Moving to the present time and over the past few years, advantages in addressing the tasks of transmitting both quantum and classical information together compared to independent solutions have been discovered, from both information theoretic and coding theoretic points of view. Motivated by a number of considerations, including a generalization of the operator quantum error correction approach to infinitedimensional Hilbert space, it was also recognized that the OAQEC approach could provide a framework for error correction of hybrid classical and quantum information, though this specific line of investigation remained dormant for lack of motivating applications at the time. More recently, but still over a decade ago, the framework of "operator algebra quantum error correction" (OAQEC) was introduced. Combining our techniques with those of Brannan-Ganesan-Harris, we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions $N\subseteq M$. As an application of our result in the setting of \(\mathrm )$ over $N'$, generalising work of Werner. Along the way, we introduce an appropriate generalisation of LOCC operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von Neumann algebras. Namely, we establish a generalisation to this setting of Nielsen’s theorem on the convertibility of quantum states under local operations and classical communication (LOCC) schemes. We establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi-finite von Neumann algebras.
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