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Sequential product and Jordan product of quantum effects. (English) Zbl 1215.81009
Summary: The quantum effects for a physical system can be described by the set \({\mathcal E}({\mathcal H})\) of positive operators on a complex Hilbert space \({\mathcal H}\) that are bounded above by the identity operator \(I\). We denote the set of sharp effects by \({\mathcal P}({\mathcal H})\). For \(A,B \in{\mathcal E}({\mathcal H})\), the operation of sequential product \(A\circ B = A^{\frac{1}{2}} BA^{\frac{1}{2}}\) was proposed as a model for sequential quantum measurements. Denote by \(A*B=\frac{AB+BA}{2}\) the Jordan product of \(A,B\in{\mathcal E}({\mathcal H})\). The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on \(A*B\) imply that \(A\) and \(B\) commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if \(A \ast B\) satisfies certain associative laws, then \(AB=BA\). Moreover, \(A*B\in{\mathcal P}({\mathcal H})\) if and only if \(A\circ B\in{\mathcal P}({\mathcal H})\).

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
17A15 Noncommutative Jordan algebras
81P15 Quantum measurement theory, state operations, state preparations
62L86 Fuzziness and sequential statistical methods
Full Text: DOI
[1] Busch, P., Singh, J.: Luders theorem for unsharp quantum measurements. Phys. Lett. A 249, 10–12 (1998) · doi:10.1016/S0375-9601(98)00704-X
[2] Du, H.K., Deng, C.Y., Li, Q.H.: On the infimum problem of Hilbert space effects. Sci. China 51, 320–332 (2006) (In Chinese)
[3] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Mathematics and Its Applications, vol. 516. Kluwer Academic/Ister Science, Dordrecht/Bratislava (2000) · Zbl 0987.81005
[4] Gheondea, A., Gudder, S., Jonas, P.: On the infimum of quantum effects. J. Math. Phys. 46, 062102 (2005) · Zbl 1110.81026 · doi:10.1063/1.1904704
[5] Gheondea, A., Gudder, S.: Sequential product of quantum effect. Proc. Am. Math. Soc. 132, 503–512 (2004) · Zbl 1059.47041 · doi:10.1090/S0002-9939-03-07063-1
[6] Gudder, S., Nagy, G.: Sequentially independent effects. Proc. Am. Math. Soc. 130(4), 1125–1130 (2002) · Zbl 1016.47020 · doi:10.1090/S0002-9939-01-06194-9
[7] Gudder, S., Nagy, G.: Sequential quantum measurements. J. Math. Phys. 42(11), 5212–5222 (2001) · Zbl 1018.81005 · doi:10.1063/1.1407837
[8] Gudder, S., Greechie, R.: Sequential products on effect algebras. Rep. Math. Phys. 49(1), 87–111 (2002) · Zbl 1023.81001 · doi:10.1016/S0034-4877(02)80007-6
[9] Gudder, S.: Examples, problems, and results in effect algebras. Int. J. Theor. Phys. 35, 2365–2376 (1996) · Zbl 0868.03028 · doi:10.1007/BF02302453
[10] Gudder, S.: Lattice properties of quantum effects. J. Math. Phys. 37, 2637–2642 (1996) · Zbl 0879.47045 · doi:10.1063/1.531533
[11] Li, Y., Sun, X.H., Chen, Z.L.: Generalized infimum and sequential product of quantum effects. J. Math. Phys. 48, 102101 (2007) · Zbl 1152.81530
[12] Li, Y., Du, H.K.: A note on the infimum problem of Hilbert space effects. J. Math. Phys. 47, 102103 (2006) · Zbl 1112.58011 · doi:10.1063/1.2354330
[13] Murphy, G.J.: C algebra and Operator Theory. Academic Press, New York (1990) · Zbl 0714.46041
[14] Moreland, T., Gudder, S.: Infima of Hilbert space effects. Linear Algebra Appl. 286, 1–17 (1999) · Zbl 0964.46010 · doi:10.1016/S0024-3795(98)10119-2
[15] Shmulyan, L.Y.: An operator Hellinger integral. Mat. Sb. 49, 381–430 (1959)
[16] Williams, J.P.: On the range of a derivation. Pac. J. Math. 38, 273–279 (1971) · Zbl 0218.46039 · doi:10.2140/pjm.1971.38.273
[17] Yang, J., Du, H.K.: A note on commutativity up to a factor of bounded operators. Proc. Am. Math. Soc. 132, 1713–1720 (2004) · Zbl 1055.47005 · doi:10.1090/S0002-9939-04-07224-7
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