As with classical information, error-correcting codes enable reliable transmission of quantum information through noisy or lossy channels. In contrast to the classical theory, imperfect quantum channels exhibit a strong kind of synergy: there exist pairs of discrete memoryless quantum channels, each of zero quantum capacity, which acquire positive quantum capacity when used together. Here we show that this "superactivation" phenomenon also occurs in the more realistic setting of optical channels with attenuation and Gaussian noise. This paves the way for its experimental realization and application in real-world communications systems.
Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user and broadcast channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. In this paper, it is shown that the second conjecture suffices to prove the classical capacity of the bosonic wiretap channel, which in …
Bosonic channels describe quantum-mechanically many practical communication links such as optical, microwave, and radiofrequency. We investigate the maximum rates for the bosonic multiple access channel (MAC) in the presence of thermal noise added by the environment and when the transmitters utilize Gaussian state inputs. We develop an outer bound for the capacity region for the thermal-noise lossy bosonic MAC. We additionally find that the use of coherent states at the transmitters is capacity-achieving in the limits of high and low mean input photon numbers. Furthermore, we verify that coher…
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…
The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its Holevo information is not superadditive, and that a coherent-state encoding achieves capacity. The capacity of far-field, free-space optical communications is given as an example.
Secret-sharing building blocks based on quantum broadcast communication are studied. The confidential capacity region of the pure-loss bosonic broadcast channel is determined, both with and without key assistance, and an achievable region is established for the lossy bosonic broadcast channel. If the main receiver has a transmissivity of η<1/2, then confidentiality solely relies on the key-assisted encryption of the one-time pad. We also address conference key agreement for the distillation of two keys, a public key and a secret key. A regularized formula is derived for the key-agreement capac…
We consider a line with noise in the simplest case. Loss does not add noise. Amplification via phase insensitive amplifiers do add noise. A lower bound of this capacity is the quantum analog to the Shannon capacity of a linear channel with additive white Gaussian noise, namely the difference of the Von Neumann entropy of the signal plus noise at the output of the line and the entropy of the noise alone. We show that this expression is indeed the capacity for the case of an amplifier with infinitesimal gain $G = 1+ε$, and for a cascade of an amplifier with arbitrary gain and a large loss, such …
We prove an upper bound on the private capacity of the single-mode noiseless bosonic wiretap channel. Combined with a previous lower bound, we obtain the low photon-number asymptotic expression for the private capacity. We then show that the multiple-mode noiseless bosonic wiretap channel is equivalent to parallel single-mode channels, hence the single-mode bounds can be applied. Finally, we consider multiple-spatial-mode propagation through atmospheric turbulence, and derive a private-capacity lower bound that only requires second moments of the channel matrix.
We show that when coherent-state encoding is employed in conjunction with coherent detection, the Bosonic broadcast channel is equivalent to a classical degraded Gaussian broadcast channel whose capacity region is dual to that of the classical Gaussian multiple-access channel. We further show that if a minimum output-entropy conjecture holds true, then the ultimate classical information capacity of the Bosonic broadcast channel can be achieved by a coherent-state encoding. We provide some evidence in support of the conjecture.
We analyze the role of nonlinear Hamiltonians in bosonic channels. We show that the information capacity as a function of the channel energy is increased with respect to the corresponding linear case, although only when the energy used for driving the nonlinearity is not considered as part of the energetic cost and when dispersive effects are negligible.
A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest o…
We determine the ultimate classical information capacity of a linear time-invariant bosonic channel with additive phase-insensitive Gaussian noise. This channel can model fiber-optic communication at power levels below the threshold for significant nonlinear effects. We provide a general continuous-time result that gives the ultimate capacity for such a channel operating in the quasimonochromatic regime under an average power constraint. This ultimate capacity is compared with corresponding results for heterodyne and homodyne detection over the same channel.
