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arXiv:1003.0727 · Journal of Physics A Mathematical and Theoretical

On the comparison of volumes of quantum states

Deping Ye

Published Jul 1, 2010• 7 citations
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Abstract

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.

DOI
10.1088/1751-8113/43/31/315301

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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 (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 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.
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.