We investigate the semigroup structure of bosonic Gaussian quantum channels. Particular focus lies on the sets of channels which are divisible, idempotent or Markovian (in the sense of either belonging to one-parameter semigroups or being infinitesimal divisible). We show that the non-compactness of the set of Gaussian channels allows for remarkable differences when comparing the semigroup structure with that of finite dimensional quantum channels. For instance, every irreversible Gaussian channel is shown to be divisible in spite of the existence of Gaussian channels which are not infinitesimal divisible. A simpler and known consequence of non-compactness is the lack of generators for certain reversible channels. Along the way we provide new representations for classes of Gaussian channels: as matrix semigroup, complex valued positive matrices or in terms of a simple form describing almost all one-parameter semigroups.
Using the tool of quantum characteristic functions of n-mode states in the boson Fock space Γ(C_n) we construct a semigroup of quantum information channels. This leads to a special class of one-parameter semigroups of such channels. These semigroups are concrete but their generators have unbounded operator coefficients. These one-parameter semigroups are also quantum dynamical semigroups and the form of the generators involve additional features which do not appear in the standard GKSL form. A heuristic discussion of the form of these generators is included. In the wake of this analysis many o…
Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels, while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an effi…
In the present work we review and refine some results about fixed points of semigroups of quantum channels. Noncommutative potential theory enables us to show that the set of fixed points of a recurrent semigroup is a W*-algebra; aside from the intrinsic interest of this result, it brings an improvement in the study of fixed points by means of absorption operators (a noncommutative generalization of absorption probabilities): under the assumption of absorbing recurrent space (hence allowing non-trivial transient space) we can provide a description of the fixed points set and a probabilistic ch…
We study the conditions under which a semigroup is obtained upon convex combinations of channels. In particular, we study the set of Pauli and generalized Pauli channels. We find that mixing only semigroups can never produce a semigroup. Counter-intuitively, we find that for a convex combination to yield a semigroup, most of the input channels have to be noninvertible.
We characterize all Gaussian dynamical semigroups in continuous variables quantum systems of n-bosonic modes which have a thermal Gibbs state as a stationary solution. This is performed through an explicit relation between the diffusion and dissipation matrices, which characterize the semigroup dynamic, and the covariance matrix of the thermal equilibrium state. We also show that Alicki's quantum detailed-balance condition, based on a Gelfand-Naimark-Segal inner product, allows the determination of the temperature dependence of the diffusion and dissipation matrices, and the identification of …
Lie-Semigroup Structures for Reachability and Control of Open Quantum Systems: Viewing Markovian Quantum Channels as Lie Semigroups and GKS-Lindblad Generators as Lie Wedge
We analyze the set ${\cal A}_N^Q$ of mixed unitary channels represented in the Weyl basis and accessible by a Lindblad semigroup acting on an $N$-level quantum system. General necessary and sufficient conditions for a mixed Weyl quantum channel of an arbitrary dimension to be accessible by a semigroup are established. The set ${\cal A}_N^Q$ is shown to be log--convex and star-shaped with respect to the completely depolarizing channel. A decoherence supermap acting in the space of Lindblad operators transforms them into the space of Kolmogorov generators of classical semigroups. We show that fo…
In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as…
We study the memory property of the channels obtained by convex combinations of Markovian channels that are not necessarily quantum dynamical semigroups (QDSs). In particular, we characterize the geometry of the region of (non-)Markovian channels obtained by the convex combination of the three Pauli channels, as a function of deviation from the semigroup form in a family of channels. The regions are highly convex, and interestingly, the measure of the non-Markovian region shrinks with greater deviation from the QDS structure for the considered family, underscoring the counterintuitive nature o…
In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space $Γ(\mathbb{C}^n)$ over $\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian …
We show that the discrete-time evolution of an open quantum system generated by a single quantum channel $T$ can be embedded in the discrete-time evolution of an enlarged closed quantum system, i.e. we construct a unitary dilation of the discrete-time quantum-dynamical semigroup $(T^n)_{n \in \mathbb N_0}$. In the case of a cyclic channel $T$, the auxiliary space may be chosen (partially) finite-dimensional. We further investigate discrete-time quantum control systems generated by finitely many commuting quantum channels and prove a similar unitary dilation result as in the case of a single ch…
We demonstrate a method for finding the decoherence-subalgebra $\mathcal{N}(\mathcal{T})$ of a Gaussian quantum Markov semigroup on the von Neumann algebra $\mathcal{B}(Γ(\mathbb{C}^d))$ of all bounded operator on the Fock space $Γ(\mathbb{C}^d)$ on $\mathbb{C}^d$. We show that $\mathcal{N}(\mathcal{T})$ is a type I von Neumann algebra $L^\infty(\mathbb{R}^{d_c};\mathbb{C})\bar{\otimes}\mathcal{B}(Γ(\mathbb{C}^{d_f}))$ determined, up to unitary equivalence, by two natural numbers $d_c,d_f\leq d$. This result is illustrated by some applications and examples.