This paper strengthens the interpretation and understanding of the classical capacity of the pure-loss bosonic channel, first established in [Giovannetti et al., Physical Review Letters 92, 027902 (2004), arXiv:quant-ph/0308012]. In particular, we first prove that there exists a trade-off between communication rate and error probability if one imposes only a mean-photon number constraint on the channel inputs. That is, if we demand that the mean number of photons at the channel input cannot be any larger than some positive number N_S, then it is possible to respect this constraint with a code …
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.
Previous work on the classical information capacities of bosonic channels has established the capacity of the single-user pure-loss channel, bounded the capacity of the single-user thermal-noise channel, and bounded the capacity region of the multiple-access channel. The latter is a multi-user scenario in which several transmitters seek to simultaneously and independently communicate to a single receiver. We study the capacity region of the bosonic broadcast channel, in which a single transmitter seeks to simultaneously and independently communicate to two different receivers. It is known that…
The bosonic quantum channels have recently attracted a growing interest, motivated by the hope that they open a tractable approach to the generally hard problem of evaluating quantum channel capacities. These studies, however, have always been restricted to memoryless channels. Here, it is shown that the classical capacity of a bosonic Gaussian channel with memory can be significantly enhanced if entangled symbols are used instead of product symbols. For example, the capacity of a photonic channel with 70%-correlated thermal noise of one third the shot noise is enhanced by about 11% when using…
Bosonic channels are important in practice as they form a simple model for free-space or fiber-optic communication. Here we consider a single-sender two-receiver pure-loss bosonic broadcast channel and determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they are allowed arbitrary public classical communication. We show how the state merging protocol leads to achievable rates in this setting, giving an inner bound on the capacity region. We also evaluate an outer bound on the region by using the…
In this paper, using the notion of nonlinear coherent states, we define a deformed bosonic dephasing channel modelling the impact of a Kerr medium on a quantum state, as it occurs, for instance, in quantum communication based on optical fibers. We show that, in certain regimes, the Kerr nonlinearity is able to compensate the dephasing. In addition, our studies reveal that the quantum capacity of the deformed bosonic dephasing channel can be greater than that of the undeformed, standard bosonic dephasing channel for certain nonlinearity parameters.
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 study the quantum capacity of continuous variable dephasing channel, which is a notable example of non-Gaussian quantum channel. We prove that a single letter formula applies. We then consider input energy restriction and show that by increasing it, the capacity saturates to a finite value. The optimal input state is found to be diagonal in the Fock basis and with a distribution that is a discrete version of a Gaussian. Relations between its mean/variance and dephasing rate/input energy are put forward. We also show that quantum capacity decays exponentially for large values of dephasing ra…
Bosonic qubits encoded in continuous-variable systems provide a promising alternative to two-level qubits for quantum computation and communication. So far, photon loss has been the dominant source of errors in bosonic qubits, but the significant reduction of photon loss in recent bosonic qubit experiments suggests that dephasing errors should also be considered. However, a detailed understanding of the combined photon loss and dephasing channel is lacking. Here, we show that, unlike its constituent parts, the combined loss-dephasing channel is non-degradable, pointing towards a richer structu…
We address the classical capacity of a quantum bosonic memory channel with additive noise, subject to an input energy constraint. The memory is modeled by correlated noise emerging from a Gauss-Markov process. Under reasonable assumptions, we show that the optimal modulation results from a "quantum water-filling" solution above a certain input energy threshold, similar to the optimal modulation for parallel classical Gaussian channels. We also derive analytically the optimal multimode input state above this threshold, which enables us to compute the capacity of this memory channel in the limit…
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…
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 …
We study the problem of designing optical receivers to discriminate between multiple coherent states using coherent processing receivers---i.e., one that uses arbitrary coherent feedback control and quantum-noise-limited direct detection---which was shown by Dolinar to achieve the minimum error probability in discriminating any two coherent states. We first derive and re-interpret Dolinar's binary-hypothesis minimum-probability-of-error receiver as the one that optimizes the information efficiency at each time instant, based on recursive Bayesian updates within the receiver. Using this viewpoi…
