36 research outputs found
Categories of Quantum and Classical Channels (extended abstract)
We introduce the CP*-construction on a dagger compact closed category as a
generalisation of Selinger's CPM-construction. While the latter takes a dagger
compact closed category and forms its category of "abstract matrix algebras"
and completely positive maps, the CP*-construction forms its category of
"abstract C*-algebras" and completely positive maps. This analogy is justified
by the case of finite-dimensional Hilbert spaces, where the CP*-construction
yields the category of finite-dimensional C*-algebras and completely positive
maps.
The CP*-construction fully embeds Selinger's CPM-construction in such a way
that the objects in the image of the embedding can be thought of as "purely
quantum" state spaces. It also embeds the category of classical stochastic
maps, whose image consists of "purely classical" state spaces. By allowing
classical and quantum data to coexist, this provides elegant abstract notions
of preparation, measurement, and more general quantum channels.Comment: In Proceedings QPL 2012, arXiv:1407.842
A Functorial Construction of Quantum Subtheories
We apply the geometric quantization procedure via symplectic groupoids
proposed by E. Hawkins to the setting of epistemically restricted toy theories
formalized by Spekkens. In the continuous degrees of freedom, this produces the
algebraic structure of quadrature quantum subtheories. In the odd-prime finite
degrees of freedom, we obtain a functor from the Frobenius algebra in
\textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum
mechanics.Comment: 19 page
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
Inversion, Iteration, and the Art of Dual Wielding
The humble ("dagger") is used to denote two different operations in
category theory: Taking the adjoint of a morphism (in dagger categories) and
finding the least fixed point of a functional (in categories enriched in
domains). While these two operations are usually considered separately from one
another, the emergence of reversible notions of computation shows the need to
consider how the two ought to interact. In the present paper, we wield both of
these daggers at once and consider dagger categories enriched in domains. We
develop a notion of a monotone dagger structure as a dagger structure that is
well behaved with respect to the enrichment, and show that such a structure
leads to pleasant inversion properties of the fixed points that arise as a
result. Notably, such a structure guarantees the existence of fixed point
adjoints, which we show are intimately related to the conjugates arising from a
canonical involutive monoidal structure in the enrichment. Finally, we relate
the results to applications in the design and semantics of reversible
programming languages.Comment: Accepted for RC 201
A Bestiary of Sets and Relations
Building on established literature and recent developments in the
graph-theoretic characterisation of its CPM category, we provide a treatment of
pure state and mixed state quantum mechanics in the category fRel of finite
sets and relations. On the way, we highlight the wealth of exotic beasts that
hide amongst the extensive operational and structural similarities that the
theory shares with more traditional arenas of categorical quantum mechanics,
such as the category fdHilb. We conclude our journey by proving that fRel is
local, but not without some unexpected twists.Comment: In Proceedings QPL 2015, arXiv:1511.0118