PLAYING (QUANTUM) GAMES WITH OPERATOR SPACESDavid Pérez-GarcíaUniversidad Complutense de MadridBilbao 8-Oct-2011

OUR GROUP 2 postdoctoral research positions opened in

our group “Mathematics and Quantum Information”.

STRUCTURE OF THE TALK The object under study: 2P1R games Examples. Why are they important?

Complexity theory I. Inaproximability results. Complexity theory II. Parallel computation. Position based cryptography. Certifiable random number generation.

A bit (just one bit!) of history. Where are the maths?

Our contribution. Operator Spaces.

2P1R GAMES

2P1R GAMES

x y

a b

1. A set of possible questions for Alice and Bob (denoted by x,y resp.).2. A known probability distribution for the questions.3. A known boolean function V(x,y,a,b) which decides, based on

questions and answers a,b, whether they witn (=1) or loose (=0) the game.

4. A limitation in the communication between Alice and Bob.

2P1R GAMES

x y

a b

The value of the game is the largest probability of wining the game while optimizing over the possible strategies of Alice and Bob. It is assumed that Alice and Bob have free communication BEFORE the game to coordinate an strategy . Hence strategies can involve shared randomness (classical value of the game) or quantum entanglement (quantum value of the game) depending on the resources of Alice and Bob.

2P1R GAMESx y

a b

What is an strategy? A probability distribution p(ab|xy)

Which are the possible strategies in the classical case?

)|()|()()|(

ybpxapqxyabp BA

And in the quantum one?

ybxaFEtrxyabp )|(

.0],[

,

,

0,

)(

yb

xa

Hb

yb

Ha

xa

yb

xa

FE

yIdF

xIdE

FE

HS

EXAMPLES. INAPROXIMABILITY

EXAMPLES. INAPROXIMABILITY RESULTSTheorem (PCP theorem (Arora-Safra, 92)+ Parallel

repetition (Raz, 94)):Unless P=NP, given e>0 and a polynomial algorithm to determine the classical value of a game, there exist games for which the value is 1 and the algorithm outputs a value <e.

0 1

e

EXAMPLES. INAPROXIMABILITY RESULTS

It is the mother of most inaproximability results. For instance:

Theorem (Hastad, 1999):Unless P=NP, given e>0 and a polynomial algorithm to determine the MAX-CLIQUE of a graph, there exist graphs of n vertices for which

en 1

algorithm theofoutput CLIQUE-MAX

Note that MAX-CLIQUE is always less or equal than n (!!)

EXAMPLES. INAPROXIMABILITY RESULTS

Connection with 2P1R games. Via LABEL-COVER.

Given a bipartite graph

And a set of colors

And a set of valid configurations for each edge

Find a coloring of the graph which maximizes the number of edges with a valid configuration.

),( EWUV

}invalid valid,{: ,),( eCEwue

EXAMPLES. INAPROXIMABILITY RESULTS

Connection with 2P1R games. Via LABEL-COVER.

Colors

Number of valid edges = 4Solution to LABEL-COVER = 5

EXAMPLES. INAPROXIMABILITY RESULTS

Connection with 2P1R games. Via LABEL-COVER.

Given and instance of LABEL-COVER, we define a 2P1R game by:

Questions = vertices (from U to Alice and from W to Bob) with uniform probability between the pairs which conform an edge (and 0 in the rest).

Answers = colors.

They win the game if they give a valid coloring for the edge which is asked.

EXAMPLES. PARALLEL COMPUTATION

EXAMPLES. PARALLEL COMPUTATION

Given a boolean function f(x,y), minimize c in:

x y

c bits of communication

f(x,y) f(x,y)

EXAMPLES. POSITION BASED CRYPTOGRAPHY.(CHANDRAN ET AL, 2009)

EXAMPLES.POSITION BASED CRYPTOGRAPHY1D for simplicity

x y

a b

AIM: That only someone in position P could answer with probability 1 to the challenge. It would allow unconditional secure communication !!!

Coordinated

Position P

EXAMPLES.POSITION BASED CRYPTOGRAPHYRelation with 2P1R games. Since the verifiers act coordinated, we can assume there is just one of them. Based on answering times, we have:

x y

a b

Communication “independent-one-way”

EXAMPLES.POSITION BASED CRYPTOGRAPHYHence, the aim is to find a challenge which can be won always with arbitrary communication (all classical challenges are like that) but not with “independent-one-way” communication.

x y

a b

The honest case is the one of arbitrary communication, since there is only a single prover.

EXAMPLES.POSITION BASED CRYPTOGRAPHYThis is impossible classically. Both models of communication are the same. To see it, just copy and send the received question. In the quantum case (with no entanglement) it is indeed possible (Buhrman et al., 2010). The key idea lies on the fact that it is NOT possible to copy quantum states by the NO-CLONING theorem.

Question: Is it also possible when a polynomial amount of entanglement is allowed?

Partial answers (Beigi et al., Burhman et al, 2011): LINEAR = YES, EXPONENTIAL = NO.

EXAMPLES. RANDOM NUMBER GENERATION(PIRONIO ET AL., 2010)

EXAMPLE.RANDOM NUMBER GENERATIONPROBLEM: How to construct an apparatus which generates perfect random numbers (and hence secret) in a certifiable way?

