Information and Measurement

Howard L. Resnikoff

Jincheng Zhang

The notion of information in quantum mechanics has traditionally been viewed through the lens of entropy, mutual information, and measurement back‑action. In this work we introduce a new concept, Quantum Measurement Information Pressure (QMIP), which quantifies the effective mechanical or energetic pressure exerted by the informational change that occurs during a quantum measurement. QMIP is defined as the derivative of the measurement‑induced informational entropy with respect to an effective measurement volume or a parameter that governs the coupling between the system and the measuring apparatus. We present a detailed mathematical formulation of QMIP, discuss its physical interpretation in terms of work extraction and feedback control, and outline potential experimental realizations in quantum feedback systems. By bridging the gap between information theory and physical forces at the quantum scale, QMIP offers a fresh perspective on the thermodynamic cost of quantum measurement and suggests pathways towards high‑impact applications, possibly deserving recognition at the Nobel level.

Stanley Gudder

Hiroaki Terashima

Additive measures for information and disturbance in quantum measurements of a system are defined from well-known multiplicative measures such as estimation and operation fidelities using a logarithm. This is motivated by the fact that information and disturbance are naturally assumed to be additive while performing independent measurements on separable systems. Although the additivity makes no remarkable difference when information and disturbance are separately considered, it can change measurements that only introduce minimal disturbance relative to the amount of information. Such optimal measurements are shown for additive information and disturbance measures with a tradeoff relationship.

Adam V Preston

This thesis will focus on deriving informationally-optimal quantum state tomographic measurements on single-qubit systems, with the definition of informationally optimal to be defined as those measurements which maximize the average information gain. The informationally-optimal measurements that will be covered include projective measurements (formally building off of the work of [1] and putting it on firm, information-theoretic foundations), and more general types of quantum measurements called directional (also known as rank-one) positive operator-valued measure (POVM) measurements, and, finally, adaptive directional POVM measurements. For projective measurements, we build on the work of Wootters and Fields ([1]) and show, via analytical methods and maximizing the average information gained from the measurements, that orthogonal projective measurements are optimal. We then go on to define what we call directional POVMs, which are POVMs whose elements are given by weighted projection operators. We consider these measurements in the context of static quantum state tomography (QST), wherein we use the same choice of measurement throughout our analysis/experiment. By examining these directional POVMs acting on single qubit systems, we show that the most informationally optimal of these types of measurements are informationally complete directional POVMs that are both minimal, informationally complete (MIC) and symmetric, informationally complete (SIC) POVMs, simultaneously. Finally, we deal with the case of adaptive directional POVM measurements.

Ruth E. Kastner