We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for discrete quantum measurement systems with limited classical common randomness. The main coding theorem provides the achievable rate region of a lossy measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an i.i.d. predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination state via an entanglement-breaking channel with limited communication rate and common randomness.1
We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for discrete quantum measurement systems with limited classical common randomness. The main coding theorem provides the achievable rate region of a lossy measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an i.i.d. predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination state via an entanglement-breaking channel with limited communication rate and common randomness.1
We consider the rate-limited quantum-to-classical optimal transport in terms of output-constrained rate-distortion coding for both finite-dimensional and continuous-variable quantum-to-classical systems with limited classical common randomness. The main coding theorem provides a single-letter characterization of the achievable rate region of a lossy quantum measurement source coding for an exact construction of the destination distribution (or the equivalent quantum state) while maintaining a threshold of distortion from the source state according to a generally defined distortion observable. The constraint on the output space fixes the output distribution to an IID predefined probability mass function. Therefore, this problem can also be viewed as information-constrained optimal transport which finds the optimal cost of transporting the source quantum state to the destination classical distribution via a quantum measurement with limited communication rate and common randomness. We develop a coding framework for continuous-variable quantum systems by employing a clipping projection and a dequantization block and using our finite-dimensional coding theorem. Moreover, for the Gaussian quantum systems, we derive an analytical solution for rate-limited Wasserstein distance of order 2, along with a Gaussian optimality theorem, showing that Gaussian measurement optimizes the rate in a system with Gaussian quantum source and Gaussian destination distribution. The results further show that in contrast to the classical Wasserstein distance of Gaussian distributions, which corresponds to an infinite transmission rate, in the Quantum Gaussian measurement system, the optimal transport is achieved with a finite transmission rate due to the inherent noise of the quantum measurement imposed by Heisenberg’s uncertainty principle.
The rate-limited optimal transport problem is introduced for the continuous-variable quantum measurement systems in the form of output-constrained rate-distortion coding. The main coding theorem provides a single-letter characterization of the achievable rate region for lossy quantum-to-classical source coding that transforms a sufficiently large tensor product of IID continuous-variable quantum states from a quantum source to a sequence of IID samples from a classical continuous destination distribution with a prescribed distortion level. The evaluation of rate region is performed for the systems with quantum Gaussian source and Gaussian destination distribution. We establish a Gaussian observable optimality theorem for such systems and provide an analytical formulation of the rate-limited quantum-classical Wasserstein distance in the case of isotropic and one-mode Gaussian quantum systems.11The proofs of the theorems and the details of the results are provided in the extended online version [1] available at https://arxiv.org/abs/2305.10004 for further reference. This work was supported in part by NSF grants CCF 2007878 and CCF 2132815.
We derive quantum counterparts of two key theorems of classical information theory, namely, the rate-distortion theorem and the source-channel separation theorem. The rate-distortion theorem gives the ultimate limits on lossy data compression, and the source-channel separation theorem implies that a two-stage protocol consisting of compression and channel coding is optimal for transmitting a memoryless source over a memoryless channel. In spite of their importance in the classical domain, there has been surprisingly little work in these areas for quantum information theory. In this paper, we prove that the quantum rate-distortion function is given in terms of the regularized entanglement of purification. We also determine a single-letter expression for the entanglement-assisted quantum rate-distortion function, and we prove that it serves as a lower bound on the unassisted quantum rate-distortion function. This implies that the unassisted quantum rate-distortion function is nonnegative and generally not equal to the coherent information between the source and distorted output (in spite of Barnum's conjecture that the coherent information would be relevant here). Moreover, we prove several quantum source-channel separation theorems. The strongest of these are in the entanglement-assisted setting, in which we establish a necessary and sufficient condition for transmitting a memoryless source over a memoryless quantum channel up to a given distortion.
We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.
In this work, we address the lossy quantum-classical (QC) source coding problem, where the task is to compress the classical information about a quantum source, obtained after performing a measurement, below the Shannon entropy of the measurement outcomes, while incurring a bounded reconstruction error. We propose a new formulation, namely, "rate-channel theory", for the lossy QC source coding problem based on the notion of a backward (posterior) channel. We employ a single-letter posterior channel to capture the reconstruction error in place of the single-letter distortion observable. The formulation requires the reconstruction of the compressed quantum source to satisfy a block error constraint as opposed to the average single-letter distortion criterion in the rate-distortion setting. We also develop an analogous formulation for the classical variant with respect to a corresponding posterior channel. Furthermore, we characterize the asymptotic performance limit of the lossy QC and classical source coding problems in terms of single-letter quantum mutual information and mutual information quantities of the given posterior channel, respectively. We provide examples for the above formulations.