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Stochastics

Quantum detailed balance conditions with time reversal: three-level systems

Franco Fagnola, Veronica Umanità

Published Jun 9, 2011• 1 citation
Publisher

Abstract

The classical detailed balance, or reversibility, condition admits several non-commutative extensions. We discuss the most appropriate in view of our previous results and the complete case study of three-level systems.
DOI
10.1080/17442508.2010.518707

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Quantum detailed balance is formulated in terms of elementary transitions, in close analogy to detailed balance in a classical Markov chain on a finite set of points. An elementary transition is taken to be a pure state of two copies of the quantum system, as a quantum analogue of an ordered pair of classical points representing a classical transition from the first to the second point. This form of quantum detailed balance is shown to be equivalent to standard quantum detailed balance with respect to a reversing operation, thus providing a new conceptual foundation for the latter. Aspects of parity in quantum detailed balance are clarified in the process. The connection with the Accardi-Cecchini dual and the KMS dual (or Petz recovery map) is also elucidated.
In the last chapter, we saw that it is theoretically possible to switch a logically reversible gate without dissipating any energy at all (recall the proof of the Landauer-Shannon result). If there is no dissipation, then no information is ever being discarded; therefore, if we know the state of the system completely at any instant of time, we should be able to infer its state at any previous instant. This is consistent with the equations of quantum mechanics (e.g., the Schro¨dinger equation, or the Pauli equation) which are reversible in time. This property of quantum mechanical systems, known as “time reversal symmetry”, guarantees that if we know the state of a system at a time t0, we can tell what the state was at any time t < t0. Because of this fundamental connection between physical reversibility, logical reversibility, and the equations of quantum mechanics, we would expect logically reversible gates to behave as non-classical (quantum mechanical) computing machinery.∗ This will bring us to the topic of quantum computing and quantum logic gates, but before we get there, let us explore reversible logic in a little more detail.
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)].