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arXiv:quant-ph/0112068

Range Theorems for Quantum Probability and Entanglement

Itamar Pitowsky

Published Dec 12, 2001• 18 citations
arXivHTMLPDF

Abstract

We consider the set of all matrices of the form $p_{ij}=tr[W(E_{i}\otimes F_{j})]$ where $E_{i}$, $F_{j}$ are projections on a Hilbert space $H$, and $W$ is some state on $H\otimes H$. We derive the basic properties of this set, compare it with the classical range of probability, and note how its properties may be related to geometric measures of entanglement.

DOI
10.48550/arxiv.quant-ph/0112068

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We consider the set of all matrices of the form pij=tr[W(Ei⊗Fj)]p_{ij}=tr[W(E_{i}\otimes F_{j})]pij​=tr[W(Ei​⊗Fj​)] where EiE_{i}Ei​, FjF_{j}Fj​ are projections on a Hilbert space HHH, and WWW is some state on H⊗HH\otimes HH⊗H. We derive the basic properties of this set, compare it with the classical range of probability, and note how its properties may be related to geometric measures of entanglement.
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.
Abstract. We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let M = Mm(C) and N = Mn(C) be full matrix algebras with m ≥ n, fix an arbitrary state ω of N, and let E(ω) be the set of all states of M ⊗N that extend ω. The space E(ω) contains states of rank r for every r = 1, 2,..., m · rank ω, and it has a filtration into compact subspaces E 1 (ω) ⊆ E 2 (ω) ⊆ · · · ⊆ E m·rank ω = E(ω), where E r (ω) is the set of all states of E(ω) having rank ≤ r. We show first that for every r, there is a real-analytic manifold V r, homogeneous under a transitive action of a compact group G r, which parameterizes E r (ω). The unique G r-invariant probability measure on V r promotes to a probability measure P r,ω on E r (ω), and P r,ω assigns probability 1 to states of rank r. The resulting probability space (E r (ω),P r,ω) represents “choosing a rank r extension of ω at random”. Main result: For every r = 1, 2,..., [rank ω/2], states of (E r (ω), P r,ω) are almost surely entangled. 1.
The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born’s rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: “first context and then probability.” We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference. A projection of the classical contextual probabilistic model to the hyperbolic Hilbert space (a module over the commutative two dimensional Clifford algebra) has some similarities with the projection to the complex Hilbert space. However, in the hyperbolic quantum mechanics the principle of superposition is violated. Our realistic (but contextual!) approach to quantum mechanics does not contradict to various “no‐go theorems”, e.g., von Neumann, Bell, Kochen‐Specker. We emphasize that our projection of the classical probabilistic model to the complex Hilbert space is based on two incompatible observables (“reference observables”), e.g., the position and the momentum, or the position and the energy. Only these two observables can be considered as objective properties of quantum systems.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.