We investigate the quantum-secure covert-communication capabilities of lossy thermal-noise bosonic channels, the quantum-mechanical model for many practical channels. We determine the expressions for the covert capacity of these channels: $L_{\text{no-EA}}$, when Alice and Bob share only a classical secret, and $L_{\text{EA}}$, when they benefit from entanglement assistance. We find that entanglement assistance alters the fundamental scaling law for covert communication. Instead of $L_{\text{no-EA}}\sqrt{n}-r_{\text{no-EA}}(n)$, $r_{\text{no-EA}}(n)=o(\sqrt{n})$, entanglement assistance allows…
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…
The trade-off capacity region of a quantum channel characterizes the optimal net rates at which a sender can communicate classical, quantum, and entangled bits to a receiver by exploiting many independent uses of the channel, along with the help of the same resources. Similarly, one can consider a trade-off capacity region when the noiseless resources are public, private, and secret key bits. In [Phys. Rev. Lett. 108, 140501 (2012)], we identified these trade-off rate regions for the pure-loss bosonic channel and proved that they are optimal provided that a longstanding minimum output entropy …
Upper bounds on the secret-key-agreement capacity of a quantum channel serve as a way to assess the performance of practical quantum-key-distribution protocols conducted over that channel. In particular, if a protocol employs a quantum repeater, achieving secret-key rates exceeding these upper bounds is a witness to having a working quantum repeater. In this paper, we extend a recent advance [Liuzzo-Scorpo et al., arXiv:1705.03017] in the theory of the teleportation simulation of single-mode phase-insensitive Gaussian channels such that it now applies to the relative entropy of entanglement me…
We prove that several known upper bounds on the classical capacity of thermal and additive noise bosonic channels are actually strong converse rates. Our results strengthen the interpretation of these upper bounds, in the sense that we now know that the probability of correctly decoding a classical message rapidly converges to zero in the limit of many channel uses if the communication rate exceeds these upper bounds. In order for these theorems to hold, we need to impose a maximum photon number constraint on the states input to the channel (the strong converse property need not hold if there …
Determining the ultimate classical information carrying capacity of electromagnetic waves requires quantum-mechanical analysis to properly account for the bosonic nature of these waves. Recent work has established capacity theorems for bosonic single-user, broadcast, and wiretap channels, under the presumption of two minimum output entropy conjectures. Despite considerable accumulated evidence that supports the validity of these conjectures, they have yet to be proven. Here we show that the preceding minimum output entropy conjectures are simple consequences of an Entropy Photon-Number Inequal…
We identify the quantum channels corresponding to the interaction of a Gaussian quantum state with an already formed Schwarzschild black hole. Using recent advances in the classification of one-mode bosonic Gaussian channels we find that (with one exception) the black hole Gaussian channels lie in the non-entanglement breaking subset of the lossy channels C(loss), amplifying channels C(amp) and classical-noise channels B2. We show that the channel parameters depend on the black hole mass and the properties of the potential barrier surrounding it. This classification enables us to calculate the…
No advantage from classical assistance for energy-constrained bosonic dephasing channels
Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo（量子计算研究所与物理和天文学系，滑铁卢大学）；Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo（量子计算研究所与应用数学系，滑铁卢大学）
The most natural way to describe an information-carrying system containing a specific noise is an additive white Gaussian-noise (AWGN) channel. In bosonic quantum systems (especially the Gaussian case), although the classical information capacity for a phase-insensitive and thermal-noise channel is additive based on a proof of the minimum output entropy conjecture, several open questions remain. By generalizing the Gaussian noise model from thermal noise to general Gaussian noise, we rigorously revisit and calculate these strong upper bounds on the information capacity for single-mode with gen…
We discuss a Bosonic channel model with memory effects. It relies on a multi-mode squeezed (entangled) environment's state. The case of lossy Bosonic channels is analyzed in detail. We show that in the absence of input energy constraints the memory channels are equivalent to their memoryless counterparts. In the case of input energy constraint we provide lower and upper bounds for the memory channel capacity.
