Probing quantum state space: does one have to learn everything to learn something?

TL;DR

This paper explores whether partial measurements can suffice for specific quantum state tasks, analyzing the geometry of state space and establishing criteria for when full state determination is necessary.

Contribution

It introduces a framework linking quantum state space geometry to measurement sufficiency, providing criteria to identify when informationally complete measurements are required.

Findings

Criteria for when informational completeness is necessary

Bounds on measurement outcomes for specific tasks

Analysis of physically relevant examples

Abstract

Determining the state of a quantum system is a consuming procedure. For this reason, whenever one is interested only in some particular property of a state, it would be desirable to design a measurement setup that reveals this property with as little effort as possible. Here we investigate whether, in order to successfully complete a given task of this kind, one needs an informationally complete measurement, or if something less demanding would suffice. The first alternative means that in order to complete the task, one needs a measurement which fully determines the state. We formulate the task as a membership problem related to a partitioning of the quantum state space and, in doing so, connect it to the geometry of the state space. For a general membership problem we prove various sufficient criteria that force informational completeness, and we explicitly treat several physically relevant examples. For the specific cases that do not require informational completeness, we also determine bounds on the minimal number of measurement outcomes needed to ensure success in the task.