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On the transient and recurrent parts of a quantum Markov semigroup

100%

Umanità V.

Banach Center Publications

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2006

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tom 73

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nr 1

415-428

EN

We define the transient and recurrent parts of a quantum Markov semigroup 𝓣 on a von Neumann algebra 𝓐 and we show that, when 𝓐 is σ-finite, we can write 𝓣 as the sum of such semigroups. Moreover, if 𝓣 is the countable direct sum of irreducible semigroups each with a unique faithful normal invariant state ρₙ, we find conditions under which any normal invariant state is a convex combination of ρₙ's.

2

On two quantum versions of the detailed balance condition

63%

Fagnola F., Umanità V.

Banach Center Publications

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2010

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tom 89

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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

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

63%

Fagnola F., Umanità V.

Banach Center Publications

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2011

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tom 96

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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.

Ograniczanie wyników

3 Banach Center Publications

3 Umanità V.

2 Fagnola F.

1 2011

1 2010

1 2006

On the transient and recurrent parts of a quantum Markov semigroup

On two quantum versions of the detailed balance condition

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