Suppose $N \subset M$ is an inclusion of $II_1$-factors of finite index. If $N$ can be generated by a finite set of elements, then there exist finite generating sets $X$ for $N$ and $Y$ for $M$ such that $δ_0(X) \geq δ_0(Y)$, where $δ_0$ denotes Voiculescu's microstates (modified) free entropy dimension. Moreover given $ε>0$ one has $δ_0(F) \geq δ_0(G) \geq ([M:N]^{-2} -ε) \cdot (δ_0(F) -1) + 1 - ε$ for certain generating sets $F$ for $N$ and $G$ for $M$.
Based on the fact that, for a subfactor $N$ of a II$_1$ factor $M,$ the first non-trivial Jones index is 2 and then $M$ is decomposed as the crossed product of $N$ by an outer action of ${\mathbb{Z}}_2,$ we study pairs $ \{N, uNu^* \}$ from a view point of entropy for two subalgebras of $M$ with a connection to the entropy for automorphisms, where the inclusion of II$_1$ factors $ N \subset M$ is given as $M$ is the crossed product of $N$ by a finite group of outer automorphisms and $u$ is a unitary in $M.$
Given a quadruple of finite index subfactors we explicitly compute the Pimsner-Popa probabilistic constant for the pair of intermediate subfactors and relate it with the corresponding Connes-Størmer relative entropy between them. This generalizes an old result of Pimsner and Popa.
In this paper, we introduce Frobenius von Neumann algebras and study quantum convolution inequalities. In this framework, we unify quantum Young's inequality on quantum symmetries such as subfactors, and fusion bi-algebras studied in quantum Fourier analysis. Moreover, we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case. We also prove quantum smooth entropic convolution inequalities. We obtain the positivity of comultiplications of subfactor planar algebras, which is stronger than the quantum Schur product theorem. All these inequalities pr…
We obtain an estimate of Voiculescu's (modified) free entropy dimension for generators of a ${II}_1$-factor $\mc{M}$ with a subfactor $\mc{N}$ containing an abelian subalgebra $\mc{A}$ of finite multiplicity. It implies in particular that the interpolated free group subfactors of finite Jones index do not have abelian subalgebras of finite multiplicity or Cartan subalgebras.
We obtain an estimate of free entropy of generators in a type ${II}_1$-factor $\mc{M}$ which has a subfactor $\mc{N}$ of finite index with a subalgebra $\mc{P}=\mc{P}_1\vee\mc{P}_2\subset\mc{N}$ where $\mc{P}_1=\mc{R}_1'\cap\mc{P}$, $\mc{P}_2=\mc{R}_2'\cap\mc{P}$ are diffuse, $\mc{R}_1,\mc{R}_2\subset\mc{P}$ are mutually commuting hyperfinite subfactors, and an abelian subalgebra $\mc{A}\subset\mc{N}$ such that the correspondence $_\mc{P}L^2(\mc{N},τ)_\mc{A}$ is $\mc{M}$-weakly contained in a subcorrespondence $_\mc{P}H_\mc{A}$ of $_\mc{P}L^2(\mc{M},τ)_\mc{A}$, generated by $v$ vectors. The (m…
We use the free entropy defined by D. Voiculescu to prove that the free group factors can not be decomposed as closed linear spans of noncommutative monomials in elements of nonprime subfactors or abelian $*$-subalgebras, if the degrees of monomials have an upper bound depending on the number of generators. The resulting estimates for the hyperfinite and abelian dimensions of free group factors settle in the affirmative a conjecture of L. Ge and S. Popa (for infinitely many generators).
For certain generating sets of the subfactor pair $M\subset M\rtimes G$ where $G$ is a finite abelian group we prove an approximate inequality between their non-microstates free entropy dimension, resembling the Shreier formula for ranks of finite index subgroups of free groups. As an application, we give bounds on free entropy dimension of generating sets of crossed products of the form $M\rtimes(\mathbb{Z}/2\mathbb{Z})^{\oplus\infty}$ for a large class of algebras $M$.
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 Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. 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 study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $σ$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $σ$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $σ$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in an…
We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities; this entropy decay turns out to be implied by the usual log-Sobolev inequality adding some regularity co…
Talagrand's inequalities make a link between two fundamentals concepts of probability: transport of measures and entropy. The study of the counterpart of these inequalities in the context of free probability has been initiated by Biane and Voiculescu and later extended by Hiai, Petz and Ueda for convex potentials. In this work, we prove a free analogue of a result of Bobkov and Götze in the classical setting, thus providing free transport-entropy inequalities for a very natural class of measures appearing in random matrix theory. These inequalities are weaker than the ones of Hiai, Petz and Ue…
By using Araki's relative entropy, Lieb's convexity and the theory of singular integrals, we compute the mutual information associated with free fermions, and we deduce many results about entropies for chiral CFT's which are embedded into free fermions, and their extensions. Such relative entropies in CFT are here computed explicitly for the first time in a mathematical rigorous way. Our results agree with previous computations by physicists based on heuristic arguments; in addition we uncover a surprising connection with the theory of subfactors, in particular by showing that a certain dualit…
We consider three von Neumann entropy inequalities: subadditivity; Pinsker's inequality for relative entropy; and the monotonicity of relative entropy. For these we state conditions for equality, and we prove some new error bounds away from equality, including an improved version of Pinsker's inequality.
Biane proved the free analog of the logarithmic Sobolev inequality for probability measures on the real line by means of random matrix approximation procedure. We show that the same method can be applied to reprove Biane and Voiculescu's free analog of Talagrand's transportation cost inequality for measures on the real line. Furthermore, we prove the free analogs of the logarithmic Sobolev inequality and the transportation cost inequality for measures on 1-dimensional torus as well by extending the method to special unitary random matrices.
In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\mathbf{A}=(A_1,...,A_n)$ and $\mathbf{B}=(B_1,...,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\mathbf{A}|\mathbf{B}):=\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\,, $$ and then give upper and lower bounds for $S_q^f(\mathbf{A}|\mathbf{B})$ as an extensio…
Trace inequalities are general techniques with many applications in quantum information theory, often replacing classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki-Lieb-Thirring and Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then, using the Haag…
In this paper, we investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on relative operator entropies still hold in these broader settings. In addition, we improve the lower and upper bounds of the relative operator $(α, β)$-entropy established by Nikoufar which refined the bounds for the relative operator entropy obtained by Fujii and Kamei.
