Download presentation

Presentation is loading. Please wait.

Published byMadisyn Frary Modified over 4 years ago

1
Random non-local games Andris Ambainis, Artūrs Bačkurs, Kaspars Balodis, Dmitry Kravchenko, Juris Smotrovs, Madars Virza University of Latvia

2
Non-local games Referee asks questions a, b to Alice, Bob; Referee asks questions a, b to Alice, Bob; Alice and Bob reply by sending x, y; Alice and Bob reply by sending x, y; Alice, Bob win if a condition P a, b (x, y) satisfied. Alice, Bob win if a condition P a, b (x, y) satisfied. Alice Bob Referee ab xy

3
Example 1 [CHSH] Winning conditions for Alice and Bob Winning conditions for Alice and Bob (a = 0 or b = 0) x = y. (a = 0 or b = 0) x = y. (a = b = 1) x y. (a = b = 1) x y. Alice Bob Referee ab xy

4
Example 2 [Cleve et al., 04] Alice and Bob attempt to “prove” that they have a 2-coloring of a 5-cycle; 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. Referee may ask one question about color of some vertex to each of them.

5
Example 2 Referee either: asks i th vertex to both Alice and Bob; they win if answers equal. asks i th vertex to both Alice and Bob; they win if answers equal. Asks the i th vertex to Alice, (i+1) st to Bob, they win if answers different. Asks the i th vertex to Alice, (i+1) st to Bob, they win if answers different.

6
Non-local games in quantum world Shared quantum state between Alice and Bob: Shared quantum state between Alice and Bob: Does not allow them to communicate; Does not allow them to communicate; Allows to generate correlated random bits. Allows to generate correlated random bits. Alice Bob Corresponds to shared random bits in the classical case.

7
Example:CHSH game Winning condition: (a = 0 or b = 0) x = y. (a = 0 or b = 0) x = y. (a = b = 1) x y. (a = b = 1) x y. Winning probability: 0.75 classically. 0.75 classically. 0.85... quantumly. 0.85... quantumly. A simple way to verify quantum mechanics. Alice Bob Referee ab xy

8
Example: 2-coloring game Alice and Bob claim to have a 2-coloring of n- cycle, n- odd; Alice and Bob claim to have a 2-coloring of n- cycle, n- odd; 2n pairs of questions by referee. 2n pairs of questions by referee. Winning probability: classically. classically. quantumly. quantumly.

9
Random non-local games a, b {1, 2,..., N}; a, b {1, 2,..., N}; x, y {0, 1}; x, y {0, 1}; Condition P(a, b, x, y) – random; Condition P(a, b, x, y) – random; Computer experiments: quantum winning probability larger than classical. Alice Bob Referee ab xy

10
XOR games For each (a, b), exactly one of x = y and x y is winning outcome for Alice and Bob. For each (a, b), exactly one of x = y and x y is winning outcome for Alice and Bob.

11
The main results Let N be the number of possible questions to Alice and Bob. Let N be the number of possible questions to Alice and Bob. Classical winning probability p cl satisfies Classical winning probability p cl satisfies Quantum winning probability p q satisfies Quantum winning probability p q satisfies

12
Another interpretation Value of the game = p win – (1-p win ). Value of the game = p win – (1-p win ). Quantum advantage: Quantum advantage: Corresponds to Bell inequality violation

13
Related work Randomized 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.

14
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;

15
Methods: quantum Tsirelson’s theorem, 1980: Alice’s strategy - vectors u 1,..., u N, ||u 1 || =... = ||u N || = 1. Alice’s strategy - vectors u 1,..., u N, ||u 1 || =... = ||u N || = 1. Bob’s strategy - vectors v 1,..., v N, ||v 1 || =... = ||v N || = 1. Bob’s strategy - vectors v 1,..., v N, ||v 1 || =... = ||v N || = 1. Quantum advantage Quantum advantage

16
Random matrix question What is the value of What is the value of for a random 1 matrix A? for a random 1 matrix A? Can be upper-bounded using ||A||=(2+o(1)) √N

17
Upper bound =

18
Upper bound theorem Theorem For a random A, Corollary The advantage achievable by a quantum strategy in a random XOR game is at most

19
Lower bound There exists u: There exists u: There are many such u: a subspace of dimension f(n), for any f(n)=o(n). There are many such u: a subspace of dimension f(n), for any f(n)=o(n). Combine them to produce u i, v j : Combine them to produce u i, v j :

20
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

21
Modified Marčenko-Pastur law Let e 1, e 2,..., e N be the standard basis. Theorem With probability 1-O(1/N), the projection of e i to the subspace spanned by singular values j c √N is

22
Classical results Let N be the number of possible questions to Alice and Bob. Let N be the number of possible questions to Alice and Bob. Theorem Classical winning probability p cl satisfies Theorem Classical winning probability p cl satisfies

23
Methods: classical Alice’s strategy - numbers u 1,..., u N {-1, 1}. Alice’s strategy - numbers u 1,..., u N {-1, 1}. Bob’s strategy - numbers Bob’s strategy - numbers v 1,..., v N {-1, 1}. v 1,..., v N {-1, 1}. Classical advantage Classical advantage

24
Classical upper bound Chernoff bounds + union bound: Chernoff bounds + union bound: with probability 1-o(1). with probability 1-o(1).

25
Classical lower bound Random 1 matrix A; Operations: flip signs in all entries in one column or row; Goal: maximize the sum of entries.

26
Greedy strategy Choose u 1,..., u N one by one. 1...1 1 1... k-1 rows that are already chosen 2 0 -2... 0 1 1... 1 1... Choose the option which maximizes agreements of signs

27
Analysis On average, the best option agrees with fraction of signs. 1 1... 1 2 0 -2... 0 1... Choose the option which maximizes agreements of signs If the column sum is 0, it always increases.

28
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 2 0 -2... 0 1... Choose the option which maximizes agreements of signs Expected distance from origin = 1.27... √N.

29
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:

30
Comparison Random XOR game: Random XOR game: Biggest gap for XOR games: Biggest gap for XOR games:

31
Open problems 1. 1. We have What is the exact order? 2. 2. Gaussian A ij ? Different probability distributions? 3. 3. Random games for other classes of non-local games?

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google