nLab quantum set

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Idea

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

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 qRelqRel, and in the latter case, we obtain the opposite of the category qSetqSet. Note that qRelqRel 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 XX is a family of nonzero finite-dimensional Hilbert spaces over \mathbb{C} that is indexed by a set At(X)\mathrm{At}(X), which may be empty, finite, or infinite.

X={X α} αAt(X). 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 At(X)\mathrm{At}(X) are called the atoms of XX.

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

Definition

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

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

Definition

Consider quantum sets XX and YY according to Def. . Then:

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

    (X+Y) α{X α if αAt(X), Y α if αAt(Y). (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×YX \times Y is defined by At(X×Y)At(X)×At(Y)\mathrm{At}(X \times Y) \coloneqq \mathrm{At}(X) \times \mathrm{At}(Y) and

    (X×Y) (α,β)X αY β, (X \times Y)_{(\alpha, \beta)} \coloneqq X_\alpha \otimes Y_\beta,

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

We have that Inc(A+B)=Inc(A)+Inc(B)\mathrm{Inc}(A + B) = \mathrm{Inc}(A) + \mathrm{Inc}(B) and that Inc(A×B)=Inc(A)×Inc(B)\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 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 qSetqSet or qRelqRel defined below.

Remark

A similar nomenclatural clash occurs for Rel 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 qRelqRel and qSetqSet, X+YX + Y is the coproduct of XX and YY, and X×YX \times Y is the designated monoidal product. In qRelqRel, X+YX+Y is also the product of XX and YY. In qSetqSet, the product of XX and YY is not easily definable and may be notated X*YX \ast Y.

The category qRelqRel

Definition

We define the dagger-compact category qRelqRel.

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

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

    R α,βL(X α,Y β), R_{\alpha,\beta} \subseteq L(X_\alpha, Y_\beta),

    where L(,𝒦)L(\mathcal{H},\mathcal{K}) is the space of all linear maps from \mathcal{H} to 𝒦\mathcal{K}. These morphisms are called relations.

  3. The composition SR:XZS \circ R \colon X \to Z of morphisms R:XYR \colon X \to Y and S:YZS \colon Y \to Z is given by

    (SR) α,γspan{srrR α,β,sS β,γfor someβAt(Y)}. (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 id X:XX\mathrm{id}_X\colon X \to X is defined by

    (id X) α,β{span{1} ifα=β, {0} ifαβ. (\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:XYR\colon X \to Y is defined by

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

    where r r^\dagger is the Hermitian adjoint.

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

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

    (R×R) (α,α),(β,β)span{rrrR α,β,rR α,β}. (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 11 is defined by At(1){}\mathrm{At}(1) \coloneqq \{\bullet\} and 1 1_\bullet \coloneqq \mathbb{C}.

  9. The braiding σ X,Y:X×YY×X\sigma_{X,Y}\colon X \times Y \to Y \times X is defined by

    (σ X,Y) (α,β),(β,α){span{σ X α,Y β} if(α,β)=(α,β), {0} if(α,β)(α,β), (\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 σ X α,Y β\sigma_{X_\alpha, Y_\beta} denotes the braiding in Hilb Hilb . The associator and unitors are defined similarly.

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

    (X *) αX α *, (X^*)_\alpha \coloneqq X_\alpha^*,

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

Furthermore, qRelqRel 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 XX and YY is their disjoint union X+YX + Y (see Def. ). The zero object 00 is defined by At(0)=\mathrm{At}(0) = \emptyset.

Proposition

The dagger-compact category qRelqRel is enriched over suplattices with

RSR α,βS α,β R \leq S \qquad \Longleftrightarrow \qquad R_{\alpha, \beta} \subseteq S_{\alpha, \beta}

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

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

In effect, RelRel is an enriched dagger-compact subcategory of qRelqRel.

Definition

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

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

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

    Inc(R)={ if(α,β)R, 0 if(α,β)R, \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(,)L(\mathbb{C}, \mathbb{C}) with \mathbb{C} in the obvious way.

Proposition

The “inclusion” functor Inc:RelqRel\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 qSetqSet

Definition

We define the symmetric monoidal category qSetqSet to be the wide subcategory of qRelqRel whose morphisms are maps. A map is a morphism f:XYf\colon X \to Y such that f fid Xf^\dagger \circ f \geq \mathrm{id}_X and ff id Yf \circ f^\dagger \leq \mathrm{id}_Y.

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

Theorem

The symmetric monoidal category qSetqSet is complete, cocomplete, and closed. In other words, it is a Bénabou cosmos. It also has an epi-mono factorization system.

