What proportion of $n m \times n m$ positive-definite fixed-trace symmetric (Hermitian) matrices remain positive-definite under a certain operation?

Question

Given the class of $n m \times n m$ positive-definite (symmetric or Hermitian) fixed-trace (say, 1), $n,m\geq 2$, what "proportion" of the class remains positive-definite if either the $n^2$ blocks of size $m \times m$ or the $m^2$ blocks of size $n \times n$ are transposed in place?

Further, how might the answer to the question depend upon $k$, where the rank of the matrices under consideration is fixed at $k \leq n m$?

Answer 1

Let us denote the matrices in question by $\rho_{nm}$ and, first, consider the use of the measure generated by the Hilbert-Schmidt metric \begin{equation} \mbox{d} s^2_{HS}=\frac{1}{2} \mbox{Tr}[(\mbox{d} \rho_{nm})^2]. \end{equation}

Then, A. Lovas and A. Andai

LovasAndaiPaper MR3673324

have formally answered the question for the case of symmetric $\rho_{22}$, obtaining the value $\frac{29}{64}$ for the proportion ("separability probability") of this nine-dimensional set of "two-rebit density matrices" that remain positive-definite under the indicated operation of "partial transposition".

In their Conclusions, Lovas and Andai write: ``The structure of the unit ball in operator norm of $2\times 2$ matrices plays a critical role in separability probability of qubit-qubit and rebit-rebit quantum systems. It is quite surprising that the space of $2\times 2$ real or complex matrices seems simple, but to compute the volume of the set \begin{equation*} \Big\{\begin{pmatrix}a & b\\ c& e\end{pmatrix} \Big\vert\ a, b, c, e \in \mathbb{K}, \Big| \Big|{\begin{pmatrix} a & b\\ c& e\end{pmatrix}} \Big| \Big| <1,\ \ \Big| \Big|{\begin{pmatrix} a & \varepsilon b\\ \frac{c}{\varepsilon}& e \end{pmatrix}} <1 \Big| \Big|\Big\} \end{equation*} for a given parameter $\varepsilon\in [0,1]$, which is the value of the function $\chi_{d}(\varepsilon)$, is a very challenging problem. The gist of our considerations is that the behavior of the function $\chi_{d}(\varepsilon)$ determines the separability probabilities with respect to the Hilbert-Schmidt measure.'' (The operator norm $ \Big| \Big| \hspace{.15in} \Big| \Big|$ is the largest singular value or Schatten-$\infty$ norm.)

The function $\chi_{1}(\varepsilon)$--found employing an auxiliary "defect function"--which is used for the determination of the $\frac{29}{64}$ is given by \begin{equation} \label{BasicFormula} \tilde{\chi}_1 (\varepsilon ) = 1-\frac{4}{\pi^2}\int\limits_\varepsilon^1 \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s \end{equation} \begin{equation} = \frac{4}{\pi^2}\int\limits_0^\varepsilon \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s . \end{equation}
Let us note that $\tilde{\chi}_1 (\varepsilon )$ has a closed form, \begin{equation} \label{poly} \frac{2 \left(\varepsilon ^2 \left(4 \text{Li}_2(\varepsilon )-\text{Li}_2\left(\varepsilon ^2\right)\right)+\varepsilon ^4 \left(-\tanh ^{-1}(\varepsilon )\right)+\varepsilon ^3-\varepsilon +\tanh ^{-1}(\varepsilon )\right)}{\pi ^2 \varepsilon ^2}, \end{equation} where the polylogarithmic function is defined by the infinite sum \begin{equation*} \text{Li}_s (z) = \sum\limits_{k=1}^\infty \frac{z^k}{k^s}, \end{equation*} for arbitrary complex $s$ and for all complex arguments $z$ with $|z|<1$.

Lovas and Andai left unanswered the (two-qubit) matter of (15-dimensional) Hermitian $\rho_{22}$.

In MasterLovasAndai MR3767844

Slater was able to construct--though yet without formalized proof--the much simpler \begin{equation} \label{BasicFormula2} \tilde{\chi}_2 (\varepsilon ) = \frac{1}{3} \varepsilon^2 (4-\varepsilon^2) \end{equation} leading to the two-qubit separability probability of $\frac{8}{33}$.