We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, $Φ= e^{{\cal L}t}$,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together wit…
Finite-time Markovian channels, unlike their infinitesimal counterparts, do not form a convex set. As a particular instance of this observation, we consider the problem of mixing the three Pauli channels, conservatively assumed to be quantum dynamical semigroups, and fully characterize the resulting ``Pauli simplex.'' We show that neither the set of non-Markovian (completely positive indivisible) nor Markovian channels is convex in the Pauli simplex, and that the measure of non-Markovian channels is about 0.87. All channels in the Pauli simplex are P divisible. A potential application in the c…
A conjecture for the universal constraints for relaxation rates of a quantum dynamical semigroup is proposed. It is shown that it holds for several interesting classes of semigroups, e.g. unital semigroups and semigroups derived in the weak coupling limit from the proper microscopic model. Moreover, proposed conjecture is supported by numerical analysis. This conjecture has several important implications: it allows to provide universal constraints for spectra of quantum channels and provides necessary condition to decide whether a given channel is consistent with Markovian evolution.
We construct relativistic quantum Markov semigroups from covariant completely positive maps. We proceed by generalizing a step in Stinespring's dilation to a general system of imprimitivity and basing it on Poincaré group. The resulting noise channels are relativistically consistent and the method is applicable to any fundamental particle, though we demonstrate it for the case of light-like particles. The Krauss decomposition of the relativistically consistent completely positive identity preserving maps (our set up is in Heisenberg picture) enables us to construct the covariant quantum Markov…
In the study of Markovian processes, one of the principal achievements is the equivalence between the $Φ$-Sobolev inequalities and an exponential decrease of the $Φ$-entropies. In this work, we develop a framework of Markov semigroups on matrix-valued functions and generalize the above equivalence to the exponential decay of matrix $Φ$-entropies. This result also specializes to spectral gap inequalities and modified logarithmic Sobolev inequalities in the random matrix setting. To establish the main result, we define a non-commutative generalization of the carré du champ operator, and prove a …
In this paper, we analyze the evolution of the generalized Pauli channels governed by the memory kernel master equation. We provide necessary and sufficient conditions for the memory kernel to give rise to the legitimate (completely positive and trace-preserving) quantum evolution. In particular, we analyze a class of kernels generating the quantum semi-Markov evolution, which is a natural generalization of the Markovian semigroup. Interestingly, the convex combination of Markovian semigroups goes beyond the semi-Markov case. Our analysis is illustrated with several examples.
Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this articl…
In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamiołkowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaus…
We compute the $\mathcal S^p \to \mathcal S^p$ norm of a general Gaussian gauge-covariant multi-mode channel for any $1\leq p<\infty$, where $\mathcal S^p$ is a Schatten space. As a consequence, we verify the Gaussian optimizer conjecture and the multiplicativity conjecture in these cases.
We prove that whenever the coherent information of a one-mode Gaussian channel is non-zero its supremum is achieved for the infinite input power. This is a well established fact for the zero added classical noise, whereas the nonzero case has not been studied in detail. The presented analysis fills the gap for three canonical classes of one-mode Gaussian channels: the lossy, amplifying and additive classical noise channel class. For the remaining one-mode Gaussian channel classes the coherent information is known to vanish.
In the paper, we give a characterization of arbitrary $n$-mode Gaussian coherence breaking channels (GCBCs) and show the tensor product of a GCBC and arbitrary a Gaussian channel maps all input states into product states. The inclusion relation among GCBCs, Gaussian positive partial transpose channels (GPPTC), entanglement breaking channels (GEBC) and Gaussian classical-quantum channels (GCQC) is displayed. Finally, we prove that Gaussian classical ($χ$-) capacity for the tensor products of GCBCs and arbitrary Gaussian channels is additive.
