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arXiv:2505.13871 · International Journal of Theoretical Physics

Remarks on Orthogonality Spaces

John Harding, Remi Salinas Schmeis

Published Jul 23, 2025• 1 citation
arXivHTMLPDF
DOI
10.1007/s10773-025-06062-x

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The recently proposed complexification of Hamiltonians which keeps the spectra real (and is usually called PT symmetry) is re-interpreted here as a certain natural linear-algebraic alternative to Hermiticity. The juxtaposition is mediated by the Feshbachian projection on a model space which reduces the difference just to a sign of a propagator in the effective Hamiltonian. The problem of norms and orthogonality is clarified and a few perturbation aspects are mentioned. A remarkable simplicity of the secular polynomial is conjectured. The mechanism of the PT symmetry breaking is clarified via a separable approximation scheme.
In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.