Pub Date : 2024-07-11DOI: 10.1134/s0081543824010206
M. E. Shirokov
Abstract
General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization.
{"title":"On Local Continuity of Characteristics of Composite Quantum Systems","authors":"M. E. Shirokov","doi":"10.1134/s0081543824010206","DOIUrl":"https://doi.org/10.1134/s0081543824010206","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> General methods of local continuity analysis of characteristics of infinite-dimensional composite quantum systems are considered. A new approximation technique for obtaining local continuity conditions for various characteristics of quantum systems is proposed and described in detail. This technique is used to prove several general results (a Simon-type dominated convergence theorem, a theorem on the preservation of continuity under convex mixtures, etc.). Local continuity conditions are derived for the following characteristics of composite quantum systems: the quantum conditional entropy, the quantum (conditional) mutual information, the one-way classical correlation and its regularization, the quantum discord and its regularization, the entanglement of formation and its regularization, and the constrained Holevo capacity of a partial trace and its regularization. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611889","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010085
R. N. Gumerov, R. L. Khazhin
Abstract
For composite quantum systems, we consider quantum channels that uniquely determine the channels of the subsystems. Such channels of composite systems are called generating channels. Examples of generating channels are given by tensor products of two quantum channels of subsystems and their convex combinations. The paper deals with the properties of generating channels. In particular, we show that these channels form a convex compact set in the norm topology. We prove a criterion for a quantum channel to be generating. For composite systems consisting of two qubits, we construct generating phase-damping channels. For subsystems, these channels generate both phase-damping channels and depolarizing channels. Examples of nongenerating phase-damping channels are also presented.
{"title":"Generating Quantum Channels","authors":"R. N. Gumerov, R. L. Khazhin","doi":"10.1134/s0081543824010085","DOIUrl":"https://doi.org/10.1134/s0081543824010085","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> For composite quantum systems, we consider quantum channels that uniquely determine the channels of the subsystems. Such channels of composite systems are called generating channels. Examples of generating channels are given by tensor products of two quantum channels of subsystems and their convex combinations. The paper deals with the properties of generating channels. In particular, we show that these channels form a convex compact set in the norm topology. We prove a criterion for a quantum channel to be generating. For composite systems consisting of two qubits, we construct generating phase-damping channels. For subsystems, these channels generate both phase-damping channels and depolarizing channels. Examples of nongenerating phase-damping channels are also presented. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"90 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611872","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010188
A. S. Trushechkin
Abstract
Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup.
{"title":"Kinetic State and Emergence of Markovian Dynamics in Exactly Solvable Models of Open Quantum Systems","authors":"A. S. Trushechkin","doi":"10.1134/s0081543824010188","DOIUrl":"https://doi.org/10.1134/s0081543824010188","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Typically, the theory of open quantum systems studies the dynamics of the reduced state (density operator) of the system. However, in the early stages of evolution, it is impossible to separate the reservoir dynamics from the system dynamics. Among the consequences of this fact is the violation of the positivity of solutions of some quantum master equations for the reduced density operator. In this paper we study the joint dynamics of the system and reservoir at an early stage of evolution and the pre-relaxation of the joint state to a so-called kinetic state. A kinetic state of the system and reservoir is characterized by the fact that it is completely determined by the reduced density operator of the system alone. Only after the formation of a kinetic state, it becomes possible to describe the evolution of the reduced density operator of the system in terms of a semigroup. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"8 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611880","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010139
D. A. Kronberg
Abstract
We study the properties of postselective transformations of quantum states, that is, transformations for which some classical results are declared “successful” while the rest are discarded. We demonstrate that for every postselective transformation there exists a distinguished orthonormal basis for which the transformation reduces to probabilistic blocking of the basis states followed by a deterministic transformation. We also describe a generalization of an arbitrary postselective transformation that corresponds to its partial version with a given success probability.
{"title":"On the Structure of Postselective Transformations of Quantum States","authors":"D. A. Kronberg","doi":"10.1134/s0081543824010139","DOIUrl":"https://doi.org/10.1134/s0081543824010139","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the properties of postselective transformations of quantum states, that is, transformations for which some classical results are declared “successful” while the rest are discarded. We demonstrate that for every postselective transformation there exists a distinguished orthonormal basis for which the transformation reduces to probabilistic blocking of the basis states followed by a deterministic transformation. We also describe a generalization of an arbitrary postselective transformation that corresponds to its partial version with a given success probability. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"28 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611876","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010140
Kh. Sh. Meretukov, A. E. Teretenkov
Abstract
We develop a consistent perturbative technique for obtaining a time-local linear master equation based on projection methods in the case where the projection operator depends on time. Then we introduce a generalization of the Kawasaki–Gunton projection operator, which allows us to use this technique to derive, generally speaking, nonlinear master equations in the case of arbitrary ansatzes consistent with some set of observables. Most of the results obtained are of a very general nature, but when discussing them, we put emphasis on the application of these results to the theory of open quantum systems.
