Necessary and sufficient conditions on measurements of quantum channels

Quantum supermaps are a higher-order genera- lization of quantum maps, taking quantum maps to quantum maps. It is known that any completely positive and trace non-increasing (CPTNI) map can be performed as part of a quantum measurement. By providing an explicit counterexample we show that, instead, not every quantum supermap sending a quantum channel to a CPTNI map can be realized in a measurement on quantum channels. We find that the supermaps that can be implemented in this way are exactly those transforming quantum channels into CPTNI maps even when tensored with the identity supermap. We link this result to the fact that the principle of causality fails in the theory of quantum supermaps.

[1]  Hans Halvorson,et al.  Deep beauty : understanding the quantum world through mathematical innovation , 2011 .

[2]  Xin Wang,et al.  Using and reusing coherence to realize quantum processes , 2018, Quantum.

[3]  Giulio Chiribella,et al.  Quantum speedup in the identification of cause–effect relations , 2018, Nature Communications.

[4]  Bob Coecke,et al.  Terminality implies non-signalling , 2014, QPL.

[5]  G. Gour,et al.  Quantum resource theories , 2018, Reviews of Modern Physics.

[6]  Giulio Chiribella,et al.  Entanglement, non-Markovianity, and causal non-separability , 2017 .

[7]  Bob Coecke,et al.  Reconstructing quantum theory from diagrammatic postulates , 2018, Quantum.

[8]  Gilad Gour,et al.  Comparison of Quantum Channels by Superchannels , 2018, IEEE Transactions on Information Theory.

[9]  Paolo Perinotti,et al.  Theoretical framework for higher-order quantum theory , 2019, Proceedings of the Royal Society A.

[10]  Paolo Perinotti,et al.  Causal Structures and the Classification of Higher Order Quantum Computations , 2016, 1612.05099.

[11]  Victor W. K. Mak,et al.  An Operational Approach , 2011 .

[12]  Miguel Herrero-Collantes,et al.  Quantum random number generators , 2016, 1604.03304.

[13]  Matty J. Hoban,et al.  A channel-based framework for steering, non-locality and beyond , 2017, 1708.00750.

[14]  Martin B. Plenio,et al.  Quantifying Operations with an Application to Coherence. , 2018, Physical review letters.

[15]  A. Holevo.  Statistical structure of quantum theory , 2001 .

[16]  Giulio Chiribella,et al.  Quantum communication in a superposition of causal orders , 2018, ArXiv.

[17]  Brendan Fong,et al.  Additive monotones for resource theories of parallel-combinable processes with discarding , 2015, QPL.

[18]  G. D’Ariano,et al.  Transforming quantum operations: Quantum supermaps , 2008, 0804.0180.

[19]  Mark M. Wilde,et al.  Entanglement cost and quantum channel simulation , 2018, Physical Review A.

[20]  Mario Berta,et al.  Entanglement cost of quantum channels , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[21]  Mark M. Wilde,et al.  Quantifying the magic of quantum channels , 2019, New Journal of Physics.

[22]  John T. Lewis,et al.  An operational approach to quantum probability , 1970 .

[23]  Gus Gutoski,et al.  Toward a general theory of quantum games , 2006, STOC '07.

[24]  Ryuji Takagi,et al.  Application of the Resource Theory of Channels to Communication Scenarios. , 2020, Physical review letters.

[25]  K. Życzkowski.  Quartic quantum theory: an extension of the standard quantum mechanics , 2008, 0804.1247.

[26]  G. D’Ariano,et al.  Probabilistic theories with purification , 2009, 0908.1583.

[27]  Andrew J. P. Garner,et al.  Resource theories of multi-time processes: A window into quantum non-Markovianity , 2019, Quantum.

[28]  Aleks Kissinger,et al.  Generalised compositional theories and diagrammatic reasoning , 2015, 1506.03632.

[29]  G Chiribella,et al.  Quantum circuit architecture. , 2007, Physical review letters.

[30]  Some Sankar Bhattacharya,et al.  Indefinite causal order enables perfect quantum communication with zero capacity channels , 2018, New Journal of Physics.

[31]  Yeong-Cherng Liang,et al.  A resource theory of quantum memories and their faithful verification with minimal assumptions , 2017, 1710.04710.

[32]  Carlo Maria Scandolo,et al.  Information-theoretic foundations of thermodynamics in general probabilistic theories , 2019, 1901.08054.

[33]  S. Massar,et al.  Error filtration and entanglement purification for quantum communication (17 pages) , 2004, quant-ph/0407021.

[34]  John Watrous,et al.  The Theory of Quantum Information , 2018 .

[35]  G. D’Ariano,et al.  Informational derivation of quantum theory , 2010, 1011.6451.

[36]  Instruments and channels in quantum information theory , 2004, quant-ph/0409019.

