Product numerical range in a space with tensor product structure

Abstract

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.

17 pages, 4 figures. Original preprint "Local numerical range: a versatile tool in the theory of quantum information" [arXiv:0905.3646v1] was broadened and split into two papers: "Restricted numerical range: a versatile tool in the theory of quantum information", and "Product numerical range in a space with tensor product structure"

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• Norms and Cones in the Theory of Quantum Entanglement
• Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators
• Separability gap and large deviation entanglement criterion
• Real numerical shadow and generalized B-splines
• Quantum fidelity of symmetric multipartite states
• Duality of Entanglement Norms
• Decoherence-Free Communication over Multiaccess Quantum Channels
• Joint separable numerical range and bipartite entanglement witness
• Local certification of unitary operations