Joint numerical ranges, quantum maps, and joint numerical shadows
Copyright policy )Joint numerical ranges, quantum maps, and joint numerical shadows
We associate with k hermitian N\times N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian N\times N matrices defines a complex N\times N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and the numerical ranges, on the other hand. In particular, we show that under the identity resolution assumption on Kraus operators defining the quantum map, the dynamics shrinks numerical ranges.
12 latex pages, 3 figures in eps, revised version, several improvements
- Polish Academy of Learning Poland
- Nicolaus Copernicus University Torun Poland
- Jagiellonian University Poland
- Nicolaus Copernicus University Poland
Quantum state spaces, operational and probabilistic concepts, qutrit, Geometric constructions in real or complex geometry, FOS: Physical sciences, double flip channel, Hermitian operators, decaying channel, quantum information, joint numerical shadow, FOS: Mathematics, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Operator Algebras (math.OA), quantum map, qubit, Mathematical Physics, linear projection, Numerical Analysis, Quantum Physics, Algebra and Number Theory, Mathematics - Operator Algebras, Mathematical Physics (math-ph), joint numerical range, Quantum information, communication, networks (quantum-theoretic aspects), Numerical range, numerical radius, quantum state, Hermitian and normal operators (spectral measures, functional calculus, etc.), affine equivalence, Geometry and Topology, Quantum Physics (quant-ph)
Quantum state spaces, operational and probabilistic concepts, qutrit, Geometric constructions in real or complex geometry, FOS: Physical sciences, double flip channel, Hermitian operators, decaying channel, quantum information, joint numerical shadow, FOS: Mathematics, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Discrete Mathematics and Combinatorics, Operator Algebras (math.OA), quantum map, qubit, Mathematical Physics, linear projection, Numerical Analysis, Quantum Physics, Algebra and Number Theory, Mathematics - Operator Algebras, Mathematical Physics (math-ph), joint numerical range, Quantum information, communication, networks (quantum-theoretic aspects), Numerical range, numerical radius, quantum state, Hermitian and normal operators (spectral measures, functional calculus, etc.), affine equivalence, Geometry and Topology, Quantum Physics (quant-ph)
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