nLab quantum relation

Idea

The notion of quantum relations of Weaver 2010, Weaver & Kuperberg 2011 is a “quantum” or noncommutative geometry/noncommutative measure theory-version of the basic notion of relations on sets.

Concretely, they define a quantum relation on a von Neumann algebra $\mathcal{A}$ equipped with an embedding $\mathcal{A} \subset \mathcal{B}\mathscr{H}$ (into the bounded operators on some Hilbert space) as an operator bimodule over the commutant of $\mathcal{A}$ which is $\ast$-closed in the weak operator topology, and explain why this (non-obvious) definition is sensible; for instance it has good application to quantum error correction.

According to Kornell 2020, and in mild paraphrase (following the discussion at dependent linear type and quantum circuits via dependent linear types):

• quantum sets are indexed sets of finite-dimensional Hilbert spaces, hence finite-rank Hilbert-vector bundles over a discrete topological spaces regarded as a sets, and regarded as equipped with the external tensor product $\boxtimes$ of vector bundles;

• a quantum relation between quantum sets $(x \colon X) \times \mathscr{H}_x$ and $(x' \colon X') \times \mathscr{H}'_x$ is a monomorphism from a quantum set $\big((x,x') \colon X \times X'\big) \times \mathscr{R}_{(x,x')}$ to $\mathscr{H}^\ast_\bullet \boxtimes \mathscr{H}'_\bullet$:

  $$(x,x') \colon X \times X' \;\;\; \vdash \;\;\; \mathscr{R}_{(x,y)} \hookrightarrow \mathscr{H}^\ast_{(x,y)} \otimes \mathscr{H}^'_{(x,y)}$$

With composition the evident matrix multiplication (Kornell 2020 (5)), quantum relations between quantum sets form a category $qRel$, which is a dagger-compact category.

As such, this serves as categorical semantics for quantum programming languages like Quipper equipped with term recursion, via quantum CPOs (Kornell, Lindenhovius & Mislove 2021).

Related concepts

• quantum set

• quantum CPO

References

• Nik Weaver, Quantum relations [arXiv:1005.0354]

• Nik Weaver, Greg Kuperberg, A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations, Memoirs of the AMS 215 (2011) 1010 [ams:memo-215-1010]

• Andre Kornell, Quantum Sets, J. Math. Phys. 61 102202 (2020) [doi:10.1063/1.5054128]

  (cf. quantum set)

• Andre Kornell, Discrete quantum structures [arXiv:2004.04377]

• Andre Kornell, Discrete quantum structures I: Quantum predicate logic, J. Noncommut. Geom. (2023) [doi:10.4171/jncg/531]

• Andre Kornell, Bert Lindenhovius, Michael Mislove, §2 in: Quantum CPOs, EPTCS 340 (2021) 174-187 [arXiv:2109.02196, doi:10.4204/EPTCS.340.9]

  (in the context of quantum CPOs)