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Keiji Matsumoto

Publications and source records attributed to Keiji Matsumoto.

At least 19 recordsLinked to original sources
arXiv2026

The algebro-geometric aspect of the iterated limit of a quaternary of means of four terms

We study the iterated limit of a quaternary of means of four terms through the period map from the family of cyclic fourfold coverings of the complex projective line branching at six points to the three-dimensional complex ball $\mathbb{B}_3$ embedded into the Siegel upper half-space of degree four. We construct four automorphic forms on $\mathbb{B}_3$ expressing the inverse of the period map, and give an equality between one of them and a period integral, which is an analogy of Jacobi's formula between a theta constant and an elliptic integral. We find a transformation of $\mathbb{B}_3$ such that the quaternary of means appears by its actions on the four automorphic forms. These results enable us to express the iterated limit by the Lauricella hypergeometric series of type $D$ in three variables.

math.AG↗
arXiv2025

The monodromy representation of a hypergeometric system in $m$ variables of rank $p^m$

We study the monodromy representation of the hypergeometric system $\mathcal{F}_{C}^{p,m}(a,B)$ in $m$ variables of rank $p^m$ with parameters $a$ and $B$. This system can be regarded as a multi-variable model of the generalized hypergeometric equation of rank $p$. We construct $m+1$ loops which generate the fundamental group of the complement of the singular locus of $\mathcal{F}_{C}^{p,m}(a,B)$, and we show that they satisfy certain relations as elements of the fundamental group. We produce circuit matrices along these loops with respect to a fundamental system of solutions to $\mathcal{F}_C^{p,m}(a,B)$ under certain non-integrality conditions on parameters $a$ and $B$.

math.CA↗
arXiv2025

More on Intractability of Thermalization: (almost) i.i.d. inputs and finite lattices

This work is an extention of Shiraishi and Matsumoto [10], and discusses the computational complexity of the long-term average of local observables in one-dimensional lattices with shift-invariant nearest-neighbor interactions for simple initial states. As shown in the previous paper, the problem is generally intractable. In this paper we refine the statement further. First, we consider restriction of the initial state, where the state of all the sites are the same except for a single site. We show this version of the problem is also undecidable (RE-complete). Then we turn to the case where the lattice size is finite: depening on the defitiniton of the input size, this version of problem is either EXPSPACE-complete or PSPACE-complete.

quant-ph↗
arXiv2024

A system of hypergeometric differential equations in $m$ variables of rank $p^m$

We define a hypergeometric series in $m$ variables with $p+(p-1)m$ parameters, which reduces to the generalized hypergeometric series $_pF_{p-1}$ when $m=1$, and to Lauricella's hypergeometric series $F_C$ in $m$ variables when $p=2$. We give a system of hypergeometric differential equations annihilating the series. Under some non-integral conditions on parameters, we give an Euler type integral representation of the series, and linearly independent $p^m$ solutions to this system around a point near to the origin. We show that this system is of rank $p^m$, and determine its singular locus.

math.CA↗
arXiv2022

Information storing yields a point-asymmetry of state space in general probabilistic theories

It is known that the high-dimensional quantum state space is notoriously complicated in contrast with the beautiful Bloch ball of the qubit. We examined the mechanism behind this fact in the frame work of general probabilistic theory (GPT), and found rather general quantitative relations between the geometry of the state space and its information storing capability. The main result is the information-asymmetry identity, which (up to the constant term) equates the {\it Minkowski measure of asymmetry} with the {\it information storability} which, in addition to its own operational meaning, serves as an upper bound to common information measures such as semi-classical capacity. As a consequence, the asymmetry measure is lower-bounded by information storing capability of the state space, so the increase in the latter enhances the former. Coming back to the quantum systems, the $d$-level state space cannot be symmetric "because" it can store more than a single bit of information. Also, the Holevo capacity of any quantum channel with point-symmetric image is at most a single bit. In the course of the research, we applied Shannon theory to GPT, producing a couple of new results. Also presented is a new geometrical proof of known upper bounds to information measures.

quant-ph↗
arXiv2022

Analogies of Jacobi's formula

By considering Schwarz's map for the hypergeometric differential equation with parameters $(a,b,c)=(1/6,1/2,1)$ or $(1/12,5/12,1)$, we give some analogies of Jacobi's formula $\vartheta_{00}(τ)^2= F(1/2,1/2,1;λ(τ))$, where $\vartheta_{00}(τ)$ and $λ(τ)$ are the theta constant and the lambda function defined on the upper-half plane, and $F(a,b,c;z)$ is the hypergeometric series defined on the unit disk. As applications of our formulas, we give several functional equations for $F(a,b,c;z)$.