When classical information is sent over a quantum channel, attaining the ultimate limit to channel capacity requires the receiver to make joint measurements over long codeword blocks. For a pure-state channel, we construct a receiver that can attain the ultimate capacity by applying a single-shot unitary transformation on the received quantum codeword followed by simultaneous (but separable) projective measurements on the single-modulation-symbol state spaces. We study the ultimate limits of photon-information-efficient communications on a lossy bosonic channel. Based on our general results fo…
We consider quantum key distribution (QKD) and entanglement distribution using a single-sender multiple-receiver pure-loss bosonic broadcast channel. We determine the unconstrained capacity region for the distillation of bipartite entanglement and secret key between the sender and each receiver, whenever they are allowed arbitrary public classical communication. A practical implication of our result is that the capacity region demonstrated drastically improves upon rates achievable using a naive time-sharing strategy, which has been employed in previously demonstrated network QKD systems. We s…
We introduce a general model for a lossy bosonic memory channel and calculate the classical and the quantum capacity, proving that coherent state encoding is optimal. The use of a proper set of collective field variables allows to unravel the memory, showing that the n-fold concatenation of the memory channel is unitarily equivalent to the direct product of n single-mode lossy bosonic 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-Ma…
The additivity of the minimal output entropy and that of the $χ$-capacity are known to be equivalent for finite-dimensional irreducibly covariant channels. In this paper we formulate a list of conditions allowing to establish similar equivalence for infinite-dimensional covariant channels with constrained input. This is then applied to Bosonic Gaussian channels with quadratic input constraint to extend the classical capacity results of the recent paper \cite{ghg} to the case where the complex structures associated with the channel and with the constraint operator need not commute. In particula…
The maximum rates for reliably transmitting classical information over Bosonic multiple-access channels (MACs) are derived when the transmitters are restricted to coherent-state encodings. Inner and outer bounds for the ultimate capacity region of the Bosonic MAC are also presented. It is shown that the sum-rate upper bound is achievable with a coherent-state encoding and that the entire region is asymptotically achievable in the limit of large mean input photon numbers.
Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel such as classica…
Thermal attenuator channels model the decoherence of quantum systems interacting with a thermal bath, e.g., a two-level system subject to thermal noise and an electromagnetic signal travelling through a fiber or in free-space. Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the o…
We introduce a new form for the bosonic channel minimal output entropy conjecture, namely that among states with equal input entropy, the thermal states are the ones that have slightest increase in entropy when sent through a infinitesimal thermalizing channel. We then detail a strategy to prove the conjecture through variational techniques. This would lead to the calculation of the classical capacity of a communication channel subject to thermal noise. Our strategy detects input thermal ensembles as possible solutions for the optimal encoding of the channel, lending support to the conjecture.…
A general attenuator $Φ_{λ, σ}$ is a bosonic quantum channel that acts by combining the input with a fixed environment state $σ$ in a beam splitter of transmissivity $λ$. If $σ$ is a thermal state the resulting channel is a thermal attenuator, whose quantum capacity vanishes for $λ\leq 1/2$. We study the quantum capacity of these objects for generic $σ$, proving a number of unexpected results. Most notably, we show that for any arbitrary value of $λ>0$ there exists a suitable single-mode state $σ(λ)$ such that the quantum capacity of $Φ_{λ,σ(λ)}$ is larger than a universal constant $c>0$. Our …
We establish the classical capacity of optical quantum channels as a sharp transition between two regimes---one which is an error-free regime for communication rates below the capacity, and the other in which the probability of correctly decoding a classical message converges exponentially fast to zero if the communication rate exceeds the classical capacity. This result is obtained by proving a strong converse theorem for the classical capacity of all phase-insensitive bosonic Gaussian channels, a well-established model of optical quantum communication channels, such as lossy optical fibers, …
Upper bounds for private communication over quantum channels can be derived by adopting channel simulation, protocol stretching, and relative entropy of entanglement. All these ingredients have led to single-letter upper bounds to the secret key capacity which can be directly computed over suitable resource states. For bosonic Gaussian channels, the tightest upper bounds have been derived by employing teleportation simulation over asymptotic resource states, namely the asymptotic Choi matrices of these channels. In this work, we adopt a different approach. We show that teleporting over an anal…
