Strict positivity and D-majorization

References

• Brandão F, Horodecki M, Ng N, et al. The second laws of quantum thermodynamics. Proc Natl Acad Sci USA. 2015;112:3275–3279.
   PubMed Web of Science ®Google Scholar
• Dall'Arno M, Buscemi F, Scarani V. Extension of the Alberti–Ulhmann criterion beyond qubit dichotomies. Quantum. 2020;4:233.
   Web of Science ®Google Scholar
• Gour G, Müller M, Narasimhachar V, et al. The resource theory of informational nonequilibrium in thermodynamics. Phys Rep. 2015;583:1–58.
   Web of Science ®Google Scholar
• Horodecki M, Oppenheim J. Fundamental limitations for quantum and nanoscale thermodynamics. Nat Commun. 2013;4:2059.
   PubMed Web of Science ®Google Scholar
• Albert V. Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states. Quantum. 2019;3:151.
   Web of Science ®Google Scholar
• Burgarth D, Chiribella G, Giovannetti V, et al. Ergodic and mixing quantum channels in finite dimensions. New J Phys. 2013;15:073045.
   Web of Science ®Google Scholar
• Ohya M. Quantum ergodic channels in operator algebras. J Math Anal Appl. 1981;84:318–327.
   Web of Science ®Google Scholar
• Davies E. Quantum stochastic processes II. Commun Math Phys. 1970;19:83–105.
   Web of Science ®Google Scholar
• Sanz M, Pérez-García D, Wolf M, et al. A quantum version of Wielandt's inequality. IEEE Trans Inform Theory. 2010;56:4668–4673.
   Web of Science ®Google Scholar
• Blume-Kohout R, Ng H, Poulin D, et al. Extension of the Alberti–Ulhmann criterion beyond qubit dichotomies. Phys Rev A. 2010;82:062306.
   Web of Science ®Google Scholar
• Gupta V, Mandayam P, Sunder V. The functional analysis of quantum information theory. Cham: Springer; 2015.
   Google Scholar
• Cirillo G, Ticozzi F. Decompositions of Hilbert spaces, stability analysis and convergence probabilities for discrete-time quantum dynamical semigroups. J Phys A. 2015;48:085302.
   Google Scholar
• Frigerio A. Quantum dynamical semigroups and approach to equilibrium. Lett Math Phys. 1977;2:79–87.
   Web of Science ®Google Scholar
• Spohn H. Approach to equilibrium for completely positive dynamical semigroups of N-level systems. Rep Math Phys. 1976;10:189–194.
   Google Scholar
• Spohn H. An algebraic condition for the approach to equilibrium of an open N-level system. Lett Math Phys. 1977;2:33–38.
   Web of Science ®Google Scholar
• Veinott A. Least d-majorized network flows with inventory and statistical applications. Manage Sci. 1971;17:547–567.
   Web of Science ®Google Scholar
• Ruch E, Schranner R, Seligman T. The mixing distance. J Chem Phys. 1978;69:386–392.
   Web of Science ®Google Scholar
• vom Ende F, Dirr G. The d-majorization polytope; 2019. Available from: https://arxiv.org/abs/1911.01061
   Google Scholar
• Marshall A, Olkin I, Arnold B. Inequalities: theory of majorization and its applications. 2nd ed. New York: Springer; 2011.
   Google Scholar
• Holevo A. Quantum systems, channels, information: a mathematical introduction. Berlin: DeGruyter; 2012 (De Gruyter studies in mathematical physics 16).
   Google Scholar
• Choi MD. Completely positive linear maps on complex matrices. Linear Algebra Appl. 1975;10:285–290.
   Web of Science ®Google Scholar
• Pérez-García D, Wolf M, Petz D, et al. Contractivity of positive and trace–preserving maps under Lp–norms. J Math Phys. 2006;47:083506.
   Web of Science ®Google Scholar
• Heinosaari T, Ziman M. The mathematical language of quantum theory: from uncertainty to entanglement. Cambridge: Cambridge University Press; 2012.
   Google Scholar
• vom Ende F, Dirr G. Unitary dilations of discrete-time quantum-dynamical semigroups. J Math Phys. 2019;60:122702.
   Web of Science ®Google Scholar
• Li CK, Poon YT. Interpolation by completely positive maps. Linear Multilinear Algebra. 2011;59:1159–1170.
   Web of Science ®Google Scholar
• Farenick D. Irreducible positive linear maps on operator algebras. Proc Am Math Soc. 1996;124:3381–3390.
