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Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements

Carlton M. Caves, Christopher A. Fuchs, Manne, K, Joseph M. Renes

Published Jun 26, 2003• 77 citations
OpenAlex

Abstract

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.

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We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.
We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for d=2d=2d=2 from a simple requirement: the probabilities assigned to measurement outcomes must not depend on whether a system is considered on its own or as a subsystem of a larger one. In line with Gleason's original theorem, our approach assigns probabilities only to projection-valued measures, while other known extensions rely on considering more general classes of measurements. This extension of Gleason's theorem to two-dimensional systems is shown to remain valid for some foil theories of quantum theory.
The postulates of quantum theory are rather abstract in comparison with those of other physical theories such as special relativity. This thesis considers two tools for investigating this discrepancy and makes a connection between them. The first of these tools, Gleason-type theorems, illustrates the interplay between postulates concerning observables, states and probabilities of measurement outcomes, demonstrating that they need not be entirely independent. Gleason’s original and remarkable result applied to observables described by projection-valued measures; however, the theorem does not hold in dimension two. Busch generalised the idea to observables described by positive operator measures, proving a result which holds for all separable Hilbert spaces. We show that Busch’s assumptions may be weakened without affecting the result. The manner in which we weaken the assumptions brings them closer to Gleason’s original treatment of projection-valued measures. We will then demonstrate the connection between Gleason-type theorems and Cauchy’s functional equation, a connection which yields an alternative proof of Busch’s result. The second tool we consider is the family of general probabilistic theories which offers a means of comparing quantum theory with reasonable alternatives. We identify a general probabilistic theory which reproduces the set of non-local correlations achievable in quantum theory, a property often thought to be particular to quantum theory. Finally, we connect these two tools by determining the class of general probabilistic theories which admit a Gleason-type theorem.
Gleason's theorem is often cited as establishing the Born rule from the structure of Hilbert space, yet its original proof is mathematically sophisticated and rarely accessible to physicists. In this article we present a simple route to the dimensional distinction at the heart of Gleason's result, using the generalized Bloch representation of quantum states. More precisely, we consider a restricted but instructive class of probability assignments obtained by replacing the linear dependence of the Born expression on a Bloch-space scalar product with a real function f. For qubits, infinitely many continuous non-Born assignments of this form satisfy normalization for every projective measurement. In dimension greater than 3, by contrast, the geometry of the measurement simplex imposes additional relations among the relevant scalar products. Requiring normalization for arbitrary states then leads to a Cauchy-type functional equation, which, together with boundedness, forces f to be linear. Our argument provides a complementary geometric illustration of why the freedom present in dimension two disappears in higher dimensions, reaching this conclusion directly from the geometry of the measurement simplices and normalization, without explicitly invoking the intertwining of projectors across different orthogonal resolutions of the identity.
This paper reports three almost trivial theorems that nevertheless appear to have significant import for quantum foundations studies. 1) A Gleason-like derivation of the quantum probability law, but based on the positive operator-valued measures as the basic notion of measurement (see also Busch, quant-ph/9909073). Of note, this theorem also works for 2-dimensional vector spaces and for vector spaces over the rational numbers, where the standard Gleason theorem fails. 2) A way of rewriting the quantum collapse rule so that it looks almost precisely identical to Bayes rule for updating probabilities in classical probability theory. And 3) a derivation of the tensor-product rule for combining quantum systems (and with it the very notion of quantum entanglement) from Gleason-like considerations for local measurements on bipartite systems along with classical communication.