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Yuuya Yoshida

Publications and source records attributed to Yuuya Yoshida.

14 recordsLinked to original sources
arXiv2026

Optimal quantum locally differentially private mechanisms in the high-privacy regime

We optimize the trade-off between privacy and utility in the high-privacy regime. We adopt local differential privacy (LDP) and its quantum extension, quantum local differential privacy (QLDP), for privacy protection, and investigate utility functions including the Holevo information (which reduces to the mutual information in the classical case) and the error exponents in symmetric and asymmetric hypothesis testing. Under the LDP and QLDP constraints, these utility functions have classical and quantum optimal values, which we denote simply by $C$ and $Q$, respectively, in this abstract. In this paper, we provide optimal LDP and QLDP mechanisms achieving the classical and quantum optimal values in the high-privacy regime, and prove that the asymptotic ratio $Q/C$ in this regime takes the same value regardless of the utility function. Our results reveal quantum advantages (specifically, $Q/C\ge3/2$) for the above utility functions when the private data take at least three possible values. In addition, conditional on a mathematical conjecture concerning equi-isoclinic tight fusion frames (EITFFs), our QLDP mechanisms also exhibit quantum advantages in a moderate-privacy regime.

quant-ph↗
arXiv2024

On the number of representations of integers as differences between Piatetski-Shapiro numbers

For $α>1$, set $β=1/(α-1)$. We show that, for every $1<α<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^α}\rfloor - \lfloor{m^α}\rfloor$ is equal to $βα^{-β}ζ(β)d^{β-1} + o(d^{β-1})$ as $d\to\infty$, where $ζ$ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets $(l,m,n)$ of positive integers such that $l<x$ and $\lfloor{l^α}\rfloor + \lfloor{m^α}\rfloor = \lfloor{n^α}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^α}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^α}\rfloor+\lfloor{n_2^α}\rfloor=\lfloor{n_3^α}\rfloor+\lfloor{n_4^α}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_α(N^{4-α})$ when $1<α\le4/3$.

math.NT↗
arXiv2024

Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers

For all $α_1,α_2\in(1,2)$ with $1/α_1+1/α_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor{n_1^{α_1}}\rfloor+\lfloor{n_2^{α_2}}\rfloor$ is equal to $Γ(1+1/α_1)Γ(1+1/α_2)Γ(1/α_1+1/α_2)^{-1}N^{1/α_1+1/α_2-1} + o(N^{1/α_1+1/α_2-1})$ as $N\to\infty$, where $Γ$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when $α_1,α_2\in(1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.

math.NT↗
arXiv2023

Some remarks on the $[x/n]$-sequence

After the work of Bordellès, Dai, Heyman, Pan and Shparlinki (2018) and Heyman (2019), several authors studied the averages of arithmetic functions over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper, we give three remarks on this topic. Firstly, we improve the result of Wu and Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic progressions by using a variant of Dirichlet's hyperbola method. Secondly, we prove an asymptotic formula for the number of primitive lattice points with coordinates of the form $[x/n]$, for which we introduce a certain averaging trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh divisor problem. We derive asymptotic formulas for such "multiplicative" Titchmarsh divisor problems for "small" arithmetic functions and the Euler totient function with the von Mangoldt function. However, it turns out that the average of the Euler totient function over the $[x/p]$-sequence seems rather difficult and we propose a hypothetical asymptotic formula for this average.

math.NT↗
arXiv2021

Mathematical comparison of classical and quantum mechanisms in optimization under local differential privacy

Let $\varepsilon>0$. An $n$-tuple $(p_i)_{i=1}^n$ of probability vectors is called $\varepsilon$-differentially private ($\varepsilon$-DP) if $e^\varepsilon p_j-p_i$ has no negative entries for all $i,j=1,\ldots,n$. An $n$-tuple $(ρ_i)_{i=1}^n$ of density matrices is called classical-quantum $\varepsilon$-differentially private (CQ $\varepsilon$-DP) if $e^\varepsilonρ_j-ρ_i$ is positive semi-definite for all $i,j=1,\ldots,n$. Denote by $\mathrm{C}_n(\varepsilon)$ the set of all $\varepsilon$-DP $n$-tuples, and by $\mathrm{CQ}_n(\varepsilon)$ the set of all CQ $\varepsilon$-DP $n$-tuples. By considering optimization problems under local differential privacy, we define the subset $\mathrm{EC}_n(\varepsilon)$ of $\mathrm{CQ}_n(\varepsilon)$ that is essentially classical. Roughly speaking, an element in $\mathrm{EC}_n(\varepsilon)$ is the image of $(p_i)_{i=1}^n\in\mathrm{C}_n(\varepsilon)$ by a completely positive and trace-preserving linear map (CPTP map). In a preceding study, it is known that $\mathrm{EC}_2(\varepsilon)=\mathrm{CQ}_2(\varepsilon)$. In this paper, we show that $\mathrm{EC}_n(\varepsilon)\not=\mathrm{CQ}_n(\varepsilon)$ for every $n\ge3$, and estimate the difference between $\mathrm{EC}_n(\varepsilon)$ and $\mathrm{CQ}_n(\varepsilon)$ in a certain manner.

quant-ph↗
arXiv2020

Maximum Dimension of Subspaces with No Product Basis

Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, and $\mathcal{F}$ be a field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is called a product vector if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some $u^{[1]}\in\mathcal{F}^{d_1},\ldots,u^{[n]}\in\mathcal{F}^{d_n}$. A basis composed of product vectors is called a product basis. In this paper, we show that the maximum dimension of subspaces of $\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ with no product basis is equal to $d_1d_2\cdots d_n-2$ if either (i) $n=2$ or (ii) $n\ge3$ and $\#\mathcal{F}>\max\{d_i : i\not=n_1,n_2\}$ for some $n_1$ and $n_2$. When $\mathcal{F}=\mathbb{C}$, this result is related to the maximum number of simultaneously distinguishable states in general probabilistic theories (GPTs).

