Worst case analysis of nonlocal games Andris Ambainis

Worst case analysis of non-local games Andris Ambainis, Artūrs Bačkurs, Kaspars Balodis, Agnis Škuškovniks, Juris Smotrovs, Madars Virza “COMPUTER SCIENCE APPLICATIONS AND ITS RELATIONS TO QUANTUM PHYSICS”, INVESTING IN YOUR FUTURE project of the European Social Fund Nr. 2009/0216/1 DP/1. 1. 1. 2. 0/09/APIA/VIAA/044 SOFSEM 2013 Špindlerův Mlýn Česká republika 28. 01. 2013.

Overview • • • Motivation Some preliminaries “Worst-case” equal to “Average-case” “Worst-case” different from “Average-case” Games without common data

Motivation • Non-local games • Computer scientist’s way of looking at: • • Local vs. Non-local; Quantum vs. Classial; • “Worst-case” vs. “Average-case” (intro) • Worst-case and average-case complexity? • • «Real life» problems Crypograpy • Connection to input data

Preliminaries nonlocal games P 1 P 2 p 1 0 0 p 2 0 1 p 3 1 0 p 4 1 1

Average and worst case scenarios • Maximum game value for fixed distribution : • Classical: • Quantum: • In the most studied examples is uniform distrubution We will call it «average-case» • Maximum game value for any distribution: worst-case • Classical: • Quantum:

Games with worst case equal to average case x 1 CHSH game Ali ce Input: x 1, x 2 {0, 1} Output: a 1, a 2 {0, 1} a 1 Re Bo b a 2 fer x 2 ee Rules: • No communication after inputs received • Players win, • If given x 1=x 2=1, they output a 1 a 2=1 • If given x 1=0 or x 2=0, they output a 1 a 2=0 With classical resources, Pr[a 1 a 2 = x 1 x 2] ≤ 0. 75 x 1 x 2 a 1 a 2 00 0 01 0 10 0 11 1 But, with entangled quantum state 00 – 11 • Pr[a 1 a 2 = x 1 x 2] = cos 2( /8) = ½ + ¼√ 2 = 0. 853…

CHSH game: worst-case Games with worst case equal to average case Average-case: x x 1 2 p=. 25 Worst-case: p 1=? p 2=? p 3=? p 4=? Correct Answer a 1 a 2 Satisfy 00 0 + 01 0 00 0 + 10 0 00 0 + 11 1 00 0 - Correct x 1 x 2 Answer a 1=0 a 2=0 a 1=0 a 2 = x 2 a 1 = x 1 a 2=0 a 1 = x 1 a 2=! x 2 00 0 + 01 1 01 0 00 0 + 01 1 - 00 0 + 10 0 00 0 + 10 1 - 11 0 + 11 1 00 0 01 1 + 10 1 + - In quantum case the same result is achieved on every input. This leads to worst-case game values: -

Games with worst case different from average case n-party AND (n. AND) game x 1 x 2. . . xn XOR 00. . . 00 0 00. . . 01 0 00. . . 10 0 . . . 11. . . 10 0 11. . . 11 1

Games with worst case different from average case Input: x 1, x 2 {1, . . . , m} Output: a 1, a 2 {1, . . . , m} Rule: Worst-case quantum: Average-case: Equal (EEm) game

Games without common data • For known fixed probability distribution: • «Not allowed to share common randomness» equivalent to • «Allowed to share common randomness» We just fix the best value for common randomness • In the worst-case: • For many games: unable to win with p>0. 5 • Consider a game:

Conclusion • We have introduced and studied worst-case scenarios for nonlocal games • We analyzed and compared game values of worst-case and average-case scenarios • Worst-case equal to average-case game value: • • CHSH game Mermin-Ardehali game Odd cycle game Magic square game • Worst-case not equal to average-case game value: • We introduce new games that have this property (by modifying classical ones) • • • Equal-Equal game N-party AND game N-party MAJORITY game

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