We consider information transmission through a noiseless quantum channel, where the information is encoded into massive indistinguishable particles: bosons or fermions. We study the situation in which the particles are noninteracting. The encoding input states obey a set of physically motivated constraints on the mean values of the energy and particle number. In such a case, the determination of both classical and quantum capacity reduces to a constrained maximization of entropy. In the case of noninteracting bosons, signatures of Bose-Einstein condensation can be observed in the behavior of the capacity. A major motivation for these considerations is to compare the information-carrying capacities of channels that carry bosons with those that carry fermions. We show analytically that fermions generally provide higher channel capacity, i.e., they are better suited for transferring bits as well as qubits, in comparison to bosons. This holds for a large range of power-law potentials, and for moderate to high temperatures. Numerical simulations seem to indicate that the result holds for all temperatures. Also, we consider the low-temperature behavior for the three-dimensional box and harmonic trap, and again we show that the fermionic capacity is higher than the bosonic one for sufficiently low temperatures.
We consider information transmission through a noiseless quantum channel, where the information is encoded into massive indistinguishable particles: bosons or fermions. We study the situation in which the particles are noninteracting. The encoding input states obey a set of physically motivated constraints on the mean values of the energy and particle number. In such a case, the determination of both classical and quantum capacity reduces to a constrained maximization of entropy. In the case of noninteracting bosons, signatures of Bose-Einstein condensation can be observed in the behavior of the capacity. A major motivation for these considerations is to compare the information-carrying capacities of channels that carry bosons with those that carry fermions. We show analytically that fermions generally provide higher channel capacity, i.e., they are better suited for transferring bits as well as qubits, in comparison to bosons. This holds for a large range of power-law potentials, and for moderate to high temperatures. Numerical simulations seem to indicate that the result holds for all temperatures. Also, we consider the low-temperature behavior for the three-dimensional box and harmonic trap, and again we show that the fermionic capacity is higher than the bosonic one for sufficiently low temperatures.
We consider the capacity of classical information transfer for noiseless quantum channels carrying a finite average number of massive bosons and fermions. The maximum capacity is attained by transferring the Fock states generated from the grand-canonical ensemble. Interestingly, the channel capacity for a Bose gas indicates the onset of Bose-Einstein condensation, by changing its qualitative behavior at the criticality, while for a channel carrying weakly attractive fermions, it exhibits the signatures of Bardeen-Cooper-Schrieffer transition. We also show that, for noninteracting particles, fermions are better carriers of information than bosons.
In this thesis we study the information transmission through Gaussian quantum channels. Gaussian quantum channels model physical communication links, such as free space communication or optical fibers and therefore, may be considered as the most relevant quantum channels. One of the central characteristics of any communication channel is its capacity. In this work we are interested in the classical capacity, which is the maximal number of bits that can be reliably transmitted per channel use. An important lower bound on the classical capacity is given by the Gaussian capacity, which is the maximal transmission rate with the restriction that only Gaussian encodings are allowed: input messages are encoded in so-called Gaussian states for which the mean field amplitudes are Gaussian distributed. We focus in this work mainly on the Gaussian capacity for the following reasons. First, Gaussian encodings are easily accessible experimentally. Second, the difficulty of studying the classical capacity, which arises due to an optimization problem in an infinite dimensional Hilbert space, is greatly reduced when considering only Gaussian input encodings. Third, the Gaussian capacity is conjectured to coincide with the classical capacity, even though this longstanding conjecture is unsolved until today. We start with the investigation of the capacities of the single-mode Gaussian channel. We show that the most general case can be reduced to a simple, fiducial Gaussian channel which depends only on three parameters: its transmissivity (or gain), the added noise variance and the squeezing of the noise. Above a certain input energy threshold, the optimal input variances are given by a quantum water-filling solution, which implies that the optimal modulated output state is a thermal state. This is a quantum extension (or generalization) of the well-known classical water-filling solution for parallel Gaussian channels. Below the energy threshold the solution is given by a transcendental equation and only the less noisy quadrature is modulated. We characterize in detail the dependence of the Gaussian capacity on its channel parameters. In particular, we show that the Gaussian capacity is a non-monotonous function of the noise squeezing and analytically specify the regions where it exhibits one maximum, a maximum and a minimum, a saddle point or no extrema. Then, we investigate the case of n-mode channels with noise correlations (i.e. memory), where we focus in particular on the classical additive noise channel. We consider memory models for which the noise correlations can be unraveled by a passive symplectic transformation. Therefore, we can simplify the problem to the study of the Gaussian capacity in an uncorrelated basis, which corresponds to the Gaussian capacity of n single-mode channels with a common input energy constraint. Above an input energy threshold the solutions is given by a global quantum water-filling solution, which implies that all modulated single-mode output states are thermal states with the same temperature. Below the threshold the channels are divided into three sets: i) those that are excluded from information transmission, ii) those for which only the less noisy quadrature is modulated, and iii) those for which the quantum water-filling solution is satisfied. As an example we consider a Gauss-Markov correlated noise, which in the uncorrelated basis corresponds to a collection of single-mode classical additive noise channels. When rotating the collection of optimal single-mode input states back to the original, correlated basis the optimal multi-mode input state becomes a highly entangled state. We then compare the performance of the optimal input state with a simple coherent state encoding and conclude that one gains up to 10% by using the optimal encoding. Since the preparation of the optimal input state may be very challenging we consider sub-optimal Gaussian-matrix product states (GMPS) as input states as well. GMPS have a known experimental setup and, though being heavily entangled, can be generated sequentially. We demonstrate that for the Markovian correlated noise as well as for a non-Markovian noise model in a wide range of channel parameters, a nearest-neighbor correlated GMPS achieves more than 99.9% of the Gaussian capacity. At last, we introduce a new noise model for which the GMPS is the exact optimal input state. Since GMPS are known to be ground states of quadratic Hamiltonians this suggests a starting point to develop links between optimization problems of quantum communication and many body physics.
The classical capacity of a noisy optical channel is the maximum rate at which information can be transmitted reliably, given a constraint on the mean input energy. For the ubiquitous bosonic Gaussian channels, which include standard models of loss and noise in fibre and free-space optical links, determining this rate requires addressing whether entanglement between channel uses or non-Gaussian signals can outperform Gaussian encodings. More precisely, the question is whether the energy-constrained Holevo information is additive across arbitrarily many channel uses and coincides with its Gaussian counterpart. We answer these fundamental questions in the affirmative for every single-mode bosonic Gaussian channel at every input energy. Namely, we prove that independently prepared Gaussian states with Gaussian-distributed random displacements achieve the capacity, including in the low-energy regime where squeezing the signals competes with encoding information in their displacement. Our proof connects an entanglement inequality to a sharp, additive lower bound on output entropy. We then characterise communication above capacity: under a strict total photon-number cutoff, the probability of correct decoding decreases exponentially with the number of channel uses, and we derive explicit bounds on the decay exponent. When only the mean photon number is constrained, a simple energy-rescaling formula determines exactly the entire asymptotic rate--error trade-off: the capacity at mean photon number N and allowed average decoding error ε equals the vanishing-error capacity at mean photon number N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.
The optimal rate at which information can be sent through a quantum channel when the transmitted signal must simultaneously carry some minimum amount of energy is characterized. To do so, we introduce the quantum-classical analog of the capacity-power function, and we generalize results in classical information theory for transmitting classical information through noisy channels. We show that the capacity-power function for a classical-quantum channel, for both unassisted and private protocol, is concave, and we also prove additivity for unentangled and uncorrelated ensembles of input signals for such channels. This implies that we do not need regularized formulas for calculation. We show that these properties also hold for all noiseless channels when we restrict the set of input states to be pure quantum states. For general channels, we geometrically prove that the capacity-power function is piecewise concave, with further support from numerical simulations. Further, we connect channel capacity to properties of random quantum states. In particular, we obtain analytical expressions for the capacity-power function for the case of noiseless channels using properties of random quantum states under an energy constraint and concentration phenomena in large Hilbert spaces.
On the base of entropy defect principle the models are considered of narrow-band quantum communication channels and compared with the classic description of information channels. The classic model is shown to be applicable in a wide range of channels' parameters. But in the case of fermionic channels it is necessary to take into account that values of signal power and channel capacity are limited. These maximum values can be approximated by the simple expressions. Bosonic channels are free of these limitations. The rough quantum model of narrow-band bosonic channels gives too high values of capacity at low communication rate as compared with accurate one.