Non-Zero Mean Quantum Wishart Distribution Of Random Quantum States And Application

Abstract

Random quantum states are useful in various areas of quantum information science. Distributions of random quantum states using Gaussian distributions have been used in various scenarios in quantum information science. One of this is the distribution of random quantum states derived using the Wishart distibution usually used in statistics. This distribution of random quantum states using the Wishart distribution has recently been named as the quantum Wishart distribution \cite{Han}. The quantum Wishart distribution has been found for non-central distribution with a general covariance matrix and zero mean matrix. Here, we find out the closed form expression for the distribution of random quantum states pertaining to non-central Wishart distribution with any general rank one mean matrix and a general covariance matrix for arbitrary dimensions in both real and complex Hilbert space. We term this as the non-zero mean quantum Wishart distribution. We find out the method for the desired placement of its peak position in the real and complex Hilbert space for arbitrary dimensions. We also show an application of this via a fast and efficient algorithm for the random sampling of quantum states, mainly for qubits where the target distribution is a well behaved arbitrary probability distribution function occurring in the context of quantum state estimation experimental data .

Figures (8)

• Figure 1: Plot of the cross-section of the probability distribution of random quantum states according to non-zero mean quantum Wishart distribution in two Bloch plane with number of columns 18 and 10 but for the same mean value $\mu=1$.
• Figure 2: Plot of the cross-section of the probability distribution of random quantum states according to non-zero mean quantum Wishart distrbution in two Bloch plane, for different values of the mean $\mu$ denoted by the legends in the plot, such that one has a non-zero density on the boundary of the Bloch plane here.
• Figure 3: (a)This plot cross-section shows sub-optimal matching between proposal and target distribution for cross-hair measurement conjugate prior. Here number of columns is 13, the proportion of Uniform distribution is 0.0001.(b)This plot cross-section shows good matching between proposal and target distribution for cross-hair measurement conjugate prior. Here number of columns is 18, the proportion of Uniform distribution is 0.0001.
• Figure 4: Plots of the matching of the target distribution (dark blue) with the proposal distribution (light blue) after an appropriate rotation. The right figure shows the achieved close match.
• Figure 5: Fig.(a) Plot of sampled points of the trine target distribution.