001110101001010101…..

could have a copy of 001110101001010101…..

But in quantum mechanics copying is not allowed !!!

EXAMPLE.RANDOM NUMBER GENERATION

Theorem (Pironio et al., Colbeck et al., 2010): If (after many rounds in the game) one gets a value strictly larger than the classical one, there is a classical “deterministic” post-processing of the outputs a, b which produces numbers which are perfectly random and secret.

x y

a b

DONE EXPERIMENTALLY ALREADY !!!

EXAMPLE.RANDOM NUMBER GENERATION

The key is, hence, the existence of quantum strategies which are NOT classical. This guarantees an intrinsic randomness.

)|( xyabp Classical

Quantum

Non-signaling

HISTORICAL NOTE

The existence of this intrinsic randomness is precisely what Einstein did not believe in his criticism of quantum mechanics in the 30’s.

The first experiment showing that, indeed, this randomness does ocurr was done by A. Aspect in the 80’s.

The experiment was based precisely on the analysis of the value of a particular game, known as CHSH (Clauser, Horne, Shimony, Holt).

OUR CONTRIBUTION

THE PROBLEM WE WANT TO ATTACK

Estimate D.Parameters: Number of questions = N Number of answers = MSize (dimension) of the quantum system = d

D

Quantum strategies

Classical strategies

)|( xyabp

How large can D be?

valueclassical valuequantum

D

VIOLATION OF A BELL INEQUALITY. FORMAL DEFINITION OF D.

Bell inquality T

abxy

xyabc xyabpTTV )|(sup)( classical p

abxy

xyab

dq xyabpTTV )|(sup)(

ddim

quantumq

(T)V(T)V

sup1c

dq

T D

BELL INEQUALITY= 2P1R GAME (generalized)

),,,(),(Txyab yxbaVyx

OPERATIONAL INTERPRETATION OF D

Bell Inequality T

p-1p1D

Where p is the maximum (classical) noise which a quantum strategy can withstand before getting classical.

It is hence desirable to have a large D. How does D scale with the parameters N,M, d?

MAIN THEOREMTheorem (Junge, Palazuelos, Pérez-García, Villanueva, Wolf, CMP + PRL, 2010).D can be arbitrarily large, This requires: N= D^2M= EXP(D)d= D^2

Theorem (Junge, Palazuelos, CMP, 2011).D can be arbitrarily large, This requires: N= D^2, M= D^2, d= D^2

Recent improvements

Theorem (Buhrman et al, 2011).D can be arbitrarily large, This requires: N= D, M= EXP(D), d= D.

AN INTERESTING APPLICATIONTheorem (Junge, Palazuelos, Pérez-García, Villanueva, Wolf, CMP + PRL, 2010).There exist quantum estrategies with EXP(N) questions, N answers and dimension of the quantum system N which need the communication of log(N) bits to be simulated classically.

Quantum entanglement can save communication!!!

THE TOOLS: OPERATOR SPACES

OPERATOR SPACESAn operator space is a complex vector space E with a sequence of norms defined on such that:

Given a C*-algebra, there exists a unique norm which makes a C*-algebra. With these norms, A is an operator space.

)(EM n

mnnm MmMnbxaaxb

},max{mnmn

baba

)(AM n

OPERATOR SPACES

In particular:

Given

),(max

kk C

iinAx max

OPERATOR SPACESThe morphisms in this category are the completely bounded maps:

nncbuuFEu sup,:

)()(:1 FMEMuu nnnn

CB(E,F) is an operator space via

))(,()),(( FMECBFECBM nn

In particular E* is an operator space

CONNECTION WITH THE 2P1R GAMESTheorem (Junge, Palazuelos, Pérez-García, Villanueva, Wolf, CMP + PRL, 2010).Given a 2P1R game (or more generally a Bell inequality)

The classical value is given (with the order a,x,b,y) by the norm:

The quantum value, by the norm:

xyabT

)),(),,(( *NMNM BBB

)),(),,(( *NMNM CBCBCB

Operator Spaces are the natural mathematical framework to analyze 2P1R games.Some references in this direction:1. D. Pérez-García, M.M. Wolf, C. Palazuelos, I. Villanueva, M. Junge,

Unbounded violations of Bell inequalities, Comm. Math. Phys. 279, 455 (2008)

2. M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M.M. Wolf, Operator Space theory: a natural framework for Bell inequalities, Phys. Rev. Lett. 104, 170405 (2010).

3. M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M.M. Wolf, Unbounded violations of bipartite Bell Inequalities via Operator Space theory, Comm. Math. Phys. 300, 715–739 (2010).

4. M. Junge, M. Navascués, C. Palazuelos, D. Pérez-García, V.B. Scholtz, R.F. Werner, Connes' embedding problem and Tsirelson's problem, J. Math. Phys. 52, 012102 (2011)

5. A.Salles, D. Cavalcanti, A. Acín, D. Pérez-García., M.M. Wolf, Bell inequalities from multilinear contractions, Quant. Inf. Comp. 10, 0703-0719 (2010) .

6. M. Junge, C. Palazuelos, Large violation of Bell inequalities with low entanglement, Comm. Math. Phys. 306 (3), 695-746 (2011) .

WHAT WE LEARNT