Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables

Pironio, Stefano; Navascués, Miguel; Acín, Antonio

doi:10.1137/090760155

Cited by 184 publications

(306 citation statements)

References 24 publications

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“…Furthermore, we can find a whole range of quantum strategies (not equilibria) where the joint payoff of the players is strictly higher than classically possible. Finally, we show that these explicit strategies are very close to the optimal ones, by providing an upper bound that corresponds to the second level of the SDP hierarchy in [19][20][21][22]. In Fig.…”

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confidence: 72%

Nonlocality and Conflicting Interest Games

et al. 2015

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Nonlocality enables two parties to win specific games with probabilities strictly higher than allowed by any classical theory. Nevertheless, all known such examples consider games where the two parties have a common interest, since they jointly win or lose the game. The main question we ask here is whether the nonlocal feature of quantum mechanics can offer an advantage in a scenario where the two parties have conflicting interests. We answer this in the affirmative by presenting a simple conflicting interest game, where quantum strategies outperform classical ones. Moreover, we show that our game has a fair quantum equilibrium with higher payoffs for both players than in any fair classical equilibrium. Finally, we play the game using a commercial entangled photon source and demonstrate experimentally the quantum advantage. Nonlocality is one of the most important and elusive properties of quantum mechanics, where two spatially separated observers sharing a pair of entangled quantum bits can create correlations that cannot be explained by any local realistic theory. More precisely, Bell [1] showed that there exist scenarios where correlations between any local hidden variables can be shown to satisfy specific constraints (known as Bell inequalities), while these constraints can nevertheless be violated by correlations created by quantum systems.An equivalent way of describing Bell test scenarios is in the language of nonlocal games. The best-known example is the CHSH game [2]: Alice and Bob, who are spatially separated and cannot communicate, receive an input bit x and y respectively and must output bits a and b respectively, such that the outputs are different if both input bits are equal to 1, and the same otherwise. It is well known that the probability over uniform inputs that they jointly win this game when they a priori share classical resources is 0.75, while if they share and appropriately measure a pair of maximally entangled qubits, they can jointly win the game with probability cos 2 π/8 > 0.75. The classical value 0.75 corresponds to the upper bound of a Bell inequality and the CHSH game provides an example of a Bell inequality violation, since there exist quantum strategies that violate this bound.Looking at Bell inequalities through the lens of games has been very useful in practice, including in cryptography [3,4] and quantum information [5], where, for example, quantum mechanics offers stronger than classical security guarantees in quantum key distribution or verification protocols. Recently, Brunner and Linden made the connection between Bell test scenarios and games with incomplete information more explicit and provided examples of such games where quantum mechanics offers an advantage [6]. A game with incomplete information (or Bayesian game) is a game where the two parties receive some input unknown to the other party [7]. We remark that without more restrictions, quantum mechanics only offers advantages for incomplete information games, i.e., when the parties receive inputs or, in other word...

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confidence: 72%

Nonlocality and Conflicting Interest Games

et al. 2015

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“…stands for the inner product in H. For increasing orders of d, the SDPs define a growing hierarchy, whose solution converges to the optimum of Eq. (6) [Pironio et al, 2010]. It is worth noting that in some cases, such as in the ground-state energy problem of bosonic systems, the hierarchy already converges at first-order relaxations [Navascués et al, 2013].…”

Section: Relaxations Of Polynomial Optimization Problems Of Noncommutmentioning

confidence: 99%

“…The same idea of using a hierarchy of SDPs applies to the solution of polynomials of noncommuting variables [Pironio et al, 2010, Navascués et al, 2012. The sequence converges provided some conditions on the operators, and in some cases it is possible to conclude that the minimum has been reached after performing only a finite number of tests [Navascués et al, 2008].…”

Section: Introductionmentioning

confidence: 99%

Algorithm 950

Wittek

2015

ACM Trans. Math. Softw.

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A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting problems that creates the relaxation to be solved by SDPA -a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve problems such as finding the ground state energy or testing quantum correlations.

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“…[PNA10]. However, as a consequence of the Nichtnegativstellensatz 3.4, if D S is the ball B or the polydisc D then we do not need sequences of SDPs, a single SDP suffices: the first step in the noncommutative SDP hierarchy is already exact.…”

Section: Semidefinite Programming (Sdp)mentioning

confidence: 99%

“…Let us mention just a few. A nice survey on applications to control theory, systems engineering and optimization is given by Helton, McCullough, Oliveira, Putinar [HMdOP08], applications to quantum physics are explained by Pironio, Navascués, Acín [PNA10] who also consider computational aspects related so noncommutative sum of squares. For instance, optimization of nc polynomials has direct applications in quantum information science (to compute upper bounds on the maximal violation of a generic Bell inequality [PV09]), and also in quantum chemistry (e.g.…”

Section: Introductionmentioning

confidence: 99%

Constrained Polynomial Optimization Problems with Noncommuting Variables

Cafuta¹,

Klep²,

Povh³

2012

SIAM J. Optim.

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Abstract. In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows:(1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) -this sharply contrasts the commutative case where sequences of SDPs are needed; (3) the dual solution to this "single" SDP can be exploited to extract eigenvalue optimizers with an algorithm based on two ingredients:• solution to a truncated nc moment problem via flat extensions;• Gelfand-Naimark-Segal (GNS ) construction. The implementation of these procedures in our computer algebra system NCSOStools is presented and several examples pertaining to matrix inequalities are given to illustrate our results.

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Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables

Cited by 184 publications

References 24 publications

Nonlocality and Conflicting Interest Games

Nonlocality and Conflicting Interest Games

Algorithm 950

Constrained Polynomial Optimization Problems with Noncommuting Variables

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