Free Fisher Information Research Articles

Bi-free entropy with respect to completely positive maps

AbstractIn this paper, a notion of non-microstate bi-free entropy with respect to completely positive maps is constructed thereby extending the notions of non-microstate bi-free entropy and free entropy with respect to a completely positive map. By extending the operator-valued bi-free structures to allow for more analytical arguments, a notion of conjugate variables is constructed using both moment and cumulant expressions. The notions of free Fisher information and entropy are then extended to this setting and used to show minima of the Fisher information and maxima of the non-microstate bi-free entropy at bi-R-diagonal elements.

A dual and conjugate system for q-Gaussians for all q

We provide a concrete formula for a dual system as well as for a conjugate system of q-Gaussians represented on the q-deformed Fock space. Moreover, using this formula, we prove the existence of a free Gibbs potential and that the non-microstates free Fisher information is finite for any q with −1<q<1, which is an improvement on a previous result of Y. Dabrowski. We also indicate how our results can be extended to the more general setting of mixed qij-relations.

Fractional free convolution powers

The extension $k \mapsto \mu^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory. In this paper we give two proofs of the monotonicity of the free entropy and free Fisher information of the (normalized) free convolution power in this continuous setting, and also establish an intriguing variational description of this process.

Maximal correlation and monotonicity of free entropy and of Stein discrepancy

We introduce the maximal correlation coefficient R(M1,M2) between two noncommutative probability subspaces M1and M2 and show that the maximal correlation coefficient between the sub-algebras generated by sn:=x1+…+xnand sm:=x1+…+xm equals m∕n for m≤n, where (xi)i∈N is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo–Kagan–Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.

Free Stein irregularity and dimension

We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray-von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a ∗-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.

Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent 23, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [8].We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

On finite free Fisher information for eigenvectors of a modular operator

Suppose M is a von Neumann algebra equipped with a faithful normal state φ and generated by a finite set G=G⁎, |G|≥2. We show that if G consists of eigenvectors of the modular operator Δφ with finite free Fisher information, then the centralizer Mφ is a II1 factor and M is either a type II1 factor or a type IIIλ factor, 0<λ≤1, depending on the eigenvalues of G. Furthermore, (Mφ)′∩M=C, Mφ does not have property Γ, and M is full provided it is type IIIλ, 0<λ<1.

Operator-valued free Fisher information of random matrices

The free Fisher information of an operator random matrix is studied. When the covariance of a random matrix is a conditional expectation, the free Fisher information of such a matrix is the double of this conditional expectation's Watatani index.

REMARKS ON THE FREE RELATIVE ENTROPY AND THE FREE FISHER INFORMATION DISTANCE

In this paper, we shall introduce the free Fisher information distance which is inspired by the estimation-theoretic representation of the free relative entropy investigated by Verdú. We shall see the free analogue of the logarithmic Sobolev inequality with respect to a centered semicircle law and also the semicircular approximation of the free Poisson law.

A note about proving non- Γ under a finite non-microstates free Fisher information assumption

We prove that if X 1 , … , X n ( n > 1 ) are self-adjoints in a W ∗ -probability space with finite non-microstates free Fisher information, then the von Neumann algebra W ∗ ( X 1 , … , X n ) they generate doesn't have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.

A log-Sobolev type inequality for free entropy of two projections

We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections.

REMARKS ON A SEMICIRCULAR PERTURBATION OF THE FREE FISHER INFORMATION

In this paper, we shall give the representation of a semicircular perturbation of the free Fisher information [Formula: see text] by the mean of the conditional variance [Formula: see text], where S is a standard semicircular element freely independent of X, and ε &gt; 0. Using this representation, we will give alternative proofs of the free Fisher information inequality, in which the free analogue of Stam's inequality can be obtained as a special case, and of the free entropy power inequality in an infinitesimal approach to the free entropy.

A Free Logarithmic Sobolev Inequality on the Circle

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Operator-valued free Fisher information and modular frames

We introduce the operator-valued free Fisher information for a random variable in an operator-valued noncommutative probability space and point out its relations to the amalgamated freeness. Using M. Frank and D. Larson’s modular frame notion we can construct the conjugate variable for an operator-valued semicircle variable with conditional expectation covariance. Then we obtain its free Fisher information and show it is equal to the index of the conditional expectation. At last the conjugate variable with respect to a modular frame operator for a semicircle variable is also constructed.

Some applications of free Fisher information on frame theory

We introduce the notion of operator-valued free Fisher information with respect to a positive map of a random variable in an operator-valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations. Then we can apply this notion on the study of frame theory, especially on the disjointness problem of modular frames arising from conditional expectations.

Free fisher information and amalgamated freeness

The notion of operator-valued free Fisher information was introduced. It is a generalization of free Fisher information which was defined by D. Voiculescu on tracial von Neumann algebras. It is proved that the operator-valued free Fisher information is closely related to amalgamated freeness, i. e., the operator-valued free Fisher information of some random variables is additive if and only if these random variables are a free family with amalgamation over a subalgebra. Cramer-Rao inequality in operator-valued settings is also obtained.

Free Fisher information for non-tracial states

We extend Voiculescu's microstates-free definitions of free Fisher information and free entropy to the non-tracial framework. We explain the connection between these quantities and free entropy with respect to certain completely positive maps acting on the core of the non-tracial non-commutative probability space. We give a condition on free Fisher information of an infinite family of variables, which guarantees factoriality of the von Neumann algebra they generate.

COLLECTIVE POTENTIAL FOR LARGE-N HAMILTONIAN MATRIX MODELS AND FREE FISHER INFORMATION

We formulate the planar "large N limit" of matrix models with a continuously infinite number of matrices directly in terms of U(N) invariant variables. Noncommutative probability theory is found to be a good language to describe this formulation. The change of variables from matrix elements to invariants induces an extra term in the Hamiltonian, which is crucial in determining the ground state. We find that this collective potential has a natural meaning in terms of noncommutative probability theory: it is the "free Fisher information" discovered by Voiculescu. This formulation allows us to find a variational principle for the classical theory described by such large N limits. We then use the variational principle to study models more complex than the one describing the quantum mechanics of a single Hermitian matrix (i.e. go beyond the so-called D = 1 barrier). We carry out approximate variational calculations for a few models and find excellent agreement with known results where such comparisons are possible. We also discover a lower bound for the ground state by using the noncommutative analog of the Cramer–Rao inequality.

Free diffusions, free entropy and free Fisher information

Motivated by the stochastic quantization approach to large N matrix models, we study solutions to free stochastic differential equations dX t=dS t− 1 2 f(X t) dt where S t is a free brownian motion. We show existence, uniqueness and Markov property of solutions. We define a relative free entropy as well as a relative free Fisher information, and show that these quantities behave as in the classical case. Finally we show that, in contrast with classical diffusions, in general the asymptotic distribution of the free diffusion does not converge, as t→∞, towards the master field (i.e., the Gibbs state).

Free Fisher Information�with Respect to a Completely Positive Map�and Cost of Equivalence Relations

Given a family of isometries \(\) in a tracial von Neumann algebra M, a unital subalgebra B⊂M and a completely-positive map \(\) we define the free Fisher information \(\) of \(\) relative to B and η. Using this notion, we define the free dimension \(\) of \(\) relative to B, id.