Bosonic Gaussian Channels Research Articles

Majorization Theory Approach to the Gaussian Channel Minimum Entropy Conjecture

A long-standing open problem in quantum information theory is to find the classical capacity of an optical communication link, modeled as a Gaussian bosonic channel. It has been conjectured that this capacity is achieved by a random coding of coherent states using an isotropic Gaussian distribution in phase space. We show that proving a Gaussian minimum entropy conjecture for a quantum-limited amplifier is actually sufficient to confirm this capacity conjecture, and we provide a strong argument towards this proof by exploiting a connection between quantum entanglement and majorization theory.

Gaussian matrix-product states for coding in bosonic communication channels

The communication capacity of Gaussian bosonic channels with memory has recently attracted much interest. Here, we investigate a method to prepare the multimode entangled input symbol states for encoding classical information into these channels. In particular, we study the usefulness of a Gaussian matrix product state (GMPS) as an input symbol state, which can be sequentially generated although it remains heavily entangled for an arbitrary number of modes. We show that the GMPS can achieve more than 99.9% of the Gaussian capacity for Gaussian bosonic memory channels with a Markovian or non-Markovian correlated noise model in a large range of noise correlation strengths. Furthermore, we present a noise class for which the GMPS is the exact optimal input symbol state of the corresponding channel. Since GMPS are ground states of particular quadratic Hamiltonians, our results suggest a possible link between the theory of quantum communication channels and quantum many-body physics.

Operator-sum representation for bosonic Gaussian channels

Operator-sum or Kraus representations for single-mode bosonic Gaussian channels are developed, and several of their consequences explored. The fact that the two-mode metaplectic operators acting as unitary purification of these channels do not, in their canonical form, mix the position and momentum variables is exploited to present a procedure which applies uniformly to all families in the Holevo classification. In this procedure the Kraus operators of every quantum-limited Gaussian channel can be simply read off from the matrix elements of a corresponding metaplectic operator. Kraus operators are employed to bring out, in the Fock basis, the manner in which the antilinear, unphysical matrix transposition map when accompanied by injection of a threshold classical noise becomes a physical channel, denoted $\mathcal{D}(\ensuremath{\kappa})$ in the Holevo classification. The matrix transposition channels $\mathcal{D}(\ensuremath{\kappa})$, $\mathcal{D}({\ensuremath{\kappa}}^{\ensuremath{-}1})$ turn out to be a dual pair in the sense that their Kraus operators are related by the adjoint operation. The amplifier channel with amplification factor $\ensuremath{\kappa}$ and the beam-splitter channel with attenuation factor ${\ensuremath{\kappa}}^{\ensuremath{-}1}$ turn out to be mutually dual in the same sense. The action of the quantum-limited attenuator and amplifier channels as simply scaling maps on suitable quasiprobabilities in phase space is examined in the Kraus picture. Consideration of cumulants is used to examine the issue of fixed points. The semigroup property of the amplifier and attenuator families leads in both cases to a Zeno-like effect arising as a consequence of interrupted evolution. In the cases of entanglement-breaking channels a description in terms of rank 1 Kraus operators is shown to emerge quite simply. In contradistinction, it is shown that there is not even one finite rank operator in the entire linear span of Kraus operators of the quantum-limited amplifier or attenuator families, an assertion far stronger than the statement that these are not entanglement breaking channels. A characterization of extremality in terms of Kraus operators, originally due to Choi, is employed to show that all quantum-limited Gaussian channels are extremal. The fact that almost every noisy Gaussian channel can be realized as a product of a pair of quantum-limited channels is used to construct a discrete set of linearly independent Kraus operators for these noisy Gaussian channels, including the classical noise channel, and these Kraus operators have a particularly simple structure.

Robustness of Non-Gaussian Entanglement against Noisy Amplifier and Attenuator Environments

The recently developed Kraus representation for bosonic Gaussian channels is employed to study analytically the robustness of non-Gaussian entanglement against evolution under noisy attenuator and amplifier environments, and compare it with the robustness of Gaussian entanglement. Our results show that some non-Gaussian states with one ebit of entanglement are more robust than all Gaussian states, even the ones with arbitrarily large entanglement, a conclusion of direct consequence to the recent conjecture by Allegra et al. [Phys. Rev. Lett. 105, 100503 (2010)].

