Pub Date : 2022-04-30DOI: 10.1142/s0219025722500084
K. Parthasarathy
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.
{"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}
Pub Date : 2022-04-28DOI: 10.1142/s0219025723500133
E. Essaky, M. Hassani, C. E. Rhazlane
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.
本文证明了后向随机进化包含(简称BSEI)的形式为begin{align*}%label{BSDI3} begin{cases} dY_t+AY_tdt In G(t,Y_t,Z_t)dt+Z_tdW_t，quad t In [0, t] Y_t =xi， end{cases} end{align*}，其中$W=(W_t)_{t In [0, t]}$是标准布朗运动，$ a $是UMD Banach空间$E$上$C_0$-半群的产生子，$ $ xi$是来自$L^p(Omega，mathscr{F}_T;E)$的终结条件，$p>1$和$G$是满足一定条件的集值函数。研究了空间中值为鞅类型$2$的过程的情况。
{"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}
Pub Date : 2022-04-23DOI: 10.1142/s0219025722500096
Carlos Diaz-Aguilera, Tulio Gaxiola, Jorge Santos, Carlos Vargas
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.
{"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}
Pub Date : 2022-04-18DOI: 10.1142/s0219025722500060
L. Accardi, Abdallah Dhahri, Y. Lu
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.
{"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}
Pub Date : 2022-04-18DOI: 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}
Pub Date : 2022-03-23DOI: 10.1142/s0219025723500030
Matthijs Vernooij
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.
{"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}
Pub Date : 2022-03-03DOI: 10.1142/s0219025723500212
G. Androulakis, Tiju Cherian John
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.
{"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}
Pub Date : 2022-03-02DOI: 10.1142/s0219025722500035
L. Accardi, Y. Lu
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.
{"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}
Pub Date : 2022-02-27DOI: 10.1142/s0219025722400069
K. Parthasarathy
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.
{"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}
Pub Date : 2022-02-21DOI: 10.1142/s0219025722500047
Botao Long, W. W
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.
{"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}
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