Quantum Measures on Finite Effect Algebras with the Riesz Decomposition Properties
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One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure \(\lambda \) on the tensor product \(E\otimes E\) can determine a quantum measure \(\mu \) on a finite effect algebra \(E\) with the RDP such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\). Furthermore, some conditions for a grade-2 additive measure \(\mu \) on a finite effect algebra \(E\) are provided to guarantee that there exists a unique diagonally positive symmetric signed measure \(\lambda \) on \(E\otimes E\) such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\). At last, it is showed that any grade-\(t\) quantum measure on a finite effect algebra \(E\) with the RDP is essentially established by the values on a subset of \(E\).
KeywordsQuantum interference Effect algebra Quantum measure Tensor product
The authors are grateful to the anonymous referees who suggest us to note and compare the research for the fuzzy measures in [11, 26] with the research for quantum measures. The authors are indebted to the anonymous referees’ kindly comments to improve the readability of our manuscript. The authors also would like to thank Pulumannová for the useful discussions regarding the history theories. This work is partially supported by National Science Foundation of China (Grant Nos. 11201279, 11271237, 61273311 and 61373150), a research grant from Education Department of Shaanxi Province (No. 12JK0875).