nLab quantum set

Idea

Quantum sets are a generalization of sets to the context of noncommutative geometry. They have many equivalent definitions.

• A quantum set is a $C^\ast$-algebra $\mathcal{A}$ that is the $c_0$-direct sum of full matrix algebras:

  $$\mathcal{A} \cong \bigoplus_{i \in I}^{c_0} M_{n_i}(\mathbb{C}) \,.$$

• A quantum set is a von Neumann algebra $\mathcal{M}$ that is an $\ell^\infty$-direct sum of full matrix algebras.

  $$\mathcal{M} \cong \bigoplus_{i \in I}^{\ell^\infty} M_{n_i}(\mathbb{C}) \,.$$

• A quantum set $X$ is an indexed family of nonzero finite-dimensional Hilbert spaces:

  $$X = \{\mathcal{H}_i\}_{i \in I} \,.$$

These definitions do not define equal classes of objects, but they can be extended to define equivalent categories in two natural ways. Using the second definition of quantum sets, we can take the morphisms to be quantum relations or, inequivalently, to be unital normal $*$-homomorphisms. In the former case, we obtain the category $qRel$, and in the latter case, we obtain the opposite of the category $qSet$. Note that $qRel$ is equivalent to its own opposite. Quantum sets can be further generalized to “nontracial” quantum sets. This article will use the third definition of quantum sets because this definition avoids operator topologies, making it more accessible to a wider audience.

Basic definitions

Definition

A quantum set $X$ is a family of nonzero finite-dimensional Hilbert spaces over $\mathbb{C}$ that is indexed by a set $\mathrm{At}(X)$, which may be empty, finite, or infinite.

$$X = \{X_\alpha\}_{\alpha \in \mathrm{At}(X)} \,.$$

Very little changes if we allow zero-dimensional Hilbert spaces in this definition; doing so is mathematically more natural but less intuitive. The elements of $\mathrm{At}(X)$ are called the atoms of $X$.

Quantum sets are viewed as a generalization of sets by identifying each set $A$ with the quantum set $\mathrm{Inc}(A)$.

Definition

For each set A, we define the quantum set $\mathrm{Inc}(A)$ by $\mathrm{At}(\mathrm{Inc}(A)) = A$ and $\mathrm{Inc}(A)_\alpha = \mathbb{C}$ for all $\alpha \in A$.

The following basic operations generalize the familiar basic operations on sets.

Definition

Consider quantum sets $X$ and $Y$ according to Def. . Then:

• The disjoint union $X + Y$ is defined by $\mathrm{At}(X + Y) \coloneqq \mathrm{At}(X) + \mathrm{At}(Y)$ and

  $$(X + Y)_\alpha \coloneqq \begin{cases} X_\alpha & \text{if }\;\alpha \in \mathrm{At}(X), \\ Y_\alpha & \text{if }\;\alpha \in \mathrm{At}(Y). \end{cases}$$

• The Cartesian product $X \times Y$ is defined by $\mathrm{At}(X \times Y) \coloneqq \mathrm{At}(X) \times \mathrm{At}(Y)$ and

  $$(X \times Y)_{(\alpha, \beta)} \coloneqq X_\alpha \otimes Y_\beta.$$

We have that $\mathrm{Inc}(A + B) = \mathrm{Inc}(A) + \mathrm{Inc}(B)$ and that $\mathrm{Inc}(A \times B) = \mathrm{Inc}(A) \times \mathrm{Inc}(B)$.

The Cartesian product of quantum sets is so named because it generalizes the Cartesian product of sets. The former is not the categorical product in $qRel$, just as the latter is not the categorical product in $Rel$. It is also not the categorical product in $qSet$.

In both $qRel$ and $qSet$, $X + Y$ is the coproduct of $X$ and $Y$, and $X \times Y$ is their monoidal product. In $qRel$, $X+Y$ is also the product of $X$ and $Y$. In $qSet$, the product of $X$ and $Y$ is not easily definable and may be notated $X \ast Y$.

The category $qRel$

The material in this section is mostly from Kornell 2020. The morphisms were first defined in Kuperberg & Weaver 2012 and investigated in Weaver 2012; see the article on quantum relations.