math.CA↗
arXiv2022

Imaginarity-free quantum multiparameter estimation

Multiparameter quantum estimation is made difficult by the following three obstacles. First, incompatibility among different physical quantities poses a limit on the attainable precision. Second, the ultimate precision is not saturated until you discover the optimal measurement. Third, the optimal measurement may generally depend on the target values of parameters, and thus may be impossible to perform for unknown target states. We present a method to circumvent these three obstacles. A class of quantum statistical models, which utilizes antiunitary symmetries or, equivalently, real density matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.

quant-ph↗
arXiv2022

An analogy of Jacobi's formula and its applications

We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.

math.CA↗
arXiv2021

A Geometrical Approach to Quantum Estimation Theory

This post is the author's doctoral dissertation back in 1997. The dissertation covers following four kinds of problems: First, it studies achievable Cramer-Rao type bounds of various multi-parameter pure state models. Second, it relates CR-bouds, both of mixed state and pure states, with Berry-Uhlmann curvature. Roughly, it characterize how the statistical model differs from classical probability distribution family. Though the relation is rather qualitative for mixed states, the quantitative relation is obtained for pure state models. Third topic is relation between Berry-Uhlmann geometry and Amari-Nagaoka's quantum information geometry. Forth, various problems in quantum physics, uncertainty relations, measurement of temperature, time reversal symmetry, etc., are discussed using estimation theory.

quant-ph↗
arXiv2021

Asymptotic Reversibility of Thermal Operations for Interacting Quantum Spin Systems via Generalized Quantum Stein's Lemma

For quantum spin systems in any spatial dimension with a local, translation-invariant Hamiltonian, we prove that asymptotic state convertibility from a quantum state to another one by a thermodynamically feasible class of quantum dynamics, called thermal operations, is completely characterized by the Kullback-Leibler (KL) divergence rate, if the state is translation-invariant and spatially ergodic. Our proof consists of two parts and is phrased in terms of a branch of the quantum information theory called the resource theory. First, we prove that any states, for which the min and max Rényi divergences collapse approximately to a single value, can be approximately reversibly converted into one another by thermal operations with the aid of a small source of quantum coherence. Second, we prove that these divergences collapse asymptotically to the KL divergence rate for any translation-invariant ergodic state. We show this via a generalization of the quantum Stein's lemma for quantum hypothesis testing beyond independent and identically distributed (i.i.d.) situations. Our result implies that the KL divergence rate serves as a thermodynamic potential that provides a complete characterization of thermodynamic convertibility of ergodic states of quantum many-body systems in the thermodynamic limit, including out-of-equilibrium and fully quantum situations.

quant-ph↗
arXiv2021

Undecidability in quantum thermalization

The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

cond-mat.stat-mech↗
arXiv2019

Relative twisted homology and cohomology groups associated with Lauricella's $F_D$

We introduce relative twisted homology and cohomology groups associated with Euler type integrals of solutions to Lauricella's system $F_D(a,b,c)$ of hypergeometric differential equations. We define an intersection form between relative twisted homology groups and that between relative twisted cohomology groups, and show their compatibility. We prove that the relative twisted homology group is canonically isomorphic to the space of local solutions to $F_D(a,b,c)$ for any parameters $a,b,c$. Through this isomorphism, we study $\cF_D(a,b,c)$ by the relative twisted homology and cohomology groups and the intersection forms without any conditions on $a,b,c$.

math.AG↗
arXiv2019

Reversibility of distance measures of states with some focus on total variation distance

Consider a classical system, which is in the state described by probability distribution $p$ or $q$, and embed these classical informations into quantum system by a physical map $Γ$, $ρ=Γ(p)$ and $σ=Γ(q)$. Intuitively, the pair $\{p_ρ^{M},p_σ^{M}\}$ of the distributions of the data of the measurement $M$ on the pair $\{ρ,σ\}$ should contain strictly less information than the pair $\{p,q\}$ provided the pair $\{ρ,σ\}$ is non-commutative. Indeed, this statement had been shown if the information is measured by $f$-divergence such that $f$ is operator convex. In the paper, the statement is extended to the case where $f$ is strictly convex. Also, we disprove the assertion for the total variation distance $\Vert p-q\Vert_{1}$, the $f$-divergence with $f(r)=|1-r|$: if $\{ρ,σ\}$ satisfies some not very restrictive conditions, $\Vert p_ρ^{M}-p_σ^{M}\Vert_{1}$ equals $\Vert p-q\Vert_{1}$. Here we present sufficient condition for general case, and necessary and sufficient condition for qubit states.

