Non-monotonicity of Trace Distance Under Tensor Products

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• 作者：Jonas Maziero
• 关键词：Quantum distance measures ; Trace distance ; Monotonicity under tensor products
• 刊名：Brazilian Journal of Physics
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：45
• 期：5
• 页码：560-566
• 全文大小：1,257 KB
• 参考文献：1.M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)MATH
  2.M.M. Wilde, Quantum Information Theory (Cambridge University Press, Cambridge, 2013)CrossRef MATH
  3.C.A. Fuchs, Distinguishability and accessible information in quantum theory, arXiv:quant-ph/-601020
  4.P. Neumann, N. Mizuochi, F. Rempp, P. Hemmer, H. Watanabe, S. Yamasaki, V. Jacques, T. Gaebel, F. Jelezko, J. Wrachtrup, Multipartite entanglement among single spins in diamond. Science. 320, 1326 (2008)CrossRef ADS
  5.P. Walther, K.J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, A. Zeilinger, Experimental one-way quantum computing. Nature. 434, 169 (2005)CrossRef ADS
  6.J. Zhang, R. Laflamme, D. Suter, Experimental implementation of encoded logical qubit operations in a perfect quantum error correcting code. Phys. Rev. Lett. 109, 100503 (2012)CrossRef ADS
  7.V. Scarani, S. Iblisdir, N. Gisin, A. Acín, Quantum cloning. Rev. Mod. Phys. 77, 1225 (2005)CrossRef ADS MATH
  8.A.K. Pati, M.M. Wilde, A.R.U. Devi, A.K. Rajagopal, Sudha, Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memor. Phys. Rev. A. 86, 042105 (2012)CrossRef ADS
  9.M. Zwolak, C.J. Riede, W.H. Zurek, Amplification, redundancy, and quantum Chernoff information. Phys. Rev. Lett. 112, 140406 (2014)CrossRef ADS
  10.N. Boulant, T.F. Havel, M.A. Pravia, D.G. Cory, Robust method for estimating the Lindblad operators of a dissipative quantum process from measurements of the density operator at multiple time points. Phys. Rev. A. 67, 042322 (2003)CrossRef ADS
  11.A. Gilchrist, N.K. Langford, M.A. Nielsen, Distance measures to compare real and ideal quantum processes. Phys. Rev. A. 71, 062310 (2005)CrossRef ADS
  12.C.A. Fuchs, J. van de Graaf, Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory. 45, 1216 (1999)CrossRef MathSciNet MATH
  13.N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002)CrossRef ADS
  14.S.-J. Gu, Fidelity approach to quantum phase transitions. Int. J. Mod. Phys. B. 24, 4371 (2010)CrossRef ADS MATH
  15.A. Borrás, M. Casas, A.R. Plastino, A. Plastino, Entanglement and the lower bounds on the speed of quantum evolution. Phys. Rev. A. 74, 022326 (2006)CrossRef ADS
  16.P.J. Jones, P. Kok, Geometric derivation of the quantum speed limit. Phys. Rev. A. 82, 022107 (2010)CrossRef ADS MathSciNet
  17.M.M. Taddei, B.M. Escher, L. Davidovich, R.L. de Matos Filho, Quantum speed limit for physical processes. Phys. Rev Lett. 110, 050402 (2013)CrossRef ADS
  18.S. Deffner, E. Lutz, Quantum speed limit for non-Markovian dynamics. Phys. Rev. Lett. 111, 010402 (2013)CrossRef ADS
  19.A. del Campo, I.L. Egusquiza, M.B. Plenio, S.F. Huelga, Quantum speed limits in open system dynamics. Phys. Rev. Lett. 110, 050403 (2013)CrossRef ADS
  20.Y.-J. Zhang, W. Han, Y.-J. Xia, J.-P. Cao, H. Fan, Quantum speed limit for arbitrary initial states. Sci. Rep. 4, 4890 (2014)ADS
  21.F. Caruso, V. Giovannetti, C. Lupo, S. Mancini, Quantum channels and memory effects. Rev. Mod. Phys. 86, 1203 (2014)CrossRef ADS
  22.R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)CrossRef ADS MathSciNet MATH
  23.L. Aolita, F. de Melo, L. Davidovich, Open-system dynamics of entanglement: a key issues review. Rep. Prog. Phys. 78, 042001 (2015)CrossRef ADS
  24.L.C. Céleri, J. Maziero, R.M. Serra, Theoretical and experimental aspects of quantum discord and related measures. Int. J. Quant. Inf. 9, 1837 (2011)CrossRef MATH
  25.K. Modi, A. Brodutch, H. Cable, T. Paterek, V. Vedral, The classical-quantum boundary for correlations: Discord and related measures. Rev. Mod. Phys. 84, 1655 (2012)CrossRef ADS
  26.A. Streltsov. Quantum Correlations Beyond Entanglement and Their Role in Quantum Information Theory ( Springer, Berlin, 2015)MATH
  27.T. Baumgratz, M. Cramer, M.B. Plenio, Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)CrossRef ADS
  28.J. ?berg, Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014)CrossRef
  29.D. Girolami, Observable measure of quantum coherence in finite dimensional systems. Phys. Rev. Lett. 113, 170401 (2014)CrossRef ADS
  30.C.-s. Yu, Y. Zhang, H. Zhao, Quantum correlation via quantum coherence. Quant. Inf. Process. 13, 1437 (2014)CrossRef ADS MathSciNet
  31.K.M.R. Audenaert, Comparisons between quantum state distinguishability measures. Quant. Inf. Comp. 14, 31 (2014)MathSciNet
  32.D. Bures, An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite w ?/sup>?algebras. Trans. Amer. Math. Soc. 135, 199 (1969)MathSciNet
  33.A. Uhlmann, The “transition probability-in the state space of a ?/sup>?algebra. Rep. Math. Phys. 9, 273 (1976)CrossRef ADS MathSciNet MATH
  34.M. Bina,
• 作者单位：Jonas Maziero (1)
  
  1. Departamento de Física, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil
  
• 刊物类别：Physics and Astronomy
• 出版者：Springer New York
• ISSN：1678-4448

文摘

The trace distance (TD) possesses several of the good properties required for a faithful distance measure in the quantum state space. Despite its importance and ubiquitous use in quantum information science, one of its questionable features, its possible non-monotonicity under taking tensor products of its arguments (NMuTP), has been hitherto unexplored. In this article, we advance analytical and numerical investigations of this issue considering different classes of states living in a discrete and finite dimensional Hilbert space. Our results reveal that although this property of TD does not show up for pure states and for some particular classes of mixed states, it is present in a non-negligible fraction of the regarded density operators. Hence, even though the percentage of quartets of states leading to the NMuTP drawback of TD and its strength decrease as the system’s dimension grows, this property of TD must be taken into account before using it as a figure of merit for distinguishing mixed quantum states. Keywords Quantum distance measures Trace distance Monotonicity under tensor products