Abstract
Many problems in information theory can be reduced to optimizations over matrices, where the rank of the matrices is constrained. We establish a link between rank-constrained optimization and the theory of quantum entanglement. More precisely, we prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, consequently, we construct a complete hierarchy of semidefinite programs for solving the original problem. This hierarchy not only provides a sequence of certified bounds for the rank-constrained optimization problem, but also gives pretty good and often exact values in practice when the lowest level of the hierarchy is considered. We demonstrate that our approach can be used for relevant problems in quantum information processing, such as the optimization over pure states, the characterization of mixed unitary channels and faithful entanglement, and quantum contextuality, as well as in classical information theory including the maximum cut problem, pseudo-Boolean optimization, and the orthonormal representation of graphs. Finally, we show that our ideas can be extended to rank-constrained quadratic and higher-order programming.
Highlights
We prove that a large class of rank-constrained semidefinite programs can be written as a convex optimization over separable quantum states and, we construct a complete hierarchy of semidefinite programs for solving the original problem
Many efforts have been devoted to so-called semidefinite programs (SDPs), which is a class of highly tractable convex optimization problems
We demonstrate that quantum information theory does benefit from ideas of optimization theory, but the results obtained in this field can be used to study mathematical problems from a fresh perspective
Summary
The mathematical theory of optimization has become a vital tool in various branches of science. Well-known examples are the characterization of quantum correlations for a fixed dimension [8,9], the determination of the faithfulness of quantum entanglement [10], the groundstate energy in spin glasses [11], and compressed sensing tomography [12] These nonconvex optimization problems share a common structure: they can be formulated as SDPs with an extra rank constraint. In order to demonstrate the usefulness of our method, we first show that the optimization over pure quantum states or unitary matrices in quantum information can be naturally written as a rank-constrained optimization This provides a complete characterization of faithful entanglement [10,16] and of mixed unitary channels [17,18].
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