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Computation of sandwiched relative α-entropy of two n-mode Gaussian states 两个n模高斯态的夹心相对α-熵的计算
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-04-30 DOI: 10.1142/s0219025722500084
K. Parthasarathy
{"title":"Computation of sandwiched relative α-entropy of two n-mode Gaussian states","authors":"K. Parthasarathy","doi":"10.1142/s0219025722500084","DOIUrl":"https://doi.org/10.1142/s0219025722500084","url":null,"abstract":"A formula for the sandwiched relative [Formula: see text]-entropy [Formula: see text] for [Formula: see text], of two [Formula: see text]-mode Gaussian states [Formula: see text], [Formula: see text] in the boson Fock space [Formula: see text] is presented. This computation extensively employs the [Formula: see text]-parametrization of Gaussian states in [Formula: see text] introduced in Ref.10.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"12 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88641266","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: 1
Backward Stochastic Evolution Inclusions in UMD Banach Spaces UMD Banach空间中的倒向随机演化包涵体
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-04-28 DOI: 10.1142/s0219025723500133
E. Essaky, M. Hassani, C. E. Rhazlane
{"title":"Backward Stochastic Evolution Inclusions in UMD Banach Spaces","authors":"E. Essaky, M. Hassani, C. E. Rhazlane","doi":"10.1142/s0219025723500133","DOIUrl":"https://doi.org/10.1142/s0219025723500133","url":null,"abstract":"In this paper, we prove the existence of a mild $L^p$-solution for the backward stochastic evolution inclusion (BSEI for short) of the form begin{align*}%label{BSDI3} begin{cases} dY_t+AY_tdtin G(t,Y_t,Z_t)dt+Z_tdW_t,quad tin [0,T] Y_T =xi, end{cases} end{align*} where $W=(W_t)_{tin [0,T]}$ is a standard Brownian motion, $A$ is the generator of a $C_0$-semigroup on a UMD Banach space $E$, $xi$ is a terminal condition from $L^p(Omega,mathscr{F}_T;E)$, with $p>1$ and $G$ is a set-valued function satisfying some suitable conditions. The case when the processes with values in spaces that have martingale type $2$, has been also studied.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"7 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88337154","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
Combinatorics of NC-probability spaces with independent constants 具有独立常数的nc概率空间的组合学
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-04-23 DOI: 10.1142/s0219025722500096
Carlos Diaz-Aguilera, Tulio Gaxiola, Jorge Santos, Carlos Vargas
{"title":"Combinatorics of NC-probability spaces with independent constants","authors":"Carlos Diaz-Aguilera, Tulio Gaxiola, Jorge Santos, Carlos Vargas","doi":"10.1142/s0219025722500096","DOIUrl":"https://doi.org/10.1142/s0219025722500096","url":null,"abstract":"The boolean and monotone notions of independence lack the property of independent constants. We address this problem from a combinatorial point of view (based on cumulants defined from weights on set-partitions, in the general framework of operator-valued probability spaces). We show that if the weights are singleton inductive (SI), then all higher-order cumulants involving constants vanish, just as in the free and classical case. Our combinatorial considerations lead rather directly to mild variations of boolean and monotone probability theories which are closely related to the usual notions. The SI-boolean case is related to c-free and Fermi convolutions. We also describe some standard combinatorial aspects of the SI-boolean and cyclic-boolean lattices, such as their Möbius functions, featuring well-known combinatorial integer sequences.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"52 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138524130","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
A note on the infinite-dimensional quantum Strassen’s theorem 关于无限维量子Strassen定理的注解
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-04-18 DOI: 10.1142/s0219025722500060
L. Accardi, Abdallah Dhahri, Y. Lu
{"title":"A note on the infinite-dimensional quantum Strassen’s theorem","authors":"L. Accardi, Abdallah Dhahri, Y. Lu","doi":"10.1142/s0219025722500060","DOIUrl":"https://doi.org/10.1142/s0219025722500060","url":null,"abstract":"In Ref. [3], the quantum Strassen’s theorem has been extended to the infinite-dimensional case. This theorem consists in the solution of the coupling problem for two states on the algebra of bounded operators on two Hilbert spaces [Formula: see text], [Formula: see text] with the additional constraint that the coupling state has support in a pre-assigned sub-space of [Formula: see text]. In this paper, we give an alternative proof of the main theorem in Ref. [3] that allows such extension.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"39 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88253832","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
Quantum Ising model with generalized competing XY-interactions on a Cayley tree Cayley树上具有广义竞争xy相互作用的量子Ising模型
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-04-18 DOI: 10.1142/s0219025722500102
F. Mukhamedov, S. Gheteb
