2.1 Quantum sets and binary relations

In this section, we introduce quantum sets, binary relations and functions between quantum sets, and order relations on quantum sets. We recall the convention that we only use the modifier ‘quantum’ to noncommutative generalizations of structures such as quantum sets, quantum graphs or quantum posets. We do not use the modifier for noncommutative generalizations of notions on structures that coincide with the original notion when restricting to the classical case, such as binary relations and orders. We refer to Kornell (Reference Kornell2020) and Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022) for a detailed discussion including proofs of the statements in these subsections.

The most transparent formulation of the definition of $\mathbf{qRel}$ , and of quantum sets in particular, is in terms of a known categorical construction. For this, we must first define the category $\mathbf{FdOS}$ .

The space $L(X,Y)$ is actually a finite-dimensional Hilbert space via the inner product $(a,b)\mapsto \mathrm{Tr}(a^\dagger b)$ , where $a^\dagger :Y\to X$ denotes the hermitian adjoint of $a\in L(X,Y)$ . Hence, the homset $\mathbf{FdOS}(X,Y)$ is a complete modular ortholattice, where the order on the homset is explicitly given by $A\leq B$ if $A$ is a subspace of $B$ . Since composition in $\mathbf{FdOS}(X,Y)$ preserves suprema, $\mathbf{FdOS}$ is enriched over the category $\mathbf{Sup}$ of complete lattices and suprema-preserving functions;Footnote ⁶ any such category is also called a quantaloid. Products and coproducts in a quantaloid $\mathbf Q$ coincide and are also called sums. The free sum-completion of $\mathbf Q$ can be described by the quantaloid $\mathrm{Matr}(\mathbf Q)$ , whose objects are $\mathbf{Set}$ -indexed families of objects $(X_i)_{i\in I}$ of $\mathbf Q$ , and whose morphisms $R:(X_i)_{i\in I}\to (Y_j)_{j\in J}$ are ‘matrices’ whose $(i,j)$ -component $R(i,j)$ is a $\mathbf Q$ -morphism $X_i\to Y_j$ .Footnote ⁷ Composition is defined by ‘matrix multiplication’: we define $S\circ R$ for $S:(Y_j)_{j\in J}\to (Z_k)_{k\in K}$ by $(S\circ R)(i,k)=\bigvee _{j\in J}S(j,k)\cdot R(i,j)$ for each $i\in I$ and $k\in K$ , where $\cdot$ denotes the composition of morphisms in $\mathbf Q$ . The $(i,i')$ -component of the identity morphism on an object $(X_i)_{i\in I}$ is the identity morphism of $X_i$ if $i=i'$ and is $0$ otherwise. The order on the homsets of $\mathrm{Matr}(\mathbf Q)$ is defined componentwise.

The atomic quantum set ${\mathcal H}_d$ can be used to represent a qudit, that is, a $d$ -level quantum system. In particular, a qubit can be represented by ${\mathcal H}_2$ . Given an arbitrary quantum set $\mathcal X$ , its atoms intuitively represent the superselection sectorsFootnote ⁸ of the discrete quantum system represented by $\mathcal X$ . For any quantum set $\mathcal X$ , we can consider the C^*-algebra $A=\bigoplus _{X\in \mathrm{At}({\mathcal X})}L(X)$ , in which case these representations are precisely the representations $A\to L(X)$ for $X\in \mathrm{At}({\mathcal X})$ . Formally, $\mathrm{At}({\mathcal X})$ is an index set for the quantum set $\mathcal X$ , For this reason, we use the notation $X\,\, \propto {\mathcal X}$ to express that $X$ is an atom of $\mathcal X$ .

To summarize, to any quantum set $\mathcal X$ , we associate an ordinary set $\mathrm{At}({\mathcal X})$ , whose elements are the atoms of $\mathcal X$ , so $X\,\, \propto {\mathcal X}$ if and only if $X\in \mathrm{At}({\mathcal X})$ . Conversely, to any ordinary set $M$ consisting of finite-dimensional Hilbert spaces, we associate the unique quantum set $\mathcal Q M$ whose set of atoms $\mathrm{At}(\mathcal Q M) = \{H\in M\mid \dim (H) \gt 0\}$ .

Several set-theoretic concepts can also be generalized to the quantum setting. We call a quantum set $\mathcal X$ empty if $\mathrm{At}({\mathcal X})=\emptyset$ ; we call $\mathcal X$ finite if $\mathrm{At}({\mathcal X})$ is finite; we write ${\mathcal X}\subseteq \mathcal Y$ if $\mathrm{At}({\mathcal X})\subseteq \mathrm{At}(\mathcal Y)$ , in which case $\mathcal X$ is called a subset of $\mathcal Y$ . The union ${\mathcal X}\cup \mathcal Y$ is defined to be the quantum set $\mathcal Q(\mathrm{At}({\mathcal X})\cup \mathrm{At}(\mathcal Y))$ , and the Cartesian product ${\mathcal X}\times \mathcal Y$ of quantum sets is defined to be the quantum set $\mathcal Q\{X\otimes Y:X\,\, \propto {\mathcal X},Y\,\, \propto \mathcal Y\}$ , where $\otimes$ denotes the Hilbert space tensor product.

