This article provides an elementary introduction to Gaussian channels and their capacities. We review results on the classical, quantum, and entanglement assisted capacities and discuss related entropic quantities as well as additivity issues. Some of the known results are extended. In particular, it is shown that the quantum conditional entropy is maximized by Gaussian states and that some implications for additivity problems can be extended to the Gaussian setting.
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 address the question of the multiplicativity of the maximal p-norm output purities of bosonic Gaussian channels under Gaussian inputs. We focus on general Gaussian channels resulting from the reduction of unitary dynamics in larger Hilbert spaces. It is shown that the maximal output purity of tensor products of single-mode channels under Gaussian inputs is multiplicative for any p>1 for products of arbitrary identical channels as well as for a large class of products of different channels. In the case of p=2 multiplicativity is shown to be true for arbitrary products of generic channels act…
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.
A classification of one-mode Gaussian channels is given up to canonical unitary equivalence. A complementary to the quantum channel with additive classical Gaussian noise is described providing an example of one-mode Gaussian channel which is neither degradable nor anti-degradable.
We show that the following nontrivial necessary precondition for an entanglement evolution equation for pure Gaussian states under one-sided Gaussian channels holds. Suppose a Gaussian quantum channel acts on one mode of a pure entangled multi-mode Gaussian input state. Then, for a fixed channel, either all output states are entangled or none of them are. In other words, if the input state is Gaussian, pure and entangled, the separability after a one-sided Gaussian quantum channel does not depend on the input state, but only on the channel. Furthermore, a simple linear-algebraic separability c…
The optimization of entanglement evolution for two-mode Gaussian pure states under one-side Gaussian map is studied. Even there isn't complete information about the one-side Gaussian noisy channel, one can still maximize the entanglement distribution by testing the channel with only two specific states.
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…
We prove multiplicativity of maximal output $p$ norm of classical noise channels and thermal noise channels of arbitrary modes for all $p>1$ under the assumption that the input signal states are Gaussian states. As a direct consequence, we also show the additivity of the minimal output entropy and that of the energy-constrained Holevo capacity for those Gaussian channels under Gaussian inputs. To the best of our knowledge, newly discovered majorization relation on symplectic eigenvalues, which is also of independent interest, plays a central role in the proof.
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…
A noisy Gaussian channel is defined as a channel in which an input field mode is subjected to random Gaussian displacements in phase space. We introduce the quantum fidelity of a Gaussian channel for pure and mixed input states, and we derive a universal scaling law of the fidelity for pure initial states. We also find the maximum fidelity of a Gaussian channel over all input states. Quantum cloning and continuous-variable teleportation are presented as physical examples of Gaussian channels to which the fidelity results can be applied.
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…
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 set of quantum Gaussian channels acting on one bosonic mode can be classified according to the action of the group of Gaussian unitaries. We look for bounds on the classical capacity for channels belonging to such a classification. Lower bounds can be efficiently calculated by restricting to Gaussian encodings, for which we provide analytical expressions.
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…
For a quantum channel of additive Gaussian noise with loss, in the general case of $n$ copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive.
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…
We give necessary and sufficient conditions for a Gaussian quantum channel to have a dilation involving a passive, i.e., number-preserving unitary. We then establish a normal form of such channels: any passively dilatable channel is the result of applying passive unitaries to the input and output of a Gaussian additive channel. The latter combine the state of the system with that of the environment by means of a multi-mode beamsplitter.
A complete degradability analysis of one-mode Gaussian Bosonic 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 can not be used to transfer quantum information.
We address the question of optimality of entangled input states in quantum Gaussian memory channels. For a class of such channels, that can be traced back to the memoryless setting, we state a criterion which relate the optimality of entangled inputs to the symmetry properties of the channels' action. Several examples of channel models belonging to this class are discussed.
By elementary matrix algebra we show that every real $2n \times 2n$ matrix admits a dilation to an element of the real symplectic group $Sp (2(n+m))$ for some nonnegative integer $m.$ Our methods do not yield the minimum value of $m,$ for which such a dilation is possible. After listing some of the main properties of Gaussian states in $L^2 (\mathbb{R}^n),$ we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari, Holevo and Wolf \cite{3}.
