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1. A New Formulation of Lossy Quantum-Classical and Classical Source Coding based on a Posterior Channel

   https://doi.org/10.1109/ISIT54713.2023.10206859  

   Sohail, Mohammad Aamir; Anwar Atif, Touheed; Pradhan, S. Sandeep (June 2023, IEEE)

   —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 singleletter 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 singleletter 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. 

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2. Faithful Simulation of Distributed Quantum Measurements with Applications in Distributed Rate-Distortion Theory

   Mohsen Heidari, Touheed Anwar (June 2019, IEEE International Symposium on Information Theory)

   We investigate faithful simulation of distributed quantum measurements as an extension of Winter's measurement compression theorem. We characterize a set of communication and common randomness rates needed to provide faithful simulation of distributed measurements. To achieve this, we introduce binning and mutual packing lemma for distributed quantum measurements. These techniques can be viewed as the quantum counterpart of their classical analogues. Finally, using these results, we develop a distributed quantum-to-classical rate distortion theory and characterize a rate region analogous to Berger-Tung's in terms of single-letter quantum mutual information quantities. 

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3. Quantum State Compression with Polar Codes

   https://doi.org/10.1109/ISIT57864.2024.10619556  

   Weinberg, Jack; Mandal, Avijit; Pfister, Henry D (July 2024, IEEE)

   In the quantum compression scheme proposed by Schumacher, Alice compresses a message that Bob decompresses. In that approach, there is some probability of failure and, even when successful, some distortion of the state. For sufficiently large blocklengths, both of these imperfections can be made arbitrarily small while achieving a compression rate that asymp- totically approaches the source coding bound. However, direct implementation of Schumacher compression suffers from poor circuit complexity. In this paper, we consider a slightly different approach based on classical syndrome source coding. The idea is to use a linear error-correcting code and treat the state to be compressed as a superposition of error patterns. Then, Alice can use quantum gates to apply the parity-check matrix to her message state. This will convert it into a superposition of syndromes. If the original superposition was supported on correctable errors (e.g., coset leaders), then this process can be reversed by decoding. An implementation of this based on polar codes is described and simulated. As in classical source coding based on polar codes, Alice maps the information into the “frozen” qubits that constitute the syndrome. To decompress, Bob utilizes a quantum version of successive cancellation coding. 

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4. Estimating the Rate-Distortion Function by Wasserstein Gradient Descent 37th Conference on Neural Information Processing Systems (NeurIPS)

   Yang, Yibo; Eckstein, Stephan; Nutz, Marcel; Mandt, Stephan (November 2023, 37th Conference on Neural Information Processing Systems (NeurIPS))

   A. Oh; T. Naumann; A. Globerson; K. Saenko; M. Hardt; S. Levine (Ed.)

   In the theory of lossy compression, the rate-distortion (R-D) function R(D) describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining R(D) for a given data source establishes the fundamental performance limit for all compression algorithms. We propose a new method to estimate R(D) from the perspective of optimal transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support of the reproduction distribution in advance, our Wasserstein gradient descent algorithm learns the support of the optimal reproduction distribution by moving particles. We prove its local convergence and analyze the sample complexity of our R-D estimator based on a connection to entropic optimal transport. Experimentally, we obtain comparable or tighter bounds than state-of-the-art neural network methods on low-rate sources while requiring considerably less tuning and computation effort. We also highlight a connection to maximum-likelihood deconvolution and introduce a new class of sources that can be used as test cases with known solutions to the R-D problem. 

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5. The role of randomness in quantum state certification with unentangled measurements

   Liu, Yuhan; Acharya, Jayadev (June 2024, Proceedings of Machine Learning Research)

   Agrawal, Shipra; Roth, Aaron (Ed.)

   We study quantum state certification using unentangled quantum measurements, namely measurements which operate only on one copy of the state at a time. When there is a common source of randomness available and the unentangled measurements are chosen based on this randomness, prior work has shown that copies are necessary and sufficient. We show a separation between algorithms with and without randomness. We develop a lower bound framework for both fixed and randomized measurements that relates the hardness of testing to the well-established Lüders rule. More precisely, we obtain lower bounds for randomized and fixed schemes as a function of the eigenvalues of the Lüders channel which characterizes one possible post-measurement state transformation. 

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