Generating Pseudo-Random Discrete Probability Distributions

Abstract

The generation of pseudo-random discrete probability distributions is of paramount importance for a wide range of stochastic simulations spanning from Monte Carlo methods to the random sampling of quantum states for investigations in quantum information science. In spite of its significance, a thorough exposition of such a procedure is lacking in the literature. In this article we present relevant details concerning the numerical implementation and applicability of what we call the iid, normalization, and trigonometric methods for generating an unbiased probability vector p = (p 1 , · · · , p d ). An immediate application of these results regarding the generation of pseudo-random pure quantum states is also described.

Non-monotonicity of Trace Distance Under Tensor Products

“…Ref. [49]) for each value of the system dimension (with d = 2, · · · , 20). The error, in each case, is computed by comparing the trace distance obtained via diagonalization with the LAPACK subroutines and the value of TD obtained using its analytical expression.…”

Fortran Code for Generating Random Probability Vectors, Unitaries, and Quantum States

“…Then we define the components of the RPV: p j = sin 2 θ j−1 Π d−1 k=j cos 2 θ k (for j = 1, · · · , d − 1) and p d = sin 2 θ d−1 [46]. To get rid from the bias existing in the generated RPVs we use a random permutation of {1, 2, · · · , d} to shuffle its components [47].…”

“…At last we use shuffling of the components of p to obtain an unbiased RPV [47]. A somewhat related method, which is used here as the standard one for the RPVG, was proposed by Życzkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) in the Appendix A of Ref.…”

“…In the inset is shown the 2D scatter plot for the first two components (p1, p2) of five thousand RPVs with d = 3 and produced using the ZHSL (red) or the Normalization (green) method (in the last case the points are (1 − p1, 1 − p2)). opt_rpvg = "iid") and just normalizing the distribution, i.e., p j := r j /( d k=1 r k ) [47]. A related sampling method was put forward in Ref.…”

Random Sampling of Quantum States: a Survey of Methods

“…Now we recall that if we have access to a random number generator yielding random numbers with uniform distribution in [0, 1], an unbiased random discrete probability distribution (RDPD) [37,38] can be generated as follows [39]. First we create a biased RDPD generating q 1 in the interval [0, 1] and…”