Subnormalized states and trace-nonincreasing maps

Abstract

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M_N induced by partial trace of mixed quantum states distributed uniformly with respect to HS-measure in $M_{N^2}$.

LaTeX, 21 pages, 4 Encapsuled PostScript figures, 1 table

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• Generating random density matrices
• Quartic quantum theory: an extension of the standard quantum mechanics
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• Symmetries of quantum evolutions
• Generalized Measure of Quantum Fisher Information
• Quantum Hidden Markov Models based on Transition Operation Matrices
• Jarzynski-like Equality of Nonequilibrium Information Production Based on Quantum Cross Entropy
• Universal Properties of Partial Quantum Maps