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.

References in corpus (7)

• Quantum channels and their entropic characteristics
• Separability of diagonal symmetric states: a quadratic conic optimization problem
• A new class of states with positive partial transposition
• The PPT${}^2$ conjecture holds for all Choi-type maps
• A graphical calculus for integration over random diagonal unitary matrices
• Can entanglement hide behind triangle-free graphs?
• Unitary conjugation channels with continuous random phases

Cited by in corpus (16)

• Detecting positive quantum capacities of quantum channels
• The PPT${}^2$ conjecture holds for all Choi-type maps
• A class of Bell diagonal entanglement witnesses in ${\mathbb{C}}^4\otimes {\mathbb{C}}^4$: optimization and the spanning property
• Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators
• A graphical calculus for integration over random diagonal unitary matrices
• Can entanglement hide behind triangle-free graphs?
• A class of optimal positive maps in $M_n$
• Ergodic theory of diagonal orthogonal covariant quantum channels
• Quantum versus classical $P$-divisibility
• Random covariant quantum channels
• A Fisher information-based incompatibility criterion for quantum channels
• Entanglement in the Dicke subspace
• Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)
• Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products
• Zero-error communication under discrete-time Markovian dynamics
• Rank-based entanglement detection for bound entangled states