Abstract
The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In particular, if dim H=n, then on ℒ(H) there is a unique discrete state, namely m(A)=dim A/dim H, A∈ℒ(H).
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Dedicated to Pekka J. Lahti on the occasion of his 60th birthday.
The paper has been supported by the Center of Excellence SAS–Physics of Information–I/2/2005, the grant VEGA No. 2/6088/26 SAV, by Science and Technology Assistance Agency under the contract APVV-0071-06, Bratislava, Slovakia.
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Buhagiar, D., Chetcuti, E. & Dvurečenskij, A. On Gleason’s Theorem without Gleason. Found Phys 39, 550–558 (2009). https://doi.org/10.1007/s10701-008-9265-6
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DOI: https://doi.org/10.1007/s10701-008-9265-6