One-Shot Lossy Quantum Data Compression

• Published in: IEEE transactions on information theory Vol. 59; no. 12; pp. 8057 - 8076
• Main Authors: Datta, Nilanjana, Renes, Joseph M., Renner, Renato, Wilde, Mark M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.12.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Asymptotic methods; Classical and quantum physics: mechanics and fields; Codes; Coding, codes; Data compression; Distortion measurement; Entanglement assistance; Entropy; Exact sciences and technology; hypothesis testing; Information theory; Information, signal and communications theory; lossy quantum data compression; max-information; min- and max-entropy; Physics; Probability distribution; Quantum entanglement; Quantum information; quantum rate distortion; Rate distortion theory; Rate-distortion; relative entropy; Signal and communications theory; Telecommunications and information theory; Lossy compression; hypothesis testing; Entanglement assistance; Data compression; quantum rate distortion; Information quantity; Rate distortion theory; Entropy; Hypothesis test; lossy quantum data compression; max-information; Information source; Coding; Minimax method; relative entropy; min- and max-entropy; Quantum information; Memoryless source
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2283723

We provide a framework for one-shot quantum rate distortion coding, in which the goal is to determine the minimum number of qubits required to compress quantum information as a function of the probability that the distortion incurred upon decompression exceeds some specified level. We obtain a one-shot characterization of the minimum qubit compression size for an entanglement-assisted quantum rate-distortion code in terms of the smooth max-information, a quantity previously employed in the one-shot quantum reverse Shannon theorem. Next, we show how this characterization converges to the known expression for the entanglement-assisted quantum rate distortion function for asymptotically many copies of a memoryless quantum information source. Finally, we give a tight, finite blocklength characterization for the entanglement-assisted minimum qubit compression size of a memoryless isotropic qubit source subject to an average symbolwise distortion constraint.