Worst case analysis - PowerPoint Presentation

Slide1

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”,project of the European Social FundNr. 2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044

INVESTING IN YOUR FUTURE

SOFSEM 2013

Špindler

ův Mlýn Česká republika 28.01.2013.Slide2

Overview

MotivationSome preliminaries“Worst-case” equal to “Average-case”

“Worst-case” different from “Average-case”Games without common dataSlide3

Motivation

Non-local gamesComputer 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» problemsCrypograpyConnection to input dataSlide4

Preliminaries -nonlocal games

N cooperating players:

A1, A2, ..., ANThey try to maximize game valueBefore the

game the players may share a common source of correlated random data:

Classical model: common random variable

Quantum model:

Input

:

x

= (x

1

, x

2, ..., xN); Probability distribution: Output: a = (a1, a2, ..., aN); Ai  ai In this talk: ai  {0; 1} Winning condition: V(a | x)Game value :



P1

P2p100p201p310p411Slide5

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:Slide6

CHSH game

Input: x1,

x2 {0,1} Output: a1, a

2 {0,1} Rules:No communication after inputs received

Players win, If given x1

=x2=1, they output a1

a2=1If given x1=0 or x2=0, they output

a

1



a

2

=0

With classical resources, Pr[a1a2 = x1x2] ≤ 0.75But, with entangled quantum state 00 –

11

Pr[a1

a2 = x

1x2] = cos2(/8) = ½ + ¼√2 = 0.853…RefereeBob

Alice

x

1

x

2

a

1

a

2

x

1

x

2

a

1

a

2

00

0

01

0

10

0

11

1

Games with worst case

equal to

average case

0.5

 Slide7

CHSH game:worst-case

Average-case:Worst-case:

In quantum case the same result is achieved on every input. This leads to worst-case game values:

x1 x2

Correct

Answer

a1 a2

a

1



a

2

Satisfy

0000 0

0

+

01

00 0

0

+

10

0

0 0

0

+

11

1

0 0

0

-

x

1

x

2

Correct

Answer

a

1

=0

a

2

=0



a

1

=0

a

2

=

x

2



a1=x1a2=0a1=x1a2=! x20 000 0 0+0 00+0 00+0 11-0 100 00+0 11-0 00+0 00+1 000 00+0 00+1 01-1 10+1 110 00-0 11+1 01+1 01+

Games with worst case

equal to average case

p=.25

p

1

=

?

p

2

=

?

p

3

=

?

p

4

=

?Slide8

n-party AND(

nAND) gameInput: x1, ... ,x

n {0,1} Output: a1, ... ,an 

{0,1} Rule:Average-case: By using trivial «all zero» strategy players win on all but one input string.Respective game value:

Worst-case:

x

1

x

2

...x

n

XOR

00...00

0

00...01000...10

0...

...11...10011...111

Games with worst case

different from

average caseSlide9

Equal-Equal(EEm

) gameInput: x1, x2

{1,...,m} Output: a1, a2 {1,...,m}

Rule:Worst-case: Worst-case quantum: Average-case:

Games with worst case

different from

average caseSlide10

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 randomnessIn the worst-case:

For many games: unable to win with p>0.5Consider a game:Slide11

Conclusion

We have introduced and studied worst-case scenarios for nonlocal gamesWe analyzed and compared game values of

worst-case and average-case scenariosWorst-case equal to average-case game value:CHSH gameMermin-Ardehali game

Odd cycle gameMagic square gameWorst-case not equal

to average-case game value:We introduce new games that have this property (by modifying classical ones)Equal-Equal gameN-party AND game

N-party MAJORITY gameSlide12

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