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arXiv:0710.1571 · Journal of Mathematical Physics

Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive

Stanisław J. Szarek, Elisabeth M. Werner, Karol Życzkowski

Published Mar 1, 2008• 31 citations

Abstract

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.

DOI: 10.1063/1.2841325

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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.

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 investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamiołkowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the classical maps and an extended space of the discrete probability distributions.

This paper aims to study the α-volume of , an arbitrary subset of the set of N × N density matrices. The α-volume is a generalization of the Hilbert–Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the α-volume of in terms of its Hilbert–Schmidt volume. The analogous estimates between the Bures volume and the α-volume are also established. We employ our results to obtain bounds for the α-volume of the sets of separable quantum states and of states with positive partial transpose. Hence, our asymptotic results provide answers for questions listed in Życzkowski et al (1998 Phys. Rev. A 58 883) for large N in the sense of α-volume.

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

M_{N^2}

.

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.

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamio̷lkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.