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arXiv:1010.1485 · Journal of Physics A Mathematical and Theoretical

How often is a random quantum statek-entangled?

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

Published Dec 22, 2010• 26 citations
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Abstract

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.

DOI
10.1088/1751-8113/44/4/045303

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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-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 analyze the metric properties of conditioned quantum state spaces These spaces are the convex sets of density matrices that, when partially traced over m degrees of freedom, respectively yield the given n × n density matrix η . For the case n = 2, the volume of equipped with the Hilbert–Schmidt measure can be conjectured to be a simple polynomial of the radius of η in the Bloch-ball. Remarkably, for we find numerically that the probability to find a separable state in is independent of η (except for η pure). For , the same holds for , the probability to find a state with a positive partial transpose in . These results are proven analytically for the case of the family of 4 × 4 X -states, and thoroughly numerically investigated for the general case. The important implications of these findings for the clarification of open problems in quantum theory are pointed out and discussed.