Structure of uniformly continuous quantum Markov semigroups

Deschamps, Julien; Fagnola, Franco; Sasso, Emanuela; Umanità, Veronica

doi:10.1142/s0129055x16500033

Cited by 44 publications

(61 citation statements)

References 36 publications

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“…Remark 6. The results in this proposition are in some points parallel to what discussed in [17] for continuous time Markov semigroups: what is intrinsically different here in our paper is the presence of a supplementary decomposition due to the period, which cannot appear in continuous time.…”

Section: Reducible Mapssupporting

confidence: 82%

“…Then we turn to the reducible case, where we exploit the fact that the two algebras are atomic, to deduce ad-hoc decompositions of the invariant states, relations with the Kraus operators, a better description of the conditional expectations and of the cyclic behavior of the channel (Proposition 8 and Theorem 2). These results are strictly related to the studies in [8], [17] and the decomposition appearing in the last part of [44]. Finally, in Section 4, we apply our results to analyze a remarkable family of quantum channels, i.e.…”

Section: Introductionmentioning

confidence: 63%

“…Some aspects of this rigidity were already known and were object of interest in many papers in the last years, related to different problems: e.g. the structure of the invariant states and irreducible decompositions [8,13], decoherence free algebra and environmental decoherence [14,17], the notion of sufficiency in quantum statistics [30,36,34], periodicity and ergodic properties [12]. In finite dimension, the structure of the channel and its spectrum, cycles and multiplicative properties were investigated in [43,44].…”

Section: Introductionmentioning

confidence: 99%

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On Period, Cycles and Fixed Points of a Quantum Channel

Carbone

Jenčová

2019

Ann. Henri Poincaré

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We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.Point (ii) already appeared in [14] (see also references therein) and was the original representation of the decoherence free algebra used in [9] when introducing environmental decoherence.The following results are well known.Proposition 4. Assume that there is a faithful normal invariant state for Φ. Then (i) F is a von Neumann subalgebra.(ii) The restriction Φ| N is a *-automorphism.Proof. See e.g.[39] and references therein. Maps with a faithful invariant stateIn this section, we assume that there is a faithful normal state ρ ∈ S(H) for Φ. In this case, there is another special subalgebra investigated in the literature, e.g. [7,27], appearing in some asymptotic splitting, usually called the reversible subalgebra and denoted by M r . We describe the reversible subalgebra, following [7], [27] or [29]. Let S be the closure of the semigroup of channels {Φ n , n ∈ N} in the point-ultraweak topology and defineWe will show below, in Theorem 1, that the equality M r = N holds for channels on B(H) (or more generally on atomic von Neumann algebras).Due to the presence of a faithful normal invariant state, for any ϕ ∈ B(H) * , the set {Φ n * ϕ, n ∈ N} is weakly relatively compact, equivalently, the set S consists of normal operators and is a compact semitopological semigroup ([27, Proposition 2.1]). Further, S contains a minimal ideal M(S) which is a compact topological group. Let F be the unit of this group, then M(S) = F •S and F is a normal conditional expectation preserving the invariant state ρ such that T F = F T for all T ∈ S. Finally, M r is a von Neumann algebra and the minimal ideal M(S) acts as a compact group of *-automorphisms on M r ([7, Theorem 1.2 and Corollary 1.3]). This last fact trivially implies, in particular, that M r ⊆ N and that equality holds in finite dimension, but the infinite dimensional case is quite more delicate and tricky. Now let X ∈ B(H) and let O 0 (X) := {Φ k (X), k ∈ N} be the orbit of X under {Φ k } k≥0 . Then the weak*-closureŌ 0 (X) is the orbit of X under S, O 0 (X) = O(X) := {T (X), T ∈ S} and we can define the stable subspace as M s := {X ∈ B(H), 0 ∈ O(X)}.The following lemma can be deduced from [29, Theorem 2.1], [7, Theorem 1.2] and [27, Proposition 2.2]. Since we did not find an explicit and comprehensive statement in the literature, we reconstruct here the detailed result that we need and the proof for the convenience of the reader.Lemma 1. Let F be the normal conditional expectation introduced before.

