Universal recovery map for approximate Markov chains

A central question in quantum information theory is to determine how well lost information can be reconstructed. Crucially, the corresponding recovery operation should perform well without knowing the information to be reconstructed. In this work, we show that the quantum conditional mutual information measures the performance of such recovery operations. More precisely, we prove that the conditional mutual information I(A:C|B) of a tripartite quantum state ρABC can be bounded from below by its distance to the closest recovered state RB→BC(ρAB), where the C-part is reconstructed from the B-part only and the recovery map RB→BC merely depends on ρBC. One particular application of this result implies the equivalence between two different approaches to define topological order in quantum systems.

[1]  Iordanis Kerenidis,et al.  Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications , 2012, SIAM J. Comput..

[2]  Dave Touchette,et al.  Quantum Information Complexity and Amortized Communication , 2014, ArXiv.

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

[4]  B. M. Fulk.  MATH , 1992 .

[5]  Mark M. Wilde,et al.  Quantum Markov chains, sufficiency of quantum channels, and Rényi information measures , 2015, ArXiv.

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

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

[8]  D. Petz.  Monotonicity of quantum relative entropy revisited , 2002, quant-ph/0209053.

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

[10]  Mark M. Wilde,et al.  Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy , 2013, Communications in Mathematical Physics.

[11]  Mario Berta,et al.  Monotonicity of quantum relative entropy and recoverability , 2014, Quantum Inf. Comput..

[12]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[13]  Isaac H. Kim.  Conditional Independence in Quantum Many-Body Systems , 2013 .

[14]  E. Knill,et al.  Reversing quantum dynamics with near-optimal quantum and classical fidelity , 2000, quant-ph/0004088.

[15]  A. Winter,et al.  Robustness of Quantum Markov Chains , 2006, quant-ph/0611057.

[16]  A. Isar,et al.  ABOUT QUANTUM-SYSTEMS , 2004 .

[17]  Mark M. Wilde,et al.  Fidelity of recovery, geometric squashed entanglement, and measurement recoverability , 2014, 1410.1441.

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

[19]  A. Lichnerowicz.  Proof of the Strong Subadditivity of Quantum-Mechanical Entropy , 2018 .

[20]  E. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[21]  Andreas Winter,et al.  Squashed Entanglement, k-Extendibility, Quantum Markov Chains, and Recovery Maps , 2018 .

[22]  Aram Wettroth Harrow,et al.  Product-state approximations to quantum ground states , 2013, STOC '13.

[23]  Matthias Christandl,et al.  Entanglement of the Antisymmetric State , 2009, 0910.4151.

[24]  D. Petz.  Sufficient subalgebras and the relative entropy of states of a von Neumann algebra , 1986 .

[25]  Mario Berta,et al.  Renyi generalizations of the conditional quantum mutual information , 2014, ArXiv.

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

[27]  Jaikumar Radhakrishnan,et al.  A lower bound for the bounded round quantum communication complexity of set disjointness , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[28]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[29]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[30]  M. Hayashi.  Asymptotics of quantum relative entropy from a representation theoretical viewpoint , 1997, quant-ph/9704040.

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

[32]  F. Verstraete,et al.  Lieb-Robinson bounds and the generation of correlations and topological quantum order. , 2006, Physical review letters.

[33]  D. Petz.  SUFFICIENCY OF CHANNELS OVER VON NEUMANN ALGEBRAS , 1988 .

[34]  F. Hiai,et al.  The proper formula for relative entropy and its asymptotics in quantum probability , 1991 .

[35]  R. Schumann.  Quantum Information Theory , 2000, quant-ph/0010060.

[36]  Aram W. Harrow,et al.  Strengthened monotonicity of relative entropy via pinched Petz recovery map , 2015, 2016 IEEE International Symposium on Information Theory (ISIT).

[37]  M. Wilde.  Quantum Information Theory: Noisy Quantum Shannon Theory , 2013 .

[38]  C. Fuchs.  Distinguishability and Accessible Information in Quantum Theory , 1996, quant-ph/9601020.

[39]  Lin Zhang,et al.  Conditional Mutual Information and Commutator , 2012, 1212.5023.

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

[41]  R. Renner,et al.  Quantum Conditional Mutual Information and Approximate Markov Chains , 2014, Communications in Mathematical Physics.

[42]  I. Hirschman,et al.  A convexity theorem for certain groups of transformations , 1952 .

[43]  Aram Wettroth Harrow,et al.  Quantum de Finetti Theorems Under Local Measurements with Applications , 2012, Communications in Mathematical Physics.

[44]  C. Caramanis.  What is ergodic theory , 1963 .

[45]  Andreas Winter,et al.  Squashed Entanglement, $$\mathbf {k}$$k-Extendibility, Quantum Markov Chains, and Recovery Maps , 2014, 1410.4184.

[46]  Mario Berta,et al.  The Fidelity of Recovery Is Multiplicative , 2015, IEEE Transactions on Information Theory.

[47]  Stanley Gudder,et al.  Quantum Markov chains , 2008 .

[48]  M. Einsiedler,et al.  Ergodic Theory: with a view towards Number Theory , 2010 .

[49]  Mark M. Wilde,et al.  Recoverability in quantum information theory , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[50]  M. Sion.  On general minimax theorems , 1958 .

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

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

[53]  Fernando G S L Brandão,et al.  Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution. , 2014, Physical review letters.

[54]  Serge Fehr,et al.  On quantum Rényi entropies: A new generalization and some properties , 2013, 1306.3142.