We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps.We extend the existing topological structures between scalar-valued information channels to operator-valued ones and associate them with topologies on the set of unital completely positive maps between a commutative C*-algebra and the C*-algebra of bounded weakly measurable operator-valued functions over a compact Hausdorff space.Given a measure on the input alphabet space, we introduce the notion of an almost everywhere defined operator-valued information channel and provide a characterization result.
We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps.We extend the existing topological structures between scalar-valued information channels to operator-valued ones and associate them with topologies on the set of unital completely positive maps between a commutative C*-algebra and the C*-algebra of bounded weakly measurable operator-valued functions over a compact Hausdorff space.Given a measure on the input alphabet space, we introduce the notion of an almost everywhere defined operator-valued information channel and provide a characterization result.
Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
Almost-idempotent quantum channels and approximate C∗-algebras
Let Φ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that Φ is η-idempotent, namely, ∥Φ2−Φ∥cb≤η, and construct an associated ε-C∗ algebra (of almost-invariant observables) for ε=O(η). This type of structure has the axioms of a unital C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε. We prove that any finite-dimensional ε-C∗ algebra A is O(ε)-isomorphic to some genuine C∗ algebra B. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When A comes from a finite-dimensional η-idempotent UCP map Φ, the O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗ into a decoding channel, producing a state on B, and an encoding channel.
Abstract For a state ω on a C ∗ -algebra A , we characterize all states ρ in the weak* closure of the set of all states of the form ω∘φ , where φ is a map on A of the form φ(x)=∑i=1nai∗xai,∑i=1nai∗ai=1 ( ai∈A , n∈N ). These are precisely the states ρ that satisfy ∥ρ∣J∥≤∥ω∣J∥ for each ideal J of A . The corresponding question for normal states on a von Neumann algebra R (with the weak* closure replaced by the norm closure) is also considered. All normal states of the form ω∘ψ , where ψ is a quantum channel on R (that is, a map of the form ψ(x)=∑jaj∗xaj , where aj∈R are such that the sum ∑jaj∗aj converge to 1 in the weak operator topology) are characterized. A variant of this topic for hermitian functionals instead of states is investigated. Maximally mixed states are shown to vanish on the strong radical of a C ∗ -algebra and for properly infinite von Neumann algebras the converse also holds.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H) of bounded operators on a separable Hilbert space H ) to the space of all operators on H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ can be uniquely extended to a bounded linear operator on the Banach space of all G -bounded operators on H provided that the operation Φ is G -limited: the predual operation Φ∗ maps the set of positive trace class operators ρ with finite value of TrρG into itself. Assuming that G has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the G -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
Given a positive operator-valued measure ν acting on the Borel sets of a locally compact Hausdorff space X, with outcomes in the algebra B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H, one can consider ν-integrable functions X→B(H) that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.
We study no-signalling correlations, defined over a quadruple of second countable compact Hausdorff spaces. Using operator-valued information channels over abstract alphabets, we define the subclasses of local, quantum spatial and quantum commuting measurable no-signalling correlations. En route, we establish measurable versions of the Stinespring's Dilation Theorem. We define values of measurable non-local games of local, quantum spatial and quantum commuting type, as well as inner versions thereof, and show how the asymptotic values of a finite non-local game can be viewed as special cases of the corresponding inner values of a measurable game, canonically associated with the given finite game.