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arXiv:quant-ph/0304041 · Journal of Physics A Mathematical and General

Bures volume of the set of mixed quantum states

Hans-J rgen Sommers, Karol Życzkowski

Published Sep 15, 2003• 163 citations
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Abstract

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 .

DOI
10.1088/0305-4470/36/39/308

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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.
We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parametrizations of Boya et al (1998 Preprint quant-ph/9810084) of the n × n density matrices, in terms of squared components of the unit ( n - 1)-sphere and the n × n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, thereby obtaining `Bures prior probability distributions' over the two- and three-state systems. Then, as a first step in extending these results to n >3, we determine that the `Hall normalization constant' ( C n ) for the marginal Bures prior probablity distribution over the ( n - 1)-dimensional simplex of the n eigenvalues of the n × n density matrices is, for n = 4, equal to 71 680/ 2 . Since we also find that C 3 = 35/ , it follows that C 4 is simply equal to 2 11 C 3 / . ( C 2 itself is known to equal 2/ .) The constant C 5 is also found. It too is associated with a remarkably simple decomposition, involving the product of the eight consecutive prime numbers from 3 to 23. We also preliminarily investigate several cases n >5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n . This would have diverse applications, including quantum inference and universal quantum coding.
The aim of this paper is to derive explicit formulae for the computation of the Riemannian Bures metric g on the manifold of (finite-dimensional) nonsingular density matrices . This Riemannian metric introduced by Uhlmann generalizes the Fubini-Study metric to mixed states and is the infinitesimal version of the Bures distance. Several formulae are known for computing the Bures metric in low dimensions. The formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures. The first equations we give are, essentially, of the form , where is a matrix of invariants of . A further formula, , is adapted to the local orthogonal decomposition at generic points.