A Completeness Theorem of Quantum Set Theory

Arghya Chongdar

We develop a theory of integration with respect to quantum instruments through two complementary approaches: a vector measure formulation based on Bartle's integration theory and a tensor product construction. As a principal application, we establish a CP map-valued Lyapunov theorem by characterizing the convexity of the range of non-atomic completely positive instruments via the associated integration map. This extends the work of Plosker and Ramsey~\cite{Plosker_Ramsey} from POVMs to completely positive instruments and gives a general characterization of the phenomenon exhibited in their setting. We also establish a correspondence between completely positive instruments and completely positive maps, prove a Krein--Milman type theorem for the $C^*$-convex set of unital completely positive instruments, and show that the tensor product construction connects the theory of CP instruments with the integration theory for POVMs due to Farenick \emph{et al}(\cite{douglus_plosker_ramsey_povmintegration_1}).

Arthur I. Fine

Arthur I. Fine

Oliver Rudolph

Recently, Schreckenberg investigated symmetries in the context of history quantum theories. In the case that the space of histories is given by the set of projectors on some finite dimensional Hilbert space he obtained, first, a complete characterization of all physical symmetries of history quantum theories — an analog of Wigner’s theorem — and, second, a complete mathematical characterization of symmetries of single decoherence functionals. In this paper we extend Schreckenberg’s results to the case where the underlying space of histories is given by the set of projectors on some infinite dimensional Hilbert space.

Hans Maassen, Burkhard Kümmerer

The no-broadcasting theorem in quantum information says that a set of states on a quantum system admits a common broadcasting (copying) operation if and only if their density matrices belong to a commuting family. We discuss and prove this theorem, as well as the closely related no-cloning theorem in the context of quantum probability theory, i.e. in the category of (finite dimensional) C-star-algebras with unital completely positive maps.

Daniel Cariello

In order to compute the Schmidt decomposition of A ∈ Mk⊗ Mm, we must consider an associated self-adjoint map. Here, we show that if A is positive under partial transposition (PPT) or symmetric with positive coefficients (SPCs) or invariant under realignment, then its associated self-adjoint map is completely reducible. We give applications of this fact in quantum information theory. We recover some theorems (recently proved for PPT and SPC matrices), and we prove them for matrices invariant under realignment using theorems of the Perron-Frobenius theory. We also provide a new proof of the fact that if Ckcontains k mutually unbiased bases, then there exists another orthonormal basis which is mutually unbiased with these k bases. We study other types of matrices that could have the same property. We consider a collection of linear transformations acting on Mk⊗ Mk, which contains the partial transpositions and the realignment map. For each linear transformation, we consider the set of matrices in Mk⊗ Mk≃ M(k2) that are positive and remain positive, or invariant, under the action of this linear transformation. Within this family of sets, we have the set of PPT matrices, the set of SPC matrices and the set of matrices invariant under realignment. We show that these three sets are the only sets of this family, such that the associated self-adjoint map of each matrix is completely reducible. We also show that every matrix invariant under realignment is PPT in M2⊗ M2and we present a counterexample in Mk⊗ Mkand k ≥ 3.