Definition

We define the dagger-compact category $qRel$.

1. An object $X$ is a quantum set (see Def. ).

2. A morphism $R \colon X \to Y$ is a choice of subspaces

   $$R_{\alpha,\beta} \subseteq L(X_\alpha, Y_\beta),$$

   where $L(\mathcal{H},\mathcal{K})$ is the space of all linear maps from $\mathcal{H}$ to $\mathcal{K}$.

3. The composition $S \circ R \colon X \to Z$ of morphisms $R \colon X \to Y$ and $S \colon Y \to Z$ is given by

   $$(S \circ R)_{\alpha,\gamma} \coloneqq \mathrm{span} \{s r \mid r \in R_{\alpha, \beta},\; s \in S_{\beta, \gamma}\;\text{for some}\; \beta \in \At(Y)\}.$$

4. The identity morphism $\mathrm{id}_X\colon X \to X$ is defined by

   $$(\mathrm{id}_{X})_{\alpha, \beta} \coloneqq \begin{cases} \mathrm{span}\{1\} & \text{if}\;\alpha = \beta,\\ \{0\} & \text{if}\; \alpha \neq \beta.\end{cases}$$

5. The dagger of a morphism $R\colon X \to Y$ is defined by

   $$(R^\dagger)_{\beta, \alpha} \coloneqq \{r^\dagger \mid r \in R_{\alpha, \beta}\},$$

   where $r^\dagger$ is the Hermitian adjoint.

6. The monoidal product of objects $X$ and $X'$ is the Cartesian product $X \times X'$ (see Def. ).

7. The monoidal product of morphisms $R\colon X \to Y$ and $R'\colon X' \to Y'$ is defined by

   $$(R \times R')_{(\alpha, \alpha'),(\beta, \beta')} \coloneqq \mathrm{span}\{r \otimes r' \mid r \in R_{\alpha, \beta},\; r' \in R'_{\alpha', \beta'}\}.$$

8. The monoidal unit $1$ is defined by $\mathrm{At}(1) \coloneqq \{\bullet\}$ and $1_\bullet \coloneqq \mathbb{C}$.

9. The braiding $\sigma_{X,Y}\colon X \times Y \to Y \times X$ is defined by

   $$(\sigma_{X,Y})_{(\alpha,\beta),(\beta', \alpha')} \coloneqq \begin{cases} \mathrm{span}\{\sigma_{X_\alpha, Y_\beta}\} & \text{if}\; (\alpha, \beta) = (\alpha', \beta'),\\ \{0\} & \text{if}\;(\alpha, \beta) \neq (\alpha', \beta'), \end{cases}$$

   where $\sigma_{X_\alpha, Y_\beta}$ denotes the braiding in $Hilb$. The associator and unitors are defined similarly.

10. The dual of an object $X$ is the dual quantum set $X^*$, which is defined by $\mathrm{At}(X^*) \coloneqq \mathrm{At}(X)$ and

    $$(X^*)_\alpha \coloneqq X_\alpha^*,$$

    where $\mathcal{H}^*$ is the dual Hilbert space.

Furthermore, $qRel$ has all coproducts. It is a semiadditive dagger category with infinitary dagger biproducts and an infinitary distributive monoidal category with a unitary distributor. The dagger biproduct of $X$ and $Y$ is their disjoint union $X + Y$ (see Def. ). The zero object $0$ is defined by $\mathrm{At}(0) = \emptyset$.

Proposition

The dagger-compact category $qRel$ is enriched over suplattices with

$$R \leq S \qquad \Longleftrightarrow \qquad R_{\alpha, \beta} \subseteq S_{\alpha, \beta}$$

for all $\alpha \in \mathrm{At}(X)$ and $\beta \in \mathrm{At}(Y)$, where $R, S \colon X \to Y$. In other words, $qRel$ is a quantaloid.

The dagger-compact category $Rel$ is enriched over suplattices too. It is an allegory, while $qRel$ fails to be an allegory only because the relevant modular law fails. Nevertheless, $qRel(X,Y)$ is always a modular lattice.

