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Metric on the Space of Quantum Processes 量子过程空间的度量
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010164
E. A. Pankovets, L. E. Fedichkin
{"title":"Metric on the Space of Quantum Processes","authors":"E. A. Pankovets, L. E. Fedichkin","doi":"10.1134/s0081543824010164","DOIUrl":"https://doi.org/10.1134/s0081543824010164","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider a metric <span>(D)</span> describing the difference between real (noisy) and ideal processes that is based on the operator norm of the maximum deviation between the final real and ideal states of a quantum system. We discuss the properties as well as geometric and experimental interpretations of the metric. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Some Properties of the Fractional Derivative of the Brownian Local Time 论布朗局部时间分数衍生物的一些特性
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010115
I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev
{"title":"On Some Properties of the Fractional Derivative of the Brownian Local Time","authors":"I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev","doi":"10.1134/s0081543824010115","DOIUrl":"https://doi.org/10.1134/s0081543824010115","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the properties of the fractional derivative <span>(D_alpha l(t,x))</span> of order <span>(alpha&lt;1/2)</span> of the Brownian local time <span>(l(t,x))</span> with respect to the variable <span>(x)</span>. This derivative is understood as the convolution of the local time with the generalized function <span>(|x|^{-1-alpha})</span>. We show that <span>(D_alpha l(t,x))</span> appears naturally in Itô’s formula for the process <span>(|w(t)|^{1-alpha})</span>. Using the martingale technique, we also study the limit behavior of <span>(D_alpha l(t,x))</span> as <span>(ttoinfty)</span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Minimum of the Wehrl Entropy for a Locally Compact Abelian Group 论局部紧凑阿贝尔群的韦尔熵最小值
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010097
Evgeny I. Zelenov
{"title":"On the Minimum of the Wehrl Entropy for a Locally Compact Abelian Group","authors":"Evgeny I. Zelenov","doi":"10.1134/s0081543824010097","DOIUrl":"https://doi.org/10.1134/s0081543824010097","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A construction of the Wehrl entropy is proposed for an arbitrary locally compact abelian group <span>(G)</span>. It is proved that the Wehrl entropy is not less than a certain nonnegative integer, which is an invariant of the group <span>(G)</span>. The minimum of the Wehrl entropy is attained on coherent states. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Violation of Bell’s Inequalities in Jordan Triples and Jordan Algebras 乔丹三元组和乔丹代数中贝尔不等式的违反
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s008154382401019x
Jan Hamhalter, Ekaterina A. Turilova
{"title":"Violation of Bell’s Inequalities in Jordan Triples and Jordan Algebras","authors":"Jan Hamhalter, Ekaterina A. Turilova","doi":"10.1134/s008154382401019x","DOIUrl":"https://doi.org/10.1134/s008154382401019x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We formulate and prove Bell’s inequalities in the realm of JB<span>(^*)</span> triples and JB<span>(^*)</span> algebras. We show that the maximal violation of Bell’s inequalities occurs in any JBW<span>(^*)</span> triple containing a nonassociative <span>(2)</span>-Peirce subspace. Moreover, we show that the violation of Bell’s inequalities in a nonmodular JBW<span>(^*)</span> algebra and in an essentially nonmodular JBW<span>(^*)</span> triple is generic. We describe the structure of maximal violators and its relation to the spin factor. In addition, we present a synthesis of available results based on a unified geometric approach. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Extreme Points of Sets in Operator Spaces and State Spaces 论算子空间和状态空间中集合的极值点
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010024
G. G. Amosov, A. M. Bikchentaev, V. Zh. Sakbaev
{"title":"On Extreme Points of Sets in Operator Spaces and State Spaces","authors":"G. G. Amosov, A. M. Bikchentaev, V. Zh. Sakbaev","doi":"10.1134/s0081543824010024","DOIUrl":"https://doi.org/10.1134/s0081543824010024","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere <span>(S_1(mathcal H))</span> of a Hilbert space <span>(mathcal H)</span>. For a measure on <span>(S_1(mathcal H))</span>, we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital <span>(mathrm C^*)</span>-algebra are characterized in terms of extreme points. We also study extreme points <span>(mathrm{extr}(mathcal E^1))</span> of the unit ball <span>(mathcal E^1)</span> of a normed ideal operator space <span>(langlemathcal E,|kern1pt{cdot}kern1pt|_{mathcal E}rangle)</span> on <span>(mathcal H)</span>. If <span>(Uinmathrm{extr}(mathcal E^1))</span> for some unitary operator <span>(Uinmathcal{B}(mathcal H))</span>, then <span>(Vinmathrm{extr}(mathcal E^1))</span> for all unitary operators <span>(Vinmathcal{B}(mathcal H))</span>. In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Jensen Gap and Capacity of a Shifted Depolarizing Quantum Channel 论偏移去极化量子通道的詹森间隙和容量
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010048
E. L. Baitenov
{"title":"On Jensen Gap and Capacity of a Shifted Depolarizing Quantum Channel","authors":"E. L. Baitenov","doi":"10.1134/s0081543824010048","DOIUrl":"https://doi.org/10.1134/s0081543824010048","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the problem of maximizing the Jensen gap with respect to the probability distribution in a fairly general case, and prove a theorem on the optimal distribution. Using the results obtained, we calculate the one-shot capacity of a certain family of non-unital quantum channels. We show that in sufficiently large dimensions the channel admits one of two modes of an optimal input ensemble depending on the parameters. We also prove that both the fulfillment and the violation of the entanglement-breaking property are possible in any dimension depending on the parameters of the channel. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Tightness-Type Properties of the Space of Weakly Additive Functionals 论弱相加函数空间的严密性特征
