Paschke Dilations

  title={Paschke Dilations},
  author={Abraham Westerbaan and Bas Westerbaan},
The Stinespring Dilation Theorem[17] entails that every normal completely positive linear map (NCPmap) φ : A → B(H ) is of the form A π // B(K ) V ∗( ·)V // B(H ) where V : H → K is a bounded operator and π a normal unital ∗-homomorphism (NMIU-map). Stinespring’s theorem is fundamental in the study of quantum information and quantum computing: it is used to prove entropy inequalities (e.g. [10]), bounds on optimal cloners (e.g. [20]), full completeness of quantum programming languages (e.g. [16… Expand
Quantum channels as a categorical completion
  • Mathieu Huot, S. Staton
  • Computer Science, Mathematics
  • 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2019
A categorical foundation for the connection between pure and mixed states in quantum information and quantum computation is proposed and it is proved that the category of all quantum channels is a canonical completion of the categories of pure quantum operations (with ancilla preparations). Expand
Pure Maps between Euclidean Jordan Algebras
We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of Kraus rank one channels ($A\mapsto B^*AB$). We show that thisExpand
A computer scientist's reconstruction of quantum theory
The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in orderExpand
An effect-theoretic reconstruction of quantum theory
An often used model for quantum theory is to associate to every physical system a C*-algebra. From a physical point of view it is unclear why operator algebras would form a good description ofExpand
From probability monads to commutative effectuses
  • B. Jacobs
  • Mathematics, Computer Science
  • J. Log. Algebraic Methods Program.
  • 2018
It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate- and state-transformers, normalisation and conditioning of states. Expand
Stinespring's construction as an adjunction
Given a representation of a unital $C^*$-algebra $\mathcal{A}$ on a Hilbert space $\mathcal{H}$, together with a bounded linear map $V:\mathcal{K}\to\mathcal{H}$ from some other Hilbert space, oneExpand
Purity through Factorisation
A construction is given that identifies the collection of pure processes within a theory containing both pure and mixed processes, and defines a pure subcategory in the framework of symmetric monoidal categories. Expand
Picture-perfect Quantum Key Distribution
We provide a new way to bound the security of quantum key distribution using only two high-level, diagrammatic features of quantum processes: the compositional behavior of complementary measurementsExpand
The Category of von Neumann Algebras
This dissertation includes an extensive introduction to the basic theory of $C^*$-algebras and von Neumann algebraes and its category of completely positive normal contractive maps. Expand
Bennett and Stinespring, Together at Last
A universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the restriction affine completion of a monoidal restriction category quotiented by well-pointedness, which shows how both mixed quantum theory and classical computation rest on entirely reversible foundations. Expand


A universal property for sequential measurement
We study the sequential product the operation p∗q=pqp on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J.Expand
The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation
A continuity theorem for Stinespring's dilation is proved: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which areClose in operator norm, with dimension-independent bounds. Expand
An Introduction to Effectus Theory
This text is an account of the basics of effectus theory, which includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. Expand
Quantum Alternation: Prospects and Problems
This work proposes a notion of quantum control in a quantum programming language which permits the superposition of finitely many quantum operations without performing a measurement, and shows that adding such a quantum IF statement to the QPL programming language simplifies the presentation of several quantum algorithms. Expand
Inner Product Modules Over B ∗ -Algebras
This paper is an investigation of right modules over a B*algebra B which posses a B-valued "inner product" respecting the module action. Elementary properties of these objects, including theirExpand
Algebraic Effects, Linearity, and Quantum Programming Languages
A new elementary algebraic theory of quantum computation, built from unitary gates and measurement is presented, and an equational theory for a quantum programming language is extracted from thegebraic theory. Expand
Dagger Compact Closed Categories and Completely Positive Maps: (Extended Abstract)
  • P. Selinger
  • Computer Science, Mathematics
  • Electron. Notes Theor. Comput. Sci.
  • 2007
This work presents a graphical language for dagger compact closed categories, and sketches a proof of its completeness for equational reasoning, and gives a general construction, the CPM construction, which associates to each Dagger compact closed category its ''category of completely positive maps'', and shows that the resulting category is again dagger compactclosed. Expand
Completely positive maps and entropy inequalities
It is proved that the relative entropy for a quantum system is non-increasing under a trace-preserving completely positive map. The proof is based on the strong sub-additivity property of theExpand
1. Modules 2. Multipliers and morphisms 3. Projections and unitaries 4. Tensor products 5. The KSGNS construction 6. Stabilisation or absorption 7. Full modules, Morita equivalence 8. Slice maps andExpand