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Applicable Analysis

On the quantum potential

Robert Carroll

Published Oct 6, 2005• 4 citations
Publisher

Abstract

We survey various origins and expressions for the quantum potential, expanding and extending the treatment given in a previous paper [R. Carroll, quant-ph 0401082].
DOI
10.1080/0036810412531282970

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These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns `doing quantum mechanics using only pictures of lines, squares, triangles and diamonds'. This picture calculus can be seen as a very substantial extension of Dirac's notation, and has a purely algebraic counterpart in terms of so-called Strongly Compact Closed Categories (introduced by Abramsky and I in quant-ph/0402130 and [4]) which subsumes my Logic of Entanglement quant-ph/0402014. For a survey on the `what', the `why' and the `hows' I refer to a previous set of lecture notes quant-ph/0506132. In a last section we provide some pointers to the body of technical literature on the subject.
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Abstract This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. After an introduction in section 1, section 2 reviews the basic definitions and observables are considered in section 3. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in section 4. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, Lüders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author’s previous articles Gudder (2020 arXiv:2005.08117 [quant-ph], arXiv:2005.13642 [quant-ph], arXiv:2009.07371 [quant-ph]).