On the centre of strong graded monads, and relative monads for
quantum effects
Quan Long
Abstract:
My talk will be based on two of my recent works.
1. On the Centre of Strong Graded Monads
The notion of the centre of a monad has been studied previously, and in
this work we generalise it to the setting of strong graded monads.
Intuitively, central effects are those that commute with all other
effects. In particular, the order in which central effects occur does
not affect the result of the overall computation. This notion turns out
to be surprisingly useful when analysing behaviours arising from
concurrent and quantum computational effects.
In the talk I will first present several examples illustrating this
phenomenon, and then discuss the categorical construction of the centre
in the graded setting. This is closely related to the motivation for
this visit: identifying the correct conjectures for a universal
property of the centre, potentially involving the premonoidal centre in
the graded case and the Kleisli category of a graded monad.
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2. Relative Monads for Quantum Effects
It is well known that ordinary monads have difficulties modelling
genuine quantum effects, in particular superoperators. Several
approaches have been proposed, such as Quantum Lambda Calculus and
models based on categories like vNA^{op}_{MIU}, but these have not been
formulated using relative monads.
Relative monads, which generalise monads and are closely related to
arrows and Fred categories, provide a natural categorical framework for
modelling superoperators. In particular, their structure allows quantum
effects to be expressed without requiring an endofunctor on a single
category.
This work was developed as part of Adjoint School 2025, where we
explored how relative monads can provide a suitable structure for
modelling quantum computational effects.