Open Systems & Information Dynamics最新文献

{"title":"Environment Decoherence of Quantum Exclusion Semigroups in Terms of Quantum Bernoulli Noises","authors":"Jinshu Chen, Shexiang Hai","doi":"10.1142/s123016122450001x","DOIUrl":"https://doi.org/10.1142/s123016122450001x","url":null,"abstract":"<p>In this paper, we study the environment induced decoherence of quantum exclusion semigroups constructed by quantum Bernoulli noises. We first study the structure of the decoherence-free algebra and discuss some of its properties, and give the conditions for its possible mutual inclusions with the set of fixed points. Then we describe the structure of the set of fixed points and use the new results obtained here to discuss the environment induced decoherence properties of the semigroup.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Lindbladian Dynamics Generates Inter-system Connectedness in Psychological Phenomena","authors":"R. Englman","doi":"10.1142/s1230161224500057","DOIUrl":"https://doi.org/10.1142/s1230161224500057","url":null,"abstract":"<p>Exploring a general stratagem for the way that Nature operates, one finds that in a variety of phenomena of one’s experience these come into existence (only) thanks to the special way that two disparate systems, being parts of an open system, are coupled together. This connectedness is shown here to come about through a Lindbladian operator, such that it leads to the exclusion of many more possibilities absent in the connectedness. An instance of this phenomenon, here subsumed under the term “inter-system connectedness” and known to occur in the reduction of quantum mechanical wave-packets, is here proposed in the emergence of consciousness resulting from and linked to neural activity. For this, a dual Hilbert-space formulation of mental activities is proposed, which then enables a semi-quantitative explanation of R. N. Shepard’s seminal findings for reaction times in recalls. Affinities with and differences from Khrennikov’s recent widely scoped treatise on these subjects are described.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nice Error Basis and Quantum Channel","authors":"B. V. Rajarama Bhat, Purbayan Chakraborty, Uwe Franz","doi":"10.1142/s1230161224500033","DOIUrl":"https://doi.org/10.1142/s1230161224500033","url":null,"abstract":"<p>The Weyl operators give a convenient basis of <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> which is also orthonormal with respect to the Hilbert-Schmidt inner product. The properties of such a basis can be generalised to the notion of a nice error basis (NEB), as introduced by E. Knill [3]. We can use an NEB of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> to construct an NEB for <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Lin</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span>, the space of linear maps on <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>. Any linear map will then correspond to a <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">×</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span> coefficient matrix in the basis decomposition with respect to such an NEB of <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Lin</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span>. Positivity, complete (co)positivity or other properties of a linear map can be characterised in terms of such a coefficient matrix.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"177 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics","authors":"Frederik vom Ende","doi":"10.1142/s1230161224500045","DOIUrl":"https://doi.org/10.1142/s1230161224500045","url":null,"abstract":"<p>We generalize a recent result stating that all analytic quantum dynamics can be represented exactly as the reduction of unitary dynamics generated by a time- dependent Hamiltonian. More precisely, we prove that the partial trace over analytic paths of unitaries can approximate any Lipschitz-continuous quantum dynamics arbitrarily well. Equivalently, all such dynamics can be approximated by analytic Kraus operators. We conclude by discussing potential improvements and generalizations of these results, their limitations, and the general challenges one has to overcome when trying to relate dynamics to quantities on the system–environment level.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"249 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An Energy Estimation Benchmark for Quantum Computing Hardware","authors":"Andreas J. C. Woitzik, Lukas Hoffmann, Andreas Buchleitner, Edoardo G. Carnio","doi":"10.1142/s1230161224500069","DOIUrl":"https://doi.org/10.1142/s1230161224500069","url":null,"abstract":"<p>Certifying the performance of available quantum computing hardware requires standardized tests. We propose a simple energy estimation benchmark to scrutinize actual output against gate and readout errors reported for the IBM Quantum System One platform, where we collected unprecedented amount of data from several devices, mostly from the one in Ehningen, Germany. We observe oscillations of the benchmark data with a characteristic timescale of two hours, as well as outliers. This suggests that current mitigation techniques need to be adapted to account for nontrivial time dependencies of the devices’ output.