Private algebras in quantum information and infinite-dimensional complementarity (Q2786613)

The authors combine two interesting notions used in quantum privacy: private quantum code (extended to a possible infinite-dimensional variant) and complementarity (between private quantum codes and a special type of operator algebra called here ``private algebra''), both unified in a general framework for ``quantum error correction at the level of von Neumann algebras''.NEWLINENEWLINEA quantum channel is defined as a special map \({\mathcal E}:M\longrightarrow {\mathcal B}(S)\), where \({\mathcal B}(S)\) is the space of all bounded linear operators on a Hilbert space \(S\), and \(M\) is a von Neumann algebra. It is associated with a subalgebra \(N\) called ``private'' (Section 3) and a definition of complementary channel (Section 4). The main result of paper is Theorem 4.7 which in particular proves that: ``a von Neumann subalgebra \(N\) is private (correctable) for \(\mathcal E\) with respect to a subspace \(P\in {\mathcal P}(S)\) if and only if it is correctable (private) for any complement of \(\mathcal E\) with respect to \(P\).'' This result is applied (Section 5) to linear bosonic quantum channels, a special type of Gaussian quantum channels.NEWLINENEWLINEThe paper is very mathematized, hard to be understand by readers without knowledge in operators theory. But results and proofs are enough important to be considered and followed in the field.