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Supermaps on generalised theories

Matt Wilson, James Hefford, Timothée Hoffreumon

Abstract

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the recently proposed higher-order processes on boxworld can be obtained as a particular instance of categorical supermaps, and put forward a stable definition of higher-order real quantum theory.

Supermaps on generalised theories

Abstract

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the recently proposed higher-order processes on boxworld can be obtained as a particular instance of categorical supermaps, and put forward a stable definition of higher-order real quantum theory.
Paper Structure (11 sections, 14 theorems, 96 equations)

This paper contains 11 sections, 14 theorems, 96 equations.

Table of Contents

  1. Generalised Theories
  2. Instruments
  3. Supermaps on Generalised Theories with Channel-State Duality
  4. Categorical Supermaps
  5. The Yoneda Lemma for Compact Closed Categories
  6. A Yoneda Lemma for Categorical Supermaps
  7. Categorical supermaps on concrete profunctors
  8. Conclusion
  9. Categorical Supermaps Satisfy NSWSE
  10. Proof of Lemma \ref{['lem:important']}
  11. Strong Profunctors and the Yoneda Lemma

Key Result

Theorem 1

Let $\mathcal{C}$ be a generalised theory with a channel-state duality, then:

Theorems & Definitions (52)

  • Theorem 1: Yoneda Lemma for Categorical Supermaps
  • Theorem 2: Recovery of Known Supermap Approaches
  • Corollary 1
  • Definition 1: Generalised Theories
  • Example 1: Classical theory
  • Example 2: Quantum theory
  • Example 3: Quantum theory on general subsystems
  • Example 4: Boxworld (the theory of non-signalling black boxes)
  • Example 5: Quantum theory over other semirings
  • Example 6: Hypercubic theories
  • ...and 42 more