Monte Carlo sampling from the quantum state space. I

TL;DR

This paper adapts Monte Carlo sampling methods to efficiently generate high-quality samples of quantum states, respecting positivity constraints, for applications like state space integration and volume estimation.

Contribution

It introduces adapted Monte Carlo strategies for quantum state sampling that account for positivity constraints, enabling reliable quantum information computations.

Findings

Successful implementation of sampling methods for two-qubit states

Application to size and credibility of error regions

Calculation of the volume of separable states

Abstract

High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the standard strategies of rejection sampling, importance sampling, and Markov-chain sampling can be adapted to this context, where the samples must obey the constraints imposed by the positivity of the statistical operator. For a comparison of these sampling methods, we generate sample points in the probability space for two-qubit states probed with a tomographically incomplete measurement, and then use the sample for the calculation of the size and credibility of the recently-introduced optimal error regions [see New J. Phys. 15 (2013) 123026]. Another illustration is the computation of the fractional volume of separable two-qubit states.