arXiv subjects

Hiroyuki Osaka

Publications and source records attributed to Hiroyuki Osaka.

At least 19 recordsLinked to original sources

arXiv2026

Alternative-mean trace divergences: geometry, data processing, and barycenters

Let $f:(0,\infty)\to(0,\infty)$ be a nontrivial normalized operator monotone function and set $s=f'(1)$. We introduce the alternative-mean trace functional $$ \altPhi_f(A,B) :=\Tr(A\nabla_s B) -\Tr\!\left( f(A^{-1}\sharp B)\,A\,f(A^{-1}\sharp B) \right). $$ on the positive definite cone. We prove that $\altPhi_f$ is a quantum divergence in the sense of Bhatia--Gaubert--Jain whose diagonal Hessian induces a positive multiple of the Bures--Wasserstein Riemannian metric. We also establish the sharp comparison $$ s(1-s)d_{\rm BW}(A,B)^2 \le \altPhi_f(A,B) \le (1-s+s^2)d_{\rm BW}(A,B)^2. $$

math-ph↗

arXiv2026

Some applications of Choi polynomials of linear maps

This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in $M_m(\mathbb{C}) \otimes M_n(\mathbb{C})$. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.

quant-ph↗

arXiv2025

Optimizing Entanglement Manipulation via Algebraic-Geometric Decompositions and Semidefinite Programming Hierarchies

In the study of distributed quantum information processing, it is a fundamental problem to optimize local operations in the implementation of non-local quantum operations assisted by limited entanglement. We develop an algebraic-geometric framework that systematically simplifies optimization over separable (SEP) channels -- widely used as approximations of local operations -- and strengthens the Doherty--Parrilo--Spedalieri (DPS) hierarchy for solving such problems. We apply this framework to computing maximum success probability for exactly implementing a broad range of different non-local operations under SEP channels. First, we present a unified generalization of previous analytical results on the entanglement cost. Via the generalization, we resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. Second, we numerically determine the trade-off between the strength of entanglement and the success probability of implementing various operations -- such as entanglement distillation, non-local unitary channels, measurements, and state verification.

quant-ph↗

arXiv2024

Limit of iteration of the induced Aluthge transformations of centered operators

Aluthge transform is a well-known mapping defined on bounded linear operators. Especially, the convergence property of its iteration has been studied by many authors. In this paper, we discuss the problem for the induced Aluthge transforms which is a generalization of the Aluthge transform defined in 2021. We give the polar decomposition of the induced Aluthge transformations of centered operators and show its iteration converges to a normal operator. In particular, if $T$ is an invertible centered matrix, then iteration of any induced Aluthge transformations converges. Using the canonical standard form of matrix algebras we show that the iteration of any induced Aluthge transformations with respect to the weighted arithmetic mean and the power mean converge. Those observation are extended to the $C^*$-algebra of compact operators on an infinite dimensional Hilbert space, and as an application we show the stability of $\mathcal{AN}$ and $\mathcal{AM}$ properties under the iteration of the induced Aluthge transformations. We also provide concrete forms of their limit points for centered matrices and several examples. Moreover, we discuss the limit point of the induced Aluthge transformation with respect to the power mean in the injective $II_1$-factor $\mathcal{M}$ and determine the form of its limit for some centered operators in $\mathcal{M}$.

math.FA↗

arXiv2023

A new estimation of the quantum Chernoff bound

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-St\"ormer type inequalities for perspective functions.

quant-ph↗

arXiv2022

Tracially sequentially-split ${}^$-homomorphisms between $C^$-algebras II

We study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly interested how regularity properties in the Elliott classification program pass from $B$ to $A$. Among them, we show that the strict comparison property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.

math.OA↗

arXiv2021

Stable rank for inclusions of Banach algebras

We give a formula for the stable rank of inclusions of unital Banach algebras in the sense of finite Watatani index. As an application we show that the stable rank of $\ell^1$-algebras of Disk algebras by any action of finite groups is 2.

math.FA↗

arXiv2021

Characterization of $k$-positive maps

We present a general characterization of k-positivity for a positive map in terms of the estimation of the Ky Fan norm of the matrix constructed from the Kraus operators of the associated completely positive map. Combining this with the result given by Takasaki and Tomiyama we construct a family of positive maps between matrix algebras of different dimensions depending on a parameter. The estimate bounds on the parameter to obtain the $k$-positivity are better than those derived from the spectral conditions considered by Chru\'sci\'nski and Kossakowski. We further look with special attention at the case where we give the precise bound for the regions of decomposability.

quant-ph↗

arXiv2020

The order-$n$ minors of certain $(n+k) \times n$ matrices

We determine sufficient conditions for certain classes of $(n+k) \times n$ matrices $E$ to have all order-$n$ minors to be nonzero. For a special class of $(n+1) \times n$ matrices $E,$ we give the formula for the order-$n$ minors. As an application we construct subspaces of $\C^m \otimes \C^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $k$ and which has a basis of Schmidt rank $k$ for $k=2,3,4$.

math.FA↗

arXiv2020

Tracially sequentially-split ${}^$-homomorphisms between $C^$-algebras

We define a tracial analogue of the sequentially split $*$-homomorphism between $C^*$-algebras of Barlak and Szabó and show that several important approximation properties related to the classification theory of $C^*$-algebras pass from the target algebra to the domain algebra. Then we show that the tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$ gives rise to a tracial version of sequentially split $*$-homomorphism from $A\rtimes_αG$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \to P$ of a finite Watatani index generates a tracial version of sequentially split map. By doing so, we provide a unified approach to permanence properties related to tracial Rokhlin property of operator algebras.