{"title":"On Time-Dependent Projectors and a Generalization of the Thermodynamical Approach in the Theory of Open Quantum Systems","authors":"Kh. Sh. Meretukov, A. E. Teretenkov","doi":"10.1134/s0081543824010140","DOIUrl":"https://doi.org/10.1134/s0081543824010140","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We develop a consistent perturbative technique for obtaining a time-local linear master equation based on projection methods in the case where the projection operator depends on time. Then we introduce a generalization of the Kawasaki–Gunton projection operator, which allows us to use this technique to derive, generally speaking, nonlinear master equations in the case of arbitrary ansatzes consistent with some set of observables. Most of the results obtained are of a very general nature, but when discussing them, we put emphasis on the application of these results to the theory of open quantum systems. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611878","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010176
A. E. Rastegin
Abstract
In quantum information theory, one needs to consider systems with incomplete information. To estimate a quantum system as an information resource, one uses various characteristics of non-classical correlations. Currently, much attention is paid to coherence quantifiers averaged over a set of specially selected states. In particular, mutually unbiased bases, symmetric informationally complete measurements, and some of their generalizations are of importance in this regard. The aim of the present study is to derive uncertainty relations for coherence quantifiers based on divergences of the Tsallis type. The obtained inequalities concern quantifiers averaged over a set of mutually unbiased bases and a set of states that form an equiangular tight frame.
{"title":"Uncertainty Relations for Coherence Quantifiers of the Tsallis Type","authors":"A. E. Rastegin","doi":"10.1134/s0081543824010176","DOIUrl":"https://doi.org/10.1134/s0081543824010176","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> In quantum information theory, one needs to consider systems with incomplete information. To estimate a quantum system as an information resource, one uses various characteristics of non-classical correlations. Currently, much attention is paid to coherence quantifiers averaged over a set of specially selected states. In particular, mutually unbiased bases, symmetric informationally complete measurements, and some of their generalizations are of importance in this regard. The aim of the present study is to derive uncertainty relations for coherence quantifiers based on divergences of the Tsallis type. The obtained inequalities concern quantifiers averaged over a set of mutually unbiased bases and a set of states that form an equiangular tight frame. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"15 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611933","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010073
E. Vl. Bulinskaya
Abstract
A model of a branching random walk on (mathbb Z^d) with a periodic set of sources of reproduction and death of particles is studied. For this model, an asymptotic description of the propagation of the population of particles with time is obtained for the first time. The intensities of the sources may be different. The branching regime is assumed to be supercritical, and the tails of the jump distribution of the walk are assumed to be “light.” The main theorem establishes the Hausdorff-metric convergence of the properly normalized random cloud of particles that exist in the branching random walk at time (t) to the limit set, as (t) tends to infinity. This convergence takes place for almost all elementary outcomes of the event meaning nondegeneracy of the population of particles under study. The limit set in (mathbb R^d), called the asymptotic shape of the population, is found in an explicit form.
{"title":"Propagation of Branching Random Walk on Periodic Graphs","authors":"E. Vl. Bulinskaya","doi":"10.1134/s0081543824010073","DOIUrl":"https://doi.org/10.1134/s0081543824010073","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A model of a branching random walk on <span>(mathbb Z^d)</span> with a periodic set of sources of reproduction and death of particles is studied. For this model, an asymptotic description of the propagation of the population of particles with time is obtained for the first time. The intensities of the sources may be different. The branching regime is assumed to be supercritical, and the tails of the jump distribution of the walk are assumed to be “light.” The main theorem establishes the Hausdorff-metric convergence of the properly normalized random cloud of particles that exist in the branching random walk at time <span>(t)</span> to the limit set, as <span>(t)</span> tends to infinity. This convergence takes place for almost all elementary outcomes of the event meaning nondegeneracy of the population of particles under study. The limit set in <span>(mathbb R^d)</span>, called the asymptotic shape of the population, is found in an explicit form. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"58 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611871","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010152
Oleg V. Morzhin, Alexander N. Pechen
Abstract
We consider an open qutrit system in which the evolution of the density matrix (rho(t)) is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the dissipation superoperator) controls. To control the qutrit, we propose to use not only coherent control but also generally time-dependent decoherence rates which are adjusted by the so-called incoherent control. In our approach, the incoherent control makes the decoherence rates time-dependent in a specific controlled manner and within a clear physical mechanism. We consider the problem of maximizing the Hilbert–Schmidt overlap between the final state (rho(T)) of the system and a given target state (rho_{text{target}}), as well as the problem of minimizing the squared Hilbert–Schmidt distance between these states. For both problems, we perform their realifications, derive the corresponding Pontryagin functions, adjoint systems (with two variants of transversality conditions for the two terminal objectives), and gradients of the objectives, and adapt the one-, two-, and three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze first the operation of the methods and second the obtained control processes, in respect of considering the environment as a resource via incoherent control.
