Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance

One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.

Figure 1

Variance of the squared Bures distance between fixed density matrix $\sigma$ and random density matrix $\rho$ for various choices of $\sigma$: (a) a pure state, (b) a maximally mixed state, and (c) a state with eigenvalues 0.08, 0.10, 0.12, 0.28, and 0.42. The dimension of the two density matrices is set to be $n=5$ in all cases and the dimension $m$ of the auxiliary subsystem associated with $\rho$ varies from $n$ to $n+7$.

Figure 2

Variance of the squared Bures distance between two independent random density matrices with (a) $n=3$, (b) $n=4$, and (c) $n=5$. In both cases various combinations of $m_1$ and $m_2$ values have been considered, as depicted on the horizontal axes.

Figure 3

Comparison of the approximate PDF and the PDF obtained from Monte Carlo simulations for the squared Bures distance between (a) a random density matrix $\rho$ and a pure state $\sigma$ with $(n,m)=(5,7)$, (b) a random density matrix $\rho$ and a maximally mixed state $\sigma$ with $(n,m)=(5,7)$, and (c) and (d) two random density matrices ${\rho }_1$ and ${\rho }_2$ with $(n,m_1,m_2)=(3,5,7)$ and $(5,6,8)$, respectively. The solid lines represent the gamma-distribution-based approximation and the dashed lines are for Gaussian-distribution-based approximation. The black circles represent the PDFs obtained from Monte Carlo simulations.

Figure 4

Comparison of the PDF approximation and the PDF obtained from coupled-kicked-top simulations for the squared Bures distance between (a) a random density matrix $\rho$ and a pure state $\sigma$ with $(n,m)=(21,25)$, (b) a random density matrix $\rho$ and a maximally mixed state $\sigma$ with $(n,m)=(21,25)$, and (c) and (d) two random density matrices ${\rho }_1$ and ${\rho }_2$ with $(n,m_1,m_2)=(21,23,25)$ and $(23,25,27)$, respectively. The solid lines represent the gamma-distribution-based approximation and the dashed lines are for Gaussian-distribution-based approximation. The circles represent the PDF obtained from coupled-kicked-top simulations.

Figure 5

Comparison of the approximated PDF and spin-chain simulations between (a) a random density matrix $\rho$ and a pure state $\sigma$ with $(n,m)=(15,22)$, (b) a random density matrix $\rho$ and a maximally mixed state $\sigma$ with $(n,m)=(21,22)$, and (c) and (d) two random density matrices ${\rho }_1$ and ${\rho }_2$ with $(n,m_1,m_2)=(14,18,15)$ and (11,42,30), respectively. The solid lines represent the gamma-distribution approximation and the dashed lines are for the Gaussian distribution approximation. The diamonds have been obtained from spin-chain simulations.