A Rate-Distortion Perspective on Quantum State Redistribution

Abstract

We consider a rate-distortion version of the quantum state redistribution task, where the error of the decoded state is judged via an additive distortion measure; it thus constitutes a quantum generalisation of the classical Wyner-Ziv problem. The quantum source is described by a tripartite pure state shared between Alice ($A$, encoder), Bob ($B$, decoder) and a reference ($R$). Both Alice and Bob are required to output a system ($\tilde{A}$ and $\tilde{B}$, respectively), and the distortion measure is encoded in an observable on $\tilde{A}\tilde{B}R$. It includes as special cases most quantum rate-distortion problems considered in the past, and in particular quantum data compression with the fidelity measured per copy; furthermore, it generalises the well-known state merging and quantum state redistribution tasks for a pure state source, with per-copy fidelity, and a variant recently considered by us, where the source is an ensemble of pure states [ZBK & AW, Proc. ISIT 2020, pp. 1858-1863 and ZBK, PhD thesis, UAB 2020, arXiv:2012.14143]. We derive a single-letter formula for the rate-distortion function of compression schemes assisted by free entanglement. A peculiarity of the formula is that in general it requires optimisation over an unbounded auxiliary register, so the rate-distortion function is not readily computable from our result, and there is a continuity issue at zero distortion. However, we show how to overcome these difficulties in certain situations.

11 pages, IEEE two-column format, final accepted version (IEEE Trans. Inf. Theory)

References in corpus (21)

• Quantum information can be negative
• "Squashed Entanglement" - An Additive Entanglement Measure
• A Sharp Fannes-type Inequality for the von Neumann Entropy
• Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
• The mother of all protocols: Restructuring quantum information's family tree
• Continuity of quantum conditional information
• The exact cost of redistributing multipartite quantum states
• A family of quantum protocols
• Classical data compression with quantum side information
• Optimal quantum source coding with quantum information at the encoder and decoder
• The quantum capacity with symmetric side channels
• The private classical capacity with a symmetric side channel and its application to quantum cryptography
• Continuity of quantum channel capacities
• An additive and operational entanglement measure: conditional entanglement of mutual information
• Quantum rate distortion coding with auxiliary resources
• Quantum-to-classical rate distortion coding
• Channel Simulation and Coded Source Compression
• On the distributed compression of quantum information
• Distributed Compression of Correlated Classical-Quantum Sources or: The Price of Ignorance
• Entanglement-Assisted Quantum Data Compression
• Noisy quantum state redistribution with promise and the Alpha-bit