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A tensor product matrix approximation problem in quantum physics. (English) Zbl 1118.15027
Summary: We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products.
We discuss this approximation problem for a composite system with two subsystems and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.

MSC:
15A90 Applications of matrix theory to physics (MSC2000)
15A69 Multilinear algebra, tensor calculus
52A41 Convex functions and convex programs in convex geometry
90C25 Convex programming
65F30 Other matrix algorithms (MSC2010)
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