[37]  Mark M. Wilde,et al.  Resource theory of entanglement for bipartite quantum channels , 2019, 1907.04181.

[38]  L. Hardy.  Foliable Operational Structures for General Probabilistic Theories , 2009, 0912.4740.

[39]  K. Melchor Quick Hall.  Beyond States , 2019, Naming a Transnational Black Feminist Framework.

[40]  Adrian Kent,et al.  No signaling and quantum key distribution. , 2004, Physical review letters.

[41]  Sina Salek,et al.  Enhanced Communication with the Assistance of Indefinite Causal Order. , 2017, Physical review letters.

[42]  Gilad Gour,et al.  Dynamical resource theory of quantum coherence , 2019, Physical Review Research.

[43]  Mario Berta,et al.  Thermodynamic Capacity of Quantum Processes. , 2018, Physical review letters.

[44]  G. D’Ariano,et al.  Theoretical framework for quantum networks , 2009, 0904.4483.

[45]  Gilles Brassard,et al.  Quantum cryptography: Public key distribution and coin tossing , 2014, Theor. Comput. Sci..

[46]  Lucien Hardy,et al.  Reconstructing Quantum Theory , 2013, 1303.1538.

[47]  C. Giarmatzi,et al.  Causal and causally separable processes , 2015, Rethinking Causality in Quantum Mechanics.

[48]  A. Winter,et al.  Resource theory of coherence: Beyond states , 2017, 1704.03710.

[49]  Gilad Gour,et al.  How to Quantify a Dynamical Quantum Resource. , 2019, Physical review letters.

[50]  L. Hardy.  Reformulating and Reconstructing Quantum Theory , 2011, 1104.2066.

[51]  Yunchao Liu,et al.  Operational resource theory of quantum channels , 2019, Physical Review Research.

[52]  M. Ziman.  Process positive-operator-valued measure: A mathematical framework for the description of process tomography experiments , 2008, 0802.3862.

[53]  Sina Salek,et al.  Resource theories of communication , 2020 .

[54]  Giulio Chiribella,et al.  Quantum from principles , 2015, ArXiv.

[55]  R Raussendorf,et al.  A one-way quantum computer. , 2001, Physical review letters.

[56]  B. Valiron,et al.  Quantum computations without definite causal structure , 2009, 0912.0195.

[57]  Alexander Barvinok,et al.  A course in convexity , 2002, Graduate studies in mathematics.

[58]  Gilad Gour,et al.  Entanglement of a bipartite channel , 2019, Physical Review A.

[59]  Mark M. Wilde,et al.  Entropy of a Quantum Channel: Definition, Properties, and Application , 2018, 2020 IEEE International Symposium on Information Theory (ISIT).

[60]  Andreas Winter,et al.  Resource theories of quantum channels and the universal role of resource erasure , 2019, 1904.04201.

[61]  Č. Brukner,et al.  Quantum correlations with no causal order , 2011, Nature Communications.

[62]  C. Branciard,et al.  Communication through coherent control of quantum channels , 2018, Quantum.

[63]  Mark M. Wilde,et al.  Quantum Information Theory , 2013 .

[64]  Giulio Chiribella,et al.  Resource theories of communication with quantum superpositions of processes , 2019, ArXiv.

[65]  Kaifeng Bu,et al.  Quantifying the resource content of quantum channels: An operational approach , 2018, Physical Review A.

[66]  H. Briegel,et al.  Measurement-based quantum computation on cluster states , 2003, quant-ph/0301052.

[67]  Stefano Gogioso,et al.  Categorical Probabilistic Theories , 2017, QPL.

[68]  Andrew Chi-Chih Yao,et al.  Quantum cryptography with imperfect apparatus , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[69]  Giulio Chiribella,et al.  Optimal quantum networks and one-shot entropies , 2016, ArXiv.

[70]  J. Preskill,et al.  Causal and localizable quantum operations , 2001, quant-ph/0102043.

[71]  Howard Barnum,et al.  Post-Classical Probability Theory , 2012, 1205.3833.

[72]  S. Pirandola,et al.  General Benchmarks for Quantum Repeaters , 2015, 1512.04945.

[73]  Robert W. Spekkens,et al.  A mathematical theory of resources , 2014, Inf. Comput..

[74]  M. Paternostro,et al.  Non-Markovian quantum processes: Complete framework and efficient characterization , 2015, 1512.00589.

[75]  Umesh Vazirani,et al.  Fully device-independent quantum key distribution. , 2012, 1210.1810.

[76]  V. Sunder,et al.  The Functional Analysis of Quantum Information Theory: A Collection of Notes Based on Lectures by Gilles Pisier, K. R. Parthasarathy, Vern Paulsen and Andreas Winter , 2015 .

[77]  Earl Campbell,et al.  Quantifying magic for multi-qubit operations , 2019, Proceedings of the Royal Society A.