Abstract
We consider the properties of a specific distribution of mixed quantum states of arbitrary dimension that can be biased towards a specific mean purity. In particular, we analyze mixtures of Haar-random pure states with Dirichlet-distributed coefficients. We analytically derive the concentration parameters required to match the mean purity of the Bures and Hilbert--Schmidt distributions in any dimension. Numerical simulations suggest that this value recovers the Hilbert--Schmidt distribution exactly, offering an alternative and intuitive physical interpretation for ensembles of Hilbert--Schmidt-distributed random quantum states. We then demonstrate how substituting these Dirichlet-weighted Haar mixtures in place of the Bures and Hilbert--Schmidt distributions results in measurable performance advantages in machine-learning-based quantum state tomography systems and Bayesian quantum state reconstruction. Finally, we experimentally characterize the distribution of quantum states generated by both a cloud-accessed IBM quantum computer and an in-house source of polarization-entangled photons. In each case, our method can more closely match the underlying distribution than either Bures or Hilbert--Schmidt distributed states for various experimental conditions.
Highlights
Ensembles of random density matrices have found broad applicability in quantum information science [1,2,3,4,5,6,7]
The findings summarized in Eqs. (19), (20) represent major contributions of our present investigation, revealing quantitatively how the parameter α of the MA distribution can be tuned to obtain an equal, higher, or lower mean purity compared to well-known fiducial density matrix measures
From a purely theoretical standpoint, the idea of performing inference with a prior that does not correspond to the actual distribution of validation states seems unwarranted; after all, why should one intentionally select a prior that does not match the quantum states under investigation? And as we explored in Sec
Summary
Ensembles of random density matrices have found broad applicability in quantum information science [1,2,3,4,5,6,7]. An additional ensemble of mixed quantum states based on sums of nonorthogonal Haar-random pure states has been explored as a prior for Bayesian quantum state reconstruction techniques, where the coefficients of these ensembles are distributed according to the symmetric Dirichlet distribution [17,18,19]. Motivated initially by both its computational simplicity and amenability to tuning effective rank [17], 2643-1564/2021/3(4)/043145(14). We find that we can more closely match the distributions of these systems using Dirichlet-distributed mixtures than by standard methods
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