Abstract
Let H and be finite-dimensional Hilbert spaces, T: B(H) → B() be a coarse-graining and D 1, D 2 be density matrices on H . In this Letter the consequences of the existence of a coarse-graining β : B() → B(H) satisfying βT(D s )=D s are given. (This means that T is sufficient for D 1 and D 2.) It is shown that D s =∑ p=1 r λ s (p) SH s H (p)RH(p) (s=1,2) should hold with pairwise orthogonal summands and with commuting factors and with some probability distributions λ s (p) for 1 ≤ p ≤ r (s=1,2). This decomposition allows to deduce the exact condition for equality in the strong subadditivity of the von Neumann entropy.
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References
Accardi, L. and Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245–273.
Accardi, L. and Liebscher, V.: Markovian KMS-states for one-dimensional spin chains, Infin. Dimensions. Anal. Quantum Probab. Relat. Topics 2 (1999), 645–661.
Barnum, H. and Knill, E.: Reversing quantum dynamics with near optimal quantum and classical fidelity, J. Math. Phys. 43 (2002), 2097–2106.
Hayden, P., Jozsa, R., Petz, D. and Winter, A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality, to be published in Comm. Math. Phys.
Koashi, M. and Imoto, N.: Operations that do not disturb partially known quantum states, Phys. Rev. A 66 (2002), 022318.
Lieb, E. H. and Ruskai, M. B.: Proof of the strong subadditivity of quantum mechanical entropy, J. Math. Phys. 14 (1973), 1938–1941.
Nielsen, M. A. and Chuang, I. L.: Quantum Computation and Quantum Information, Cambridge University Press, 2000.
Ohya, M. and Petz, D.: Quantum Entropy and Its Use, Springer-Verlag, Heidelberg, 1993.
Ohno, H.: Translation-invariant quantum Markov states, to be published in Interdisc. Inf. Sci.
Petz, D.: A dual in von Neumann algebras, Quart. J. Math. Oxford 35 (1984), 475–483.
Petz, D.: Sufficiency of channels over von Neumann algebras, Quart. J. Math. Oxford, 39 (1988), 907–1008.
Petz, D.: Characterization of sufficient observation channels, In: Mathematical Methods in Statistical Mechanics, Leuven University Press, 1989, pp. 167–178.
Petz, D.: Entropy of Markov states, Riv. Math. Pura Appl. 14 (1994), 33–42.
Petz, D.: Monotonicity of quantum relative entropy revisited, Rev. Math. Phys. 15 (2003), 79–91.
Ruskai, M. B.: Inequalities for quantum entropy: A review with conditions with equality, J. Math. Phys. 43 (2002), 4358–4375.
Strasser, H.: Mathematical Theory Of Statistics. Statistical Experiments and Asymptotic Decision Theory, Walter de Gruyter, Berlin, 1985.
Strătilă, S.: Modular Theory in Operator Algebras, Abacuss Press, Tunbridge Wells, 1981.
Takesaki, M.: Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972), 306–321.
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Mosonyi, M., Petz, D. Structure of Sufficient Quantum Coarse-Grainings. Letters in Mathematical Physics 68, 19–30 (2004). https://doi.org/10.1007/s11005-004-4072-2
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DOI: https://doi.org/10.1007/s11005-004-4072-2