Diagonal unitary and orthogonal symmetries in quantum theory

Abstract

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.

The PPT$$^2$$ Conjecture Holds for All Choi-Type Maps

“…In Sect. 2, we provide several equivalent statements of the PPT 2 conjecture and also review some useful facts about diagonal unitary covariant maps; all of them are proven in [28]. In Sect.…”

“…These maps, dubbed respectively, Diagonal Unitary Covariant (DUC) and Conjugate Diagonal Unitary Covariant (CDUC), were studied at length in [28].…”

“…In this section, we will briefly review several key aspects from the theory of diagonal unitary and orthogonal covariant maps between matrix algebras. For a more comprehensive discussion and proofs of the results stated here, the readers are referred to our previous work [28,.…”

“…This can be seen by considering extremal rays of the cones (see [28,Theorem 5.13]) generated by vectors of full support.…”

“…We conjecture that answer is the same as for (C)DUC maps. We recall (see [28,Lemma 9.3]) the composition rule for DOC maps.…”