Completely positive projections and biproducts Heunen, Chris; Kissinger, Aleks; Selinger, Peter
Electronic proceedings in theoretical computer science,
12/2014, Letnik:
171, Številka:
Proc. QPL 2013
Journal Article
Odprti dostop
The recently introduced CP*–construction unites quantum channels and classical systems, subsuming the earlier CPM–construction in categorical quantum mechanics. We compare this construction to two ...earlier attempts at solving this problem: freely adding biproducts to CPM, and freely splitting idempotents in CPM. The CP*–construction embeds the former, and embeds into the latter, but neither embedding is an equivalence in general.
We aim to characterize the category of injective *-homomorphisms between commutative C*-subalgebras of a given C*-algebra
A
. We reduce this problem to finding a weakly terminal commutative ...subalgebra of
A
, and solve the latter for various C*-algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural generalization of the Mackey–Piron programme: which lattices are those of closed subspaces of Hilbert space? We also discuss the way this categorified generalization differs from the original question.
Two fundamental contributions to categorical quantum mechanics are presented. First, we generalize the CPM-construction, that turns any dagger compact category into one with completely positive maps, ...to arbitrary dimension. Second, we axiomatize when a given category is the result of this construction.
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hubert space, or, more generally, on the lattice of projections in a ...von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Hey ting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n x n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" A̱ of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hubert space H and probability valuations on the internal Gelfand spectrum of A̱ for A = B(H).
Arrows are an extension of the well-established notion of a monad in functional-programming languages. This paper presents several examples and constructions and develops denotational semantics of ...arrows as monoids in categories of bifunctors Cop × C → C. Observing similarities to monads – which are monoids in categories of endofunctors C → C – it then considers Eilenberg–Moore and Kleisli constructions for arrows. The latter yields Freyd categories, mathematically formulating the folklore claim ‘Arrows are Freyd categories.’
Quantum information effects Heunen, Chris; Kaarsgaard, Robin
Proceedings of ACM on programming languages,
01/2022, Letnik:
6, Številka:
POPL
Journal Article
Recenzirano
Odprti dostop
We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for ...irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.
Bennett and Stinespring, Together at Last Heunen, Chris; Kaarsgaard, Robin
Electronic proceedings in theoretical computer science,
09/2021, Letnik:
343
Journal Article, Conference Proceeding
Ordered locales Heunen, Chris; van der Schaaf, Nesta
Journal of pure and applied algebra,
July 2024, 2024-07-00, Letnik:
228, Številka:
7
Journal Article
Recenzirano
Odprti dostop
We extend the Stone duality between topological spaces and locales to include order: there is an adjunction between the category of preordered topological spaces satisfying the so-called open cone ...condition, and the newly defined category of ordered locales. The adjunction restricts to an equivalence of categories between spatial ordered locales and sober T0-ordered spaces with open cones.
Sequences of commuting quantum operators can be parallelized using entanglement. This transformation is behind some optimal quantum metrology protocols and recent results on quantum circuit ...complexity. We show that dephasing quantum maps in arbitrary dimension can also be parallelized. This implies that for general dephasing noise the protocol with entanglement is not more fragile than the corresponding sequential protocol and, conversely, the sequential protocol is not less effective than the entangled one. We derive this result using tensor networks. Furthermore, we only use transformations strictly valid within string diagrams in dagger compact closed categories. Therefore, they apply verbatim to other theories, such as geometric quantization and topological quantum field theory. This clarifies and characterizes to some extent the role of entanglement in general quantum theories.
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