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"177 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Fast Recurrent Subspace on an N-Level Quantum Energy Transport Model","authors":"Jorge R. Bolaños-Servín, Josué I. Rios-Cangas, Alfredo Uribe","doi":"10.1142/s1230161224500021","DOIUrl":"https://doi.org/10.1142/s1230161224500021","url":null,"abstract":"<p>The fast recurrent subspace (the biggest support of all invariant states) of a weak coupling limit type quantum Markov semigroup modeling a quantum transport open system of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi></math></span><span></span>-energy levels is determined. This is achieved by characterizing the structure of all the invariant states and their spectra in terms of a natural generalization of the discrete Fourier transform operator. Finally, the attraction domains and long-time behaviour of the evolution are studied on hereditary subalgebras where faithful invariant states exist.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"249 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometric versus Entropic Gaussian Correlations in an Open Quantum System of Two Bosonic Modes","authors":"Alina Stoica, Aurelian Isar","doi":"10.1142/s1230161223500208","DOIUrl":"https://doi.org/10.1142/s1230161223500208","url":null,"abstract":"<p>Different types of geometric and entropic quantum correlation quantifiers are studied for a system composed of two resonant bosonic modes embedded in a thermal bath. The description of the evolution of the correlation measures is formulated in the framework of the theory of open systems, based on completely positive quantum dynamical semigroups, by using both a geometric and entropic quantification of total nonclassical correlations of Gaussian states. We consider the special case when the initial squeezed thermal state of the system preserves its form in time. We show that time evolution of the measures strongly depends on the parameters characterising the initial state of the system (squeezing parameter and average thermal photon numbers of the two modes) and of the thermal environment (temperature of the thermal bath and dissipation rate). In the limit of large times all the considered measures asymptotically tend to zero value, corresponding to an asymptotic bimodal uncorrelated product state. We make a comparison between the behaviour of the evolution in time of the Gaussian geometric quantum correlations and Gaussian entropic quantum correlations.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"2-Point Markov Evolutions","authors":"Luigi Accardi, Ameur Dhahri","doi":"10.1142/s123016122350021x","DOIUrl":"https://doi.org/10.1142/s123016122350021x","url":null,"abstract":"<p>We study the Markov evolutions associated to the expected Markov processes.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"13 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes","authors":"Jader E. Brasil, Josué Knorst, Artur O. Lopes","doi":"10.1142/s123016122350018x","DOIUrl":"https://doi.org/10.1142/s123016122350018x","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; denote the set of &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; by &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; complex matrices. Consider continuous time quantum semigroups &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"cal\"&gt;𝒫&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mspace width=\".17em\"&gt;&lt;/mspace&gt;&lt;mi mathvariant=\"cal\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"cal\"&gt;ℒ&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is the infinitesimal generator. If we assume that &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"cal\"&gt;ℒ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, we will call &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mspace width=\".17em\"&gt;&lt;/mspace&gt;&lt;mi mathvariant=\"cal\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; a quantum Markov semigroup. Given a stationary density matrix &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ρ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=\"cal\"&gt;ℒ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, for the quantum Markov semigroup &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi mathvariant=\"cal\"&gt;𝒫&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, we can define a continuous time stationary quantum Markov process, denoted by &lt;span&gt;&lt;math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, &lt;span&gt;&lt;math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; Given an &lt;i&gt;a priori&lt;/i&gt; Laplacian operator &lt;span&gt;&lt;math altimg=\"e","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"153 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deviation from Equilibrium of QMS of Weak Coupling Limit Type with Respect to Uniform and Completely Nonequilibrium Invariant States","authors":"M. A. Cruz de la Rosa, J. C. García, F. Guererro-Poblete, A. Hernández","doi":"10.1142/s1230161223500221","DOIUrl":"https://doi.org/10.1142/s1230161223500221","url":null,"abstract":"<p>We study the deviation from equilibrium with respect to a special class of states, the so called uniform and completely nonequilibrium, which were introduced and characterized in [11]. We compute explicitly such a deviation, we obtain an equivalent expression in terms of circulant matrices and give a bound for it. Finally, as an example, we make explicit computations in the three level case.</p>","PeriodicalId":54681,"journal":{"name":"Open Systems & Information Dynamics","volume":"39 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}