### On random nonlocal games

```Random non-local games
Andris Ambainis, Artūrs Bačkurs,
Kaspars Balodis, Dmitry
Kravchenko, Juris Smotrovs,
University of Latvia
Non-local games
Alice
a
x
Bob
b
Referee
y
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.

Example 1 [CHSH]
Alice
a
x
Bob
b
Referee
y
Winning conditions for Alice and Bob
 (a = 0 or b = 0)  x = y.
 (a = b = 1)  x  y.

Example 2 [Cleve et al., 04]
Alice and Bob attempt to
“prove” that they have a
2-coloring of a 5-cycle;
some vertex to each of
them.

Example 2
Referee either:
 asks ith vertex to both
Alice and Bob; they win
 Asks the ith vertex to
Alice, (i+1)st to Bob, they
Non-local games in quantum world

Alice

Bob
Shared quantum state between Alice and
Bob:
Does not allow them to communicate;
 Allows to generate correlated random bits.

Corresponds to shared random bits
in the classical case.
Example:CHSH game
Alice
a
x
Bob
b
Referee
y
Winning condition:
Winning probability:
 (a = 0 or b = 0)  x = y.  0.75 classically.
 0.85... quantumly.
 (a = b = 1)  x  y.
A simple way to verify quantum mechanics.
Example: 2-coloring game
Alice and Bob claim to
have a 2-coloring of ncycle, n- odd;
 2n pairs of questions by
referee.

Winning probability:

classically.

quantumly.
Random non-local games
Alice
a
x
Bob
b
Referee
y
a, b  {1, 2, ..., N};
 x, y  {0, 1};
 Condition P(a, b, x, y) – random;

Computer experiments: quantum winning probability
larger than classical.
XOR games

For each (a, b), exactly one of x = y and
x  y is winning outcome for Alice and
Bob.
The main results
Let N be the number of possible questions
to Alice and Bob.
 Classical winning probability pcl satisfies


Quantum winning probability pq satisfies
Another interpretation

Value of the game = pwin – (1-pwin).

Corresponds to Bell inequality violation
Related work
Randomized constructions of non-local
games/Bell inequalities with large quantum
 Junge, Palazuelos, 2010: N questions, N
 Briet, Vidick, 2011: large advantage for
XOR games with 3 players.

Differences

JP10, 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.
Random matrix question

What is the value of
for a random 1 matrix A?
Can be upper-bounded using
||A||=(2+o(1)) √N
Upper bound
=
Upper bound theorem

Theorem For a random A,

Corollary The advantage achievable by a
quantum strategy in a random XOR game
is at most
Lower bound

There exists u:

There are many such u: a subspace
of dimension f(n), for any f(n)=o(n).

Combine them to produce ui, vj:
Marčenko-Pastur law



Let A – random N·N 1 matrix.
W.h.p., all singular values are between 0 and
(2o(1)) √N.
Theorem (Marčenko, Pastur, 1967) W.h.p., the
fraction of singular values i  c √N is
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
Classical results
Let N be the number of possible questions
to Alice and Bob.
 Theorem Classical winning probability pcl
satisfies

Methods: classical
Alice’s strategy - numbers
u1, ..., uN  {-1, 1}.
 Bob’s strategy - numbers
v1, ..., vN  {-1, 1}.

Classical upper bound

Chernoff bounds + union bound:
with probability 1-o(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.
Greedy strategy

Choose u1, ..., uN one by one.
1 -1
... ...
1 1
-1
...
-1
... 1
... ...
... -1
2 0
-2
...
-1 1
1
... 1
1 -1
-1
... -1
k-1 rows that are
0
Choose the option
which maximizes
agreements of signs
Analysis
2 0
-2
...
0
-1 1
1
... 1
1 -1
-1
... -1
Choose the option
which maximizes
agreements of signs

On average, the best option agrees with
fraction of signs.

If the column sum is 0, it always increases.
Rigorous proof
2 0
-2
...
0
-1 1
1
... 1
1 -1
-1
... -1
Choose the option
which maximizes
agreements of signs

Consider each column separately. Sum of
values performs a biased random walk, moving
away from 0 with probability
in each
step.

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:


Comparison

Random XOR game:

Biggest gap for XOR games:
Open problems
1.
2.
3.
We have
What is the exact order?
Gaussian Aij? Different probability distributions?
Random games for other classes of non-local
games?
```