Optimal unitary dilation for bosonic Gaussian channels

A generic quantum channel can be represented in terms of a unitary interaction between the information-carrying system and a noisy environment. Here, the minimal number of quantum Gaussian environmental modes required to provide a unitary dilation of a multi-mode bosonic Gaussian channel is analyzed both for mixed and pure environment corresponding to the Stinespring representation. In particular, for the case of pure environment we compute this quantity and present an explicit unitary dilation for arbitrary bosonic Gaussian channel. These results considerably simplify the characterization of these continuous-variable maps and can be applied to address some open issues concerning the transmission of information encoded in bosonic systems.

The Choi–Jamiolkowski forms of quantum Gaussian channels

We obtain explicit expressions for the Choi–Jamiolkowski (CJ) forms and operators defining a general bosonic Gaussian channel. The four principal cases are considered in Sec. III; in Sec. IV, we give a decomposition of Gaussian CJ form into product of these four principal types and provide a necessary and sufficient condition for the existence of the bounded CJ operator.

Entanglement behaviors of two-mode squeezed thermal states in two different environments

We study entanglement properties of two-mode squeezed thermal states subjected to two sources of decoherence: the common reservoirs and the bosonic memory Gaussian channel. For the former one, we find that there exist three different behaviors: no-sudden death, sudden death, and no-creation of entanglement. The range of parameters characterizing these processes is obtained. For the latter one, we obtain a threshold in the degree of squeezing above which the input states remain always entangled. Otherwise, no entanglement is allowed in bosonic Gaussian channel with memory effect. We show that a degree of memory for quantum channel can be help to increase the initial entanglement, while the mean number of added thermal photons is to fasten the decoherence process.

Entropy gain and the Choi-Jamiolkowski correspondence for infinite-dimensional quantum evolutions

We consider the entropy gain for infinite-dimensional evolutions and show that unlike in the finitedimensional case, there are many channels with positive minimal entropy gain. We obtain a new lower bound and compute the minimal entropy gain for a broad class of bosonic Gaussian channels. We mathematically formulate the Choi-Jamiolkowski (CJ) correspondence between channels and states in the infinite-dimensional case in a form close to the form used in quantum information theory. In particular, we obtain an explicit expression for the CJ operator defining a general nondegenerate bosonic Gaussian channel and compute its norm.

The entropy gain of infinite-dimensional quantum evolutions

In the present paper we study the entropy gain $H(\Phi [\rho])-H(\rho)$ for infinite-dimensional channels $\Phi$. We show that unlike finite-dimensional case where the minimal entropy gain is always nonpositive \cite{al}, there is a plenty of channels with positive minimal entropy gain. We obtain the new lower bound and compute the minimal entropy gain for a broad class of Bosonic Gaussian channels by proving that the infimum is attained on the Gaussian states.

The semigroup sturucture of Gaussian channels

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.

The semigroup structure of gaussian channels

We investigate the semigroup structure of bosonic Gaussian quantum channels. Par-ticular 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 di-visible). We show that the non-compactness of the set of Gaussian channels allows forremarkable differences when comparing the semigroup structure with that of finite di-mensional quantum channels. For instance, every irreversible Gaussian channel is shownto be divisible in spite of the existence of Gaussian channels which are not infinitesimaldivisible. A simpler and known consequence of non-compactness is the lack of generatorsfor certain reversible channels. Along the way we provide new representations for classesof Gaussian channels: as matrix semigroup, complex valued positive matrices or in termsof a simple form describing almost all one-parameter semigroups.