In the present paper we introduce and study Bosonic Gaussian classical-quantum (c-q) channels; the embedding of the classical input into quantum is always possible and therefore the classical entanglement-assisted capacity C_{ea} under appropriate input constraint is well defined. We prove a general property of entropy increase for weak complementary channel, that implies the equality C_{ea}=C (where C is the unassisted capacity) for certain class of c-q Gaussian channel under appropriate energy-type constraint. On the other hand, we show by explicit example that the inequality C<C_{ea} is not…
We prove the equivalence of an arbitrary single-mode Gaussian quantum channel and a newly defined fiducial channel preceded by a phase shift and followed by a Gaussian unitary operation. This equivalence implies that the energy-constrained classical capacity of any single-mode Gaussian channel can be calculated based on this fiducial channel, which is furthermore simply realizable with a beam splitter, two identical single-mode squeezers, and a two-mode squeezer. In a large domain of parameters, we also provide an analytical expression for the Gaussian classical capacity, exploiting its additi…
We present an algorithm for calculation of the Gaussian classical capacity of a quantum bosonic memory channel with additive Gaussian noise. The algorithm, restricted to Gaussian input states, is applicable to all channels with noise correlations obeying certain conditions and works in the full input energy domain, beyond previous treatments of this problem. As an illustration, we study the optimal input states and capacity of a quantum memory channel with Gauss-Markov noise [J. Schäfer, Phys. Rev. A 80, 062313 (2009)]. We evaluate the enhancement of the transmission rate when using these opti…
We characterise Gaussian quantum channels that are Gaussian incompatibility breaking, that is, transform every set of Gaussian measurements into a set obtainable from a joint Gaussian observable via Gaussian postprocessing. Such channels represent local noise which renders measurements useless for Gaussian EPR-steering, providing the appropriate generalisation of entanglement breaking channels for this scenario. Understanding the structure of Gaussian incompatibility breaking channels contributes to the resource theory of noisy continuous variable quantum information protocols.
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.
We show that the minimum output entropy for all single-mode Gaussian channels is additive and is attained for Gaussian inputs. This allows the derivation of the channel capacity for a number of Gaussian channels, including that of the channel with linear loss, thermal noise, and linear amplification.
The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we introduce a notion of approximate private quantum channel ($\varepsilon$-PQC) on fermionic Gaussian systems (i.e., $\varepsilon$-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for $\varepsilon$-FPQC on the fermionic Gaussian system…
We present a protocol based on continuous-variable quantum teleportation and Gaussian post- selection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian…
We present upper bounds on the quantum and private capacity of single-mode, phase-insentitive Bosonic Gaussian Channels based on degradable extensions. Our findings are state-of-the-art in the following parameter regions: low temperature and high transmissivity for the thermal attenuator, low temperature for additive Gaussian noise, high temperature and intermediate amplification for the thermal amplifier.
We show that there exist Gaussian channels which are amendable. A channel is amendable if when applied twice is entanglement breaking while there exists a unitary filter such that, when interposed between the first and second action of the map, prevents the global transformation from being entanglement breaking [Phys. Rev. A 86, 052302 (2012)]. We find that, depending on the structure of the channel, the unitary filter can be a squeezing transformation or a phase shift operation. We also propose two realistic quantum optics experiments where the amendability of Gaussian channels can be verifie…
We consider the notion of canonical attacks, which are the cryptographic analog of the canonical forms of a one-mode Gaussian channel. Using this notion, we explore the connections between the degradability properties of the channel and its security for quantum key distribution. Finally, we also show some relations between canonical attacks and optimal Gaussian cloners.
We derive an upper limit for the mixedness of single bosonic mode gaussian states propagating in dissipative channels. It is a function of the initial squeezing and temperature of the channel only. Moreover the time at which von Neumann's entropy reaches its maximum value coincides with that of complete loss of coherence, thus defining a quantum-classical transition.