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

Proposition

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

Definition

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

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

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

We have that Elm(X+Y)=Elm(X)+Elm(Y)\mathrm{Elm}(X + Y) = \mathrm{Elm}(X) + \mathrm{Elm}(Y), that Elm(X×Y)=Elm(X)×Elm(Y)\mathrm{Elm}(X \times Y) = \mathrm{Elm}(X) \times \mathrm{Elm}(Y), and that Elm(Inc(A))=A\mathrm{Elm}(\mathrm{Inc}(A)) = A. We also have a natural isomorphism Elm(X)qSet(1,X)\mathrm{Elm}(X) \cong qSet(1,X). The monoidal unit 11 is terminal in qSetqSet, so Elm(X)\mathrm{Elm}(X) is essentially the set of global elements of XX.

Theorem

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

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

Overall, qSetqSet 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 11 is terminal, so it is a semicartesian monoidal category. Furthermore, its monoidal product X×YX \times Y satisfies the uniqueness condition of the universal property that defines the categorical product.

Second, qSetqSet does not have a subobject classifier. However, every subobject of a quantum set XX has a unique characteristic morphism χ:X1+1\chi\colon X \to 1 + 1 such that χ=(!+!)f\chi = (! + !) \circ f for some f:XX+Xf \colon X \to X + X with f=id X\nabla \circ f = \mathrm{id}_X. Here, !:X1!\colon X \to 1 is the unique morphism from XX to 11 and :XX+X\nabla \colon X \to X + X is the codiagonal.

Proposition

Let XX be a quantum set. There is a one-to-one correspondence between the subsets of At(X)At(X) and the subobjects of XX in qSetqSet. Each subset AXA \subseteq X corresponds to a subobject j:WXj \colon W \to X with At(W)=AAt(W) = A and W α=X αW_\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 qRelqRel. In this context, the term “quantum” refers to noncommutative geometry, and the term “discrete” refers to C *C^*-algebras that are c 0c_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 qRelqRel is simply a reinterpretation of a straightforward definition in RelRel.

Second, in many cases, a comparable definition via internalization in qSetqSet is not available because the monoidal product in qSetqSet 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 qRelqRel, as well as Rel Rel and other allegories.

Example

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

  1. id XR\id_X \leq R,

  2. RRRR \circ R \leq R,

  3. R R=id XR^\dagger \wedge R = id_X,

where RSR \wedge S is the meet of RR and SS.

Example

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

  1. R =RR^\dagger = R,

  2. id XR\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 XX together with a map d:X×X *Inc([0,))d\colon X \times X^* \to \mathrm{Inc}([0, \infty)) such that

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

  2. Inc(+)(d×d)(id X×ε X×id X)Inc()d\mathrm{Inc}(+) \circ (d \times d) \circ (\id_X \times \varepsilon_X \times \id_X) \leq \mathrm{Inc}(\leq) \circ d,

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

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

Example

A discrete quantum monoid is a quantum set XX together with maps m:X×XXm\colon X \times X \to X and e:1Xe\colon 1 \to X such that

  1. m(m×id X)=m(id Xm)m \circ (m \times \id_X)= m \circ (\id_X \circ m),

  2. m(e×id X)=id X=m(id X×e)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)(X,m,e) such that

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

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

where X\top_X is the maximum relation 1X1 \to X. Note that a discrete quantum group is not a group object in qSetqSet.

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 * C^\ast -algebras in Podleś & Woronowicz 1990. However, these C *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 qSetqSet, and this section describes this duality.

Definition

For each quantum set XX, define the von Neumann algebra

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

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

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

Proposition

For each quantum set XX, there is a one-to-one correspondence between projection operators in l (X)l^\infty(X) and relations 1X1 \to X. Each projection operator pp corresponds to the relation Supp(p)Supp(p), which is defined by

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

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

Definition

We define W *\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 :qSetW *:l^\infty \colon qSet \to \mathrm{W}^\ast :

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

  • For each morphism f:XYf\colon X \to Y, we define l (f):l (Y)l (X)l^\infty(f)\colon l^\infty(Y) \to l^\infty(X) by Supp(l (f)(p))=f Supp(p)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 qSetqSet to products in W *\mathrm{W}^*. Thus, we have that

l (X+Y)l (X)l (Y),l (X×Y)l (X)l (Y). 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 XInc()X \to \mathrm{Inc}(\mathbb{R}) and self-adjoint operators affiliated with l (X)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 :qSetW *l^\infty\colon qSet \to \mathrm{W}^*,

  2. iI l M n i()\mathcal{M} \cong \bigoplus^{l^\infty}_{i \in I} M_{n_i}(\mathbb{C}) for some indexed family of positive integers {n i} iI\{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 opqSet^{op} and the full subcategory of hereditarily atomic von Neumann algebras. In effect, this generalizes the familiar equivalence between Set opSet^{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):

With composition the evident matrix multiplication (Kornell 2020 (5)), quantum relations between quantum sets form a category qRelqRel, 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).

References

Last revised on July 18, 2025 at 02:39:40. See the history of this page for a list of all contributions to it.