Abstract
In the paper, we study linear operators in complex Hilbert space \(\mathbb {C}^n\) that are called real-orthogonal projections, which are a generalization of standard (complex) orthogonal projections but for which only the real part of the scalar product vanishes. We compare some partial order properties of orthogonal and of real-orthogonal projections. In particular, this leads to the observation that a natural analogue of the ordering relationship defined on standard orthogonal projections leads to a non-transitive relationship between real-orthogonal projections. We prove that the set of all real-orthogonal projections in a finite-dimensional complex space is a quantum pseudo-logic, and briefly consider some potential applications of such a structure.
AMS2000 subject classification. 15A57; 81P10.
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“In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors.” Quoted directly from https://en.wikipedia.org/wiki/Semi-orthogonal_matrix.
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Acknowledgments
The authors would like to thank several attendees at the conference for valuable feedback, including Ismael Martinez-Martinez, Hans Römer and Thomas Filk.
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Matvejchuk, M., Widdows, D. (2016). Real-Orthogonal Projections as Quantum Pseudo-Logic. In: Atmanspacher, H., Filk, T., Pothos, E. (eds) Quantum Interaction. QI 2015. Lecture Notes in Computer Science(), vol 9535. Springer, Cham. https://doi.org/10.1007/978-3-319-28675-4_21
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DOI: https://doi.org/10.1007/978-3-319-28675-4_21
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