Subspace preservation, subspace locality, and gluing of completely positive maps (Q1886806)

A quantum system consisting of separate entities on separate locations is usually modelled with a Hilbert space of the form \({\mathcal H}_1\otimes {\mathcal H}_2\), where \({\mathcal H}_1\) and \({\mathcal H}_2\), respectively, represent the pure states of each system. This paper considers situations where the tensor product \({\mathcal H}_1\otimes {\mathcal H}_2\) is replaced with the direct sum \({\mathcal H}_1\oplus {\mathcal H}_2\), e.g., as in the case of a two path single particle interferometer. The gluing of two completely positive maps \(\Phi_1\) and \(\Phi_2\) acting on density operators on \({\mathcal H}_1\) and \({\mathcal H}_2\) is defined as a linear map \(\Phi\) acting on density operators on \({\mathcal H}_1 \oplus {\mathcal H}_2\), and represents the operation of the two systems on the same input. In this context, the author introduces the notions of subspace preservation and subspace locality for completely positive maps, investigates the construction of gluings satisfying locality properties, and addresses the question of whether the pair \((\Phi_1, \Phi_2 )\) can characterize a gluing.