Random Sampling of Quantum States: A Survey of Methods

TL;DR

This survey reviews various methods for generating random quantum states, comparing their procedures, advantages, and limitations, with a focus on practical numerical implementations and the underlying mathematical frameworks.

Contribution

It provides a comprehensive overview of existing techniques for random quantum state generation, including new insights into the overparametrized method and related Ginibre and Bures approaches.

Findings

The overparametrized method simplifies numerical generation of random states.

Different methods exhibit varying efficiency and suitability depending on the application.

A concentration of measure phenomenon affects the distribution of generated states.

Abstract

The numerical generation of random quantum states (RQS) is an important procedure for investigations in quantum information science. Here we review some methods that may be used for performing that task. We start by presenting a simple procedure for generating random state vectors, for which the main tool is the random sampling of unbiased discrete probability distributions (DPD). Afterwards the creation of random density matrices is addressed. In this context we first present the standard method, which consists in using the spectral decomposition of a quantum state for getting RQS from random DPDs and random unitary matrices. In the sequence the Bloch vector parametrization method is described. This approach, despite being useful in several instances, is not in general convenient for RQS generation. In the last part of the article we regard the overparametrized method (OPM) and the related Ginibre and Bures techniques. The OPM can be used to create random positive semidefinite matrices with unit trace from randomly produced general complex matrices in a simple way that is friendly for numerical implementations. We consider a physically relevant issue related to the possible domains that may be used for the real and imaginary parts of the elements of such general complex matrices. Subsequently a too fast concentration of measure in the quantum state space that appears in this parametrization is noticed.