Bures–Hall Ensemble: Spectral Densities and Average Entropies
We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures–Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Havrda–Charvát–Tsallis (HCT) entropy as a finite sum. Averages of von Neumann entropy, linear entropy and purity follow by considering appropriate limits in the average HCT expression. Based on exact evaluations of the average von Neumann entropy and the average purity, we also conjecture very simple formulae for these, which are similar to those in the Hilbert–Schmidt ensemble.
Keywords: Bures–Hall ensemble, Eigenvalue statistics, Average entropies
The density matrix formalism was introduced by von Neumann to describe statistical concepts in quantum mechanics . It plays a fundamental role in quantum mechanics and provides a natural approach to deal with mixed states [2, 3]. Given the set of finite-size density matrices, it is now well acknowledged that, there is no unique measure which can be used to describe it [3, 5, 7, 8, 9, 6, 4]. Therefore, one seeks a good and useful measure which can be associated with these density matrices. One of the ways, to do this, is to consider certain metric on the space of mixed states. Such a distance induces a measure on the space of density matrices and, consequently, on the corresponding eigenvalues [3, 5, 7, 8, 9, 6, 4]. The two popular and physically relevant choices are the Hilbert–Schmidt distance [4, 3] and the Bures distance [12, 11, 10], and lead to the respective distributions on eigenvalues. The former is Riemannian but not monotone, while the latter is both Riemannian and monotone . Interestingly, the Hilbert-Schmidt measure is also obtained by the operation of partial tracing over random pure states of bipartite systems [5, 6, 4, 13, 3, 14]. The Bures-Hall measure, on the other hand, is obtained by performing a partial trace over a symmetric superposition of two pure states, one of which is a local unitary transformed copy of the other [3, 14, 13].
The Hilbert–Schmidt fixed trace ensemble has been extensively studied from a random matrix theory perspective, and consequently the corresponding spectral statistics and behavior of the associated observables are fairly well understood. For instance, based on the knowledge of the joint probability density (jpd) of eigenvalues [15, 16, 18, 17, 8, 5], we know explicit answers for the level density and two-point correlation function, moments and cumulants of the eigenvalues and the entropy measures, asymptotic and universal behavior and also extreme eigenvalue distributions and moments [4, 24, 25, 26, 33, 18, 21, 6, 19, 20, 36, 35, 37, 38, 41, 22, 28, 30, 32, 23, 27, 29, 34, 31, 39, 40]. In comparison, the fixed trace Bures–Hall ensemble has been explored very little due to its more involved mathematical structure. The jpd of eigenvalues for this ensemble was derived by Hall in . Życzkowski, Sommers and co-workers have obtained several key results pertaining to the fixed trace Bures–Hall ensemble in the study of statistical distribution of random density matrices and the associated entropy measures [5, 9, 4, 3, 14, 13]. Borot and Nadal have obtained the purity distribution for a generalized version of the Bures–Hall fixed trace ensemble in the large dimension limit . The corresponding unrestricted trace variant has been investigated by Forrester and Kieburg in connection with the Cauchy two-matrix model . This connection was discovered by Bertola et al. while investigating the Cauchy two-matrix model . Despite these invaluable contributions, there remain several aspects related to the Bures–Hall ensemble that remain to be explored.
In this work, we investigate the spectral statistics of both unrestricted trace and fixed trace variants of the Bures–Hall ensemble. For the former, we obtain an exact result for the -point correlation function of arbitrary order in terms of a Pfaffian. This Pfaffian expression offers an alternative representation for the correlation function than the one derived by Forrester and Kieburg . We then focus on the level density and use it to obtain the corresponding exact closed-form expression for the fixed trace ensemble. This, in turn, is used to calculate the average Havrda–Charvát–Tsallis (HCT) entropy [44, 45] of random density matrices which are described by the fixed trace Bures–Hall ensemble. Appropriate limits of the HCT entropy also lead to exact expressions for the average von-Neumann entropy and the average linear entropy or, equivalently, the average purity. Based on exact evaluations, we also conjecture very simple formulae for the average von-Neumann entropy and the average purity. Finally, we validate these analytical results using numerical simulation based on Dyson’s log-gas formalism [46, 47].
The presentation scheme of the paper is as follows. In section 2 we derive exact results pertaining to the spectral statistics of unrestricted trace Bures–Hall ensemble. This is then used in section 3 to obtain an exact result for the level density in the fixed trace case. In section 4 we derive exact expressions for the average entropies. In section 5 we conclude with a brief summary of our results and also indicate directions in which this work can be extended. Appendices collect details of the derivations of the analytical results presented in this paper.
