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arXiv:quant-ph/0006087 · Physical Review A

Conceptual inadequacy of the Shannon information in quantum measurements

Časlav Brukner, Anton Zeilinger

Published Jan 18, 2001• 172 citations
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Abstract

In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement, conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurements, quantum measurements, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed.

DOI
10.1103/physreva.63.022113

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In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement, conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurements, quantum measurements, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed.
In Phys. Rev. A 63 022113 (2001), Brukner and Zeilinger claim that the Shannon information is not well defined as a measure of information in quantum mechanics, adducing arguments that seek to show that it is inextricably tied to classical notions of measurement. It is shown here that these arguments do not succeed: the Shannon information does not have problematic ties to classical concepts. In a further argument, Brukner and Zeilinger compare the Shannon information unfavourably to their preferred information measure, I(p), with regard to the definition of a notion of `total information content'. This argument is found unconvincing and the relationship betwen measures of information and notions of `total information content' investigated.
Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.
This paper introduces a short survey on recent information theories and reviews some critical notes expressed on quantum information. The severe difficulties emerging from the literature lead us to argue about the way to follow, and in a preliminary stage, we consider how to proceed in order to provide a reasonable contribution to the conceptualization of information in classical and quantum physics. We conclude that we should go toward the essential elements of the system that acquire information and should define the common components of the measurement processes. In this way we should be able to establish fundamental properties and to circumvent tricky difficulties arisen by the concept of the observer and the variety of interferences that disturb the acquisition of information. Universal experience shows how sharpness is the indispensable feature of detected signals and we calculate the discernability of observables using various mathematical formalisms. The present logical frame brings evidence on how information is not an absolute quantity, and we close with a few notes on the information relativism which modern literature tackles from the operational stance and the philosophical stance.
A generalization of Shannon’s amount of information into quantum measurements of continuous observables is introduced. A necessary and sufficient condition for measuring processes to have a non-negative amount of information is obtained. This resolves Groenewold’s conjecture completely including the case of measurements of continuous observables. As an application the approximate position measuring process considered by von Neumann and later by Davies is shown to have a non-negative amount of information.
Since its emergence, quantum mechanics has been a challenge for an understanding of reality which is based on our intuition in a classical world. Nevertheless, it has often been tried to impose this understanding of reality on quantum theory - with limited success. Instead, it might be a better alternative to redefine the meaning of physical reality. This is the objective of the paper. A consideration of the quantum measurement process, conditional probabilities and some well-known typical quantum physical experiments provides the reasoning for the following three hypotheses: (1) Prior to a first measurement, a physical system is not in a quantum state. (2) Physical reality is all that and only that about which (classical) information is available in the universe. (3) Information creation is an independent process and is not covered by the Schroedinger equation. It is the first step of the quantum measurement process and does not have a classical counterpart. The first hypothesis makes sense only if the quantum measurement process can be described without presupposing an initial state for the system under consideration. This becomes possible by the objective conditional probabilities which represent the transition probabilities between the outcomes of successive quantum measurements and have been introduced by the author in some recent papers. The second hypothesis holds as well in the classical case, but a certain incompleteness of reality is typical of quantum mechanics and the origin of many quantum phenomena. Classically, the existence of a complete reality is presumed, and hypothesis 3 has no meaning then.