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Entropy for a pair of subalgebras via automorphisms.

Marie Choda

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

This is an expository survey article based on a talk for the Spe- cial Session (Operator Algebra) in AMC2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A|B) of the Connes-Stormer relative entropy H(A|B). As one of the most elementary examples, we pick two maximal abelian *-subalgebras A, B of the n × n com- plex matrices Mn(C), and show relations between h(A|B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.

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This is an expository survey article based on a talk for the Spe- cial Session (Operator Algebra) in AMC2009. For two subalgebras A and B of a finite von Neumann algebra M, we discuss a modified version h(A|B) of the Connes-Stormer relative entropy H(A|B). As one of the most elementary examples, we pick two maximal abelian *-subalgebras A, B of the n × n com- plex matrices Mn(C), and show relations between h(A|B) and the entropy for unistochastic matrices. Also we show related results in the case of a pair of subfactors of a type II1 factor.
Let A and B be two maximal abelian *-subalgebras of the n × n complex matrices Mn(ℂ). To study the movement of the automorphisms of Mn(ℂ), we modify the Connes–Størmer relative entropy H(A|B) and the Connes relative entropy Hϕ(A|B) with respect to a state ϕ, and introduce the two kinds of the constant h(A|B) and hϕ(A|B). For the unistochastic matrix b(u) defined by a unitary u with B = uAu*, we show that h(A|B) is the entropy H(b(u)) of b(u). This is obtained by our computation of hϕ(A|B). The h(A|B) attains to the maximal value log n if and only if the pair {A, B} is orthogonal in the sense of Popa.
We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner–Popa index connects to sandwiched [Formula: see text]-Rényi relative entropy for all [Formula: see text], including Umegaki’s relative entropy at [Formula: see text]. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.
We develop a cohomological framework for entanglement entropy that unifies information theory, operator algebras, and noncommutative geometry. Entropy is a 1-cocycle in information cohomology; we generalize this to the quantum setting via Hochschild and cyclic cohomology. For an inclusion of von Neumann algebras N contained in M with a faithful normal state w admitting a cyclic and separating vector, and a conditional expectation E: M -> N (whose existence is equivalent to modular invariance of N by Takesaki's theorem), we construct the entanglement complex as the image of the transgression map tau_w on the relative cyclic complex, where A is a dense *-subalgebra of M and B = A intersect N. Our main result identifies entanglement cohomology with the kernel of the restriction map HC*(A) -> HC*(B), up to the connecting homomorphism d of the long exact sequence of the pair (A,B); when d = 0, the identification is exact. The relative modular cocycle u_t = (Dw : D(w|_N) o E)_t generates explicit entanglement classes when E is not w-invariant, with the Araki relative entropy as its infinitesimal generator. The framework accommodates Type III von Neumann algebras, where no local density matrix exists, and leads to a holographic proposal: a subspace of H^1_ent is conjecturally dual to linearized modular Hamiltonian perturbations of the bulk entanglement wedge. We verify the construction in finite-dimensional, type II_1, and type III_1 examples.
We establish duality relations for conditional entropies relative to subalgebras of arbitrary von Neumann algebras, including the Umegaki relative entropy as well as the sandwiched- and Petz-Rényi divergences. Even in finite dimensions, these results extend the familiar dualities for conditional entropies from tensor-product subsystems to general subalgebras, while also lifting the framework to the fully infinite-dimensional setting. As an application, we establish an infinite-dimensional extension of quantum Stein's lemma when the alternative hypothesis is the state space of a finite-index subalgebra, which gives an operational interpretation of the relative entropy to a subalgebra as the optimal asymptotic error exponent in composite quantum hypothesis testing-the first beyond finite dimensions.
To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor. I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis for the problems in question is collected in an appendix.
Abstract We prove the existence of a universal recovery channel that approximately recovers states on a von Neumann subalgebra when the change in relative entropy, with respect to a fixed reference state, is small. Our result is a generalization of previous results that applied to type-I von Neumann algebras by Junge at al. [ arXiv:1509.07127 ]. We broadly follow their proof strategy but consider here arbitrary von Neumann algebras, where qualitatively new issues arise. Our results hinge on the construction of certain analytic vectors and computations/estimations of their Araki–Masuda LpL_pLp​ L p norms. We comment on applications to the quantum null energy condition.