Remarks on free mutual information and orbital free entropy

Izumi, Masaki; Ueda, Yoshimichi

doi:10.1017/s0027763000027173

Cited by 10 publications

(33 citation statements)

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“…In the recent preprint [37], Zhong has studied the multiplicative version of Biane's free heat flow, using related subordination technology to prove similar smoothness properties of the law of the free unitary Brownian motion multiplied by a free unitary. These results bear on the very recent related work of Izumi and Ueda [20], as described below in Remark 1.12 below.…”

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confidence: 79%

“…Remark 1.12. Theorem 1.11 was also recently proved in the preprint [20], which was posted to the arXiv seven months after the first public version of the present manuscript. Indeed, using the Jakowski transform as in [9], Izumi and Ueda show that µ t can be identified with the spectral measure of the product of a free unitary Brownian motion with a free unitary operator whose distribution is determined by the initial projections (p, q).…”

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confidence: 54%

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Liberation of projections

Collins

Kemp

2014

Journal of Functional Analysis

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We study the liberation process for projections: (p, q) → (p t , q) = (u t pu * t , q) where u t is a free unitary Brownian motion freely independent from {p, q}. Its action on the operator-valued angle qp t q between the projections induces a flow on the corresponding spectral measures µ t ; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure µ t possesses a piecewise analytic density for any t > 0 and any initial projections of trace 1 2 . We us this to prove the Unification Conjecture for free entropy and information in this trace 1 2 setting.Let V and W be subspaces of a the finite-finite dimensional complex space C d . From elementary linear algebra, it follows that dim V ∩ W ≥ max{dim V + dim W − d, 0}. In fact, this inequality is almost surely equality:The almost surely can be interpreted in a number of ways: for example, it can be taken with respect to any reasonable probability measure on the (product) Grassmannian manifold of subspaces. We will discuss a more probabilistic interpretation shortly. When two subspaces satisfy the equality of Equation 1.1, they are said to be in general position. For convenience, we may rewrite this relation as follows. Let P, Q be the orthogonal projections onto V and W respectively, and let P ∧Q denote the orthogonal projection onto V ∩W . The dimension of a subspace is the trace of the projection onto it, so the subspaces are in general position if and only if Tr (P ∧ Q) = max{Tr P + Tr Q − d, 0}. Normalizing, letting tr = 1 d Tr , two projections (i.e. subspaces) are in general position if and only ifIn this language, a good way to express the genericity of general position is as follows: let P, Q be as above, and let U be a random unitary matrix. IfP = U P U * (i.e. the projection onto the rotation of the image of P by U ), thenP and Q are in general position almost surely. This statement is valid for all reasonable notions of random unitary matrix; for example, it holds true if U is sampled from some measure that has a continuous, strictly positive density with respect to the Haar measure. (Indeed, the subset of the Grassmannian product where general position fails to hold is a subvariety of lower dimension, and so it is easy to see that any reasonable measure will assign it probability 0.) Since any neighborhood of the identity has positive measure, it follows that rotations arbitrarily close to the identity produce projections in general position, agreeing with our intuition. Note: the same result applies as well even if P, Q are random projections, provided that the random unitary U is independent from {P, Q}.An important flow of random unitaries is given by the unitary Brownian motion. The group U d of unitary d×d matrices is a compact Lie group, and so its left-invariant Riemannian metric (given by the Hilbert-Schmidt norm on the Lie algebra u d ) gives rise to a heat kernel, which generates a Markov process: Brownian m...

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confidence: 54%

Liberation of projections

Collins

Kemp

2014

Journal of Functional Analysis

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“…This quantity χ orb has many properties in common with i * , but there is no general relationship between them so far. A few years ago, Collins and Kemp [3] gave a proof of the identity i * (CP + C(I − P ); CQ + C(I − Q)) = χ orb (P ) + χ orb (Q) − χ orb (P, Q) = −χ orb (P, Q) for two projections of traces 1 2 and the same result was subsequently proved by Izumi and Ueda in [8] with a completely independent proof. Motivated both by the ideas in [8] and the heuristic argument in [7,Section 3.2], we developed in [5] a theory of subordination for the liberation process of symmetries which allowed us to establish some partial results generalizing the equality i * = −χ orb for two projections.…”

Section: Introductionmentioning

confidence: 69%

“…This subordination relation is used in [8,Lemma 4.4] to prove that the equality i * = −χ orb holds, for any two projections, under the assumption that K(t, .) define a function of Hardy class H 3 (D) for any t > 0.…”

Section: Introductionmentioning

confidence: 99%

Free Mutual Information for Two Projections

Hamdi

2018

Complex Anal. Oper. Theory

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“…Consequently, the rank of this symmetry vanishes when the rank of the projection equals 1/2 and in this case, U t and Y 2t shares the same spectrum ( [11]). Note that this last result was subsequently generalized in [14] to the case of two projections with equal rank 1/2 and to arbitrary distributions of U 0 . Note also that at the analytic level, the moment generating functions of U t and of J t (the latter is viewed as an element of the compressed algebra) are related to each other by the Szegö map.…”

Section: Introductionmentioning

confidence: 88%

Inverse of the flow and moments of the free Jacobi process associated with one projection

Демни

Hamdi

2018

Random Matrices: Theory Appl.

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This paper is a companion to a series of papers devoted to the study of the spectral distribution of the free Jacobi process associated with a single projection. Actually, we notice that the flow derived in [12] solves a radial Löwner equation and as such, the general theory of Löwner equations implies that it is univalent in some connected region in the open unit disc. We also prove that its inverse defines the Aleksandrov-Clark measure at z = 1 of some Herglotz function which is absolutely-continuous with an essentially bounded density. As a by-product, we deduce that z = 1 belongs only to the discrete spectrum of the unitary operator whose spectral dynamics are governed by the flow. Moreover, we use a previous result due to the first author in order to derive an explicit, yet complicated, expression of the moments of both the unitary and the free Jacobi processes. The paper is closed with some remarks on the boundary behavior of the flow's inverse.

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Remarks on free mutual information and orbital free entropy

Cited by 10 publications

References 22 publications

Liberation of projections

Liberation of projections

Free Mutual Information for Two Projections

Inverse of the flow and moments of the free Jacobi process associated with one projection

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