Abstract
In the earlier paper (Molnár and Šemrl in Lett Math Phys 80:239–255, 2007), we described the structure of all spectral order automorphisms of the sets of Hilbert space effects and bounded observables in the case where the dimension of the underlying Hilbert space is at least 3. The aim of this note is to present a complete description in the missing two-dimensional case. We will see that in that case there is a one-to-one correspondence between the set of all spectral order automorphisms and the set of all bijective maps of pure states together with the set of all strictly increasing bijections of the real unit interval or the real line.
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References
Döring A., Dewitt B.: Self-adjoint operators as functions I: lattices, Galois connections, and the spectral order. Commun. Math. Phys. 328, 499–525 (2014)
Flori C.: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol. 868. Springer, Berlin (2013)
Kadison R.V.: Order properties of bounded self-adjoint operators. Proc. Am. Math. Soc. 2, 505–510 (1951)
Molnár L.: Order-automorphisms of the set of bounded observables. J. Math. Phys. 42, 5904–5909 (2001)
Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895. Springer, Berlin (2007)
Molnár L., Šemrl P.: Spectral order automorphisms of the spaces of Hilbert space effects and observables. Lett. Math. Phys. 80, 239–255 (2007)
Olson M.P.: The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice. Proc. Am. Math. Soc. 28, 537–544 (1971)
Petz D.: Quantum Information Theory and Quantum Statistics. Springer, Berlin (2008)
Šemrl P.: Comparability preserving maps on Hilbert space effect algebras. Commun. Math. Phys. 313, 375–384 (2012)
Uhlhorn U.: Representation of symmetry transformations in quantum mechanics. Ark. Fysik 23, 307–340 (1963)
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The authors were supported by the “Lendület” Program (LP2012-46/2012) of the Hungarian Academy of Sciences.
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Molnár, L., Nagy, G. Spectral Order Automorphisms on Hilbert Space Effects and Observables: The 2-Dimensional Case. Lett Math Phys 106, 535–544 (2016). https://doi.org/10.1007/s11005-016-0830-1
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DOI: https://doi.org/10.1007/s11005-016-0830-1