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

Franco Fagnola; Veronica Umanità

Banach Center Publications (2011)

- Volume: 96, Issue: 1, page 159-174
- ISSN: 0137-6934

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topFranco Fagnola, and Veronica Umanità. "Quantum detailed balance conditions with time reversal: the finite-dimensional case." Banach Center Publications 96.1 (2011): 159-174. <http://eudml.org/doc/282552>.

@article{FrancoFagnola2011,

abstract = {We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely $tr(ρ^\{1/2\}xρ^\{1/2\}_t(y)) = tr(ρ^\{1/2\}θy*θρ^\{1/2\}_t(θx*θ))$ for all x,y ∈ and t ≥ 0.
Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual quantum detailed balance condition with non-symmetric multiplications $x ↦ ρ^\{s\}xρ^\{1-s\}$ (s ∈ [0,1], s ≠ 1/2) whose generators must commute with the modular group associated with ρ. This supports our conclusion that the most appropriate non-commutative version of the classical detailed balance condition is the above standard quantum detailed balance condition with an anti-unitary time reversal.},

author = {Franco Fagnola, Veronica Umanità},

journal = {Banach Center Publications},

keywords = {quantum detailed balance; quantum Markov semigroup; Lindblad representation},

language = {eng},

number = {1},

pages = {159-174},

title = {Quantum detailed balance conditions with time reversal: the finite-dimensional case},

url = {http://eudml.org/doc/282552},

volume = {96},

year = {2011},

}

TY - JOUR

AU - Franco Fagnola

AU - Veronica Umanità

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

JO - Banach Center Publications

PY - 2011

VL - 96

IS - 1

SP - 159

EP - 174

AB - We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely $tr(ρ^{1/2}xρ^{1/2}_t(y)) = tr(ρ^{1/2}θy*θρ^{1/2}_t(θx*θ))$ for all x,y ∈ and t ≥ 0.
Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual quantum detailed balance condition with non-symmetric multiplications $x ↦ ρ^{s}xρ^{1-s}$ (s ∈ [0,1], s ≠ 1/2) whose generators must commute with the modular group associated with ρ. This supports our conclusion that the most appropriate non-commutative version of the classical detailed balance condition is the above standard quantum detailed balance condition with an anti-unitary time reversal.

LA - eng

KW - quantum detailed balance; quantum Markov semigroup; Lindblad representation

UR - http://eudml.org/doc/282552

ER -

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