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We consider the most general Gaussian quantum Markov semigroup on a one-mode Fock space, discuss its construction from the generalized GKSL representation of the generator. We prove the known explicit formula on Weyl operators, characterize irreducibility and its equivalence to a Hörmander type condition on commutators and establish necessary and s...
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... that, if the quadratic term q 2 θ+π/2 in H does not vanish, then we can not get the same conclusion. This is the generator of a deterministic translation process with drift (in the generic case where |κ| sin(2(θ − φ)) = 0) towards the point x ∞ := (ζe −iθ )/( √ 2|κ| sin(2(θ − φ))) ( Fig. 1 below). The invariant density of the classical process is clearly δ x∞ which does not induce a faithful normal state on B(h). ...
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Citations
... In this section, we briefly introduce the fundamental concepts of Gaussian states and Gaussian QMSs. Theorem 2.3 and Theorem 2.4 are taken from [AFP21]. ...
We investigate the spectrum of the generator induced on the space of Hilbert-Schmidt operators by a Gaussian quantum Markov semigroup with a faithful normal invariant state in the general case, without any symmetry or quantum detailed balance assumptions. We prove that the eigenvalues are entirely determined by those of the drift matrix, similarly to classical Ornstein-Uhlenbeck semigroups. This result is established using a quasi-derivation property of the generator. Moreover, the same spectral property holds for the adjoint of the induced generator. Finally, we show that these eigenvalues constitute the entire spectrum when the induced generator has a spectral gap.
... A big challenge when investigating open quantum systems is to characterize invariant states when they are coupled to environments that drive the system out of equilibrium (see [10][11][12]). However, in the case of Gaussian Markov systems, explicit formulas ( [13][14][15][16][17]) allow one to write invariant states and analyze them. Furthermore, necessary and sufficient conditions ( [18,19]) are available to establish whether a certain state of a bipartite system is entangled or not. ...
... It is known that Gaussian QMSs with a stable matrix Z are, in turn, stable in the sense that each initial state converges toward a unique invariant state. More precisely, by [22] The same conclusion can also be obtained from arguments based on irreducibility in [13,26] taking care of domain conditions on G. ...
We show that a bipartite Gaussian quantum system interacting with an external Gaussian environment may possess a unique Gaussian entangled stationary state and that any initial state converges toward this stationary state. We discuss dependence of entanglement on temperature and interaction strength and show that one can find entangled stationary states only for low temperatures and weak interactions.
... Much theoretical work has focused on the more tractable generators of Gaussian dynamics semigroups, where the generator L is expressed as a quadratic form in the creation and annihilation operators [38,26,18,2,34]. For those generators, the Feller property as well as properties of the spectrum and convergence results are known [18,12,13,14,21]. ...
... Next, we realize that W k+4d,1 are L s -admissible subspaces, where we recall that d denotes the degree of L s . This already proves assumptions (1) and (2) in Theorem 2.10. Since the coefficients of the polynomials p H and p j are continuous and operators of the form ...
... where µ (2) k is defined in Lemma 4.4. Therefore, L[a 2 − e 2iπt/T α 2 ] generates a Sobolev and positivity preserving quantum evolution system P t,t 0 which satisfies for all states ρ ∈ W k,1 ...
The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is in bosonic error correction schemes, where dissipation is used to drive states back to the code-space – an invariant subspace protected against certain types of errors. In this paper, we investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that (i) the generator is in GKSL form with corresponding jump operators defined as polynomials of annihilation and creation operators; and (ii) the (possibly unbounded) generator increases all moments in a controlled manner. Additionally, we show that the level sets of operators with bounded first moments are admissible subspaces of the evolution, providing the foundations for a perturbative analysis. Our results also extend to time-dependent semigroups and multi-mode systems. We apply our general framework to two settings of interest in continuous variable quantum information processing. First, we provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of quantum dynamical semigroups. Second, we provide quantitative perturbation bounds for the steady state of the quantum Ornstein-Uhlenbeck semigroup and the invariant subspace of the photon dissipation used in bosonic error correction.
... In the realm of open quantum systems, [38] explores invariant subspaces in the presence of symmetries, which are associated with projections P of the corresponding semigroups (refer also to [8]). When the projection P is a rank-one projection, the study of the uniform convergence of irreducible Markov semigroups has been explored in [2,20,22]. Furthermore, the Dobrushin ergodicity coefficient δ(T ) has been employed to examine the asymptotic stability of Markov C 0 -semigroups on abstract state spaces in [15,16]. Particularly noteworthy are the perturbation results for C 0 -semigroups of Markov operators obtained in [17,19]. ...
... It is noted that the averages of operators appear in several fields of mathematics, theoretical physics, and other fields of science [2,6,21,37]. In [13,14], convergence of averages are deeply studied on different spaces. ...
The first goal of the present paper is to study residualities of the set of uniform P-ergodic Markov semigroups defined on abstract state spaces by means of a generalized Dobrushin ergodicity coefficient. In the last part of the paper, we explore uniform mean ergodicities of Markov semigroups.
... A big challenge when investigating open quantum systems is to characterize invariant states when they are coupled to environments that drive the system out of equilibrium (see [1,9,16]). However, in the case of Gaussian Markov systems, explicit formulas ( [3,10,14,21,24]) allow one to write invariant states and analyze them. Furthermore, necessary and sufficient conditions ( [26,27]) are available to establish whether a certain state of a bipartite system is entangled or not. ...
... The same conclusion can also be obtained from arguments based on irreducibility in [3,12] taking care of domain conditions on G. ...
We show that a bipartite Gaussian quantum system interacting with an external Gaussian environment may possess a unique Gaussian entangled stationary state and that any initial state converges towards this stationary state. We discuss dependence of entanglement on temperature and interaction strength and show that one can find entangled stationary states only for low temperatures and weak interactions.