{"title":"Using and Optimizing Time-Dependent Decoherence Rates and Coherent Control for a Qutrit System","authors":"Oleg V. Morzhin, Alexander N. Pechen","doi":"10.1134/s0081543824010152","DOIUrl":"https://doi.org/10.1134/s0081543824010152","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider an open qutrit system in which the evolution of the density matrix <span>(rho(t))</span> is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation with simultaneous coherent (in the Hamiltonian) and incoherent (in the dissipation superoperator) controls. To control the qutrit, we propose to use not only coherent control but also generally time-dependent decoherence rates which are adjusted by the so-called incoherent control. In our approach, the incoherent control makes the decoherence rates time-dependent in a specific controlled manner and within a clear physical mechanism. We consider the problem of maximizing the Hilbert–Schmidt overlap between the final state <span>(rho(T))</span> of the system and a given target state <span>(rho_{text{target}})</span>, as well as the problem of minimizing the squared Hilbert–Schmidt distance between these states. For both problems, we perform their realifications, derive the corresponding Pontryagin functions, adjoint systems (with two variants of transversality conditions for the two terminal objectives), and gradients of the objectives, and adapt the one-, two-, and three-step gradient projection methods. For the problem of maximizing the overlap, we also adapt the regularized first-order Krotov method. In the numerical experiments, we analyze first the operation of the methods and second the obtained control processes, in respect of considering the environment as a resource via incoherent control. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611879","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010061
Yu. I. Bogdanov, N. A. Bogdanova, V. F. Lukichev
Abstract
Using an example of Pearson type IV distributions, we propose a procedure of completing the classical probability distribution to a quantum state. We obtain a wave function corresponding to Pearson type IV distributions and construct the corresponding set of basis functions. Then we demonstrate how the developed method applies to problems of statistical data analysis and quantum mechanics, and show the efficiency of our approach for the problem of approximating statistical distributions with heavy tails.
摘要 以皮尔逊 IV 型分布为例，我们提出了一种将经典概率分布补全为量子态的程序。我们获得了与皮尔逊 IV 型分布相对应的波函数，并构建了相应的基函数集。然后，我们演示了所开发的方法如何应用于统计数据分析和量子力学问题，并展示了我们的方法在近似重尾统计分布问题上的效率。
{"title":"The Set of Basis Functions Generated by Pearson Type IV Distributions and Its Application to Problems of Statistical Data Analysis and Quantum Mechanics","authors":"Yu. I. Bogdanov, N. A. Bogdanova, V. F. Lukichev","doi":"10.1134/s0081543824010061","DOIUrl":"https://doi.org/10.1134/s0081543824010061","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Using an example of Pearson type IV distributions, we propose a procedure of completing the classical probability distribution to a quantum state. We obtain a wave function corresponding to Pearson type IV distributions and construct the corresponding set of basis functions. Then we demonstrate how the developed method applies to problems of statistical data analysis and quantum mechanics, and show the efficiency of our approach for the problem of approximating statistical distributions with heavy tails. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611875","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}
Pub Date : 2024-07-11DOI: 10.1134/s0081543824010218
Vsevolod I. Yashin
Abstract
Conditionally unital completely positive maps are used to characterize generators of unital completely positive semigroups on (C^*)-algebras. In this work, a generalization of this notion is proposed that includes maps between different operator systems. In terms of this generalization, conditionally unital completely positive maps are infinitesimal increments of unital completely positive maps. The basic properties of conditionally unital completely positive maps are studied, the Choi–Jamiołkowski duality is established, and an Arveson-type extension theorem for completely bounded conditionally unital completely positive maps is proved in the case of maps with values in finite-dimensional (C^*)-algebras.
{"title":"Arveson’s Extension Theorem for Conditionally Unital Completely Positive Maps","authors":"Vsevolod I. Yashin","doi":"10.1134/s0081543824010218","DOIUrl":"https://doi.org/10.1134/s0081543824010218","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Conditionally unital completely positive maps are used to characterize generators of unital completely positive semigroups on <span>(C^*)</span>-algebras. In this work, a generalization of this notion is proposed that includes maps between different operator systems. In terms of this generalization, conditionally unital completely positive maps are infinitesimal increments of unital completely positive maps. The basic properties of conditionally unital completely positive maps are studied, the Choi–Jamiołkowski duality is established, and an Arveson-type extension theorem for completely bounded conditionally unital completely positive maps is proved in the case of maps with values in finite-dimensional <span>(C^*)</span>-algebras. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"53 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611887","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}
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