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arXiv:quant-ph/0108035 · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Coherent and compatible information: a basis for information analysis of quantum systems

B. A. Grishanin, Victor N. Zadkov

Published Apr 3, 2002• 2 citations
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Abstract

Relevance of key quantum information measures for analysis of quantum systems is discussed. It is argued that possible ways of measuring quantum information are based on compatibility/incompatibility of the quantum states of a quantum system, resulting in the coherent information and introduced her the compatible information measures, respectively. A sketch of an information optimization of a quantum experimental setup is proposed.

DOI
10.1117/12.464500

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Relevance of key quantum information measures for analysis of quantum systems is discussed. It is argued that possible ways of measuring quantum information are based on compatibility/incompatibility of the quantum states of a quantum system, resulting in the coherent information and introduced her the compatible information measures, respectively. A sketch of an information optimization of a quantum experimental setup is proposed.
Basic quantum information measures involved in the information analysis of quantum systems are considered. It is shown that the main quantum information measurement methods depend on whether the corresponding quantum events are compatible or incompatible. For purely quantum channels, the coherent and compatible information measures, which are qualitatively different, can be distinguished. A general information scheme is proposed for a quantum-physical experiment. In this scheme, informational optimization of an experimental setup is formulated as a mathematical problem.
Optimization of quantum measurement processes has a pivotal role in carrying out better, more accurate or less disrupting, measurements and experiments on a quantum system. Especially, convex optimization, i.e., identifying the extreme points of the convex sets and subsets of quantum measuring devices plays an important part in quantum optimization since the typical figures of merit for measuring processes are affine functionals. In this thesis, we discuss results determining the extreme quantum devices and their relevance, e.g., in quantum-compatibility-related questions. Especially, we see that a compatible device pair where one device is extreme can be joined into a single apparatus essentially in a unique way. Moreover, we show that the question whether a pair of quantum observables can be measured jointly can often be formulated in a weaker form when some of the observables involved are extreme. Another major line of research treated in this thesis deals with convex analysis of special restricted quantum device sets, covariance structures or, in particular, generalized imprimitivity systems. Some results on the structure ofcovariant observables and instruments are listed as well as results identifying the extreme points of covariance structures in quantum theory. As a special case study, not published anywhere before, we study the structure of Euclidean-covariant localization observables for spin-0-particles. We also discuss the general form of Weyl-covariant phase-space instruments. Finally, certain optimality measures originating from convex geometry are introduced for quantum devices, namely, boundariness measuring how ‘close’ to the algebraic boundary of the device set a quantum apparatus is and the robustness of incompatibility quantifying the level of incompatibility for a quantum device pair by measuring the highest amount of noise the pair tolerates without becoming compatible. Boundariness is further associated to minimum-error discrimination of quantum devices, and robustness of incompatibility is shown to behave monotonically under certain compatibility-non-decreasing operations. Moreover, the value of robustness of incompatibility is given for a few special device pairs.
The present investigation is about the quantum mechanics of multislit interference experiments. One of the cornerstones of our understanding of quantum mechanics is provided by the analysis of this type of experiments. Beginning with Bohr and Einstein, the discussion revolved mostly around single-slit diffraction and double-slit interference, probing the complementarity of path determination and the appearance of interference fringes, and contrasting the familiar with the quantum. Here, we are concerned with developing a systematic understanding and description of multislit interference experiments, i.e. setups in which a plurality of slits is illuminated. We provide a characterisation of a number of relevant observables, discussing those that are compatible and may be measured jointly, and also incompatible observables which cannot be measured jointly but instead display quantum uncertainty. \nWe begin with a discussion of a particular modification of the classic double- slit interference experiment which highlights the realisation of specific position and momentum observables which are jointly measurable. Although there are technical results regarding the coexistence of specific position and momentum observables, it may be surprising that ubiquitous experimental setups provide a preparation of such quantum states. We proceed with a discussion of the particular character of the incompatibility of certain measurements by building on an initial heuristic argument provided by Aharonov, Pendleton and Petersen. We prove, extend and discuss a formulation of un- certainty suitable for the context of multislit experiments. We conclude with a comparison of this formulation of uncertainty with an alternative uncertainty formulation developed by Uffink and Hilgevoord. Although these two uncertainty formulations are very different technically, we demonstrate that the same tradeoff is expressed independently.