In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.
In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. Then we study optimization problems over synchronous quantum correlations arising from quantum graph parameters. We introduce semidefinite programming hierarchies and unify existing bounds on quantum chromatic and quantum stability numbers by placing them in the framework of tracial polynomial optimization.
We study and extend the semidefinite programming (SDP) hierarchies introduced in Navascu\'es and V\'ertesi [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of the statistical correlations arising from finite-dimensional quantum systems. First, we introduce the dimension-constrained noncommutative polynomial optimization (NPO) paradigm, where a number of polynomial inequalities are defined and optimization is conducted over all feasible operator representations of bounded dimensionality. Important problems in device-independent and semi-device-independent quantum information science can be formulated (or almost formulated) in this framework. We present effective SDP hierarchies to attack the general dimension-constrained NPO problem (and related ones) and prove their asymptotic convergence. To illustrate the power of these relaxations, we use them to derive a number of dimension witnesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes.
We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.
We study optimization programs given by a bilinear form over noncommutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement-assisted coding for classical channels, and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascués, and Acín [SIAM J. Optim., 20 (2010), pp. 2157--2180], our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. Moreover, this allows us to give a hierarchy of SDP outer approximations for the completely positive semidefinite cone introduced by Laurent and Piovesan.
Optimization is a fundamental area in mathematics and computer science, with many real-world applications. In this thesis we study the efficiency with which we can solve certain optimization problems from two different perspectives. Firstly, we study it from the perspective of matrix factorization ranks, which comes with connections to quantum information theory. Secondly, we study it from the perspective of quantum computing. This thesis is accordingly divided into two parts, where we take these perspectives. In the first part of this thesis we study several matrix factorization ranks and their connections to quantum information theory. In Chapter 5, we first give a unified approach to lower bounding matrix factorization ranks, using polynomial optimization techniques. In Chapter 6, we exploit the connection between one particular factorization rank, the completely positive semidefinite rank, and quantum correlations to provide an explicit family of matrices with a large completely positive semidefinite factorization rank. In Chapter 8, we use the same connection to study quantum versions of classical graph parameters from the perspective of polynomial optimization (albeit in noncommutative variables). In Chapter 7 we propose and study a new measure for the amount of entanglement needed to realize quantum correlations. We can approximate this new measure using our approach to matrix factorization ranks, i.e., through polynomial optimization in noncommutative variables. In the second part of this thesis we turn our attention to the question: Can we solve optimization problems faster on a quantum computer? First, in Chapter 10, we look at the problem of evaluating a Boolean function. We give a new semidefinite programming characterization of the minimum number of quantum queries to the input that are needed to determine the corresponding function value. In Chapter 11 we revisit the semidefinite programming problem that plays a crucial role in the rest of the thesis; we provide a quantum algorithm for solving semidefinite programs. Finally, in Chapter 12, we study the general problem of solving convex optimization problems in the oracle model. We provide both upper and lower bounds on the efficiency of various reductions in the quantum setting. In particular, we show that quantum computers are more efficient than classical computers for the task of answering separation queries when access to the convex problem is given through membership queries.
Uncertainty relations are a fundamental feature of quantum mechanics. How can these relations be found systematically? Here, we develop a semidefinite programming hierarchy for additive uncertainty relations in the variances of noncommuting observables. Our hierarchy is built on the state polynomial optimization framework, also known as scalar extension. The hierarchy is complete in the sense that it converges to tight uncertainty relations. We improve upon upper bounds for all 1292 additive uncertainty relations on up to nine operators for which a tight bound is not known. The bounds are dimension-free and depend entirely on the algebraic relations among the operators. The techniques apply to a range of scenarios, including Pauli, Heisenberg-Weyl, and fermionic operators, and generalize to higher order moments and multiplicative uncertainty relations.
The characterization of possible transformations between quantum states and the quantification of entanglement are central for understanding quantum resources. Explicit results are, however, scarce since they require complicated optimizations over transformation protocols or state decompositions. We introduce a method to optimize functions of the entanglement spectrum of bipartite states over mixed-state decompositions with arbitrary accuracy. Our results are based on a convergent semidefinite programming hierarchy of optimizations over entanglement spectra using a natural polynomial basis and Schur-Weyl duality. Our approach can also be used to determine non-polynomial measures, such as the entanglement of formation, and to characterize the dimensionality of entanglement. Moreover, extracting the decomposition allows us to formulate criteria to determine state transformations under local operations and classical communication (LOCC). In particular, we provide a hierarchy of certificates of LOCC non-convertibility from pure to mixed states in any dimension and a heuristic approach for LOCC convertibility.