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Restrictions of CP-semigroups to maximal commutative subalgebras

100%
Fagnola F., Skeide M.
Banach Center Publications
|
2007
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tom 78
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nr 1
121-132
EN
We give a necessary and sufficient criterion for a normal CP-map on a von Neumann algebra to admit a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on ℬ(G).
2
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On two quantum versions of the detailed balance condition

100%
Fagnola F., Umanità V.
Banach Center Publications
|
2010
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tom 89
|
nr 1
105-119
EN
Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form $⟨x,y⟩_s: = tr(ρ^{1-s}x*ρ^{s}y)$ (s ∈ [0,1]) in order to compare the resulting quantum versions of the classical detailed balance condition. Moreover, we discuss the structure of generators of a quantum Markov semigroup which commute with the modular automorphism because this condition appears when we consider pre-scalar products with s ≠ 1/2.
3
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Quantum detailed balance conditions with time reversal: the finite-dimensional case

100%
Fagnola F., Umanità V.
Banach Center Publications
|
2011
|
tom 96
|
nr 1
159-174
EN
We classify generators of quantum Markov semigroups 𝓣 on 𝓑 (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 $tr(ρ^{1/2}xρ^{1/2}𝓣_t(y)) = tr(ρ^{1/2}θy*θρ^{1/2}𝓣_t(θx*θ))$ for all x,y ∈ 𝓑 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.
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