In effect, $Rel$ is an enriched dagger-compact subcategory of $qRel$.

Definition

We define the “inclusion” functor $\mathrm{Inc}\colon Rel \to qRel$:

• For each set $A$, we define $\mathrm{Inc}(A)$ as in Def. .

• For each relation $R\colon A \to B$, we define

  $$\mathrm{Inc}(R) = \begin{cases} \mathbb{C} & \text{if}\;(\alpha, \beta) \in R, \\ 0 & \text{if}\;(\alpha, \beta) \notin R, \end{cases}$$

  where we identify $L(\mathbb{C}, \mathbb{C})$ with $\mathbb{C}$ in the obvious way.

Proposition

The “inclusion” functor $\mathrm{Inc}\colon Rel \to qRel$ is an enriched strong monoidal dagger functor that satisfies $\mathrm{Inc}(A^*) \cong \mathrm{Inc}(A)^*$ and that is full and faithful.

The category $qSet$

The material in this section is mostly from Kornell 2020, Kornell, Lindenhovius & Mislove 2022, and Jenča & Lindenhovius 2025.

Definition

We define the symmetric monoidal category $qSet$ to be the wide subcategory of $qRel$ whose morphisms are maps. A map is a morphism $f\colon X \to Y$ such that $f^\dagger \circ f \geq \mathrm{id}_X$ and $f \circ f^\dagger \leq \mathrm{id}_Y$.

In the terminology of allegories, $\qSet \coloneqq Map(qRel)$. The maps in $qRel$, i.e., the morphisms in $qSet$, are sometimes called functions.

Theorem

The symmetric monoidal category $qSet$ is complete, cocomplete, and closed. In other words, it is a Bénabou cosmos.

The closure of $qSet$ can be deduced from general principles (see Theorem 6.16 of Jenča & Lindenhovius 2025).

Proposition

The “inclusion” functor $\mathrm{Inc}\colon Rel \to qRel$ restricts to a functor $\mathrm{Inc}\colon Set \to qSet$, which has a right adjoint $\mathrm{Elm}\colon qSet \to Set$.

Definition

We define the “elements” functor $\mathrm{Elm}\colon qSet \to Set$:

• For each quantum set $X$, we define $\mathrm{Elm}(X) = \{\alpha \in \mathrm{At}(X) \mid \mathrm{dim} (X_\alpha) = 1\}$.

• For each morphism $f \colon X \to Y$, we define $\mathrm{Elm}(f) = \{(\alpha, \beta) \in \mathrm{Elm}(X) \times \mathrm{Elm}(Y) \mid \mathrm{dim} (f_{\alpha, \beta}) = 1\}$.

We have that $\mathrm{Elm}(X + Y) = \mathrm{Elm}(X) + \mathrm{Elm}(Y)$, that $\mathrm{Elm}(X \times Y) = \mathrm{Elm}(X) \times \mathrm{Elm}(Y)$, and that $\mathrm{Elm}(\mathrm{Inc}(A)) = A$. We also have a natural isomorphism $\mathrm{Elm}(X) \cong qSet(1,X)$. The monoidal unit $1$ is terminal in $qSet$, so $\mathrm{Elm}(X)$ is essentially the set of global elements of $X$.

Quantum sets as bundles

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 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 CPO

References

• Greg Kuperberg, Nik Weaver: A von Neumann Algebra approach to quantum metrics, Mem. Amer. Math. Soc. 215 (2012) [arXiv:1005.0353, ams:memo-215-1010]

• Nik Weaver, Quantum relations, Mem. Amer. Math. Soc. 215 (2012) [arXiv:1005.0354, ams:memo-215-1010]

• Kenney De Commer, Paweł Kasprzak, Adam Skalski, Piotr M. Sołtan: Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations, Israel J. Math. 226 (2018).

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

• 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)

• Andre Kornell, Bert Lindenhovius, Michael Mislove, A category of quantum posets, Indag. Math. 33 (2022).

• Bert Lindenhovius, Gejza Jenča, Monoidal quantaloids (2025), arXiv:2504.18266.