An intuitive proof of the data processing inequality

The data processing inequality (DPI) is a fundamental feature of information theory. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. When the smooth min-entropy is used as the relevant information measure, then the DPI follows immediately from the definition of the entropy. The DPI for the von Neumann entropy is then obtained by specializing the DPI for the smooth min-entropy by using the quantum asymptotic equipartition property (QAEP). We provide a short proof of the QAEP and therefore obtain a self-contained proof of the DPI for the von Neumann entropy.

[1]  Robert König,et al.  The Operational Meaning of Min- and Max-Entropy , 2008, IEEE Transactions on Information Theory.

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

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

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

[5]  M. Headrick,et al.  Preprint typeset in JHEP style- HYPER VERSION arXiv:0704.3719 [hep-th] , 2022 .

[6]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[7]  J. VonNeumann.  Mathematische Grundlagen der Quantenmechanik , 1932 .

[8]  M. Horodecki,et al.  Quantum State Merging and Negative Information , 2005, quant-ph/0512247.

[9]  M. Ruskai,et al.  A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY , 2009, 0903.2895.

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

[11]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[12]  N. Datta.  Relative Entropies and Entanglement Monotones , 2011 .

[13]  M. Ruskai.  Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy , 2006, quant-ph/0604206.

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

[15]  Andreas Winter,et al.  Partial quantum information , 2005, Nature.

[16]  E. Lieb.  Convex trace functions and the Wigner-Yanase-Dyson conjecture , 1973 .

[17]  Michael A. Nielsen,et al.  A simple proof of the strong subadditivity inequality , 2005, Quantum Inf. Comput..

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

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

[20]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[21]  F. Herbut.  On mutual information in multipartite quantum states and equality in strong subadditivity of entropy , 2003, quant-ph/0311193.

[22]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

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

[25]  B. Simon.  Trace ideals and their applications , 1979 .

[26]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[27]  E. Lieb,et al.  A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy , 2007, math/0701352.

[28]  E. Lieb,et al.  A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity , 2007, 0710.4167.

[29]  D. Petz.  Quasi-entropies for finite quantum systems , 1986 .

[30]  T. Takayanagi,et al.  AdS/CFT and strong subadditivity of entanglement entropy , 2006, hep-th/0608213.