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Banach Center Publications

Quantum detailed balance conditions with time reversal: the finite-dimensional case

Franco Fagnola, Veronica Umanità

Published Jan 1, 2011• 1 citation
Publisher

Abstract

We classify generators of quantum Markov semigroups $\mathcal T$ on $\mathcal{B}(\mathsf h)$, with ${\mathsf{h}}$ finite-dimensional and with a faithful normal invariant state $\rho$ satisfying the standard quantum detailed balance condition with an anti

DOI
10.4064/bc96-0-10

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We classify generators of quantum Markov semigroups T\mathcal TT on B(h)\mathcal{B}(\mathsf h)B(h), with h{\mathsf{h}}h finite-dimensional and with a faithful normal invariant state ρ\rhoρ satisfying the standard quantum detailed balance condition with an anti
For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).
Generators of KMS Symmetric Markov Semigroups on B(h){\mathcal{B}({\rm h})}B(h) Symmetry and Quantum Detailed Balance
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 the 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.
In this study, we first deal with the canonical anti-commutation relation algebra in which the Bernoulli creation and annihilation operators generate the structure of ultra hyperfinite C[Formula: see text]-algebra. This observation simplifies the construction of the minimal quantum Markov semigroup construct from quantum Bernoulli noises by taking a filter through the family gamma of finite subsets of the natural numbers. As a result, we obtain a Feller quantum Markov semigroup defined on the C[Formula: see text]-algebra, which in turn is extended to the entire algebra of bounded operators. Then, one condition for us to obtain a faithful normal invariant state of this semigroup is an equilibrium state (it satisfies the detailed balance conditions). Finally, we use the quantum Wasserstein distance to discuss the W-entropy rate of this invariant state.