2 Unrestricted trace Bures–Hall ensemble
In this, with , is an -dimensional complex Ginibre random matrix having the associated probability measure
with , and is an -dimensional random unitary matrix from the measure
Here represents the flat measure given by the product of differentials of all independent components in , and is the Haar measure on the group of -dimensional unitary matrices. The parameter under this realization is given by , and therefore assumes half-integer values only. Moreover, the nonzero eigenvalues of the random matrix
are also described by the jpd (1). Additionally, possesses generic zero eigenvalues.
The jpd in (1) is related to that of the fixed trace Bures–Hall ensemble of random density matrices via a Laplace transform , as discussed in section 3. Interestingly, the above jpd also connects to the matrix model [48, 49], as was discovered by Bertola et al. while investigating the Cauchy two-matrix model . Later on, Forrester and Kieburg demonstrated the explicit relationship between the unrestricted Bures–Hall ensembles and the Cauchy two-matrix model in . This is very interesting since the former constitutes a Pfaffian point process, while the latter corresponds to a determinantal point process. Also, very recently, Muttalib–Borodin kind of deformation has been considered in the Cauchy two-matrix model and the unrestricted Bures–Hall ensemble by Forrester and Li . Furthermore, Hu and Li have shown that the partition function of the unrestricted trace Bures-Hall ensemble can be identified as the -function of BKP and DKP hierarchies .
and the result . As a consequence, we have
In the above expression, the kernels are
and in addition, when is odd,
The inverse of the normalisation factor can be obtained using de Brujin’s integration theorem  as
where is an -dimensional matrix with elements,
for , and additionally,
Given the jpd of eigenvalues, one is interested in calculating the -point correlation function, which is defined as
In the present case an exact result for can be obtained using the generalization of the de Brujin’s theorem, as derived by Kieburg [55, Appendix A.1]. The result is in terms of a Pfaffian of an -dimensional antisymmetric matrix:
In the above expression, the indices in a matrix block are the row and column indices, respectively. The kernels and appearing in the above Pfaffian are given by
for . Here E is the exponential integral function. Moreover, when is odd, we have
for . The and within the Pfaffian in (17) are as in (10), (13) and (14). In reference , the -point correlation function for the unrestricted Bures–Hall ensemble has been derived by exploiting its relationship with the Cauchy two-matrix ensemble. It involves the Pfaffian of a antisymmetric matrix with kernels involving integral over certain Meijer G-functions. While it appears difficult to demonstrate a direct equivalence of the Pfaffian result of  with the above Pfaffian result, it can be numerically verified on a case-by-case basis that they are indeed equivalent.
The level density is of special interest since the first order marginal density reveals the behavior of a generic eigenvalue of the ensemble. Furthermore, it enables one to obtain the averages of observables which are linear statistic on the eigenvalues. We use the Pfaffian-expansion result given in [56, Corollary 2.4] to obtain the following expression for the level density:
and is the -dimensional antisymmetric matrix obtained after removing the th and th rows and columns from . For , should be taken as . Moreover, as shown in A, this Pfaffian can be evaluated in terms of a restricted product as
The notation in the product means that both and do not assume the values .
In Fig. 1 we show the plots of the marginal density for various values of . The solid curves are based on the above analytical result, and the symbols have been obtained using the numerical simulation using Dyson’s log-gas formalism [46, 47], as briefly described in the B. We can see very good agreements between the analytical-expression based and numerical simulation based results.
3 Fixed trace Bures–Hall ensemble
The above jpd describes the behavior of eigenvalues () of the random density matrix
where is given in (2). We note that, in this case, the choice of variance of the Gaussian matrix elements of constituting in (3) is immaterial and hence any leads to the same jpd. Equation (25) also describes the nonzero eigenvalues of dimensional and rank random matrices
where is as in (5).
For , the distribution (25) on eigenvalues coincides with the one induced by the Bures metric over the space of random mixed states. Another approach to realize the above distribution, which is equivalent to the construction (26), is as follows . Consider a superposition of two pure states, . Here, is a random pure state belonging to a bipartite composite Hilbert space , where and are and dimensional, respectively. is a global unitary matrix of size taken from the Haar measure . The state is the state transformed by a local unitary , i.e. , where is an -dimensional unitary matrix taken from the measure . The reduced density operator is then obtained by , where represents partial trace over the subsystem . Again, the standard Bures–Hall distribution is realized for .
