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