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arXiv:2411.02339 · Quantum

Quantum detailed balance via elementary transitions

Rocco Duvenhage, Kyle Oerder, Keagan van den Heuvel

Published May 15, 2025• 0 citations
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Abstract

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.

DOI
10.22331/q-2025-05-15-1743

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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.
Markovian master equations (formally known as quantum dynamical semigroups) can be used to describe the evolution of a quantum state $\ensuremath{\rho}$ when in contact with a memoryless thermal bath. This approach has had much success in describing the dynamics of real-life open quantum systems in the laboratory. Such dynamics increase the entropy of the state $\ensuremath{\rho}$ and the bath until both systems reach thermal equilibrium, at which point entropy production stops. Our main result is to show that the entropy production at time ttt is bounded by the relative entropy between the original state and the state at time 2t2t2t. The bound puts strong constraints on how quickly a state can thermalize, and we prove that the factor of 2 is tight. The proof makes use of a key physically relevant property of these dynamical semigroups, detailed balance, showing that this property is intimately connected with the field of recovery maps from quantum information theory. We envisage that the connections made here between the two fields will have further applications. We also use this connection to show that a similar relation can be derived when the fixed point is not thermal.
The dynamics of dissipative many-body quantum systems sometimes admit an emergent classical description in terms of a Markov chain even though the corresponding state space contains highly entangled states. In particular, a bad-cavity laser is a paradigm system whose dynamics can be formulated as a Markov chain in a two-dimensional state space spanned by collective angular momentum states. In this article, we investigate the connection between a dissipative phase transition that occurs in the subradiant regime of this system in the large atom number limit, and the properties of the underlying Markov chain. In one of the phases, the Markov chain approaches the detailed-balance condition with increasing atom number NNN and hence becomes effectively time-reversible. This is caused by a collective atomic emission process that effectively reduces the Markov chain to one dimension. In the other phase, we find the emergence of time-asymmetric probability currents in the two-dimensional state space that signals a breakdown of detailed balance. This is accompanied by a macroscopic internal entropy production rate in the Markov chain that scales extensively with the atom number NNN. An observable consequence of the probability currents is a self-pulsing of the cavity light output in this phase, which can be detected as an anti-correlation dip in the intensity correlation function.
Abstract Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.