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010036
Sh. A. Ayupov, N. K. Mamadaliev
{"title":"On Tightness-Type Properties of the Space of Weakly Additive Functionals","authors":"Sh. A. Ayupov, N. K. Mamadaliev","doi":"10.1134/s0081543824010036","DOIUrl":"https://doi.org/10.1134/s0081543824010036","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study tightness-type properties such as tightness, minitightness, and local density of the space of weakly additive functionals with finite support. We also investigate some generalizations of continuous functions. Furthermore, we present an extension of the functor of weakly additive functionals with finite support to the class of strictly <span>(tau)</span>-continuous mappings. We introduce two extensions of the categories <span>(mathrm{Comp})</span> and <span>(mathrm{Tych})</span> (of compact and Tychonoff spaces, respectively). One of the main results of the paper is that the functor <span>(O_n)</span> of weakly additive functionals with finite support preserves the tightness character of infinite compact spaces. In addition, we show that the local densities of the spaces <span>(X)</span> and <span>(O_n(X))</span> coincide for any infinite compact space <span>(X)</span>. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Generalized Coherent States and Random Shift Operators 广义相干态和随机移位算子
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-07-11 DOI: 10.1134/s0081543824010127
R. Sh. Kalmetev, Yu. N. Orlov, V. Zh. Sakbaev
{"title":"Generalized Coherent States and Random Shift Operators","authors":"R. Sh. Kalmetev, Yu. N. Orlov, V. Zh. Sakbaev","doi":"10.1134/s0081543824010127","DOIUrl":"https://doi.org/10.1134/s0081543824010127","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the Chernoff averages for random generalized shift operators in the case of noncanonical commutation relations between creation and annihilation operators. We introduce the concepts of shift-dual ladder operators and generalized shift operators. As an example, we consider a one-parameter family of commutation relations for which generalized shift operators are unitary and satisfy the semigroup property on straight lines passing through the origin. For this family, we prove that the sequence of expectations of Feynman–Chernoff iterations of random shift operators converges to a limit strongly continuous semigroup. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces 与贝索夫空间结构相混合的布尔干涉-莫雷空间
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050152
Yirui Zhao, Yoshihiro Sawano, Jin Tao, Dachun Yang, Wen Yuan
{"title":"Bourgain–Morrey Spaces Mixed with Structure of Besov Spaces","authors":"Yirui Zhao, Yoshihiro Sawano, Jin Tao, Dachun Yang, Wen Yuan","doi":"10.1134/s0081543823050152","DOIUrl":"https://doi.org/10.1134/s0081543823050152","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Bourgain–Morrey spaces <span>(mathcal{M}^p_{q,r}(mathbb R^n))</span>, generalizing what was introduced by J. Bourgain, play an important role in the study related to the Strichartz estimate and the nonlinear Schrödinger equation. In this article, via adding an extra exponent <span>(tau)</span>, the authors creatively introduce a new class of function spaces, called Besov–Bourgain–Morrey spaces <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>, which is a bridge connecting Bourgain–Morrey spaces <span>(mathcal{M}^p_{q,r}(mathbb R^n))</span> with amalgam-type spaces <span>((L^q,ell^r)^p(mathbb R^n))</span>. By making full use of the Fatou property of block spaces in the weak local topology of <span>(L^{q'}(mathbb R^n))</span>, the authors give both predual and dual spaces of <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Applying these properties and the Calderón product, the authors also establish the complex interpolation of <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Via fully using fine geometrical properties of dyadic cubes, the authors then give an equivalent norm of <span>(|kern1pt{cdot}kern1pt|_{mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n)})</span> having an integral expression, which further induces a boundedness criterion of operators on <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span>. Applying this criterion, the authors obtain the boundedness on <span>(mathcal{M}dot{B}^{p,tau}_{q,r}(mathbb R^n))</span> of classical operators including the Hardy–Littlewood maximal operator, the fractional integral, and the Calderón–Zygmund operator. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Embedding of Besov Spaces of Zero Smoothness into Lorentz Spaces 论零光滑度贝索夫空间嵌入洛伦兹空间
IF 0.5 4区 数学
Proceedings of the Steklov Institute of Mathematics Pub Date : 2024-03-06 DOI: 10.1134/s0081543823050127
D. M. Stolyarov
{"title":"On Embedding of Besov Spaces of Zero Smoothness into Lorentz Spaces","authors":"D. M. Stolyarov","doi":"10.1134/s0081543823050127","DOIUrl":"https://doi.org/10.1134/s0081543823050127","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We show that the zero smoothness Besov space <span>(B_{p,q}^{0,1})</span> does not embed into the Lorentz space <span>(L_{p,q})</span> unless <span>(p=q)</span>; here <span>(p,qin (1,infty))</span>. This answers in the negative a question posed by O. V. Besov. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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