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arXiv:quant-ph/0701143 · Journal of Mathematical Physics

Subnormalized states and trace-nonincreasing maps

Valerio Cappellini, Hans-Jürgen Sommers, Karol Życzkowski

Published May 1, 2007• 9 citations
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Abstract

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set MN 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 (HS) (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 MN induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in MN2.

DOI
10.1063/1.2738359

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We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set MN 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 (HS) (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 MN induced by partial trace of mixed quantum states distributed uniformly with respect to the HS measure in MN2.
We investigate the set of completely positive, trace-non-increasing 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-non-increasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a byproduct we derive the measure in M_N induced by partial trace of mixed quantum states distributed uniformly with respect to HS-measure in MN2M_{N^2}MN2​.
We investigate the set (a) of positive, trace preserving maps acting on density matrices of size N and a sequence of its nested subsets: the sets of maps which are (b) decomposable, (c) completely positive, and (d) extended by identity impose positive partial transpose and (e) are superpositive. Working with the Hilbert–Schmidt (Euclidean) measure, we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiołkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity, high-dimensional probability, and geometry of Banach spaces, some of which are not standard.
We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x K mixed states, if the reduction takes place with respect to the first subsystem). For K=N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of NtimesNN \\times NNtimesN pure states behaves as lnN-1/2.
The set of trace-preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k -positive maps, where k = 2, ..., d . Working with the measure induced by the Hilbert–Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of ( k + 1) -positive maps forms a small fraction of the outer set of k -positive maps. These results are related to analogous bounds for the relative volume of the sets of k -entangled states describing a bipartite d × d system.
We analyse several product measures in the space of mixed quantum states. In particular, we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N × K composite system, induces a unique measure in the space of N × N mixed states (or in the space of K × K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N × N pure states behaves as ln N -1/2.