Quantum Tomography under Prior Information

TL;DR

This paper analyzes the measurement requirements for quantum state identification under prior information constraints, revealing topological factors and providing bounds for informationally complete measurements, especially for pure states.

Contribution

It introduces a topological framework to determine measurement bounds for quantum tomography with prior information, including explicit bounds for pure states.

Findings

Topological obstructions can double the measurement count needed.

Almost all measurements become informationally complete if outcomes exceed twice the Minkowski dimension.

4d-4 outcomes suffice to identify all pure states in a d-dimensional space.

Abstract

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.