Also, in this paper, counterparts were given for quaternionic [$\tilde{\chi}_4 (\varepsilon ) = \frac{1}{35} \varepsilon^4 (84-64\varepsilon^2+15 \varepsilon^4)$ yielding $\frac{26}{323}$],...density matrices.

Then, these three formulas were incorporated into a "Master Lovas-Andai" formula--the index $d$ being a form of "Dyson-index" of random matrix theory- \begin{equation} \tilde{\chi}_d (\varepsilon ) = \frac{\varepsilon^d \Gamma(d+1)^3 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon^2\right)}{\Gamma(\frac{d}{2}+1)^2}, \end{equation} where the regularized hypergeometric function is indicated.

For $\alpha=\frac{d}{2}$, the desired probabilities ($\frac{29}{64}, \frac{8}{33},\ldots$) are yielded by \begin{equation} \label{Hou1} P(\alpha) =\Sigma_{i=0}^\infty f(\alpha+i), \end{equation} where \begin{equation} \label{Hou2} f(\alpha) = P(\alpha)-P(\alpha +1) = \frac{ q(\alpha) 2^{-4 \alpha -6} \Gamma{(3 \alpha +\frac{5}{2})} \Gamma{(5 \alpha +2})}{6 \Gamma{(\alpha +1)} \Gamma{(2 \alpha +3)} \Gamma{(5 \alpha +\frac{13}{2})}}, \end{equation} and \begin{equation} \label{Hou3} q(\alpha) = 185000 \alpha ^5+779750 \alpha ^4+1289125 \alpha ^3+1042015 \alpha ^2+410694 \alpha +63000 = \end{equation} \begin{equation} \alpha \bigg(5 \alpha \Big(25 \alpha \big(2 \alpha (740 \alpha +3119)+10313\big)+208403\Big)+410694\bigg)+63000. \end{equation} (Qing-Hu Hou helped in the derivation of this formula, using "Zeilberger's algorithm" ["creative telescoping"] ConciseFormula.)

An equivalent formula--now employing $d$ rather than $\alpha=\frac{d}{2}$--was given by C. F. Dunkl in App. D of MasterLovasAndai \begin{equation} \mathcal{P}(d) =3456^{d}\frac{\left( \frac{1}{2}\right) _{d/2}% ^{3}\left( \frac{7}{6}\right) _{d/2}^{2}\left( \frac{5}{6}\right) _{d/2}^{2}\left( 2d\right) !}{\left( \frac{d}{2}\right) !\left( 3\right) _{5d}}\sum_{i\geq0,j\geq0}^{i+j\leq d/2}\frac{\left( -\frac{d}{2}\right) _{i+j}\left( \frac{d}{2}\right) _{j}\left( d\right) _{j}\left( 2+3d\right) _{i}\left( 1+d\right) _{i}}{\left( 2+\frac{5d}{2}\right) _{i+j}\left( 1+\frac{d}{2}\right) _{j}i!j!\left( -2d\right) _{i}}. \end{equation}

However, for dimensions $n$ or $m$ greater than 2, no analogous formulas are yet available.

Extensive numerical (quasirandom estimation) investigations

NumericalExact

have led to conjectures that for $n=3,m=2$ (or $n=2,m=3$) for symmetric ("rebit-retrit") density matrices the Hilbert-Schmidt probability in question is $\frac{860}{6561} =\frac{2 \cdot 5 \cdot 43}{3^8}$, and for Hermitian ("qubit-qutrit" density matrices, the corresponding probability is $\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$.

However, despite these limited results pertaining to small $n,m$, Szarek, Bengtsson and Zyczkowski

StructureBody MR2200422 (2006i:81029)

were able to formally establish--specifically in the case of the Hilbert-Schmidt measure--that for all dimensions, both in the symmetric and Hermitian scenarios, the probability for the class of rank-$nm-1$ matrices is one-half that for the class of full rank ($nm$) matrices. The proof was accomplished by showing that the set of full rank ($nm$) matrices is "pyramid-decomposable", and hence is a body of constant height.

Ruskai and Werner

RuskaiWerner MR2525543 (2010h:81031)

have established that the probability in question is zero if the rank of the $n m \times n m$ density matrix is less than or equal to $\mbox{max}(n,m)$.

