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States on Quantum Structures Versus Integrals

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Abstract

We show conditions when a state on a quantum structure E like an effect algebra, a pseudo effect algebra E satisfying some kind of the Riesz Decomposition Properties (RDP) or on an MV-algebra, a BL-algebra, a pseudo MV-algebra and a pseudo BL-algebra is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of (RDP), the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K. The same is true for states on an MV-algebra and a BL-algebra and their noncommutative variants.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

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  1. Anatolij Dvurečenskij
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Correspondence to Anatolij Dvurečenskij.

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The author thanks for the support by Center of Excellence SAS—Quantum Technologies—ERDF OP R&D Projects CE QUTE ITMS 26240120009 and meta-QUTE ITMS 26240120022, the grant VEGA No. 2/0032/09 SAV.

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Dvurečenskij, A. States on Quantum Structures Versus Integrals. Int J Theor Phys 50, 3761–3777 (2011). https://doi.org/10.1007/s10773-011-0693-2

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