It can be observed that, by introducing an auxiliary variable to replace the 1 inside the delta function, performing Laplace transform and then applying some rescaling, we are lead to the jpd given by (1) for the unrestriced Bures–Hall ensemble. Consequently, the corresponding normalization factors are also related. As shown in the C, the normalization factor in the above jpd is given by
Similar to the jpd of eigenvalues, the -level correlation function for the fixed trace ensemble can be related to that of the unrestricted trace ensemble by means of an inverse Laplace transform; see the C. We have the result
In particular, the level density for the fixed trace Bures–Hall ensemble can be obtained from unrestricted trace ensemble result using the relation
In the above equation, is given by
and when is odd, additionally, we have
Here, is the incomplete Beta function and, for compactness, we have defined
4 Entropy Measures
Given an ensemble of random density matrices, a natural question to ask is,”how far or close the associated states are to being pure or maximally mixed”? There are several entropy measures which can be used to answer this. We focus here on the Havrda-Charvát-Tsallis (HCT) [44, 45, 3] entropy, which is defined as
Here is a positive real parameter. The value of varies from 0 to . The former indicates a pure state, while the latter signifies a maximally mixed state. In the limit , the HCT entropy leads to the von Neumann entropy,
For , the HCT entropy yields the linear entropy,
The linear entropy is related to the purity as . The HCT entropy is advantageous to use, as the ensemble averages are more easily done with compared to the in the Rényi entropy .
The calculation of average entropy can be performed in two ways. In the first approach, the result for the average entropy associated with the Bures–Hall ensemble can be found by directly integrating with the fixed trace ensemble jpd (25),
Now, HCT entropy being a linear statistic, the symmetry of the eigenvalues in the jpd allows us to reduce the above average involving integrals to an average involving a single integral on the level density . We obtain
The second approach relies on mapping the average entropy calculation as an average over the level density of the unrestricted trace ensemble. This has been done in the E, and yields the result
As demonstrated in the F, after some simplification and rearrangement, both approaches lead to the same expression, which is given by
Moreover, when is odd, we have
For , (41) gives the average linear entropy , and also the average purity via the relation . The average von Neumann entropy can be obtained by carefully taking the limit , and after some rearrangement of terms, as
when is odd. Here is the digamma function with the integral representation .
|Exact||Numerical value||Exact||Numerical value|
|Exact||Numerical value||Exact||Numerical value|
In Tables 1 and 2 we compile the evaluations for average von Neumann entropy , and the average purity . For comparison, we also show the results for Hilbert–Schmidt ensemble, for which the average von Neumann entropy and average purity are given by [18, 21, 4, 20]
In figure 3 we show the average entropy results for (von Neumann entropy) and (purity) for both Bures–Hall and Hilbert–Schmidt ensembles. We investigate the behavior for fixed, varying, and fixed, varying scenarios. We find that for any , on average, the Bures–Hall measure is concentrated more towards states of higher purity than the Hilbert–Schmidt measure. This was shown in  for the square case and now we see that it holds even for the rectangular case. Additionally, it can be seen that for fixed , the difference in averages increases as is increased towards . On the other hand, for fixed , if the value is increased, then the difference approaches zero.
On examining the exact evaluations of the average von Neumann entropy and the average purity (or, equivalently, the average linear entropy) for the Bures–Hall ensemble, we come up with the conjecture that, similar to those of the Hilbert–Schmidt ensemble, they are given by very simple formulae:
Moreover, we find that
where the equality holds for . Therefore, this is in conformity with the conclusion that the Bures–Hall measure is concentrated more towards the states of higher purity compared to the Hilbert–Schmidt measure. Moreover, the other observations found in Fig. 3 for the difference of average purities are consistent with this expression.
5 Summary and Outlook
In this work we investigated statistical properties of random matrix ensemble distributed according to the Bures measure. We started our analysis with the unrestricted trace Bures–Hall ensemble and obtained a new Pfaffian based representation for the correlation function of arbitrary order. This was then used to obtain a closed-form exact result for the level density of the fixed trace Bures–Hall ensemble. Based on this, we computed the average HCT entropy and also obtained the average von-Neumann entropy, average linear entropy and average purity by considering appropriate limits. The exact evaluations of these average entropies enabled us to propose very simple conjectural expressions for the average von-Neumann entropy and average linear entropy or purity. We found that the Bures measure is concentrated more towards states of higher purity than the Hilbert–Schmidt measure even for the rectangular case.