math.OA↗

arXiv2019

Functions preserving operator means

Let $σ$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $OM_+^1$ the set of normalized positive operator monotone functions on $(0, \infty)$. In this paper, we study class of $σ$-subpreserving functions $f\in OM_+^1$ satisfying $$f(AσB) \le f(A)σf(B)$$ for all positive operators $A$ and $B$. We provide some criteria for $f$ to be trivial, i.e., $f(t)=1$ or $f(t)=t$. We also establish characterizations of $σ$-preserving functions $f$ satisfying $$f(AσB) = f(A)σf(B)$$ for all positive operators $A$ and $B$. In particular, when $\lim_{t\rightarrow 0} (1σt) =0$, the function $f$ preserves $σ$ if and only if $f$ and $1σt$ are representing functions for weighted harmonic means.

math.FA↗

arXiv2019

A factorization property of positive maps on $C^*$-algebras

The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $ α_1 \otimes α_2 \geq β$, factorizes as $β= γ\otimes α_2$ for some positive map $γ$. In particular we show that when $α_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$, and $α_2$ is a pure CP map and $β$ is a CP map so that $α_1 \otimes α_2 - β$ is also CP, then $β= γ\otimes α_2$ for some CP map $γ$. We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $α_2 = id$. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map $τ$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $τ\otimes id_k$ is decomposable for some $k \geq 2$, where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.

quant-ph↗

arXiv2019

Comparing Geometric Discord and Negativity for Bipartite States

The geometric discord $\mathcal{D}$ of a state is a measure of the quantumness of the state and the negativity $\mathcal{N}$ is a measure of the entanglement of a state. It was proved by D. Girolami and G. Adesso that for states on $\mathbb{C}^2\otimes\mathbb{C}^2$, the geometric discord is always greater than or equal to the square of the negativity and conjectured that this holds in general. S. Rana and P. Parashar showed that this relation does not hold for all states on $\mathbb{C}^2\otimes\mathbb{C}^n$ for $n>2$. We provide several analytic families of states on $\mathbb{C}^2\otimes\mathbb{C}^3$ violating this relation. Certain upper and lower bounds for $\mathcal{N}^2-\mathcal{D}$ are obtained for states on $\mathbb{C}^m\otimes\mathbb{C}^n$ for any $m, n\in\mathbb{N}$.

quant-ph↗

arXiv2018

Rokhlin property and approximate representability for inclusions of index-finite type

Let $P\subset A$ be an inclusion of unital $C\sp*$-algebras of index-finite type with a fixed conditional expectation $E:A\to P$. We introduce the weak tracial Rokhlin property and its dual notion, weak tracial approximate representability, using positive contractions in central sequence algebras; in particular, the definitions apply to projectionless algebras. Under natural simplicity, finiteness, and outerness hypotheses, we prove that these properties are exchanged by the Jones--Watatani basic construction and its dual conditional expectation. We show that the associated Rokhlin contractions produce tracially large injective completely positive order-zero maps and that our inclusion-theoretic definition recovers the weak tracial Rokhlin property for finite-group actions. The duality also yields inclusions of index-finite type with the weak tracial Rokhlin property which are not isomorphic to fixed-point inclusions arising from actions of ordinary finite groups. Explicit actions on an infinite-type UHF algebra, a simple monotracial AF algebra, and $\mc Z$ separate approximate, tracial approximate, and weak tracial approximate representability. Finally, by showing that the inclusion $P\hookrightarrow A$ is tracially sequentially-split by order zero, we obtain permanence of tracial $\mc{Z}$-absorption, $\mc{Z}$-stability, strict comparison, tracial $m$-comparison, tracial $m$-almost divisibility, and tracial nuclear dimension from $A$ to $P$.

math.OA↗

arXiv2018

The Rokhlin property for inclusions of C*-algebras

Let $P \subset A$ be an inclusion of $σ$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$; (7)C*-algebras of decomposition rank at most $n$, where $n \in Z^+$; (8) Separable simple C*-algebras that are stably isomorphic to AF algebras; (9) Separable simple C*-algebras that are stably isomorphic to AI algebras; (10) Separable simple C*-algebras that are stably isomorphic to AT algebras; (11) Separable simple C*-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; (12) Separable C*-algebras with strict comparison of positive elements. In particular, when $α: G \rightarrow \rm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $(1) \sim (12)$ are inherited to the fixed point algebra $A^α$ and the crossed product algebra $A \rtimes_αG$ from $A$.

math.OA↗

arXiv2018

On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $Φ_{α, β,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $α$ and $β$ for which the family of maps $Φ_{α, β,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $Φ_{α, β,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $α$ and $β$ for which the family of maps $Φ_{α, β,3}$ is $2$-positive and not completely positive.

math-ph↗

arXiv2017

On dualities of actions and inclusions

Following the results known in the case of a finite abelian group action on $C\sp*$-algebras we prove the following two theorems; 1. an inclusion $P\subset A$ of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property). 2. an inclusion $P\subset A$ of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors.

math.OA↗

arXiv2017

Stable rank for crossed products by actions of finite groups on C*-algebras

Let $G$ be a finite group, $A$ a unital separable finite simple nuclear C*-algebra, and $α$ an action of $G$ on $A$. Assume that $A$ absorbs the Jiang-Su algebra $\mathcal{Z}$, the extremal boundary of the trace space of $A$ is compact and finite dimensional and that $α$ fixes any tracial state of $A$. Then tsr$(A \rtimes_αG) = 1$. In particular, when $A$ has a unique tracial state, we conclude it without above conditons on a tracial state space of $A$.

math.OA↗