Information transmission through bosonic gaussian channels
Joachim Schäfer, Nicolas J. Cerf
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.
•Sep 20, 2013•1 citation