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 $l^\infty$-direct sum of full matrix algebras.

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

• 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

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

  where $\mathcal{H} \otimes \mathcal{K}$ denotes the Hilbert-space tensor product.

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. However, in contrast to how the Cartesian product of ordinary sets is their categorical product in $Set$ in the usual sense of universal properties, be warned that the Cartesian product of quantum sets is not their categorical product in either of the categories $qSet$ or $qRel$ defined below.

Remark

A similar nomenclatural clash occurs for $Rel$, consisting of sets and relations: there the cartesian product of sets as familiarly understood yields a monoidal product but not a categorical product.

In both of the symmetric monoidal categories $qRel$ and $qSet$, $X + Y$ is the coproduct of $X$ and $Y$, and $X \times Y$ is the designated 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 $qRel$

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 $\mathcal{K}$. These morphisms are called relations.

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 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 is full and faithful.

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.

The category $qSet$

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. It also has an epi-mono factorization system.

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 $X$.

Theorem

The inclusion functor $qSet \hookrightarrow qRel$ has a right adjoint $\mathrm{qPow}\colon qRel \to qSet$ with $\mathrm{qPow}(X) = [X^*, 1 + 1]$ for each quantum set $X$.

The notation $X^*$ refers to the dual of $X$, as in Def. . The notation $[X,Y]$ refers to the internal hom of $X$ and $Y$, as in Thm. . Thm. expresses that $qRel$ behaves somewhat like a power allegory. The quantum power set $qPow(X)$ is canonically a quantum poset in the sense of Example . In general, $\mathrm{qPow}(\mathrm{Inc}(A)) \ncong \mathrm{Inc}(\mathrm{Pow}(A))$, where $\mathrm{Pow}(A)$ is the power set of $A$. However, we do have that $\mathrm{Elm}(\mathrm{qPow}(\mathrm{Inc}(A)) \cong \mathrm{Pow}(A)$.

Overall, $qSet$ is unlike an elementary topos in two respects. First, its monoidal product is not its categorical product, so it is not a cartesian monoidal category. However, its monoidal unit $1$ is terminal, so it is a semicartesian monoidal category. Furthermore, its monoidal product $X \times Y$ satisfies the uniqueness condition of the universal property that defines the categorical product.

Second, $qSet$ does not have a subobject classifier. However, every subobject of a quantum set $X$ has a unique characteristic morphism $\chi\colon X \to 1 + 1$ such that $\chi = (! + !) \circ f$ for some $f \colon X \to X + X$ with $\nabla \circ f = \mathrm{id}_X$. Here, $!\colon X \to 1$ is the unique morphism from $X$ to $1$ and $\nabla \colon X \to X + X$ is the codiagonal.

Proposition

Let $X$ be a quantum set. There is a one-to-one correspondence between the subsets of $At(X)$ and the subobjects of $X$ in $qSet$. Each subset $A \subseteq X$ corresponds to a subobject $j \colon W \to X$ with $At(W) = A$ and $W_\alpha = X_\alpha$.

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

Internalization

Many classes of discrete quantum structures can be defined via internalization in the dagger-compact quantaloid $qRel$. In this context, the term “quantum” refers to noncommutative geometry, and the term “discrete” refers to $C^*$-algebras that are $c_0$-direct sums of full matrix algebras, as in section 1.

We highlight two aspects of this internalization. First, for many classes of discrete quantum structures, the original definition in noncommutative geometry is motivated by sophisticated considerations that are specific to that class. In contrast, the definition via internalization in $qRel$ is simply a reinterpretation of a straightforward definition in $Rel$.

Second, in many cases, a comparable definition via internalization in $qSet$ is not available because the monoidal product in $qSet$ is not its categorical product. Thus, this section refers to “allegorical internalization” rather than “categorical internalization.” In other words, the given definitions are meant to be interpreted in $qRel$, as well as $Rel$ and other allegories.

Example

A discrete quantum poset is a quantum set $X$ together with a relation $R\colon X \to X$ such that

1. $\id_X \leq R$,

2. $R \circ R \leq R$,

3. $R^\dagger \wedge R = id_X$,

where $R \wedge S$ is the meet of $R$ and $S$.

Example

A discrete quantum graph is a quantum set $X$ together with a relation $R\colon X \to X$ such that

1. $R^\dagger = R$,

2. $\id_X \leq R$.

These axioms internalize simple graphs with the convention that every vertex is adjacent to itself. This convention is standard in quantum information theory.

Example

A discrete quantum metric space is a quantum set $X$ together with a map $d\colon X \times X^* \to \mathrm{Inc}([0, \infty))$ such that

1. $\mathrm{Inc}(0)^\dagger \circ d = \varepsilon_X$,

2. $\mathrm{Inc}(+) \circ (d \times d) \circ (\id_X \times \varepsilon_X \times \id_X) \leq \mathrm{Inc}(\leq) \circ d$,

3. $d \circ \sigma_{X^*,X} = (d^*)^\dagger$,

where $\varepsilon_X$ is the counit and $\sigma_{X,Y}$ is the braiding. In axiom 1, the map $0: \{\bullet\} \to [0, \infty)$ has the value $0 \in [0, \infty)$.

