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On spectra of Lüders operations. (English) Zbl 1153.81410
Summary: We show that all the eigenvalues of certain generalized Lüders operations are non-negative real numbers in two cases of interest. In particular, given a commuting \(n\)-tuple \(\mathcal A=(A_1,\dots,A_n)\) consisting of positive operators on a Hilbert space \(\mathcal H\), satisfying \(\sum_{j=1}^nA_j=I\), we show that the spectrum of the Lüders operation: \[ \Lambda_{\mathcal A}: \mathcal B(\mathcal H)\ni X\mapsto \sum_{j=1}^nA_j^{1/2}XA_j^{1/2}\in \mathcal B(\mathcal H) \] is contained in \([0,\infty)\), so the only solution of the equation \(\Lambda(X)=I-X\) is the “expected” one: \(X=\tfrac 12 I\).

81R15 Operator algebra methods applied to problems in quantum theory
47N50 Applications of operator theory in the physical sciences
81P15 Quantum measurement theory, state operations, state preparations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: DOI
[1] DOI: 10.1063/1.1519669 · Zbl 1060.81009 · doi:10.1063/1.1519669
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