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IEEE Access

Hybrid Quantum Noise Model to Compute Gaussian Quantum Channel Capacity

Mouli Chakraborty, Anshu Mukherjee, Avishek Nag, Subhash Chandra

Published Jan 1, 2024• 24 citations
Publisher

Abstract

Quantum information processing leverages the principles of quantum mechanics, utilizing qubits, to improve computational and communicative tasks. In this realm, the quantum channel’s capacity is pivotal in determining the efficiency and accuracy of quantum information handling, with its performance being significantly influenced by channel noise. Our study aims to establish a holistic hybrid quantum noise model to determine the quantum channel capacity. In this paper, we formulated a mathematical expression for this capacity and conducted simulations for both Gaussian and non-Gaussian inputs. A hybrid noise model is constructed by convolution of Poisson-distributed quantum noise with classical additive white Gaussian noise. We characterized the quantum-classical noise and the received signal using Gaussian Mixture Models. The maximum amount of quantum information that can be reliably transmitted over a quantum channel (per use of the channel) is determined by its capacity, and entropy and related quantities like mutual information play a role in calculating this capacity. Our formulation of quantum channel capacity is derived from the mutual information shared between the transmitter and receiver, encompassing the entropies of the signals. The quantum channel presents a higher capacity-to-signal-to-noise ratio for Gaussian inputs than non-Gaussian ones.

DOI
10.1109/access.2024.3355789

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Quantum information processing leverages the principles of quantum mechanics, utilizing qubits, to improve computational and communicative tasks. In this realm, the quantum channel’s capacity is pivotal in determining the efficiency and accuracy of quantum information handling, with its performance being significantly influenced by channel noise. Our study aims to establish a holistic hybrid quantum noise model to determine the quantum channel capacity. In this paper, we formulated a mathematical expression for this capacity and conducted simulations for both Gaussian and non-Gaussian inputs. A hybrid noise model is constructed by convolution of Poisson-distributed quantum noise with classical additive white Gaussian noise. We characterized the quantum-classical noise and the received signal using Gaussian Mixture Models. The maximum amount of quantum information that can be reliably transmitted over a quantum channel (per use of the channel) is determined by its capacity, and entropy and related quantities like mutual information play a role in calculating this capacity. Our formulation of quantum channel capacity is derived from the mutual information shared between the transmitter and receiver, encompassing the entropies of the signals. The quantum channel presents a higher capacity-to-signal-to-noise ratio for Gaussian inputs than non-Gaussian ones.
For a continuous-input-continuous-output arbitrarily distributed quantum channel carrying classical information, the channel capacity can be computed in terms of the distribution of the channel envelope, received signal strength over a quantum propagation field and the noise spectral density. If the channel envelope is considered to be unity with unit received signal strength, the factor controlling the capacity is the noise. Quantum channel carrying classical information will suffer from the combination of classical and quantum noise. Assuming additive Gaussian-distributed classical noise and Poisson-distributed quantum noise, we formulate a hybrid noise model by deriving a joint Gaussian- Poisson distribution in this letter. For the transmitted signal, we consider the mean of signal sample space instead of considering a particular distribution and study how the maximum mutual information varies over such mean value. Capacity is estimated by maximizing the mutual information over unity channel envelope.
In this article, we are proposing a closed-form solution for the capacity of the single quantum channel. The Gaussian distributed input has been considered for the analytical calculation of the capacity. In our previous couple of papers, we invoked models for joint quantum noise and the corresponding received signals; in this current research, we proved that these models are Gaussian mixtures distributions. In this paper, we showed how to deal with both of cases, namely (I)the Gaussian mixtures distribution for scalar variables and (II) the Gaussian mixtures distribution for random vectors. Our target is to calculate the entropy of the joint noise and the entropy of the received signal in order to calculate the capacity expression of the quantum channel. The main challenge is to work with the function type of the Gaussian mixture distribution. The entropy of the Gaussian mixture distributions cannot be calculated in the closed-form solution due to the logarithm of a sum of exponential functions. As a solution, we proposed a lower bound and a upper bound for each of the entropies of joint noise and the received signal, and finally upper inequality and lower inequality lead to the upper bound for the mutual information and hence the maximum achievable data rate as the capacity. In this paper reader will able to visualize an closed-form capacity experssion which make this paper distinct from our previous works. These capacity experssion and coresses ponding bounds are calculated for both the cases: the Gaussian mixtures distribution for scalar variables and the Gaussian mixtures distribution for random vectors as well.
In this paper we are interested to model quantum signal by statistical signal processing methods. The Gaussian distribution has been considered for the input quantum signal as Gaussian state have been proven to a type of important robust state and most of the important experiments of quantum information are done with Gaussian light. Along with that a joint noise model has been invoked, and followed by a received signal model has been formulated by using convolution of transmitted signal and joint quantum noise to realized theoretical achievable capacity of the single quantum link. In joint quantum noise model we consider the quantum Poisson noise with classical Gaussian noise. We compare the capacity of the quantum channel with respect to SNR to detect its overall tendency. In this paper we use the channel equation in terms of random variable to investigate the quantum signals and noise model statistically. These methods are proposed to develop Quantum statistical signal processing and the idea comes from the statistical signal processing.
As its classical counterpart，quantum Gaussian noise is an important noise model in quantum information theory. The study of capacity of quantum Gaussian channel plays an important role in quantum communications. Based on the definition of quantum Gaussian state，the entropy of quantum Gaussian state，and Holevo bound theorem，the classical capacity of quantum Gaussian channel is given at first in the paper，where the classical capacity is defined as the capacity when only classical messages are transmitted through the quantum channel. Then，the classical capacities of quantum multiple access Gaussian channels are deduced according to the classical capacity theorem of multiple access quantum channels. For simplicity and generality，a multiple access quantum channel with two senders and one receiver is used for computation. The results can be generalized to the quantum channel with n senders and one receiver.
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 NNN and allowed average decoding error εεε equals the vanishing-error capacity at mean photon number N/(1−ε)N/(1-ε)N/(1−ε). Together, these results identify the ultimate limits of classical communication for some of the most fundamental models of optical communication.
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 classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural cases in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality on which we elaborate in depth.