The additivity problem asks if the use of entanglement can boost the information-carrying capacity of a given channel beyond what is achievable by coding with simple product states only. This has recently been shown not to be the case for phase-insensitive one-mode Gaussian channels, but remains unresolved in general. Here we consider two general classes of bosonic noise channels, which include phase-insensitive Gaussian channels as special cases: these are beamsplitters with general, potentially non-Gaussian environment states and classical noise channels with general probabilistic noise. We …
The Amosov-Holevo-Werner conjecture implies the additivity of the minimum Re'nyi entropies at the output of a channel. The conjecture is proven true for all Re'nyi entropies of integer order greater than two in a class of Gaussian bosonic channel where the input signal is randomly displaced or where it is coupled linearly to an external environment.
A formulation of the generalized minimal output entropy conjecture for Gaussian channels is presented. It asserts that, for states with fixed input entropy, the minimal value of the output entropy of the channel (i.e. the minimal output entropy increment for fixed input entropy) is achieved by Gaussian states. In the case of centered channels (i.e. channels which do not add squeezing to the input state) this implies that the minimum is obtained by thermal (Gibbs) inputs. The conjecture is proved to be valid in some special cases.
In all practical communication channels, the code word consist of Gaussian states and the measurement strategy is often a Gaussian detector such as homodyning or heterodyning. In this paper we investigate the communication performance using a phase-conjugated alphabet and joint Gaussian detection in an phase-insensitive amplifying channel. We find that a communication scheme consisting of a phase-conjugated alphabet of coherent states and a joint detection strategy significantly outperforms a standard coherent-state strategy based in individual detection. Moreover, we show that the performance…
Activation of quantum capacity is a surprising phenomenon according to which the quantum capacity of a certain channel may increase by combining it with another channel with zero quantum capacity. Superactivation describes an even more particular occurrence, in which both channels have zero quantum capacity, but their composition has a nonvanishing one. We investigate these effects for all single-mode phase-insensitive Gaussian channels, which include thermal attenuators and amplifiers, assisted by a two-mode positive-partial-transpose channel. Our result shows that activation phenomena are sp…
I show that classical capacity per unit cost of noisy bosonic Gaussian channels can be attained by employing generalized on-off keying modulation format and a projective measurement of individual output states. This means that neither complicated collective measurements nor phase-sensitive detection is required to communicate over optical channels at the ultimate limit imposed by laws of quantum mechanics in the limit of low average cost.
The classical capacity of phase-invariant Gaussian channels has been recently determined under the assumption that such channels are memoryless. In this work we generalize this result by deriving the classical capacity of a model of quantum memory channel, in which the output states depend on the previous input states. In particular we extend the analysis of [C. Lupo, et al., PRL and PRA (2010)] from quantum limited channels to thermal attenuators and thermal amplifiers. Our result applies in many situations in which the physical communication channel is affected by nonzero memory and by therm…
For a quantum channel with additive Gaussian quantum noise, at the large input energy side, we prove that the one shot capacity is achieved by the thermal noise state for all Gaussian state inputs, it is also true for non-Gaussian input in the sense of first order perturbation. For a general case of $n$ copies input, we show that up to first order perturbation, any non-Gaussian perturbation to the product thermal state input has a less quantum information transmission rate when the input energy tend to infinitive.
Einstein-Podolsky-Rosen (EPR) steering is a useful resource for secure quantum information tasks. It is crucial to investigate the effect of inevitable loss and noise in quantum channels on EPR steering. We analyze and experimentally demonstrate the influence of purity of quantum states and excess noise on Gaussian EPR steering by distributing a two-mode squeezed state through lossy and noisy channels, respectively. We show that the impurity of state never leads to sudden death of Gaussian EPR steering, but the noise in quantum channel can. Then we revive the disappeared Gaussian EPR steering …
If a quantum channel is commutable with a certain set of operators, a Choi state of the channel becomes a sufficient resource for any entanglement generation achieved by a single use of the channel with the help of local operation and classical communications (LOCC). This property, called the teleportation stretchable, could be useful to determine an upper bound of a fundamental rate-loss trade-off for optical quantum key distribution. Unfortunately, the known formulation of the teleportation-stretching expansion for lossy Gaussian channels is based on an additional assumption of the Choi stat…
A recovery map effectively cancels the action of a quantum operation to a partial or full extent. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channe…
We investigate the capacity of bosonic quantum channels for the transmission of quantum information. Achievable rates are determined from measurable moments of the channel by showing that every channel can asymptotically simulate a Gaussian channel which is characterized by second moments of the initial channel. We calculate the quantum capacity for a class of Gaussian channels, including channels describing optical fibers with photon losses, by proving that Gaussian encodings are optimal. Along the way we provide a complete characterization of degradable Gaussian channels and those arising fr…
We provide a model to study memory effects in quantum Gaussian channels with additive classical noise over an arbitrary number of uses. The correlation among different uses is introduced by contiguous two-mode interactions. Numerical results for few modes are presented. They confirm the possibility to enhance the classical information rate with the aid of entangled inputs, and show a likely asymptotic behavior that should lead to the full capacity of the channel.