... Much theoretical work has focused on the more tractable generators of Gaussian dynamics semigroups, where the generator L is expressed as a quadratic form in the creation and annihilation operators [35,22,16,2,31]. For those generators, the Feller property as well as properties of the spectrum and convergence results are known [16,11,12,13,50]. ...
The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. One such example is in bosonic error correction schemes, where dissipation is used to drive states back to the code-space -- an invariant subspace protected against certain types of errors. In this paper, we investigate perturbations of quantum dynamical semigroups that operate on continuous variable (CV) systems and admit an invariant subspace. First, we prove a generation theorem for quantum Markov semigroups on CV systems under the physical assumptions that (i) the generator has GKSL form with corresponding jump operators defined as polynomials of annihilation and creation operators; and (ii) the (possibly unbounded) generator increases all moments in a controlled manner. Additionally, we show that the level sets of operators with bounded first moments are admissible subspaces of the evolution, providing the foundations for a perturbative analysis. Our results also extend to time-dependent semigroups. We apply our general framework to two settings of interest in continuous variables quantum information processing. First, we provide a new scheme for deriving continuity bounds on the energy-constrained capacities of Markovian perturbations of Quantum dynamical semigroups. Second, we provide a quantitative analysis of the dampening of continuous-time evolutions generating a universal gate set for CAT-qubits outside their code-space.
... The classification of irreducible Gaussian QMSs in the one-dimensional case (d = 1) was done 1 showing that irreducibility holds when there are two linearly independent Kraus operators L , namely m = 2 (Ref. 1 We also show that this sufficient condition for irreducibility is also necessary (Theorem 11) in those situations in which, roughly speaking, the noise action on the open quantum system is fully non-commutative (see hypothesis FQN before Lemma 10). However, we do not expect it to be also necessary in general. ...
... The classification of irreducible Gaussian QMSs in the one-dimensional case (d = 1) was done 1 showing that irreducibility holds when there are two linearly independent Kraus operators L , namely m = 2 (Ref. 1 We also show that this sufficient condition for irreducibility is also necessary (Theorem 11) in those situations in which, roughly speaking, the noise action on the open quantum system is fully non-commutative (see hypothesis FQN before Lemma 10). However, we do not expect it to be also necessary in general. ...
... If the linear space V 2d−m coincides with C 2d , then a candidate invariant subspace (well-behaved with respect to Dom(G)) must be invariant for all creation and annihilation operators, whence it must be 0 or h either by irreducibility of the Fock representation of the CCR or by our standard argument as in the proof of Theorem 5 in Ref. 1. This leads us to the following. ...
The generator of a Gaussian quantum Markov semigroup on the algebra of bounded operator on a [Formula: see text]-mode Fock space is represented in a generalized GKLS form with an operator [Formula: see text] quadratic in creation and annihilation operators and Kraus operators [Formula: see text] linear in creation and annihilation operators. Kraus operators, commutators [Formula: see text] and iterated commutators [Formula: see text] up to the order [Formula: see text], as linear combinations of creation and annihilation operators determine a vector in [Formula: see text]. We show that a Gaussian quantum Markov semigroup is irreducible if such vectors generate [Formula: see text], under the technical condition that the domains of [Formula: see text] and the number operator coincide. Conversely, we show that this condition is also necessary if the linear space generated by Kraus operators and their iterated commutator with [Formula: see text] is fully non-commutative.
... It is important due to the fact that it is still very actively studied area. [22][23][24][25][26] We use the notations similar to Refs. 16, 27 and 28. ...
In this paper, we discuss effective quantum dynamics obtained by averaging projector with respect to free dynamics. For unitary dynamics generated by quadratic fermionic Hamiltonians, we obtain effective Heisenberg dynamics. By perturbative expansions, we obtain the correspondent effective time-local Heisenberg equations. We also discuss a similar problem for bosonic case.
... 2. Two elements z 1 , z 2 of M are called symplectically orthogonal if they satisfy z 1 , z 2 = 0. 3. Let M 1 ⊂ M be a real linear subspace. We call symplectic complement of M 1 in M , and denote it by M 1 , the set ...
... , α(d)) is a multi-index, |α| = α(1) + . . . + α(d), and the vector e T α = (e α (1) , . . . , e α(d) ). ...
... , α(d)) is a multi-index, |α| = α(1) + . . . + α(d), and the vector e α = (e α (1) , . . . , e α(d) ). ...
We demonstrate a method for finding the decoherence-free subalgebra N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ ( C d ) on C d . We show that N ( T ) is a type I von Neumann algebra L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers d c , d f ≤ d . This result is illustrated by some applications and examples.
... Moreover, this general time-independent situation could be interesting for the purposes of incoherent control [15,14] and tomography in the case of time-dependent generators [16]. Let us remark that although the GKSL equations with a quadratic generator are well-studied, there is still keen current interest in further reasearch into their properties and applications [19,20,21,22,23,17,18]. ...
... Eqs. (19) and (21) In particular, such a tensor contains the terms of the form â † iâ iâ † jâ j − â † iâ i â † jâ j , which describe the correlations of intensities of electromagnetic field [13,Subsec. 12.12.2]. ...
... Then Eqs. (19) take the form ...
We derive Heisenberg equations for arbitrary high order moments of creation and annihilation operators in the case of the quantum master equation with a multimode generator which is quadratic in creation and annihilation operators and obtain their solutions. Based on them we also derive similar equations for the case of the quantum master equation, which occur after averaging the dynamics with a quadratic generator with respect to the classical Poisson process. This allows us to show that dynamics of arbitrary finite-order moments of creation and annihilation operators is fully defined by finite number of linear differential equations in this case.
