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arXiv:1605.01210

Quantum dynamics on Orlicz spaces

Louis E. Labuschagne, W. A. Majewski

Published May 4, 2016• 3 citations
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Abstract

Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces $\langle L^{\cosh - 1}, L\log(L+1)\rangle$. The present paper therefore in some sense "completes" the picture by showing that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition, canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation technique, specifically suited to the above context, for extending the action of such maps to the appropriate intermediate spaces of the pair $\langle L^\infty,L^1\rangle$. Moreover, it is shown that quantum dynamics in the form of Markov semigroups described by some Dirichlet forms naturally extends to the context proposed in [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)].

DOI
10.48550/arxiv.1605.01210

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Quantum dynamical maps are defined and studied for quantum statistical physics based on Orlicz spaces. This complements earlier work [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)] where we made a strong case for the assertion that statistical physics of regular systems should properly be based on the pair of Orlicz spaces ⟨Lcosh⁡−1,Llog⁡(L+1)⟩\langle L^{\cosh - 1}, L\log(L+1)\rangle⟨Lcosh−1,Llog(L+1)⟩. The present paper therefore in some sense "completes" the picture by showing that even in the most general non-commutative contexts, completely positive Markov maps satisfying a natural Detailed Balance condition, canonically admit an action on a large class of quantum Orlicz spaces. This is achieved by the development of a new interpolation technique, specifically suited to the above context, for extending the action of such maps to the appropriate intermediate spaces of the pair ⟨L∞,L1⟩\langle L^\infty,L^1\rangle⟨L∞,L1⟩. Moreover, it is shown that quantum dynamics in the form of Markov semigroups described by some Dirichlet forms naturally extends to the context proposed in [W. A. Majewski, L.E. Labuschagne, Ann. H. Poincare. 15, 1197-1221, (2014)].
We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. The pair of Orlicz spaces we explicitly use are, respectively, built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states). They form a dual pair (both for classical and quantum systems). This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function. Moreover for small quantum systems, this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. The pair of Orlicz spaces we explicitly use are, respectively, built on the exponential function (for the description of regular observables) and on an entropic type function (for the corresponding states). They form a dual pair (both for classical and quantum systems). This pair has the advantage of being general enough to encompass regular observables, and specific enough for the latter Orlicz space to select states with a well-defined entropy function. Moreover for small quantum systems, this pair is shown to agree with the classical pairing of bounded linear operators on a Hilbert space, and the trace-class operators.
In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied within the recently developed Orlicz space based approach to large systems. This means that the proposed approach successfully provides a refined framework for the treatment of entropy in each of classical statistical physics, Dirac's formalism of Quantum Mechanics, large systems of quantum statistical physics, and finally also for Quantum Field Theory.
We construct a large class of non-Markovian master equations that describe the dynamics of open quantum systems featuring strong memory effects, which relies on a quantum generalization of the concept of classical semi-Markov processes. General conditions for the complete positivity of the corresponding quantum dynamical maps are formulated. The resulting non-Markovian quantum processes allow the treatment of a variety of physical systems, as is illustrated by means of various examples and applications, including quantum optical systems and models of quantum transport.
We prove a noncommutative ppp-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial σσσ-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial σσσ-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.
Abstract The closing chapter details the application of noncommutative Orlicz spaces to Quantum Physics. We specifically show that the (Köthe dual) pair of Orlicz spaces Llog⁡(L+1) and Lcosh−1 may be better suited as home for regular states and observables respectively than the more standard pair of spaces L∞ and L∞. Specifically the geometry of the space L1∩Llog⁡(L+1) is shown to be well suited to identifying those states which admit a well-defined notion of entropy. In fact the geometry of this space may be used to posit a clear definition of entropy valid for the most general von Neumann algebras. Parallel to this the space Lcosh−1 proves to be home for possibly unbounded observables which nevertheless are very regular in the sense of for example having all their moments finite. It is further pointed out that in the case of B(H) this approach yields exactly the Dirac style elementary Quantum Mechanical formalism. The technology thus developed is also used to gain a clearer understanding of relative entropy of states. The fact that the space Lcosh−1 admits unbounded observables allows for an application to the necessarily unbounded field operators of Quantum Field Theory. We specifically show that under very natural criteria, a large number of quantum fields which satisfy a so-called generalised H-bound condition automatically also satisfy an Lcosh−1 regularity condition namely that one may canonically embed the field operators into Lcosh−1 - the natural superspace for the regular observables.