Capacities of lossy bosonic channel with correlated noise

We evaluate the information capacities of a lossy bosonic channel with correlated noise. The model generalizes the one recently discussed by Pilyavets et al (2008 Phys. Rev. A 77 052324), where memory effects come from the interaction with correlated environments. Environmental correlations are quantified by a multimode squeezing parameter, which vanishes in the memoryless limit. We show that a global encoding/decoding scheme, which involves input-entangled states among different channel uses, is always preferable with respect to a local one in the presence of memory. Moreover, in a certain range of the parameters, we provide an analytical expression for the classical capacity of the channel showing that a global encoding/decoding scheme allows it to be attained. All the results can be applied to a broad class of bosonic Gaussian channels.

Adaptive continuous variable quantum key distribution based on channel estimation

In order to obtain steady key rate for a continuous variable quantum key distribution system, Alice should adaptively adjust the intensity of optical pulses she sents. In this paper, an optical fiber channel is taken as the additive bosonic quantum Gaussian channel. It is proven that a Gaussian channel transforms Gaussian states into Gaussian states. The maximum-likelihood method is used to estimate the channel parameters after balanced homodyne detection. Then, Alice can change the intensity of optical pulses adaptively according to the estimated noise level, thereby a steady key rate can be obtained.

Multi-mode bosonic Gaussian channels

A complete analysis of multi-mode bosonic Gaussian channels (BGCs) is proposed. We clarify the structure of unitary dilations of general Gaussian channels involving any number of bosonic modes and present a normal form. The maximum number of auxiliary modes that is needed is identified, including all rank deficient cases, and the specific role of additive classical noise is highlighted. By using this analysis, we derive a canonical matrix form of the noisy evolution of n-mode BGCs and of their weak complementary counterparts, based on a recent generalization of the normal mode decomposition for non-symmetric or locality constrained situations. This allows us to simplify the weak-degradability classification. Moreover, we investigate the structure of some singular multi-mode channels, like the additive classical noise channel that can be used to decompose a noisy channel in terms of a less noisy one in order to find new sets of maps with zero quantum capacity. Finally, the two-mode case is analyzed in detail. By exploiting the composition rules of two-mode maps and the fact that anti-degradable channels cannot be used to transfer quantum information, we identify sets of two-mode bosonic channels with zero capacity.

DISCRIMINATING QUANTUM CHANNELS WITH RELATIVE ENTROPY

We address the possibilities of distinguishing the quantum channels in terms of relative entropy. Particularly, depolarizing channels and bosonic Gaussian channels are considered. To some extent, the relative entropy can be treated as a measure to discriminate the quantum channels.

THE FIDELITY OF GENERAL BOSONIC CHANNELS WITH PURE STATE INPUT

We give the exact form of the fidelity for general bosonic Gaussian channels in cases where the channel's input are limited to Gaussian pure states. Some examples have been studied and the properties of the fidelity for memory channel are carefully examined.

Degradability of Bosonic Gaussian channels

The notion of weak-degradability of quantum channels is introduced by generalizing the degradability definition given by Devetak and Shor. Exploiting the unitary equivalence with beam-splitter/amplifier channels we then prove that a large class of one-mode Bosonic Gaussian channels are either weakly degradable or anti-degradable. In the latter case this implies that their quantum capacity Q is null. In the former case instead, this allows us to establish the additivity of the coherent information for those maps which admit unitary representation with single-mode pure environment.

One-mode bosonic Gaussian channels: a full weak-degradability classification

A complete degradability analysis of one-mode bosonic Gaussian channels is presented. We show that apart from the class of channels which are unitarily equivalent to the channels with additive classical noise, these maps can be characterized in terms of weak- and/or anti-degradability. Furthermore a new set of channels which have null quantum capacity is identified. This is done by exploiting the composition rules of one-mode Gaussian maps and the fact that anti-degradable channels cannot be used to transfer quantum information.

INFORMATION TRANSMISSION VIA ENTANGLED QUANTUM STATES IN GAUSSIAN CHANNELS WITH MEMORY

Gaussian quantum channels have recently attracted a growing interest, since they may lead to a tractable approach to the generally hard problem of evaluating quantum channel capacities. However, the analysis performed so far has always been restricted to memoryless channels. Here, we consider the case of a bosonic Gaussian channel with memory, and show that the classical capacity can be significantly enhanced by employing entangled input symbols instead of product symbols.

Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels

This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the maximal distance property is obtained.