One-Shot Lossy Quantum Data Compression

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.

[1]  Katalin Marton,et al.  Error exponent for source coding with a fidelity criterion , 1974, IEEE Trans. Inf. Theory.

[2]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[3]  Charles H. Bennett,et al.  Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. , 1993, Physical review letters.

[4]  Sergio Verdú,et al.  Simulation of random processes and rate-distortion theory , 1996, IEEE Trans. Inf. Theory.

[5]  R. Renner,et al.  One-shot classical-quantum capacity and hypothesis testing. , 2010, Physical review letters.

[6]  Toby Berger,et al.  Quantum rate-distortion theory for memoryless sources , 2002, IEEE Trans. Inf. Theory.

[7]  I. Devetak,et al.  Triangle of dualities between quantum communication protocols. , 2006, Physical review letters.

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

[9]  D. A. Bell,et al.  Information Theory and Reliable Communication , 1969 .

[10]  R. Renner,et al.  Min- and Max-Entropy in Infinite Dimensions , 2010, 1004.1386.

[11]  Marco Tomamichel,et al.  Duality Between Smooth Min- and Max-Entropies , 2009, IEEE Transactions on Information Theory.

[12]  R. Jozsa.  Fidelity for Mixed Quantum States , 1994 .

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

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

[15]  Andreas J. Winter,et al.  Coding theorem and strong converse for quantum channels , 1999, IEEE Trans. Inf. Theory.

[16]  Sergio Verdú,et al.  Fixed-Length Lossy Compression in the Finite Blocklength Regime , 2011, IEEE Transactions on Information Theory.

[17]  Igor Devetak,et al.  Quantum rate-distortion theory for i.i.d. sources , 2000, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[18]  M. Tomamichel.  A framework for non-asymptotic quantum information theory , 2012, 1203.2142.

[19]  Renato Renner,et al.  Smooth Max-Information as One-Shot Generalization for Mutual Information , 2013, IEEE Transactions on Information Theory.

[20]  Tomohiro Ogawa,et al.  Making Good Codes for Classical-Quantum Channel Coding via Quantum Hypothesis Testing , 2007, IEEE Transactions on Information Theory.

[21]  M. Fannes,et al.  Continuity of quantum conditional information , 2003, quant-ph/0312081.

[22]  A. Winter,et al.  The mother of all protocols: restructuring quantum information’s family tree , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  Marco Tomamichel,et al.  A Fully Quantum Asymptotic Equipartition Property , 2008, IEEE Transactions on Information Theory.

[24]  Mark M. Wilde,et al.  Quantum Rate-Distortion Coding With Auxiliary Resources , 2012, IEEE Transactions on Information Theory.

[25]  A. Winter.  Compression of sources of probability distributions and density operators , 2002, quant-ph/0208131.

[26]  Nilanjana Datta,et al.  Min- and Max-Relative Entropies and a New Entanglement Monotone , 2008, IEEE Transactions on Information Theory.

[27]  Howard Barnum.  Quantum rate-distortion coding , 1998 .

[28]  Mark M. Wilde,et al.  Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation , 2011, IEEE Transactions on Information Theory.

[29]  N. Datta,et al.  The apex of the family tree of protocols: optimal rates and resource inequalities , 2011, 1103.1135.

[30]  Aaron D. Wyner,et al.  Coding Theorems for a Discrete Source With a Fidelity CriterionInstitute of Radio Engineers, International Convention Record, vol. 7, 1959. , 1993 .

[31]  Xiao-Yu Chen,et al.  Entanglement Information Rate Distortion of a Quantum Gaussian Source , 2008, IEEE Transactions on Information Theory.

[32]  R. Renner,et al.  Generalized Entropies , 2012, 1211.3141.

[33]  Ashish V. Thapliyal,et al.  Entanglement-Assisted Classical Capacity of Noisy Quantum Channels , 1999, Physical Review Letters.

[34]  Mark M. Wilde,et al.  Quantum-to-classical rate distortion coding , 2012, ArXiv.

[35]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[36]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[37]  Igor Devetak,et al.  Channel Simulation With Quantum Side Information , 2009, IEEE Transactions on Information Theory.

[38]  Seth Lloyd,et al.  Gaussian quantum information , 2011, 1110.3234.

[39]  R. Renner,et al.  The Quantum Reverse Shannon Theorem Based on One-Shot Information Theory , 2009, 0912.3805.

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

[41]  P. Hayden,et al.  Universal entanglement transformations without communication , 2003 .