A morphism $R:{\mathcal X}\to \mathcal Y$ in $\mathbf{qRel}$ is called a binary relation or more simply a relation; as we have seen, it assigns to each $X\,\, \propto {\mathcal X}$ and each $Y\,\, \propto \mathcal Y$ an operator space $R(X,Y):X\to Y$ . Sometimes we will write $R_X^Y$ instead of $R(X,Y)$ , a notation that improves the readability of large calculations. Given our conventions, composition with $S:\mathcal Y\to \mathcal Z$ is then given by $(S\circ R)(X,Z)=\bigvee _{Y\,\, \propto \mathcal Y}S(Y,Z)\cdot R(X,Y)$ . We denote the identity morphism of $\mathcal X$ by $I_{\mathcal X}$ , whose only nonvanishing components are $I_{\mathcal X}(X,X)={\mathbb C} 1_X$ for $X\,\, \propto {\mathcal X}$ . The order on the homset $\mathbf{qRel}({\mathcal X},\mathcal Y)$ is given by $R\leq S$ if and only if $R(X,Y)\leq S(X,Y)$ for all $X\,\, \propto {\mathcal X}$ and $Y\,\, \propto \mathcal Y$ .

The following theorem succinctly summarizes several relevant properties of this category:

To make the dagger compact structure of $\mathbf{qRel}$ more explicit, we first extend the Cartesian product $\times$ of quantum sets to a monoidal product on $\mathbf{qRel}$ as follows. Given relations $R:{\mathcal X}_1\to \mathcal Y_1$ and $S:{\mathcal X}_2\to \mathcal Y_2$ , we define $R\times S$ by $(R\times S)(X_1\otimes X_2,Y_1\otimes Y_2)=R(X_1,Y_1)\otimes S(Y_1,Y_2)$ . The monoidal unit is the quantum set $\mathbf 1:=\mathcal Q\{{\mathbb C}\}$ . The dagger $R^\dagger :\mathcal Y\to {\mathcal X}$ of a relation $R:{\mathcal X}\to \mathcal Y$ is given by $R^\dagger (Y,X)=\{r^\dagger :r\in R(X,Y)\}$ , where $r^\dagger$ denotes the hermitian adjoint of the linear map $r:X\to Y$ . The dual ${\mathcal X}^*$ of a quantum set $\mathcal X$ is the quantum set $\mathcal Q\{X^*:X\,\, \propto {\mathcal X}\}$ , where $X^*$ is the Banach space dual of the Hilbert space $X$ .

We emphasize that a relation $R$ in $\mathbf{qRel}$ is not a binary relation in the traditional sense; it is not a set of ordered pairs of atoms. Binary relations between quantum sets can be regarded as a generalization of ordinary binary relations because we have a fully faithful functor $`(\!-\!):\mathbf{Rel}\to \mathbf{qRel}$ (cf. Section 2.3). We also stress the difference in terminology between quantum relations and binary relations. The term ‘quantum relation’ was introduced by Weaver and Kuperberg, who do not follow our convention on the modifier ‘quantum’. The term refers to a morphism between von Neumann algebras, whereas the terms ‘relation’ and ‘binary relation’ refer to morphisms between sets or quantum sets. These classes of morphisms form categories that are equivalent but not equal: indeed, each quantum set $\mathcal X$ corresponds to the hereditarily atomic von Neumann algebra $\ell ^\infty ({\mathcal X}):=\bigoplus _{X\,\, \propto {\mathcal X}}L(X)$ , where $\bigoplus$ denotes the $\ell ^\infty$ -sum, and this object-level correspondence extends to an equivalence between $\mathbf{qRel}$ , whose morphisms are relations, and $\mathbf{WRel}_{\mathrm{HA}}$ , whose morphisms are normal unital $\ast$ -homomorphisms [Kornell (Reference Kornell2024a), Propositions A.2.1 & A.2.2]. Since dagger categories are self-dual, we also obtain an equivalence between $\mathbf{qRel}$ and $\mathbf{WRel}_{\textrm {HA}}^{\mathrm{op}}$ , whose morphisms are quantum relations.

