General Mixed State Quantum Data Compression with and without Entanglement Assistance

We consider the most general (finite-dimensional) quantum mechanical information source, which is given by a quantum system A that is correlated with a reference system R. The task is to compress A in such a way as to reproduce the joint source state ρAR at the decoder with asymptotically high fidelity. This includes Schumacher’s original quantum source coding problem of a pure state ensemble and that of a single pure entangled state, as well as general mixed state ensembles. Here, we determine the optimal compression rate (in qubits per source system) in terms of the Koashi-Imoto decomposition of the source into a classical, a quantum, and a redundant part. The same decomposition yields the optimal rate in the presence of unlimited entanglement between compressor and decoder, and indeed the full region of feasible qubitebit rate pairs. Full version at arXiv:1912.08506 [1].

[1]  Andreas J. Winter.  Coding theorems of quantum information theory , 1999 .

[2]  R. Jozsa,et al.  On quantum coding for ensembles of mixed states , 2000, quant-ph/0008024.

[3]  Zahra Baghali Khanian,et al.  Entanglement-Assisted Quantum Data Compression , 2019, 2019 IEEE International Symposium on Information Theory (ISIT).

[4]  Benjamin Schumacher,et al.  A new proof of the quantum noiseless coding theorem , 1994 .

[5]  Jozsa,et al.  General fidelity limit for quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[6]  M. Nielsen,et al.  Information transmission through a noisy quantum channel , 1997, quant-ph/9702049.

[7]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[8]  M. Fannes.  A continuity property of the entropy density for spin lattice systems , 1973 .

[9]  Masato Koashi,et al.  Operations that do not disturb partially known quantum states , 2002 .

[10]  Dave Touchette,et al.  Incompressibility of classical distributions , 2019, ArXiv.

[11]  K. Audenaert.  A sharp continuity estimate for the von Neumann entropy , 2006, quant-ph/0610146.

[12]  A. Winter,et al.  Communications in Mathematical Physics Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality , 2022 .

[13]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[14]  Andreas J. Winter,et al.  Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints , 2015, ArXiv.

[15]  N Imoto,et al.  Compressibility of quantum mixed-state signals. , 2001, Physical review letters.

[16]  W. Stinespring.  Positive functions on *-algebras , 1955 .

[17]  M. Horodecki.  Limits for compression of quantum information carried by ensembles of mixed states , 1997, quant-ph/9712035.

[18]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[19]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.