Abstract
The entropy of a subalgebra, which has been used in quantum ergodic theory to construct a noncommutative dynamical entropy, coincides for N-level systems and Abelian subalgebras with the notion of maximal mutual information of quantum communication theory. The optimal decompositions of mixed quantum states singled out by the entropy of Abelian subalgebras correspond to optimal detection schemes at the receiving end of a quantum channel. It is then worthwhile studying in some detail the structure of the convex hull of quantum states brought about by the variational definition of the entropy of a subalgebra. In this Letter, we extend previous results on the optimal decompositions for 3-level systems.
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References
Bennett, C. H., Di Vincenzo, D. P., Smolin, J. and Wooters, W. K.: Mixed state entanglement and quantum error correction, Phys. Rev. A 54 (1996), 3824–3851.
Benatti, F.: Entropy of a subalgebra and quantum estimation, J. Math. Phys. 37 (1996), 5244–5258.
Connes, A.: C.R. Acad. Paris 301 I (1985) 1–3.
Narnhofer, H. and Thirring W.: From relative entropy to entropy, Fizika 17 (1985), 257–264.
Connes, A., Narhofer, H. and Thirring, W.: Dynamical entropy of C* algebras and von Neumann algebras, Comm. Math. Phys. 112 (1987), 691–719.
Benatti, F., Narnhofer, H. and Uhlmann, A.: Decompositions of quantum states with respect to entropy, Rep. Math. Phys. 38 (1996), 123–141.
Connes, A., Narnhofer, H. and Thirring, W.: Quantum dynamical entropy, in: H. Mitter and L. Pittner (eds), Recent Developements in Mathematical Physics, Springer, Berlin, 1987, pp. 102–136.
Benatti, F. and Grava, T.: The entropy of a subalgebra, in: C. Cecchini (ed.), Contributions in Probability, Volume in Memory of A. Frigerio, Forum Publ., Udine, 1996, pp. 57–65.
Levitin, L, B.: OSID 3 (1994), 319.
Uhlmann, A.: Optimizing entropy relative to a channel or a subalgebra, in: H. D. Doebner, P. Nattermann and W. Scherer (eds), GROUP21, Proc. XXI Int. Coll. on Group Theoretical Methods in Physics, Vol.I, World Scientific, Singapore, 1997, pp. 343–348.
Uhlmann, A.: Entropy and optimal decompositions of states relative to a maximal commutative subalgebra, q-ph/9701017.
Hill, S. and Wootters, W. K.: Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78 (1997), 5022–5025.
Wootters, W. K.: Entanglement of formation of an arbitrary state of two qubits, q-ph/9709029 v2.
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Benatti, F., Narnhofer, H. & Uhlmann, A. Optimal Decompositions with Respect to Entropy and Symmetries. Letters in Mathematical Physics 47, 237–253 (1999). https://doi.org/10.1023/A:1007537800344
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DOI: https://doi.org/10.1023/A:1007537800344