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Leibniz international proceedings in informatics

Why Classificatory Information of Geographic Regions Is Quantum Information (Vision Paper)

Thomas Bittner

Published Jan 1, 2019• 0 citations
Publisher

Abstract

This paper gives an information - theoretic argument in support of the claim that there is geographic quantum information. Quantum information is information in the sense of Shannon’s information theory, that, in addition, satisfies two characteristic postulates. The paper aims to show that if the density of information (bits per unit of space) that is possible for classificatory geographic qualities is limited, then it follows that the two characteristic postulates of quantum information are satisfied for information about those geographic qualities.

DOI
10.4230/lipics.cosit.2019.16

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This paper gives an information - theoretic argument in support of the claim that there is geographic quantum information. Quantum information is information in the sense of Shannon’s information theory, that, in addition, satisfies two characteristic postulates. The paper aims to show that if the density of information (bits per unit of space) that is possible for classificatory geographic qualities is limited, then it follows that the two characteristic postulates of quantum information are satisfied for information about those geographic qualities.
Quantum information is defined by applying the concepts of ordinary (Shannon) information theory to a quantum sample space consisting of a single framework or consistent family. A classical analogy for a spin-half particle and other arguments show that the infinite amount of information needed to specify a precise vector in its Hilbert space is not a measure of the information carried by a quantum entity with a d-dimensional Hilbert space; the latter is, instead, bounded by log2d{\mathrm{log}}_{2}dlog2​d bits (one bit per qubit). The two bits of information transmitted in dense coding are located not in one but in the correlation between two qubits, consistent with this bound. A quantum channel can be thought of as a structure or collection of frameworks, and the physical location of the information in the individual frameworks can be used to identify the location of the channel. Analysis of a quantum circuit used as a model of teleportation shows that the location of the channel depends upon which structure is employed; for ordinary teleportation it is not (contrary to Deutsch and Hayden) present in the two bits resulting from the Bell-basis measurement, but in correlations of these with a distant qubit. In neither teleportation nor dense coding does information travel backwards in time, nor is it transmitted by nonlocal (superluminal) influences. It is (tentatively) proposed that all aspects of quantum information can in principle be understood in terms of the (basically classical) behavior of information in a particular framework, along with the framework dependence of this information.
Does information play a significant role in the foundations of physics? Information is the abstraction that allows us to refer to the states of systems when we choose to ignore the systems themselves. This is only possible in very particular frameworks, like in classical or quantum theory, or more generally, whenever there exists an information unit such that the state of any system can be reversibly encoded in a sufficient number of such units. In this work, we show how the abstract formalism of quantum theory can be deduced solely from the existence of an information unit with suitable properties, together with two further natural assumptions: the continuity and reversibility of dynamics, and the possibility of characterizing the state of a composite system by local measurements. This constitutes a set of postulates for quantum theory with a simple and direct physical meaning, like the ones of special relativity or thermodynamics, and it articulates a strong connection between physics and information.
Masanes, Galley and Müller \cite{masanes2019measurement} argue that the measurement postulates of non-relativistic quantum mechanics follow from the structural postulates together with an assumption they call the "possibility of state estimation". Their argument also relies on what they term a "theory-independent characterization of measurements for single and multipartite systems". We refute their conclusion, giving explicit examples of non-quantum measurement and state update rules that satisfy all their assumptions. We also show that their "possibility of state estimation" assumption is neither necessary nor sufficient to ensure a sensible notion of state estimation within a theory whose states are described by the quantum formalism. We further show their purportedly "theory-independent" characterization assumes several properties of quantum measurements that exclude plausible alternative types of measurement. We illustrate all these points with specific alternative measurement postulates and post-measurement state update rules. We conclude that, contrary to some folklore, quantum mechanics is by no means an island in theory-space. It can consistently be extended by rules for obtaining information about quantum states other than via POVMs. Whether such rules are realised in nature, for example in linking quantum theory and gravity, is an empirical question that cannot be resolved by theoretical analysis alone.
This paper answers Bell’s question: What does quantum information refer to? It is about quantum properties represented by subspaces of the quantum Hilbert space, or their projectors, to which standard (Kolmogorov) probabilities can be assigned by using a projective decomposition of the identity (PDI or framework) as a quantum sample space. The single framework rule of consistent histories prevents paradoxes or contradictions. When only one framework is employed, classical (Shannon) information theory can be imported unchanged into the quantum domain. A particular case is the macroscopic world of classical physics whose quantum description needs only a single quasiclassical framework. Nontrivial issues unique to quantum information, those with no classical analog, arise when aspects of two or more incompatible frameworks are compared.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
Main Claim I argue in this chapter that Einstein and von Neumann meet in algebraic relativistic quantum field theory in the following metaphorical sense: algebraic quantum field theory was created in the late 1950s/early 1960s and was based on the theory of “rings of operators,” which von Neumann established in 1935–1940 (partly in collaboration with J. Murray). In the years 1936–1949, Einstein criticized standard, nonrelativistic quantum mechanics, arguing that it does not satisfy certain criteria that he regarded as necessary for any theory to be compatible with a field theoretical paradigm. I claim that algebraic quantum field theory (AQFT) does satisfy those criteria and hence that AQFT can be viewed as a theory in which the mathematical machinery created by von Neumann made it possible to express in a mathematically explicit manner the physical intuition about field theory formulated by Einstein. The argument in favor of this claim has two components: 1. Historical : An interpretation of Einstein's (semi)formal wordings of his critique of nonrelativistic quantum mechanics. This interpretation results in mathematically explicit operational independence definitions, which, I claim, express independence properties of systems that are localized in causally disjoint spacetime regions. Einstein regarded these as necessary for a theory to comply with field theoretical principles. 2. Systematic : The presentation of several propositions formulated in terms of AQFT that state that the operational independence conditions in question do in fact typically hold in AQFT.