We generalize a recently discovered example of private quantum subsystem to find private subsystems for Abelian subgroups of the n-qubit Pauli group, which exist in the absence of private subspaces. In doing so, we also connect these quantum privacy investigations with the theory of quasiorthogonal operator algebras through the use of tools from group theory and operator theory.2010 Mathematics Subject Classification. 47L90, 81P45, 81P94, 94A40. Key words and phrases. private quantum subsystem, private quantum channel, n-qubit Pauli group, completely positive map, quasiorthogonal operator algebras, conditional expectations.
We verify the Perfect-Mirsky Conjecture on the structure of the set of eigenvalues for all n × n doubly stochastic matrices in the four-dimensional case. The n = 1, 2, 3 cases have been established previously and the n = 5 case has been shown to be false. Our proof is direct and uses basic tools from matrix theory and functional analysis. Based on this analysis we formulate new conjectures for the general case.
We investigate the quantum privacy properties of an important class of quantum channels, by making use of a connection with Schur product matrix operations and associated correlation matrix structures. For channels implemented by mutually commuting unitaries, which cannot privatise qubits encoded directly into subspaces, we nevertheless identify private algebras and subsystems that can be privatised by the channels. We also obtain further results by combining our analysis with tools from the theory of quasiorthogonal operator algebras and graph theory.2010 Mathematics Subject Classification. 47L90, 81P45, 81P94, 94A40. Key words and phrases. Schur product, correlation matrix, quantum channel, operator system, private quantum code, private algebra, quasiorthogonal algebra, graph of a matrix.
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