Data-driven inference and observational completeness of quantum devices

TL;DR

This paper develops a formalism for data-driven inference of quantum states and measurements, providing algorithms and characterizations that connect observational completeness with well-known quantum measurement sets.

Contribution

It introduces a unified formalism for inferring quantum states and measurements, with explicit algorithms for qubits and a characterization of observational completeness.

Findings

Explicit convex programming algorithm for qubit inference

Complete characterization of observational completeness for systems

Spherical 2-designs are the only observationally complete sets for qubits

Abstract

Data-driven inference was recently introduced as a protocol that, upon the input of a set of data, outputs a mathematical description for a physical device able to explain the data. The device so inferred is automatically self-consistent, that is, capable of generating all given data, and least committal, that is, consistent with a minimal superset of the given dataset. When applied to the inference of an unknown device, data-driven inference has been shown to output always the "true" device whenever the dataset has been produced by means of an observationally complete setup, which plays here the same role played by informationally complete setups in conventional quantum tomography. In this paper we develop a unified formalism for the data-driven inference of states and measurements. In the case of qubits, in particular, we provide an explicit implementation of the inference protocol as a convex programming algorithm for the machine learning of states and measurements. We also derive a complete characterization of observational completeness for general systems, from which it follows that only spherical 2-designs achieve observational completeness for qubit systems. This result provides symmetric informationally complete sets and mutually unbiased bases with a new theoretical and operational justification.