We prove the longstanding conjecture stating that Gaussian thermal input states minimize the output von Neumann entropy of one-mode phase-covariant quantum Gaussian channels among all the input states with a given entropy. Phase-covariant quantum Gaussian channels model the attenuation and the noise that affect any electromagnetic signal in the quantum regime. Our result is crucial to prove the converse theorems for both the triple trade-off region and the capacity region for broadcast communication of the Gaussian quantum-limited amplifier. Our result extends to the quantum regime the Entropy…
Coherence is a basic notion for quantum states. Instead of quantum states, in this work, We establish a resource theory for quantifying the coherence of Gaussian channels. To do this, we propose the definitions of incoherent Gaussian channels and incoherent Gaussian superchannels.
We analyze the fidelity of the generalized Pauli channels governed by memory kernel master equations. It is shown that, by appropriate engineering of parameters of the corresponding memory kernel, the quantum evolution with non-local noise can have higher fidelity than the corresponding purely Markovian evolution governed by the Markovian semigroup. Similar engineering can substantially influence the evolution of quantum entanglement, entropy, and quantum coherence.
We study the degradability of fermionic Gaussian channels. Fermionic quantum channels are a central building block of quantum information processing with fermions, and the family of Gaussian channels, in particular, is relevant in the emerging field of electron quantum optics and its applications for quantum information. Degradable channels are of particular interest since they have a simple formula that characterizes their quantum capacity. We derive a simple standard form for fermionic Gaussian channels. This allows us to fully characterize all degradable $n$-mode fermionic Gaussian channels…
Recently, there have been considerable progresses on the bounds of various quantum channel capacities for bosonic Gaussian channels. Especially, several upper bounds for the classical capacity and the quantum capacity on the bosonic Gaussian channels, via a technique known as quantum entropy power inequality, have been shed light on understanding the mysterious quantum-channel-capacity problems. However, upper bounds for the private capacity on quantum channels are still missing for the study on certain universal upper bounds. Here, we derive upper bounds on the private capacity for bosonic Ga…
We study one-mode Gaussian quantum channels in continuous-variable systems by performing a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. Our study covers as particular limit the case of singular channels, thus connecting our results with their known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by derivi…
We present a framework for studying bosonic non-Gaussian channels of continuous-variable systems. Our emphasis is on a class of channels that we call photon-added Gaussian channels, which are experimentally viable with current quantum-optical technologies. A strong motivation for considering these channels is the fact that it is compulsory to go beyond the Gaussian domain for numerous tasks in continuous-variable quantum information processing such as entanglement distillation from Gaussian states and universal quantum computation. The single-mode photon-added channels we consider are obtained…
In this work we investigate the effectiveness of continuous-variable (CV) entangled states, transferred through high-loss atmospheric channels, as a means of viable quantum key distribution (QKD) between terrestrial stations and low-Earth orbit (LEO) satellites. In particular, we investigate the role played by the Gaussian CV states as compared to non-Gaussian states. We find that beam-wandering induced atmospheric losses lead to QKD performance levels that are in general quite different from those found in fixed-attenuation channels. For example, circumstances can be found where no QKD is via…
The full solution of the optimization problem giving the Gaussian capacity of the single-mode fiducial Gaussian quantum channel is provided. Since it was shown that the Gaussian capacity of an arbitrary (phase-sensitive or insensitive) single-mode Gaussian quantum channel is equal to the Gaussian capacity of this fiducial channel, the solution presented in this work can be regarded as universal. The analytical study of this solution, below and above the energy threshold, shows that the dependence of the Gaussian capacity on the environment noise squeezing is not monotonic. In particular, the c…
Quantum channels can be activated by a kind of channels whose quantum capacity is zero. This activation effect might be useful to overcome noise of channels by attaching other channels which can enhance the capacity of a given channel. In this work, we show that such an activation is possible by specific positive-partial-transpose channels for Gaussian lossy channels whose quantum capacities are known. We also test more general case involving Gaussian thermal attenuator whose the exact value of quantum capacity has been unknown so far. For a recently suggested narrow upper bound on quantum cap…
