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arXiv:1701.06069 · Journal of Mathematical Physics

Unentangled measurements and frame functions

Jiř́í Lebl, Asif Shakeel, Nolan R. Wallach

Published Jun 1, 2018• 0 citations
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Abstract

Gleason’s theorem asserts the equivalence of von Neumann’s density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author’s unentangled Gleason’s theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.

DOI
10.1063/1.5042336

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Gleason’s theorem asserts the equivalence of von Neumann’s density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author’s unentangled Gleason’s theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.
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.
Abstract Gleason’s theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I 2 factors). Here, we prove a generalisation of Gleason’s theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.
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.
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.
Abstract Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We will call this principle, wherein the deterministic elements of a theory completely characterise the probabilistic elements, intermediate determinism. In dimensions of at least three, intermediate determinism in quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason’s theorem, which is why it fails to hold in dimension two. In this work we extend the idea of intermediate determinism from properties to measurements. Under this extension intermediate determinism follows from the structure of quantum effects for separable Hilbert spaces of any dimension, including dimension two. Then, we find necessary and sufficient conditions for a general probabilistic theory to obey intermediate determinism. We show that, although related, both the no-restriction hypothesis and a Gleason-type theorem are neither necessary nor sufficient for intermediate determinism.