In this thesis, we present optimization tools for different problems in quantum information theory. First, we introduce an algorithm for quantum estate estimation. The algorithm consists of orthogonal projections on intersecting hyperplanes, which are determined by the probability distributions and the measurement operators. We show its performance, in both runtime and fidelity, considering realistic errors. Second, we present a technique for certifying quantum non-locality. Given a set of bipartite measurement frequencies, this technique finds a Bell inequality that maximizes the gap between the local hidden variable and the quantum value of a Bell inequality. Lastly, to study the quantum marginal problem, we introduce an operator and develop an algorithm, which takes as inputs a set of quantum marginals and eigenvalues, and outputs a density matrix, if exists, compatible with the prescribed data.
In this thesis, we present optimization tools for different problems in quantum information theory. First, we introduce an algorithm for quantum estate estimation. The algorithm consists of orthogonal projections on intersecting hyperplanes, which are determined by the probability distributions and the measurement operators. We show its performance, in both runtime and fidelity, considering realistic errors. Second, we present a technique for certifying quantum non-locality. Given a set of bipartite measurement frequencies, this technique finds a Bell inequality that maximizes the gap between the local hidden variable and the quantum value of a Bell inequality. Lastly, to study the quantum marginal problem, we introduce an operator and develop an algorithm, which takes as inputs a set of quantum marginals and eigenvalues, and outputs a density matrix, if exists, compatible with the prescribed data.
The first half of this thesis deals with the problem of parameter estimation in a quantum system. In quantum parameter estimation theory, the quantum Fisher information is usually considered as the ultimate precision limit, beyond which no further improvement is possible due to the inherent stochasticity of quantum measurements. On the other hand, in this thesis we will show that a better precision is achievable than predicted by a quantum Fisher information analysis. This is true if some regularity assumptions about the underlying quantum parametric model are relaxed. In such situations, the quantum Fisher information does not completely capture the best possible performance of quantum measurements, and a different approach must be followed. In the second part of the thesis, we will focus on some applications of orthogonal array theory to two notable quantum information problems: the problem of multipartite entanglement classification and the quantum marginal problem. Introduced by Rao in 1947, orthogonal arrays have been usefully applied to different fields, from cryptography and coding theory to the statistical design of experiments, software testing and quality control. Remarkably, orthogonal arrays have also found application in quantum information and, in particular, in the study of quantum entanglement. We will employ tools from orthogonal array theory to study a toy version of the multipartite entanglement classification problem. Finally, we will show how orthogonal arrays can be employed to build constructive solutions to low-dimensional quantum marginal problems.
A problem in quantum information theory is to find the experimental setup that maximizes the nonlocality of correlations with respect to some suitable measure such as the violation of Bell inequalities. There are however some complications with Bell inequalities. First and foremost it is unfeasible to determine the whole set of Bell inequalities already for a few measurements and thus unfeasible to find the experimental setup maximizing their violation. Second, the Bell violation suffers from an ambiguity stemming from the choice of the normalization of the Bell coefficients. An alternative measure of nonlocality with a direct information-theoretic interpretation is the minimal amount of classical communication required for simulating nonlocal correlations. In the case of many instances simulated in parallel, the minimal communication cost per instance is called nonlocal capacity, and its computation can be reduced to a convex-optimization problem. This quantity can be computed for a higher number of measurements and turns out to be useful for finding the optimal experimental setup. Focusing on the bipartite case, we present a simple method for maximizing the nonlocal capacity over a given configuration space and, in particular, over a set of possible measurements, yielding the corresponding optimal setup. Furthermore, we show that there is a functional relationship between Bell violation and nonlocal capacity. The method is illustrated with numerical tests and compared with the maximization of the violation of CGLMP-type Bell inequalities on the basis of entangled two-qubit as well as two-qutrit states. Remarkably, the anomaly of nonlocality displayed by qutrits turns out to be even stronger if the nonlocal capacity is employed as a measure of nonlocality.
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.
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make difficult problems more accessible. A demonstration of how to efficiently implement such an algorithm, avoiding interfacial bottlenecks, is provided, finding optimal protocols to establish entanglement through a lossy channel. The central software package developed addresses polynomial optimization problems. Many problems naturally involve only a polynomial objective and constraint polynomials. Such problems can automatically be cast into semidefinite programs that provide a hierarchy of outer approximations. The resulting problems are often so large and scale so unfavorably with respect to the variable number and degree involved that the boundary of the doable is reached quickly. However, technical progress both in hardware and algorithms has pushed this boundary - but software frameworks for polynomial optimization have not followed in the same manner, often now making them the bottleneck that before was the solver. The package PolynomialOptimization.jl developed during this thesis aims to fill the gap and provide a very resource-efficient intermediate layer together with a wide number of algorithms to reduce the problem size, and naturally supporting complex numbers and semidefinite constraints ubiquitous in quantum information problems. Its application on an entanglement distribution problem is demonstrated, showing that even relaxations with semidefinite matrices of three- and four-digit size can be solved conveniently. Finally, a new way to calculate interior-point barriers for the cone of sums-of-squares matrices in a nearly time-optimal way is developed, whose efficient implementation has the potential of further reducing resource consumption.
Abstract Marginal problems naturally arise in a variety of different fields: basically, the question is whether some marginal/partial information is compatible with a joint probability distribution. To this aim, the characterization of marginal sets via quantifier elimination and polyhedral projection algorithms is of primal importance. In this work, before considering specific problems, we review polyhedral projection algorithms with focus on applications in information theory, and, alongside known algorithms, we also present a newly developed geometric algorithm which walks along the face lattice of the polyhedron in the projection space. One important application of this is in the field of quantum non-locality, where marginal problems arise in the computation of Bell inequalities. We apply the discussed algorithms to discover many tight entropic Bell inequalities of the tripartite Bell scenario as well as more complex networks arising in the field of causal inference. Finally, we analyze the usefulness of these inequalities as nonlocality witnesses by searching for violating quantum states.
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound 22 for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable λ. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP∗=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.