Two Poorly Measured Quantum Observables as a Complete Set of Commuting Observables

TL;DR

This paper demonstrates that two poorly measured quantum observables with Poisson spectra can uniquely identify an integrable quantum state if measurement errors are sufficiently small, impacting quantum state determination and classical pattern recognition.

Contribution

It shows that imperfect measurements of two Poisson-spectra observables are sufficient for state identification, extending understanding of measurement requirements in quantum systems.

Findings

Two observables with Poisson spectra suffice for state identification with small measurement errors.

The results apply to integrable quantum systems and have implications for classical pattern recognition.

Measurement error thresholds are critical for the sufficiency of observables.

Abstract

In this article, we revisit the century-old question of the minimal set of observables needed to identify a quantum state: here, we replace the natural coincidences in their spectra by effective ones, induced by an imperfect measurement. We show that if the detection error is smaller than the mean level spacing, then two observables with Poisson spectra will suffice, no matter how large the system is. The primary target of our findings is the integrable (i.e. exactly solvable) quantum systems whose spectra do obey the Poisson statistics. We also consider the implications of our findings for classical pattern recognition techniques.