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arXiv:quant-ph/0302197 · Journal of Physics A Mathematical and General

Hilbert–Schmidt volume of the set of mixed quantum states

Karol Życzkowski, Hans-J rgen Sommers

Published Sep 15, 2003• 105 citations
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Abstract

We compute the volume of the convex ( N 2 − 1)-dimensional set ℳ N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides information about the structure of ℳ N . Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step, we analyse volumes of the unitary and orthogonal groups and of the flag manifolds.

DOI
10.1088/0305-4470/36/39/310

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We compute the volume of the convex ( N 2 − 1)-dimensional set ℳ N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides information about the structure of ℳ N . Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step, we analyse volumes of the unitary and orthogonal groups and of the flag manifolds.
We compute the volume of the ( N 2 − 1)-dimensional set N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an ( N 2 − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½ S 3 ⊂ 4 . We find also the area of the boundary of the set N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N .
We compute the volume of the N^2-1 dimensional set M_N of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N^2-1 dimensional hyper-halfsphere of radius 1/2. For N=2 we obtain the volume of the Uhlmann 3-D hemisphere, embedded in R^4. We find also the area of the boundary of the set M_N and obtain analogous results for the smaller set of all real, symmetric density matrices. An explicit formula for the Bures-Hall normalization constants is derived for an arbitrary N.
The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.
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​.