The simple conjectural expressions suggest that there is some additional structure in the eigenvalue statistics of Bures-Hall ensemble that needs to be unveiled. Moreover, it would be of interest to obtain higher order statistics for the entropies, i.e, higher moments and cumulants. This would help provide a better understanding concerning the statistics of these entropies when dealing with Bures measure.
AS acknowledges DST-INSPIRE for financial support. SK is grateful to Prof. P. J. Forrester for fruitful correspondences.
Appendix A Evaluation of Pfaffians
We need to evaluate , where is an antisymmetric matrix defined by (13) and (14). First of all, we observe that factors , can be pulled out of the Pfaffian from the th row as well as the th column, thereby giving rise to an overall factor of . Therefore, we obtain
Comparing this with the Schur’s Pfaffian identity, equation (6), we find that by setting , , we can reduce the above Pfaffian to the following product:
Now, , and . Therefore, we have
where the last step followed because the term in the second product is 1. We now use the identity . This yields
which is the required result.
Appendix B Generating eigenvalues from the Bures–Hall ensemble
Dyson showed that the jpd of eigenvalues of classical random matrix ensembles coincide with the Gibbs-Boltzmann factor associated with a log-gas system at three special values of the inverse temperature and 4, with the Boltzmann constant set to 1 [46, 47]. The ‘log’ has to do with the fact that the gas particles interact via Coulombic interaction, which is logarithmic in two-dimensional space. We can use the same idea here and, for the unrestricted ensemble case, interpret the jpd in (1) as a Gibbs-Boltzmann weight:
with . This gives the energy function as
In this case, along with the two-dimensional Coulombic interaction, there is another two-body interaction between the particles given by . Moreover, the particles are constrained to move on the positive real axis, and also feel the one-body potential . The above energy function can be implemented in the standard Metropolis-Hastings algorithm based Monte Carlo simulation to generate the stationary configurations, which yield the jpd. The generic one-eigenvalue density can be obtained by collecting the values of all and then plotting the histogram. We note that numerically we have to put a “large” cut-off for the domain of eigenvalues (position of the particles) as we cannot assign the full positive real domain. It should be large enough so that the eigenvalues density resulting from the simulation becomes negligible beyond this cut-off.
For the eigenvalues in the fixed trace case, we can use the eigenvalues generated in the above simulation, and obtain . Another option is to implement the simulation with the above energy function but restrict positions of the charges () in the domain , along with the constraint that . In this case, for the Metropolis-Hastings algorithm, we perturb positions of two charges simultaneously by amounts and , so that the fixed trace constraint remains imposed throughout the simulation, if one starts with such an initial configuration . Additionally, if position of any of the two charges fall outside as a result of perturbation, that move is rejected. In the present work, we have used the first approach as it generates the eigenvalue densities for both unrestricted and restricted trace ensembles at once.
Yet another way to obtain the eigenvalues is to use the random matrix itself using (2) and (26) for the unrestricted trace and fixed trace cases, respectively, and then diagonalize it. For the square case (), it is comparatively easier to do so since one requires generating complex Ginibre (Gaussian) matrices, and Haar-distributed unitary matrices. For the rectangular case (), the unitary matrices have to be generated from the measure , and this requires some additional work. One way to do this is to implement Monte Carlo simulation by performing random walk in the space of unitary matrices and use Metropolis-Hastings algorithm with the statistical weight , i.e. use the energy function as . The perturbation moves in the unitary matrices can be implemented using , where is an -dimensional Hermitian random matrix from the Gaussian Unitary Ensemble (GUE) [46, 47], and is a small scalar. Once the stationarity is achieved, the unitary matrices would be generated from the measure and hence can be used to construct and , which can then be diagonalized to obtain the eigenvalues. It should be noted that the eigenvalues for can be obtained easily using those of and therefore, in practice, one needs to diagonalize only.
Appendix C Relationship between level densities of fixed trace and unrestricted trace ensembles
Establishing a relationship between the fixed trace and unrestricted trace variants of the Bures–Hall ensemble is based on implementing a Laplace transform which has been used in earlier works also. We use the same idea here and introduce an auxiliary variable to replace the 1 for the fixed trace condition in the expression for -level density function for the fixed trace Bures–Hall ensemble:
It should be noted that we have extended the integration domains from to and this keeps the result of the multidimensional integral unchanged due to the delta-function constraint. We now apply the Laplace transform () and obtain
After some rearrangement, the right hand side of the above equation can be expressed in terms of the level density of the unrestricted trace ensemble as
so that the application of inverse-Laplace transform yields