The Role of Topology in Quantum Tomography

TL;DR

This paper explores how the topology of state spaces in quantum tomography influences the minimal measurement settings needed for state discrimination, providing bounds for various constrained state subsets.

Contribution

It introduces a general framework linking topology to measurement complexity and applies it to specific quantum state subsets, deriving bounds on measurement requirements.

Findings

Topology constrains measurement settings in quantum tomography.

Derived bounds for states with bounded rank, fixed spectrum, or symmetry.

Applicable to various quantum state discrimination scenarios.

Abstract

We investigate quantum tomography in scenarios where prior information restricts the state space to a smooth manifold of lower dimensionality. By considering stability we provide a general framework that relates the topology of the manifold to the minimal number of binary measurement settings that is necessary to discriminate any two states on the manifold. We apply these findings to cases where the subset of states under consideration is given by states with bounded rank, fixed spectrum, given unitary symmetry or taken from a unitary orbit. For all these cases we provide both upper and lower bounds on the minimal number of binary measurement settings necessary to discriminate any two states of these subsets.