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P LAYING ( QUANTUM ) GAMES WITH OPERATOR SPACES David Pérez-García Universidad Complutense de Madrid Bilbao 8-Oct-2011

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O UR GROUP 2 postdoctoral research positions opened in our group “Mathematics and Quantum Information”.

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S TRUCTURE 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.

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2P1R GAMES

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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.

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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.

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2P1R GAMES xy a b What is an strategy? A probability distribution p(ab|xy) Which are the possible strategies in the classical case? And in the quantum one?

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EXAMPLES. I NAPROXIMABILITY

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EXAMPLES. I NAPROXIMABILITY RESULTS Theorem (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

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E XAMPLES. I NAPROXIMABILITY 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 Note that MAX-CLIQUE is always less or equal than n (!!)

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E XAMPLES. I NAPROXIMABILITY 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.

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E XAMPLES. I NAPROXIMABILITY RESULTS Connection with 2P1R games. Via LABEL-COVER. Colors Number of valid edges = 4 Solution to LABEL-COVER = 5

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E XAMPLES. I NAPROXIMABILITY 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.

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E XAMPLES. P ARALLEL COMPUTATION

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Given a boolean function f(x,y), minimize c in: xy c bits of communication f(x,y)

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E XAMPLES. P OSITION BASED CRYPTOGRAPHY. ( CHANDRAN ET AL, 2009)

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E XAMPLES. P OSITION BASED CRYPTOGRAPHY 1D for simplicity xy 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

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E XAMPLES. P OSITION BASED CRYPTOGRAPHY Relation 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”

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E XAMPLES. P OSITION BASED CRYPTOGRAPHY Hence, 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.

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E XAMPLES. P OSITION BASED CRYPTOGRAPHY This 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.

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E XAMPLES. R ANDOM NUMBER GENERATION ( PIRONIO ET AL., 2010)

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E XAMPLE. R ANDOM NUMBER GENERATION PROBLEM: How to construct an apparatus which generates perfect random numbers (and hence secret) in a certifiable way? ….. could have a copy of ….. But in quantum mechanics copying is not allowed !!!

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E XAMPLE. R ANDOM 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. xy a b DONE EXPERIMENTALLY ALREADY !!!

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E XAMPLE. R ANDOM NUMBER GENERATION The key is, hence, the existence of quantum strategies which are NOT classical. This guarantees an intrinsic randomness. Classical Quantum Non- signaling

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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).

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O UR CONTRIBUTION

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T HE PROBLEM WE WANT TO ATTACK Estimate D. Parameters: Number of questions = N Number of answers = M Size (dimension) of the quantum system = d D Quantum strategies Classical strategies How large can D be?

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V IOLATION OF A BELL INEQUALITY. F ORMAL DEFINITION OF D. Bell inquality T BELL INEQUALITY= 2P1R GAME (generalized)

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O PERATIONAL INTERPRETATION OF D Bell Inequality T 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?

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

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A N INTERESTING APPLICATION Theorem (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!!!

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T HE TOOLS : O PERATOR SPACES

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O PERATOR S PACES An 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.

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O PERATOR SPACES In particular: Given

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O PERATOR SPACES The morphisms in this category are the completely bounded maps: CB(E,F) is an operator space via In particular E* is an operator space

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C ONNECTION WITH THE 2P1R GAMES Theorem (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:

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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, (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, (2011) 5.A.Salles, D. Cavalcanti, A. Acín, D. Pérez-García., M.M. Wolf, Bell inequalities from multilinear contractions, Quant. Inf. Comp. 10, (2010). 6.M. Junge, C. Palazuelos, Large violation of Bell inequalities with low entanglement, Comm. Math. Phys. 306 (3), (2011). W HAT WE LEARNT

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