   Web of Science ®Google Scholar
• Gaubert S, Qu Z. Checking strict positivity of kraus maps is NP-hard. Inform Process Lett. 2017;118:35–43.
   Web of Science ®Google Scholar
• Rahaman M. A new bound on quantum Wielandt inequality. IEEE Trans Inform Theory. 2020;66:147–154.
   Web of Science ®Google Scholar
• Bhatia R. Positive definite matrices. Princeton (NJ): Princeton University Press; 2007.
   Google Scholar
• Hayashi M. Quantum information: an introduction. Berlin: Springer; 2006.
   Google Scholar
• Kadison R, Ringrose J. Fundamentals of the theory of operator algebras, vol. 1: elementary theory. Providence (RI): American Mathematical Society; 1983.
   Google Scholar
• Horn R, Johnson C. Topics in matrix analysis. Cambridge: Cambridge University Press; 1991.
   Google Scholar
• Gorini V, Kossakowski A, Sudarshan E. Completely positive dynamical semigroups of N-level systems. J Math Phys. 1976;17:821–825.
   Web of Science ®Google Scholar
• Lindblad G. On the generators of quantum dynamical semigroups. Commun Math Phys. 1976;48:119–130.
   Web of Science ®Google Scholar
• Joe H. Majorization and divergence. J Math Anal Appl. 1990;148:287–305.
   Web of Science ®Google Scholar
• Schulte-Herbrüggen T, Dirr G. Exploring the limits of open quantum dynamics. I: Motivation, first results from toy models to applications; 2020. Available from: https://arxiv.org/abs/2003.06018
   Google Scholar
• Ando T. Majorization, doubly stochastic matrices, and comparison of eigenvalues. Linear Algebra Appl. 1989;118:163–248.
   Web of Science ®Google Scholar
• Sagawa T. Entropy, divergence, and majorization in classical and quantum thermodynamics; 2020. Available from: https://arxiv.org/abs/2007.09974
   Google Scholar
• Parker D, Ram P. Greed and majorization, Los Angeles, CA; 1996 (Technical report, Department of Computer Science, University of California).
   Google Scholar
• Verstraete F, Verschelde H. On quantum channels; 2002. Available from: https://arxiv.org/abs/quant-ph/0202124
   Google Scholar
• Alberti P, Uhlmann A. A problem relating to positive linear maps on matrix algebras. Rep Math Phys. 1980;18:163–176.
   Google Scholar
• Heinosaari T, Jivulescu M, Reeb D, et al. Extending quantum operations. J Math Phys. 2012;53:102208.
   Web of Science ®Google Scholar
• Li CK, Mathias R. Matrix inequalities involving a positive linear map. Linear Multilinear Algebra. 1996;41:221–231.
   Google Scholar
• Huang Z, Li CK, Poon E, et al. Physical transformations between quantum states. J Math Phys. 2012;53:102209.
   Web of Science ®Google Scholar
• Brondsted A. An introduction to convex polytopes. New York: Springer; 1983 (Graduate texts in mathematics, Vol. 90).
   Google Scholar
• Baronti M, Papini P. Convergence of sequences of sets. Methods Funct Anal Approx Theory, ISNM. 1986;76:133–155.
   Google Scholar
• Brouwer L. Über Abbildung von Mannigfaltigkeiten. Math Ann. 1911;71:97–115.
   Google Scholar
• Wolf M, Cirac J. Dividing quantum channels. Commun Math Phys. 2008;279:147–168.
   Web of Science ®Google Scholar
• Horn R, Johnson C. Matrix analysis. Cambridge: Cambridge University Press; 1987.
   Google Scholar
• Kraus F. Über konvexe Matrixfunktionen. Math Z. 1936;41:18–42.
   Google Scholar
• Bendat J, Sherman S. Monotone and convex operator functions. Trans Am Math Soc. 1955;79:58–71.
   Google Scholar
• Ando T. Concavity of certain maps on positive definite matrices and applications to hadamard products. Linear Algebra Appl. 1979;26:203–241.
   Web of Science ®Google Scholar
• Bhatia R. Matrix analysis. New York: Springer; 1997.
   Google Scholar
• Cohn P. Universal algebra. Dordrecht: Springer Netherlands; 1981 (Mathematics and its applications, Vol. 6).
   Google Scholar
• Nadler S. Hyperspaces of sets: a text with research questions. New York: M. Dekker; 1978.
   Google Scholar
• Dirr G. The C-numerical range in infinite dimensions. Linear Multilinear Algebra. 2018;68:652–678.
   Web of Science ®Google Scholar