Gaussian loss channels are of particular importance since they model realistic optical communication channels. Except for special cases, quantum capacity of Gaussian loss channels is not yet known completely. In this paper, we provide improved upper bounds of Gaussian loss channel capacity, both in the energy-constrained and unconstrained scenarios. We briefly review the Gottesman-Kitaev-Preskill (GKP) codes and discuss their experimental implementation. We then prove, in the energy-unconstrained case, that the GKP codes achieve the quantum capacity of Gaussian loss channels up to at most a co…
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-Ma…
We show how to compute or at least to estimate various capacity-related quantities for Bosonic Gaussian channels. Among these are the coherent information, the entanglement assisted classical capacity, the one-shot classical capacity, and a new quantity involving the transpose operation, shown to be a general upper bound on the quantum capacity, even allowing for finite errors. All bounds are explicitly evaluated for the case of a one-mode channel with attenuation/amplification and classical noise.
We present a full experimental characterization of continuous variable quantum communication channels established by shared entanglement together with local operations and classical communication. The resulting teleportation channel was fully characterized by measuring all elements of the covariance matrix of the shared two-mode squeezed Gaussian state. From the experimental data we determined the lower bound to the quantum channel capacity, the teleportation fidelity of coherent states and the logarithmic negativity and the purity of the shared state. Additionally, a positive secret key rate …
Gaussian channels are the typical way to model the decoherence introduced by the environment in continuous-variable quantum states. It is known that those channels can be simulated by a teleportation protocol using as a resource state either a maximally entangled state passing through the same channel, i.e., the Choi-state, or a state that is entangled at least as much as the Choi-state. Since the construction of the Choi-state requires infinite mean energy and entanglement, i.e. it is unphysical, we derive instead every physical state able to simulate a given channel through teleportation wit…
We study classical capacity regions of quantum Gaussian multiple access channels (MAC). In classical variants of such channels, whilst some capacity superadditivity-type effects such as the so called {\it water filling effect} may be achieved, a fundamental classical additivity law can still be identified, {\it viz.} adding resources to one sender is never advantageous to other senders in sending their respective information to the receiver. Here, we show that quantum resources allows violation of this law, by providing two illustrative schemes of experimentally feasible Gaussian MACs.
We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms.
We address continuous variable quantum teleportation in Gaussian quantum noisy channels, either thermal or squeezed-thermal. We first study the propagation of twin-beam and evaluate a threshold for its separability. We find that the threshold for purely thermal channels is always larger than for squeezed-thermal ones. On the other hand, we show that squeezing the channel improves teleportation of squeezed states and, in particular, we find the class of squeezed states that are better teleported in a given noisy channel. Finally, we find regimes where optimized teleportation of squeezed states …
A long-standing problem on the classical capacity of bosonic Gaussian channels has recently been resolved by proving the minimum output entropy conjecture. It is also known that the ultimate capacity quantified by the Holevo bound can be achieved asymptotically by using an infinite number of channels. However, it is less understood to what extent the communication capacity can be reached if one uses a finite number of channels, which is a topic of practical importance. In this paper, we study the capacity of Gaussian communication, i.e., employing Gaussian states and Gaussian measurements to e…
We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of 2N canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.