quant-ph↗
arXiv2019

A portable potentiometric electronic tongue leveraging smartphone and cloud platforms

Electronic tongues based on potentiometry offer the prospect of rapid and continuous chemical fingerprinting for portable and remote systems. The present contribution presents a technology platform including a miniaturized electronic tongue based on electropolymerized ion-sensitive films, microcontroller-based data acquisition, a smartphone interface and cloud computing back-end for data storage and deployment of machine learning models. The sensor array records a series of differential voltages without use of a true reference electrode and the resulting time-series potentiometry data is used to train supervised machine learning algorithms. For trained systems, inferencing tasks such as the classification of liquids are realized within less than 1 minute including data acquisition at the edge and inference using the cloud-deployed machine learning model. Preliminary demonstration of the complete electronic tongue technology stack is reported for the classification of beverages and mineral water.

cs.ET↗
arXiv2018

Irreducibility of the monodromy representation of Lauricella's $F_C$

Let $E_C$ be the hypergeometric system of differential equations satisfied by Lauricella's hypergeometric series $F_C$ of $m$ variables. We show that the monodromy representation of $E_C$ is irreducible under our assumption consisting of $2^{m+1}$ conditions for parameters. We also show that the monodromy representation is reducible if one of them is not satisfied.

math.AG↗
arXiv2018

A new quantum version of f-divergence

This paper proposes and studies new quantum version of $f$-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version ($\mathrm{D}_{f}^{\max}$, below), defined as the solution to an optimization problem, or the minimum classical $f$- divergence necessary to generate a given pair of quantum states. It turns out to be the largest quantum $f$-divergence. The closed formula of $\mathrm{D}_{f}^{\max}$ is given either if $f$ is operator convex, or if one of the state is a pure state. Also, concise representation of $\mathrm{D}_{f}^{\max}$ as a pointwise supremum of linear functionals is given and used for the clarification of various properties of the quality. Using the closed formula of $\mathrm{D}_{f}^{\max}$, we show: Suppose $f$ is operator convex. Then the\ maximum $f\,$- divergence of the probability distributions of a measurement under the state $ρ$ and $σ$ is strictly less than $\mathrm{D}_{f}^{\max}\left( ρ\Vertσ\right) $. This statement may seem intuitively trivial, but when $f$ is not operator convex, this is not always true. A counter example is $f\left( λ\right) =\left\vert 1-λ\right\vert $, which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.

quant-ph↗
arXiv2017

Considerations in the Time-Energy Uncertainty Relation from the Viewpointt of Hypothesis Testing

It is difficult to discriminate between the state of initial time $t_0$ and that of time $t_0+Δt$. The task of discriminating between the two states can be interpreted as the hypothesis testing problem as to a state $ρ$ deciding whether $ρ$ is $ρ_{t_0}$ or$ρ_{t_0+Δt}$ We show the process of optimization of this hypotheses testing and the condition that discrimination between the two states become difficult even when optimized test was executed. In addition, We consider time-enegy uncertainty relation from the viewpoint of optimized hypotheses testing. Optimization consists of two steps. The first step is to optimize the decision rule based on the outcome of the given measurement by the Neyman-Pearson theorem. The second step is to select optimum measurement in order to maximize the power of test. The optimum measurement is proved to be $\{ |0\rangle \langle 0|,1-|0\rangle \langle 0|\}$ and the power of test in the optimized test holds $γ_{Max}=1-\exp(-\frac{2n}{\hbar^2}Δt^2ΔH^2)+o(Δt^2)$ when $Δt \ll 1$ satisfies,where n is the number of the data. It will be impossible to discriminate between the two states when the power of this test declines. This condition becomes $\frac{2n}{\hbar^2}Δt^2ΔH^2\ll 1$ in the optimized hypotheses testing. It is remarkable that the optimum measurement is composed of the operators $Q_0$ and $1-Q_0$ in the proof of time-energy uncertainty relation. Namely, the time-energy uncertainty relation presents the time interval in which even the optimum measurement hardly detect modification of the initial state .

quant-ph↗