Example

A discrete quantum monoid is a quantum set $X$ together with maps $m\colon X \times X \to X$ and $e\colon 1 \to X$ such that

1. $m \circ (m \times \id_X)= m \circ (\id_X \circ m)$,

2. $m \circ (e \times \id_X) = \id_X = m \circ (\id_X \times e)$.

Example

A discrete quantum group is a quantum monoid $(X,m,e)$ such that

1. $e^\dagger \circ m \circ (\!\top_X\! \times \id_X) = \!\top_X^\dagger$,

2. $e^\dagger \circ m \circ (\id_X \times \!\top_X\!) = \!\top_X^\dagger$,

where $\top_X$ is the maximum relation $1 \to X$. Note that a discrete quantum group is not a group object in $qSet$.

These examples are all discrete in the sense that the underlying quantum space is discrete and not in the sense that the structure on that quantum space is discrete. This qualification is necessary because the underlying space of a quantum structure is often assumed to be from a larger class. For example, the underlying quantum space of a quantum graph may be a quantum measurable space, and the underlying space of a quantum group may be a quantum topological space.

The material in this section is mostly from Weaver 2012 and Kornell 2024.

Duality

Quantum sets first appeared implicitly as a class of $C^\ast$-algebras in Podleś & Woronowicz 1990. However, these $C^\ast$-algebras are not unital, and this makes the relevant category complicated to define. It is easier to work with an equivalent category of von Neumann algebras. This category is dual to $qSet$, and this section describes this duality.

Definition

For each quantum set $X$, define the von Neumann algebra

$$l^\infty(X) \coloneqq \bigoplus_{i \in I}^{l^\infty} L(X_\alpha),$$

where the notation refers to the $l^\infty$-direct sum of von Neumann algebras.

This definition generalizes $l^\infty$ sequence spaces in the sense that $l^\infty(\mathrm{Inc}(A)) = l^\infty(A, \mathbb{C})$ for every set $A$.

Proposition

For each quantum set $X$, there is a one-to-one correspondence between projection operators in $l^\infty(X)$ and relations $1 \to X$. Each projection operator $p$ corresponds to the relation $Supp(p)$, which is defined by

$$Supp(p)_{\bullet, \alpha} \coloneqq \{r \in L(\mathbb{C}, X_\alpha) \mid p_\alpha r = r\}.$$

In this context, a projection operator $p \in l^\infty(X)$ is an element that satisfies $p^2 = p$ and $p^\dagger = p$, where $a^\dagger$ denotes the canonical anti-involution of the von Neumann algebra. For $a \in l^\infty(X)$, we define $(a^\dagger)_\alpha = a_\alpha^\dagger$.

Definition

We define $\mathrm{W}^\ast$ to be the symmetric monoidal category of von Neumann algebras and unital ultraweakly continuous $\dagger$-homomorphisms with the spatial tensor product.

Theorem

There is a full and faithful strong monoidal contravariant functor $l^\infty \colon qSet \to \mathrm{W}^\ast :$

• For each quantum set $X$, we define $l^\infty(X)$ as in Def. .

• For each morphism $f\colon X \to Y$, we define $l^\infty(f)\colon l^\infty(Y) \to l^\infty(X)$ by $Supp(l^\infty(f)(p)) = f^\dagger \circ Supp(p)$.

(Any ultraweakly continuous $\dagger$-homomorphism is completely determined by its values on projection operators.)

This functor also maps coproducts in $qSet$ to products in $\mathrm{W}^*$. Thus, we have that

$$l^\infty(X + Y) \cong l^\infty(X) \oplus l^\infty(Y), \qquad \qquad l^\infty(X \times Y) \cong l^\infty(X) \otimes l^\infty(Y).$$

We also have a one-to-one correspondence between maps $X \to \mathrm{Inc}(\mathbb{R})$ and self-adjoint operators affiliated with $l^\infty(X)$.

Theorem

Let $\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})$ be a von Neumann algebra. The following are equivalent:

1. $\mathcal{M}$ is in the essential image of the contravariant functor $l^\infty\colon qSet \to \mathrm{W}^*$,

2. $\mathcal{M} \cong \bigoplus^{l^\infty}_{i \in I} M_{n_i}(\mathbb{C})$ for some indexed family of positive integers $\{n_i\}_{i \in I}$,

3. every von Neumann subalgebra of $\mathcal{M}$ is atomic,

4. every self-adjoint operator in $\mathcal{M}$ has an orthonormal basis of eigenvectors in $\mathcal{H}$.

Von Neumann algebras of this kind are sometimes called hereditarily atomic, referring to condition 3.

Overall, we have an equivalence of symmetric monoidal categories between the opposite category $qSet^{op}$ and the full subcategory of hereditarily atomic von Neumann algebras. In effect, this generalizes the familiar equivalence between $Set^{op}$ and the category of complete atomic Boolean algebras. The projection operators in a von Neumann algebra form a complete orthomodular lattice, which is a Boolean algebra iff the von Neumann algebra is commutative.

The material in this section is mostly from Kornell 2020.

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

• Piotr Podleś, Stanisław Lech Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990).

• 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).

• Andre Kornell, Discrete quantum structures II: Examples, J. Noncommut. Geom. 18 (2024).

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