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Section: Reducible Mapssupporting

confidence: 82%

Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

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On Period, Cycles and Fixed Points of a Quantum Channel

Carbone

Jenčová

2019

Ann. Henri Poincaré

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“…and we obtain a similar expression for the dephasing frequencies ω nm , see (23). We note that the obstruction will in this case be given by ∆ nml = j∈J ν j sin(ϑ j,m − ϑ j,n ) + sin(ϑ j,l − ϑ j,m ) + sin(ϑ j,n − ϑ j,l ) .…”

Section: Classical Dilations Via Jumpssupporting

confidence: 54%

Mathematical models of Markovian dephasing

Fagnola

Gough

Nurdin

et al. 2019

J. Phys. A: Math. Theor.

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We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal. With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson-Parthasarathy dilation. To this end, we make use of a seminal result of Kümmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, the Hamiltonian obstruction, which vanishes if and only if the latter admits a self-adjoint representation and quantifies the hindrance to having a classical diffusive noise model.

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“…ax do not have presence in the subspace of coherences between the blocks. Since the most general AsðHÞ is a direct sum of such NS blocks [78,80,81], we can shade gray the blocks in which J μ may not be zero [ Fig. 1(b)], dual to Ψ μ [ Fig.…”

Section: B Nonsteady Asymptotic Subspacesmentioning

confidence: 99%

Geometry and Response of Lindbladians

et al. 2016

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Markovian reservoir engineering, in which time evolution of a quantum system is governed by a Lindblad master equation, is a powerful technique in studies of quantum phases of matter and quantum information. It can be used to drive a quantum system to a desired (unique) steady state, which can be an exotic phase of matter difficult to stabilize in nature. It can also be used to drive a system to a unitarily evolving subspace, which can be used to store, protect, and process quantum information. In this paper, we derive a formula for the map corresponding to asymptotic (infinite-time) Lindbladian evolution and use it to study several important features of the unique state and subspace cases. We quantify how subspaces retain information about initial states and show how to use Lindbladians to simulate any quantum channels. We show that the quantum information in all subspaces can be successfully manipulated by small Hamiltonian perturbations, jump operator perturbations, or adiabatic deformations. We provide a Lindblad-induced notion of distance between adiabatically connected subspaces. We derive a Kubo formula governing linear response of subspaces to time-dependent Hamiltonian perturbations and determine cases in which this formula reduces to a Hamiltonian-based Kubo formula. As an application, we show that (for gapped systems) the zerofrequency Hall conductivity is unaffected by many types of Markovian dissipation. Finally, we show that the energy scale governing leakage out of the subspaces, resulting from either Hamiltonian or jump-operator perturbations or corrections to adiabatic evolution, is different from the conventional Lindbladian dissipative gap and, in certain cases, is equivalent to the excitation gap of a related Hamiltonian. DOI: 10.1103/PhysRevX.6.041031 Subject Areas: Condensed Matter Physics, Quantum Information, Statistical Physics I. MOTIVATION AND OUTLINEConsider coupling a quantum mechanical system to a Markovian reservoir which evolves initial states of the system into multiple nonequilibrium (i.e., nonthermal) asymptotic states in the limit of infinite time. After tracing out the degrees of freedom of the reservoir, the time evolution of the system is governed by a Lindbladian L [1,2] (see also ), and its various asymptotic states ρ ∞ are elements of an asymptotic subspace AsðHÞ-a subspace of OpðHÞ, the space of operators on the system Hilbert space H. The asymptotic subspace attracts all initial states ρ in ∈ OpðHÞ, is free from the decoherence effects of L, and any remaining time evolution within AsðHÞ is exclusively unitary. If AsðHÞ has no time evolution, all ρ ∞ are stationary or steady. This work provides a thorough investigation into the response and geometrical properties of the various asymptotic subspaces.On one hand, AsðHÞ that support quantum information [7][8][9][10] are promising candidates for storing, preserving, and manipulating such information, particularly when their states can be engineered to possess favorable features (e.g., topological protection [11][12][13])....

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Structure of uniformly continuous quantum Markov semigroups

Cited by 44 publications

References 36 publications

On Period, Cycles and Fixed Points of a Quantum Channel

On Period, Cycles and Fixed Points of a Quantum Channel

Mathematical models of Markovian dephasing

Geometry and Response of Lindbladians

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