Quantum Programs as Kleisli Maps

We propose a definition of purity for positive linear maps between Euclidean Jordan Algebras (EJA) that generalizes the notion of purity for quantum systems. We show that this definition of purity is closed under composition and taking adjoints and thus that the pure maps form a dagger category (which sets it apart from other possible definitions.) In fact, from the results presented in this paper, it follows that the category of EJAs with positive contractive linear maps is a dagger-effectus, a type of structure originally defined to study von Neumann algebras in an abstract categorical setting. In combination with previous work this characterizes EJAs as the most general systems allowed in a generalized probabilistic theory that is simultaneously a dagger-effectus. Using the dagger structure we get a notion of dagger-positive maps of the form f = g*g. We give a complete characterization of the pure dagger-positive maps and show that these correspond precisely to the Jordan algebraic version of the sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of dagger-positivity therefore characterizes the sequential product.

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into L 1 ( μ ) {L^1}(\mu ) or C ( X ) C(X) for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.

A note on inequalities for positive linear maps

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.

This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int construction. Along the way, we discuss some new examples of partially traced categories, mostly arising in the context of quantum computation. The second part deals with the construction of categorical models of higher-order quantum computation. We construct a concrete semantic model of Selinger and Valiron's quantum lambda calculus, which has been an open problem until now. We do this by considering presheaf categories over appropriate base categories arising from first-order quantum computation. The main technical ingredients are Day's convolution theory and Kelly and Freyd's notion of continuity of functors. We first give an abstract description of the properties required of the base categories for the model construction to work; then exhibit a specific example of base categories satisfying these properties.

Let $M$ be a $\sigma$-finite, finite von Neumann algebra with a faithful, normalized, normal trace Tr and a positive linear map $\rho$ of $M$ into itself. If $\rho$ is approximately inner with respect to the norm $||.|{|_2}$ induced by Tr, then $\rho$ is closely related to a $*$-homomorphism. In particular, if $\rho$ is unital and approximately inner, then $\rho$ is a $*$-homomorphism of $M$ into itself.

A Lie triple derivation of an associative algebra M is a linear map L : M → M L:M \to M such that \[ L [ [ X , Y ] , Z ] = [ [ L ( X ) , Y ] , Z ] + [ [ X , L ( Y ) ] , Z ] + [ [ X , Y ] , L ( Z ) ] L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ] \] for all X , Y , Z ∈ M X,Y,Z \in M . (Here [ X , Y ] = X Y − Y X [X,Y] = XY - YX and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator A ∈ M A \in M such that L ( X ) = [ A , X ] + λ ( X ) L(X) = [A,X] + \lambda (X) where λ : M → Z M \lambda :M \to {Z_M} is a linear map which annihilates brackets of operators in M.

Convex hulls of unitary orbits of normal elements in C⁎-algebras with tracial rank zero

The Banach-Stone theorem says, in the language we shall use, that if two commutative C*-algebras are isomorphic, then they are as C*-algebras. In this formulation there are two natural question: whether the theorem holds for noncommutative C*-algebras, and whether it is possible to replace isometrically isomorphic with isomorphic. A complete answer to the first question was given by Kadison in [10], and M. Cambern settled the second in the affirmative in [5]. In Section 3 we show that Cambern's result may be partially extended to noncommutative von Neumann algebras having the property P ([9, 18, 19]). When we have the result of Section 3 that a nearly isometric completely positive map of a von Neumann algebra having property P is close to a * isomorphism between the algebras, we have nearly proved that von Neumann algebras whose unit balls are close in the Hausdorff metric are unitarily equivalent by a unitary close to the identity. The only thing we have to prove then is that if q) is an isomorphism of a v o n Neumann algebra A having property P onto a v o n Neumann algebra B acting on the same Hilbert space, and 4~ satisfies, for some positive k less than one and any x in A,

Let A and B be unital JB-algebras. We study the extremal structure of the convex set S (A,B) of all identity preserving positive linear maps from A to B. We show that every unital Jordan homomorphism from A to B is an extreme point of S (A,B). An extreme point of S (A,B) need not be a homomorphism and we show that, given A, every extreme point of S (A,B) is a homomorphism for any B if, and only if, dim A ≤ 2. We also determine when S (A,B) is a simplex.

Shannon like inequalities for f-connections of positive linear maps and positive operators

We generalize two classes of D-type positive linear maps by permutations. For the first class, we discuss their indecomposability and give conditions when they are indecomposable and decomposable. For the other class, we show that they are atomic.

A a-weakly closed subspace ` of B(' X) is said to have Propett S_ if for any a-weakly closed subspace \5 of a von Neumann algebra -I, {.x E 5 0 A: R-(,(.x) (= ' for all rp E B(0.)) = X ' l, where R , is the right slice map associated with T. It is shown that semidiscrete von Neumann algebras have Property Sa, and various stability properties of the class of a-weakly closed subspaces with Property S0 are established. It is also shown that if (?111, G, a) is a W*-dynamical system such that 'k has Property S0 and C is compact abelian, then all of the spectral subspaces associated with a have Property S0. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if L , is a commutative subspace lattice with totally atomic core, and L, is an arbitrary subspace lattice, then alg(i 0 E2) algC1 0 algL?2. Introduction. Slice maps, introduced by Tomiyama, have proved to be a powerful tool in studying tensor products of C*-algebras and von Neumann algebras (see, e.g., [23,24,26-281). In this paper we use slice maps to study tensor products of a-weakly closed subspaces of von Neumann algebras, particularly tensor products of spectral subspaces, and tensor products of reflexive algebras. If ')T and 6. are von Neumann algebras, then for each qp in X*, the predual of .X, there is a unique a-weakly continuous linear map R., from 9DZ 0 9 to 6 such that R,W,(a X b) = (a, q) b (a E 91R, b E DL)l called the right slice map associated with p. Similarly, for each 4 E 9L, there is a left slice map LI, from 9DTh0 6 to 9DT. If 5 C 9Th and Y C % are a-weakly closed subspaces, and 5 8 '5 denotes the a-weak closure of the algebraic tensor product of 5 and C, it is clear that if x E 0 0 then R (x) C 9 for all p E 9<X*. If (x E 5 0 : R ,(x) Efor all qg E ' } = 5 Received by the editors February 19, 1982 and, in revised form, July 20, 1982. 1980 Mathematics Subject Classification. Primary 46L10, 46L55, 46M05, 47D25.

To implement fault-tolerant quantum computation with continuous variables, the Gottesman--Kitaev--Preskill (GKP) qubit has been recognized as an important technological element. We have proposed a method to reduce the required squeezing level to realize large scale quantum computation with the GKP qubit [Phys. Rev. X. {\bf 8}, 021054 (2018)], harnessing the virtue of analog information in the GKP qubits. In the present work, to reduce the number of qubits required for large scale quantum computation, we propose the tracking quantum error correction, where the logical-qubit level quantum error correction is partially substituted by the single-qubit level quantum error correction. In the proposed method, the analog quantum error correction is utilized to make the performances of the single-qubit level quantum error correction almost identical to those of the logical-qubit level quantum error correction in a practical noise level. The numerical results show that the proposed tracking quantum error correction reduces the number of qubits during a quantum error correction process by the reduction rate $\left\{{2(n-1)\times4^{l-1}-n+1}\right\}/({2n \times 4^{l-1}})$ for $n$-cycles of the quantum error correction process using the Knill's $C_{4}/C_{6}$ code with the concatenation level $l$. Hence, the proposed tracking quantum error correction has great advantage in reducing the required number of physical qubits, and will open a new way to bring up advantage of the GKP qubits in practical quantum computation.