Detecting quantum many-body states with imperfect measuring devices

TL;DR

This paper analyzes how imperfect measurements in quantum systems affect the state detection, revealing that larger systems tend to appear more mixed and providing methods to infer original states from coarse-grained data.

Contribution

It introduces a geometric and random-matrix approach to characterize the probability distribution of coarse-grained quantum states and their inverse states.

Findings

Probability density concentrates around the maximally mixed state as system size increases.

The inverse state of the maximally mixed state has a finite singlet component.

Monte Carlo simulations validate the analytical predictions.

Abstract

We study a coarse-graining map arising from incomplete and imperfect addressing of particles in a multipartite quantum system. In its simplest form, corresponding to a two-qubit state, the resulting channel produces a convex mixture of the two partial traces. We derive the probability density of obtaining a given coarse-grained state, using geometric arguments for two qubits coarse-grained to one, and random-matrix methods for larger systems. As the number of qubits increases, the probability density sharply concentrates around the maximally mixed state, making nearly pure coarse-grained states increasingly unlikely. For two qubits, we also compute the inverse state needed to characterize the effective dynamics under coarse-graining and find that the average preimage of the maximally mixed state contains a finite singlet component. Finally, we validate the analytical predictions by inferring the underlying probabilities from Monte-Carlo-generated coarse-grained statistics.