Minimal informationally complete measurements for probability representation of quantum dynamics

TL;DR

This paper introduces a probability-based approach to describe quantum dynamics using minimal informationally complete measurements, facilitating analysis of multi-qubit systems and classicality transitions in noisy quantum devices.

Contribution

It develops a MIC-POVM-based probability formalism for quantum dynamics, offering easier construction and stricter non-classicality criteria compared to SIC-POVMs.

Findings

Established correspondence between quantum formalism and probability representation.

Derived equations for unitary and dissipative quantum evolution in probability form.

Demonstrated classical-like transition in dissipative spin-1/2 dynamics.

Abstract

In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-$1/2$ evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum (NISQ) devices.