Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-Order Quantum Objects are Strong Profunctors

Matt Wilson, James Hefford

Abstract

We explore the sense in which the existing constructions for higher-order maps on quantum theory based on causality constraints and compositionality constraints respectively, coincide. More precisely, we construct a functor F : Caus(C) -> StProf(C1) from higher-order causal categories to the category of strong profunctors over first-order causal processes that is lax-lax duoidal, full, faithful, and strongly closed whenever C is additive. When C = CP this embedding is furthermore strong on the sequencer for duoidal categories, expressing the possibility to interpret one-way signalling (but not general non-signalling) constraints in terms of the coend calculus for profunctors. We conclude that insofar as compositional constraints can be used to express causality constraints, the profunctorial approach generalises higher-order quantum theory to a construction over general symmetric monoidal categories.

Higher-Order Quantum Objects are Strong Profunctors

Abstract

We explore the sense in which the existing constructions for higher-order maps on quantum theory based on causality constraints and compositionality constraints respectively, coincide. More precisely, we construct a functor F : Caus(C) -> StProf(C1) from higher-order causal categories to the category of strong profunctors over first-order causal processes that is lax-lax duoidal, full, faithful, and strongly closed whenever C is additive. When C = CP this embedding is furthermore strong on the sequencer for duoidal categories, expressing the possibility to interpret one-way signalling (but not general non-signalling) constraints in terms of the coend calculus for profunctors. We conclude that insofar as compositional constraints can be used to express causality constraints, the profunctorial approach generalises higher-order quantum theory to a construction over general symmetric monoidal categories.
Paper Structure (8 sections, 8 theorems, 35 equations)

This paper contains 8 sections, 8 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Higher-order quantum operations and category theory
  3. Duoidal and BV-categories
  4. Higher-order causal categories
  5. Strong profunctors
  6. A Lax Duoidal Embedding
  7. The Embedding is Full and Strongly Closed
  8. The Embedding is Strong on the Sequencer

Key Result

Proposition 3.1

The following assignment, defines a $\otimes$-lax monoidal functor $\mathcal{F}:\mathbf{Caus}(\mathbf{C}) \rightarrow \mathbf{StProf}(\mathbf{C}_1)$. Note that $X$ and $X'$ vary over only the first-order objects, i.e. those of $\mathbf{C}_1$.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 2.1
  • Definition 6
  • Remark 2.2
  • Definition 7
  • Definition 8
  • ...and 16 more