2), G(x) =

∫ x

−∞g(s)ds.




References


The Marcenko-Pastur law

Let (εij)1≤i,j<∞ be an array of independent symmetric Bernoullirandom variables with E(εij) = 0 for every (i , j) and E(ε2ij ) = 1.We consider the random matrix An = (εij)

ni,j=1.

Then

supx|F An(x)−G(x)| → 0 almost surely inAn as n→∞,




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Let An be a random matrix as before. Then

limn→∞

1√n‖An‖ = 2 a.s.




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Ad-hoc modifications on M-P law

Let An be a random matrix as before. Let |i〉 be the i th vector ofthe standard basis. Let PC be the projector on the subspacegenerated by the right singular vectors with associated singularvalues s satisfying s ≥ C

√n. Let ε > 0. Then

Pr(∣∣∣‖PC(|i〉)‖22 −G(C)

∣∣∣ > ε)= O(

1n),

with the big-O constant depending only on C and ε.The same result holds for the left singular vectors.




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Quantum upper value

The quantum upper value follows from the probabilitydistribution of the biggest singular value.




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Sketch of the quantum lower bound

Pick ε > 0.Pick the singular values sk bigger than 2− ε, and theassociated right and left singular vectors rk , lkConsider the vectors formed by their i th coordinatesUse the modified MP law to dilate them adequately.




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References

A. Ambainis, A. Backurs, K. Balodis, D. Kravcenko, R.Ozols, J. Smotrovs, M. VirzaQuantum strategies are better than classical in almost anyXOR gamearXiv:1112.3330C. H. Jimenez, C. Palazuelos, I. VillanuevaThe quantum value of XOR gamesin preparation


Documents

Grothendieck's inequality for most matrices · References Motivation Deﬁnition Values of the game Quantum (Hilbertian) value of the game We deﬁne the Quantum, or Hilbertian, value