For rank-4 ($6 \times 6$) qubit-qutrit density matrices, certain numerical evidence suggests that the associated probability might be $\frac{1}{34}$ that of the rank-6 probability (conjectured, as indicated above, to be $\frac{27}{1000}$).

Additionally, other choices of measures on the density matrices have been considered (in particular, the "Bures", an example of an operator monotone measure)

[GeometryOfQuantumStates][4] MR3752196 (extensive review of first edition MR2230995 (2007k:81001))

for which a two-rebit estimate of 0.15709623 has been obtained, and a two-qubit conjecture of $\frac{25}{341} =\frac{5^2}{11 \cdot 31}$ advanced.

For asymptotic aspects of this question, see. Chap. 9 of

[AliceBobBanach][4]  MR3699754

Answer 2

Let us consider approaching the problem from a numerical (random-matrix) viewpoint, denoting the matrices in question by $\rho_{nm}$ and, first, considering the use of the measure generated by the Hilbert-Schmidt metric \begin{equation} \mbox{d} s^2_{HS}=\frac{1}{2} \mbox{Tr}[(\mbox{d} \rho_{nm})^2]. \end{equation}

Say we are interested in the proportion of $n m \times n m$ matrices of rank $k$ that remain positive-definite under the indicated operation of partial transposition, and desire to generate a matrix in that class random with respect to the Hilbert-Schmidt measure.

A procedure to do so—based on the Wishart-Laguerre distribution—has been outlined (not yet published) by K. Zyczkowski and the group of A. Khevelidze, I. Rogojin and V. Abgaryan.

In the case of Hermitian matrices, one generates a $k \times k+ 2 (nm -k)$ matrix $G$, the entries of which are complex numbers, the real and imaginary parts being normal random variates. Then, $G G^\dagger$ is normalized to have trace 1. (If $k=n m$, this is the desired density matrix. RandomMatrix) If $k<nm$, the normalized matrix is diagonalized fo obtain an $k \times k$ matrix which is extended to an $nm \times nm$ matrix, the additional entries being zeros. Rotating this expanded matrix by a random unitary, the desired $nm \times nm$ matrix $\rho_{nm}$, random with respect to the Hilbert-Schmidt measure on the $nm \times nm$ Hermitian, positive-definite, trace 1 matrices of rank $k$ is obtained.

In the case of symmetric matrices, one generates a $k \times k+ 2 (nm -k)+1$ matrix $G$, the entries of which are normal random variates. Then, $G G^T$ is normalized to have trace 1. (If $k=n m$, this is the desired density matrix.) If $k<nm$, it is diagonalized fo obtain an $k \times k$ matrix which is extended to an $nm \times nm$ matrix, the additional entries being zeros. Rotating this expanded matrix by a random orthogonal one, the desired $nm \times nm$ matrix $\rho_{nm}$, random with respect to the Hilbert-Schmidt measure on the $nm \times nm$ symmetric, positive-definite, trace 1 matrices of rank $k$ is obtained.

We currently are implementing this procedure in two cases.

In one, we are analyzing $6 \times 6$ (rebit-retrit) density matrices of rank 4. Based on 675 million randomly generated (in the indicated manner) such matrices, we obtain an estimate of a probability of 0.00774073 of having a positive partial-transpose (equivalent to separability in this case).

Relatedly, in a previous study,

NumericalExact

we had advanced a conjecture of $\frac{860}{6561} =\frac{2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$ for the $6 \times 6$ symmetric density matrices of full rank (6). (It has been formally proven that in the $4 \times 4$ counterpart, the Hilbert-Schmidt probability is $\frac{29}{64}=\frac{29}{2^6}$ LovasAndai MasterLovasAndai .)

Now, we find it intriguing to propose a rank-4 rebit-retrit separability probability conjecture of $\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^{4}} = 0.00774$. Subject to these two conjectures, the ratio of the rank-4 to rank-6 probabilities would be $\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049$, with the common factor 43 cancelling.

Our second ongoing study is concerned with the rank-6 $(4 \times 2) \times (4 \times 2) $ Hermitian density matrices, so far--based on 149 million random matrices--obtaining estimates of 0.00129085 and 0.000054624 for the full-rank and rank-6 Hilbert-Schmidt probabilities, respectively.