Algebraic and algorithmic frameworks for optimized quantum measurements

TL;DR

This paper introduces a unified algebraic and algorithmic framework for designing optimized adaptive quantum measurements, improving figures of merit like Bell factors and compensating for detector inefficiencies.

Contribution

It develops a novel, unified approach combining matrix diagonalization and cascaded probabilistic networks for optimized quantum measurement design.

Findings

Optimized measurements improve Bell factors.

Circuits can compensate for low detector efficiency.

Framework unifies algebraic and algorithmic methods.

Abstract

Von Neumann projections are the main operations by which information can be extracted from the quantum to the classical realm. They are however static processes that do not adapt to the states they measure. Advances in the field of adaptive measurement have shown that this limitation can be overcome by "wrapping" the von Neumann projectors in a higher-dimensional circuit which exploits the interplay between measurement outcomes and measurement settings. Unfortunately, the design of adaptive measurement has often been ad hoc and setup-specific. We shall here develop a unified framework for designing optimized measurements. Our approach is two-fold: The first is algebraic and formulates the problem of measurement as a simple matrix diagonalization problem. The second is algorithmic and models the optimal interaction between measurement outcomes and measurement settings as a cascaded network of conditional probabilities. Finally, we demonstrate that several figures of merit, such as Bell factors, can be improved by optimized measurements. This leads us to the promising observation that measurement detectors which---taken individually---have a low quantum efficiency can be be arranged into circuits where, collectively, the limitations of inefficiency are compensated for.