Absolute dimensionality of quantum ensembles

TL;DR

This paper introduces a basis-independent measure of the effective dimension of quantum ensembles, enabling better understanding and simulation of high-dimensional quantum systems in quantum information processing.

Contribution

It proposes an absolute, basis-independent dimensionality concept for quantum ensembles, with analytical and numerical tools for its determination and simulation.

Findings

Developed analytical witness criteria for quantum ensemble dimension

Created semidefinite programming methods based on information capacity

Constructed optimal simulation models for noisy pure states

Abstract

The dimension of a quantum state is traditionally seen as the number of superposed distinguishable states in a given basis. We propose an absolute, i.e.~basis-independent, notion of dimensionality for ensembles of quantum states. It is based on whether a quantum ensemble can be simulated with states confined to arbitrary lower-dimensional subspaces and classical postprocessing. In order to determine the absolute dimension of quantum ensembles, we develop both analytical witness criteria and a semidefinite programming criterion based on the ensemble's information capacity. Furthermore, we construct explicit simulation models for arbitrary ensembles of pure quantum states subject to white noise, and in natural cases we prove their optimality. Also, efficient numerical methods are provided for simulating generic ensembles. Finally, we discuss the role of absolute dimensionality in high-dimensional quantum information processing.