math.CO↗
arXiv2020

Distributions of Finite Sequences Represented by Polynomials in Piatetski-Shapiro Sequences

By using the work of Frantzikinakis and Wierdl, we can see that for all $d\in\mathbb{N}$, $α\in(d,d+1)$, and integers $k\ge d+2$ and $r\ge1$, there exist infinitely many $n\in\mathbb{N}$ such that the sequence $(\lfloor{(n+rj)^α}\rfloor)_{j=0}^{k-1}$ is represented as $\lfloor{(n+rj)^α}\rfloor=p(j)$, $j=0,1,\ldots,k-1$, by using some polynomial $p(x)\in\mathbb{Q}[x]$ of degree at most $d$. In particular, the above sequence is an arithmetic progression when $d=1$. In this paper, we show the asymptotic density of such numbers $n$ as above. When $d=1$, the asymptotic density is equal to $1/(k-1)$. Although the common difference $r$ is arbitrarily fixed in the above result, we also examine the case when $r$ is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields.

math.NT↗
arXiv2020

Perfect Discrimination in Approximate Quantum Theory of General Probabilistic Theories

As a modern approach for the foundation of quantum theory, existing studies of General Probabilistic Theories gave various models of states and measurements that are quite different from quantum theory. In this paper, to seek a more realistic situation, we investigate models approximately close to quantum theory. We define larger measurement classes that are smoothly connected with the class of POVMs via a parameter, and investigate the performance of perfect discrimination. As a result, we give a sufficient condition of perfect discrimination, which shows a significant improvement beyond the class of POVMs.

quant-ph↗
arXiv2019

Asymptotic and Non-Asymptotic Analysis for Hidden Markovian Process with Quantum Hidden System

We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower bounds for the cumulant generating function of the sample mean of the data. Using these bounds, we derive the central limit theorem and large and moderate deviations for the tail probability. Then, we give the asymptotic variance is given by using the second derivative of the cumulant generating function. We also derive another expression for the asymptotic variance by considering the quantum version of the fundamental matrix. Further, we explain how to extend our results to a general probabilistic system.

quant-ph↗
arXiv2019

Perfect Discrimination of Non-Orthogonal Separable Pure States on Bipartite System in General Probabilistic Theory

We address perfect discrimination of two separable states. When available states are restricted to separable states, we can theoretically consider a larger class of measurements than the class of measurements allowed in quantum theory. The framework composed of the class of separable states and the above extended class of measurements is a typical example of general probabilistic theories. In this framework, we give a necessary and sufficient condition to discriminate two separable pure states perfectly. In particular, we derive measurements explicitly to discriminate two separable pure states perfectly, and find that some non-orthogonal states are perfectly distinguishable. However, the above framework does not improve the capacity, namely, the maximum number of states that are simultaneously and perfectly distinguishable.

quant-ph↗
arXiv2019

Asymptotic Properties for Markovian Dynamics in Quantum Theory and General Probabilistic Theories

We address asymptotic decoupling in the context of Markovian quantum dynamics. Asymptotic decoupling is an asymptotic property on a bipartite quantum system, and asserts that the correlation between two quantum systems is broken after a sufficiently long time passes. The first goal of this paper is to show that asymptotic decoupling is equivalent to local mixing which asserts the convergence to a unique stationary state on at least one quantum system. In the study of Markovian dynamics, mixing and ergodicity are fundamental properties which assert the convergence and the convergence of the long-time average, respectively. The second goal of this paper is to show that mixing for dynamics is equivalent to ergodicity for the two-fold tensor product of dynamics. This equivalence gives us a criterion of mixing that is a system of linear equations. All results in this paper are proved in the framework of general probabilistic theories (GPTs), but we also summarize them in quantum theory.

quant-ph↗
arXiv2019

Arithmetic Progressions in the Graphs of Slightly Curved Sequences

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^α$ for some $α>0$. In this paper, we prove that arbitrarily long arithmetic progressions are contained in the graph of a slightly curved sequence with small error. Furthermore, we extend Szemerédi's theorem to a theorem about slightly curved sequences. As a corollary, it follows that the graph of the sequence $\{\lfloor{n^a}\rfloor\}_{n\in A}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and every $A\subset\mathbb{N}$ with positive upper density. Using this corollary, we show that the set $\{ \lfloor{\lfloor{p^{1/b}}\rfloor^a}\rfloor \mid \text{$p$ prime} \}$ contains arbitrarily long arithmetic progressions for every $1\le a<2$ and $b>1$. We also prove that, for every $a\ge2$, the graph of $\{\lfloor{n^a}\rfloor\}_{n=1}^\infty$ does not contain any arithmetic progressions of length $3$.

math.NT↗
arXiv2019

Optimal Mechanism for Randomized Responses under Universally Composable Security Measure

We consider a problem of analyzing a global property of private data through randomized responses subject to a certain rule, where private data are used for another cryptographic protocol, e.g., authentication. For this problem, the security of private data was evaluated by a universally composable security measure, which can be regarded as $(0,δ)$-differential privacy. Here we focus on the trade-off between the global accuracy and a universally composable security measure, and derive an optimal solution to the trade-off problem. More precisely, we adopt the Fisher information of a certain distribution family as the estimation accuracy of a global property and impose $(0,δ)$-differential privacy on a randomization mechanism protecting private data. Finally, we maximize the Fisher information under the $(0,δ)$-differential privacy constraint and obtain an optimal mechanism explicitly.

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