Entropy Production for Quantum Markov Semigroups

Abstract

An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on B(h) as a numerical index measuring "how much far" they are from equilibrium. The entropy production is defined as derivative of the relative entropy of the one-step forward and backward evolution in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds.

Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance

“…The characterization of dynamics for open classical and quantum systems is an essential and fundamental topic. One approach to characterize the dynamics for open systems is entropy production [1][2][3][4][5][6], from an information theory perspective. For Markovian dynamics, the entropy production is positive, which relates to the data processing inequality [7][8][9] and complies with the second law of thermodynamics [10,11].…”

Detailed balance and entanglement

“…Connections between detailed balance and entangled states have in fact already been exploited in [10,11,12] with regards to entropy production for quantum Markov semigroups (also see [13] for related work). Here our goal is to study this connection itself more explicitly, in particular how it arises as well as one instance of how it fits into other parts of physics, specifically the area known as thermofield dynamics.…”

“…A heuristic motivation as to why one might in general expect a connection between detailed balance and entanglement is presented in Section 3. Two definitions of quantum detailed balance, one of which was also considered in [10,11,12], are discussed in Section 4. The characterization of these forms of detailed balance in terms of a certain entangled state is then described in Section 5, and proved in Section 6.…”

Balance Between Quantum Markov Semigroups

“…Non-equilibrium is only touched on. The papers on quantum detailed balance that most directly lead to the work presented in this paper are [28], [29], [30] and [25]. Of particular relevance are ideas connected to standard quantum detailed balance and maximally entangled bipartite states.…”

“…Of particular relevance are ideas connected to standard quantum detailed balance and maximally entangled bipartite states. Standard quantum detailed balance conditions were mentioned in [20], but discussed and developed in [30] and [29]. Connections to maximally entangled states were discussed in [28], [29] and [25].…”