Random non-local games

Andris Ambainis, Artūrs Bačkurs, Kaspars Balodis, Dmitry Kravchenko,

Juris Smotrovs, Madars Virza

University of Latvia

Non-local games

Referee asks questions a, b to Alice, Bob;Alice and Bob reply by sending x, y;Alice, Bob win if a condition Pa, b(x, y)

satisfied.

Alice Bob

Referee

a b

x y

Example 1 [CHSH]

Winning conditions for Alice and Bob(a = 0 or b = 0) x = y.(a = b = 1) x y.

Alice Bob

Referee

a b

x y

Example 2 [Cleve et al., 04] Alice and Bob attempt to

“prove” that they have a 2-coloring of a 5-cycle;

Referee may ask one question about color of some vertex to each of them.

Example 2Referee either: asks ith vertex to both

Alice and Bob; they win if answers equal.

Asks the ith vertex to Alice, (i+1)st to Bob, they win if answers different.

Non-local games in quantum world

Shared quantum state between Alice and Bob:Does not allow them to communicate;Allows to generate correlated random bits.

Alice Bob

Corresponds to shared random bits in the classical case.

Example:CHSH game

Winning condition:(a = 0 or b = 0) x = y.(a = b = 1) x y.

Winning probability:0.75 classically.0.85... quantumly.

A simple way to verify quantum mechanics.

Alice Bob

Referee

a b

x y

Example: 2-coloring gameAlice and Bob claim to

have a 2-coloring of n-cycle, n- odd;

2n pairs of questions by referee.

Winning probability: classically.

quantumly.

n211

21nC

Random non-local games

a, b {1, 2, ..., N};x, y {0, 1};Condition P(a, b, x, y) – random;

Computer experiments: quantum winning probability larger than classical.

Alice Bob

Referee

a b

x y

XOR gamesFor each (a, b), exactly one of x = y and

x y is winning outcome for Alice and Bob.

winsyxwinsyx

Aab 11

NNNN

N

N

AAA

AAAAAA

A

...............

...

...

21

22221

11211

The main resultsLet N be the number of possible questions

to Alice and Bob.Classical winning probability pcl satisfies

Quantum winning probability pq satisfies

Np

N cl...8325.0

21...6394.0

21

Nopq)1(1

21

Another interpretation

Value of the game = pwin – (1-pwin).

Nv

N cl...6651.1...2788.1

Novq)1(2

Quantum advantage:

...5638.1...2011.1 cl

q

vv

Corresponds to Bell inequality violation

Related workRandomized constructions of non-local

games/Bell inequalities with large quantum advantage.

Junge, Palazuelos, 2010: N questions, N answers, √N/log N advantage.

Regev, 2011: √N advantage.Briet, Vidick, 2011: large advantage for

XOR games with 3 players.

DifferencesJP10, R10, BV11:

Goal: maximize quantum-classical gap;Randomized constructions = tool to achieve

this goal;This work:

Goal: understand the power of quantum strategies in random games;

Methods: quantum

Tsirelson’s theorem, 1980:Alice’s strategy - vectors u1, ..., uN, ||

u1|| = ... = ||uN|| = 1.Bob’s strategy - vectors v1, ..., vN, ||

v1|| = ... = ||vN|| = 1. Quantum advantage

N

jijiij vuA

N1,2loswin ,1p-p

Random matrix question What is the value of

for a random 1 matrix A?

N

jijiij

uvvuA

Nji 1,21

,1max

Can be upper-bounded using ||A||=(2+o(1)) √N

Upper bound

n

jijiij vuA

1,

,

Nu

uu

u...2

1

Nv

vv

v...2

1

=

vAIuT

vAuvAIuT max

vuNo 12

Upper bound theoremTheorem For a random A,

Corollary The advantage achievable by a quantum strategy in a random XOR game is at most

NovuA

N

N

jijiij

uv ji

)1(2,1max1,

21

No )1(2

Lower bound

NoA ))1(2(

There exists u: NoAu ))1(2(

There are many such u: a subspace of dimension f(n), for any f(n)=o(n).

Combine them to produce ui, vj:

NNovuAn

jijiij

uv ji

))1(2(,max1,1

Marčenko-Pastur law Let A – random N·N 1 matrix. W.h.p., all singular values are between 0 and

(2o(1)) √N. Theorem (Marčenko, Pastur, 1967) W.h.p., the

fraction of singular values i c √N is

4

2

)1(1421

cx

odxx

Modified Marčenko-Pastur law Let e1, e2, ..., eN be the standard basis. Theorem With probability 1-O(1/N), the projection

of ei to the subspace spanned by singular values j c √N is

4

2

)1(1421

cx

odxx

Classical resultsLet N be the number of possible questions

to Alice and Bob.Theorem Classical winning probability pcl

satisfies

Np

N cl...8325.0

21...6394.0

21

Methods: classicalAlice’s strategy - numbers

u1, ..., uN {-1, 1}. Bob’s strategy - numbers v1, ..., vN {-1, 1}.Classical advantage

N

jijiij vuAN1,

2loswin1p-p

Classical upper bound

Chernoff bounds + union bound: with probability 1-o(1).

N

jijiij vuAN1,

2loswin1p-p

Npp loswin

...65.1

Classical lower bound

Random 1 matrix A; Operations: flip signs in all entries in one

column or row; Goal: maximize the sum of entries.

N

jijiijvuvuA

ji 1,1,max

Greedy strategyChoose u1, ..., uN one by one.

1 -1 -1 ... 1... ... ... ... ... 1 1 -1 ... -1

k-1 rows that are already chosen

2 0 -2 ... 0-1 1 1 ... 1

1 -1 -1 ... -1

Choose the optionwhich maximizesagreements of signs

Analysis

On average, the best option agrees with fraction of signs.

-1 1 1 ... 12 0 -2 ... 0

1 -1 -1 ... -1

Choose the optionwhich maximizesagreements of signs

21

N 2

If the column sum is 0, it always increases.

Rigorous proof

Consider each column separately. Sum of values performs a biased random walk, moving away from 0 with probability in each step.

-1 1 1 ... 12 0 -2 ... 0

1 -1 -1 ... -1

Choose the optionwhich maximizesagreements of signs

N2

21

Expected distance from origin = 1.27... √N.

Conclusion We studied random XOR games with n questions

to Alice and Bob. For both quantum and classical strategies, the

best winning probability ½. Quantumly:

Classically:No )1(1

21

...8325.0...6394.0,21

CNC

Comparison Random XOR game:

...5638.1...2011.1 cl

q

vv

...782.1...676.1, GGcl

q KKvv

Biggest gap for XOR games:

Open problems

1. We have

What is the exact order?2. Gaussian Aij? Different probability distributions?3. Random games for other classes of non-local

games?

NNvuANNn

jijiijvu ji

...65.1max...27.11,1,