{"title":"Quantum Ising model with generalized competing XY-interactions on a Cayley tree","authors":"F. Mukhamedov, S. Gheteb","doi":"10.1142/s0219025722500102","DOIUrl":"https://doi.org/10.1142/s0219025722500102","url":null,"abstract":"","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"83 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83811115","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 existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups 关于状态对称量子马尔可夫半群发生器的导数作为平方根的存在性
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-03-23 DOI: 10.1142/s0219025723500030
Matthijs Vernooij
{"title":"On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups","authors":"Matthijs Vernooij","doi":"10.1142/s0219025723500030","DOIUrl":"https://doi.org/10.1142/s0219025723500030","url":null,"abstract":"Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a tracially symmetric quantum Markov semigroup on a C*-algebra $mathcal{A}$ there exists a densely defined derivation $delta$ from $mathcal{A}$ to a Hilbert bimodule $H$ such that $L^{(2)}=delta^*circ overline{delta}$. Here we show that this construction of a derivation can in general not be generalised to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that in the finite-dimensional case the existence of such a derivation is equivalent to the existence of a positive matrix solution of a system of linear equations. We solve this system of linear equations for concrete examples using Mathematica to complete the proof.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"25 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84251681","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: 2
Relative Entropy via Distribution of Observables 通过观测分布的相对熵
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-03-03 DOI: 10.1142/s0219025723500212
G. Androulakis, Tiju Cherian John
{"title":"Relative Entropy via Distribution of Observables","authors":"G. Androulakis, Tiju Cherian John","doi":"10.1142/s0219025723500212","DOIUrl":"https://doi.org/10.1142/s0219025723500212","url":null,"abstract":"We obtain formulas for Petz-R'enyi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. Classical results on R'enyi and Kullback-Leibler divergences are applied to obtain new results and new proofs for some known results about Petz-R'enyi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz-R'enyi $alpha$-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77172382","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: 2
The quantum moment problem for a classical random variable and a classification of interacting Fock spaces 经典随机变量的量子矩问题及相互作用Fock空间的分类
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-03-02 DOI: 10.1142/s0219025722500035
L. Accardi, Y. Lu
{"title":"The quantum moment problem for a classical random variable and a classification of interacting Fock spaces","authors":"L. Accardi, Y. Lu","doi":"10.1142/s0219025722500035","DOIUrl":"https://doi.org/10.1142/s0219025722500035","url":null,"abstract":"The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem 3 of the present paper provides such a criterium. Theorem 5 deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem 3. Theorem 3 naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"21 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82715216","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: 2
Twisted convolution quantum information channels, one-parameter semigroups and their generators 扭曲卷积量子信息信道，单参数半群及其产生
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-02-27 DOI: 10.1142/s0219025722400069
K. Parthasarathy
{"title":"Twisted convolution quantum information channels, one-parameter semigroups and their generators","authors":"K. Parthasarathy","doi":"10.1142/s0219025722400069","DOIUrl":"https://doi.org/10.1142/s0219025722400069","url":null,"abstract":"Using the tool of quantum characteristic functions of n -mode states in the boson Fock space Γ( C n ) we construct a semigroup of quantum information channels. This leads to a special class of one-parameter semigroups of such channels. These semigroups are concrete but their generators have unbounded operator coefficients. These one-parameter semigroups are also quantum dynamical semigroups and the form of the generators involve additional features which do not appear in the standard GKSL form. A heuristic discussion of the form of these generators is included. In the wake of this analysis many open problems arise naturally.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88528197","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
The category of compact quantum metric spaces 紧致量子度量空间的范畴
IF 0.9 4区 数学
Infinite Dimensional Analysis Quantum Probability and Related Topics Pub Date : 2022-02-21 DOI: 10.1142/s0219025722500047
Botao Long, W. W
{"title":"The category of compact quantum metric spaces","authors":"Botao Long, W. W","doi":"10.1142/s0219025722500047","DOIUrl":"https://doi.org/10.1142/s0219025722500047","url":null,"abstract":"A compact quantum metric space is a complete order unit space endowed with a Lip-norm. We introduce a metric on the state space and give several equivalent conditions to characterize the Lipschitz morphisms and Lipschitz isomorphisms between two compact quantum metric spaces.","PeriodicalId":50366,"journal":{"name":"Infinite Dimensional Analysis Quantum Probability and Related Topics","volume":"178 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72552010","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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