Diagonal unitary and orthogonal symmetries in quantum theory

Satvik Singh1 and Ion Nechita2

1Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Punjab, India.
2Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France.

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We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

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[7] Márton Mestyán, Balázs Pozsgay, and Ian M. Wanless, "Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states", arXiv:2210.13017.

[8] Satvik Singh, "Entanglement detection in triangle-free quantum states", Physical Review A 103 3, 032436 (2021).

[9] Satvik Singh, "Can entanglement hide behind triangle-free graphs?", arXiv:2010.11891.

[10] Ion Nechita and Satvik Singh, "A graphical calculus for integration over random diagonal unitary matrices", arXiv:2007.11219.

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