Paschke Dilations

In 1973 Paschke defined a factorization for completely positive maps between C*-algebras. In this paper we show that for normal maps between von Neumann algebras, this factorization has a universal property, and coincides with Stinespring's dilation for normal maps into B(H).Comment: In Proceedings QPL 2016, arXiv:1701.0024

Unordered Tuples in Quantum Computation

It is well known that the C*-algebra of an ordered pair of qubits is M_2 (x) M_2. What about unordered pairs? We show in detail that M_3 (+) C is the C*-algebra of an unordered pair of qubits. Then we use Schur-Weyl duality to characterize the C*-algebra of an unordered n-tuple of d-level quantum systems. Using some further elementary representation theory and number theory, we characterize the quantum cycles. We finish with a characterization of the von Neumann algebra for unordered words.Comment: In Proceedings QPL 2015, arXiv:1511.0118

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 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.Comment: In Proceedings QPL 2018, arXiv:1901.0947

Dagger and Dilation in the Category of Von Neumann algebras

This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics. First up are dilations, covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the second topic: effectus theory. Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own

A Kochen-Specker system has at least 22 vectors (extended abstract)

At the heart of the Conway-Kochen Free Will theorem and Kochen and Specker's argument against non-contextual hidden variable theories is the existence of a Kochen-Specker (KS) system: a set of points on the sphere that has no 0,1-coloring such that at most one of two orthogonal points are colored 1 and of three pairwise orthogonal points exactly one is colored 1. In public lectures, Conway encouraged the search for small KS systems. At the time of writing, the smallest known KS system has 31 vectors. Arends, Ouaknine and Wampler have shown that a KS system has at least 18 vectors, by reducing the problem to the existence of graphs with a topological embeddability and non-colorability property. The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability. Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22.Comment: In Proceedings QPL 2014, arXiv:1412.810

Sign in finite fields

Often in cryptography one needs to make a consistent choice of square root in a finite field. We show that such a choice is equivalent to providing a reasonable sign function. Then we show that for $\mathbb{F}_{p^k}$ (with odd prime $p \neq 1$ and $k\neq 0$) such a sign function exists if and only if $k$ is odd

The universal property of infinite direct sums in C∗^*-categories and W∗^*-categories

When formulating universal properties for objects in a dagger category, one usually expects a universal property to characterize the universal object up to unique unitary isomorphism. We observe that this is automatically the case in the important special case of C$^*$-categories, provided that one uses enrichment in Banach spaces. We then formulate such a universal property for infinite direct sums in C$^*$-categories, and prove the equivalence with the existing definition due to Ghez, Lima and Roberts in the case of W$^*$-categories. These infinite direct sums specialize to the usual ones in the category of Hilbert spaces, and more generally in any W$^*$-category of normal representations of a W$^*$-algebra. Finding a universal property for the more general case of direct integrals remains an open problem.Comment: 11 page