Private quantum subsystems and quasiorthogonal operator algebras

Abstract

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.

Approximate Quasiorthogonality of Operator Algebras and Relative Quantum Privacy

“…In [16] the first example of a private quantum subsystem [1,8,4,3] was discovered such that no private subspaces existed for the given channel, and error-correction complementarity [21,17,12] failed. This example motivated further work and generalizations, including a framing of it in terms of operator algebra language [24,25]. With the algebra perspective we can apply the theorem above to that example.…”

“…n I n for all B ∈ B and E B (A) = Tr(A) n I n for all A ∈ A The ideal (ǫ = 0 -see next section) notion of quantum privacy we consider here is given as follows. More general notions of private algebras have been considered in the literature, often with different nomenclature as well, such as private quantum channels, decoherence-full or private subspaces and subsystems, and private algebras [1,8,4,3,21,11,17,16,12,24,25]. The distinguished special case we consider here captures many of the most naturally occurring examples from these settings, in addition to, as we shall see, allowing us to establish a tight connection with quasiorthogonality in the approximate case.…”

“…From a different direction, still with quantum information motivation, it has recently been recognized that there are connections between the study of quasiorthogonal operator algebras and work in quantum privacy; specifically, on the topic of what are variously known as private quantum channels or codes, decoherence-full or private subspaces and subsystems, and private algebras [1,8,4,3,21,11,17,16,12]. In particular, for a number of special cases of channels or algebras, quasiorthogonality has been linked with certain quantum privacy properties in those cases [24,25,22], all suggesting a deeper more general link between the topics.…”

Quantum privacy and Schur product channels

“…One more idea we need is the following: we say that a * -subalgebra A ⊆ M n (C) is quasiorthogonal to an operator system S ⊆ M n (C) if nTr(sa) = Tr(s)Tr(a) for all s ∈ S, a ∈ A. For more details on quasiorthogonality see [17] and the references therein. The following result brings these concepts together.…”

“…Our analysis is based on the underlying correlation matrix graph structure. We then combine this analysis with a connection between operator systems, another graph, and quasiorthonal operator algebra techniques [17] to obtain further general privatisation results on the channels. This paper is organised as follows.…”