Remark on multi-particle observables and entangled states with constant complexity

TL;DR

This paper investigates the complexity of preparing and measuring multi-particle quantum states, showing that constant-depth networks produce states nearly commuting with mean-field observables, with measurement complexity scaling logarithmically with particle number.

Contribution

It establishes a connection between quantum network depth and the commutation properties of resulting states, introducing bounds for states outside certain hypersurfaces in density matrix space.

Findings

States prepared by constant-depth networks asymptotically commute with mean-field observables.

Measuring non-commuting observables requires networks of depth proportional to log n.

Lower bounds are provided for the depth needed to prepare states outside specific hypersurfaces.

Abstract

We show that every density matrix of an n-particle system prepared by a quantum network of constant depth is asymptotically commuting with the mean-field observables. We introduce certain pairs of hypersurfaces in the space of density matrices and give lower bounds for the depth of a network which prepares states lying outside those pairs. The measurement of an observable which is not asymptotically commuting with the mean-field observables requires a network of depth in the order of log n, if one demands the measurement to project the state into the eigenspace of the measured observable.