2.2 Conventions

We will generally use calligraphic letters, for example, $\mathcal X$ , to name quantum sets and the corresponding capital letters, for example, $X$ , to name their atoms. In particular, if $\mathcal X$ is an atomic quantum set, then we will use $X$ to refer to its unique atom. We adopt the convention that $\mathcal H$ always denotes an atomic quantum set. As usual, the Dirac delta symbol $\delta _{a,b}$ names the complex number $1$ if $a = b$ and otherwise names $0$ . Similarly, the symbol $\Delta _{a,b}$ names the maximum binary relation on $\mathbf 1$ if $a=b$ and otherwise names the minimum binary relation on $\mathbf 1$ . Thus, $\delta _{a,b} \in \Delta _{a,b}({\mathbb C},{\mathbb C})$ . The binary relation $\Delta _{a,b}$ is a scalar in the symmetric monoidal category $\mathbf{qRel}$ Abramsky and Coecke (Reference Abramsky and Coecke2008). Hence, for each binary relation $R$ from a quantum set $\mathcal X$ to a quantum set $\mathcal Y$ , we write $\Delta _{a,b}R$ for the binary relation from $\mathcal X$ to $\mathcal Y$ obtained by composing $\Delta _{a,b} \times R$ with unitors in the obvious way. Finally, we will reserve the notation $\leq$ for the partial order on the homsets of $\mathbf{qRel}$ . For any other partial order, we use the notation $\sqsubseteq$ . We use the adjective ‘ordinary’ to emphasize that we are using a term in its standard sense. Thus, an ordinary poset is just a poset.

2.3 Quantum sets as a generalization of ordinary sets

To each ordinary set $S$ , we associate a quantum set $`S$ as follows. For each $s\in S$ , we introduce a one-dimensional Hilbert space ${\mathbb C}_s$ such that ${\mathbb C}_s\neq {\mathbb C}_t$ for $s\neq t$ . For example, we might define ${\mathbb C}_s:=\ell ^2(\{s\})$ . This allows us to define $`S$ formally as the quantum set $\mathcal Q\{{\mathbb C}_s:s\in S\}$ .

We can now extend the assignment $S\mapsto `S$ to a functor $`(\!-\!):\mathbf{Rel}\to \mathbf{qRel}$ as follows. Given a binary relation $r:S\to T$ between ordinary sets $S$ and $T$ , we define $`r:`S\to `T$ to be the relation given by

\begin{align*} `r({\mathbb C}_s,{\mathbb C}_t)=\begin{cases} L({\mathbb C}_s,{\mathbb C}_t), & (s,t)\in r, \\ 0, & \text{otherwise}. \end{cases} \end{align*}

Since $L({\mathbb C}_s,{\mathbb C}_t)$ is one dimensional, it follows easily that $`(\!-\!):\mathbf{Rel}\to \mathbf{qRel}$ is a fully faithful functor, which justifies our view that binary relations between quantum sets generalize the ordinary binary relations. Furthermore, $`(\!-\!)$ preserves the other relevant structure: it is a strong monoidal functor that preserves the dagger, and the order structure of the homsets.

The quote notation may look a bit odd, but there is a reason for this choice of notation: we want to treat the objects and morphisms of $\mathbf{Rel}$ as special cases of objects and morphisms of $\mathbf{qRel}$ , while still keeping a formal distinction. Hence, the notation for the embedding $\mathbf{Rel}\to \mathbf{qRel}$ should not be too prominent visually, and this is exactly what the quote notation achieves.

2.4 Discrete quantization via quantum sets

The following summarizes results from Kornell (Reference Kornell2020), where proofs can be found. Since $\mathbf{qRel}$ is a dagger category that is equivalent to $\mathbf{WRel}_{\mathrm{HA}}$ , we could equivalently define discrete quantization to be the process of internalizing structures in $\mathbf{qRel}$ . The basic principle is the following: given a category $\mathbf C$ whose objects and morphisms can be described in terms of sets, binary relations, and constraints expressed in the language of dagger compact quantaloids, we define a category $\mathbf{qC}$ by replacing any set in the definition of $\mathbf C$ by a quantum set, and any binary relation between sets by a binary relation between quantum sets such that the original constraints still hold. In many cases, the functor $`(\!-\!):\mathbf{Rel}\to \mathbf{qRel}$ induces a fully faithful functor $\mathbf C\to \mathbf{qC}$ , which we also notate $`(\!-\!)$ . If this fully faithful functor exists, then the objects in $\mathbf{C}$ can also be regarded as objects in $\mathbf{qC}$ . Often, $\mathbf{qC}$ has the same categorical properties as $\mathbf C$ , except that if $\mathbf C$ has a categorical product, this product translates to a symmetric monoidal product on $\mathbf{qC}$ , due to the quantum character of the objects of the latter category.

As an example, a function $f:S\to T$ between ordinary sets can be described in terms of the dagger compact quantaloid $\mathbf{Rel}$ as a morphism such that $f^\dagger \circ f\geq 1_S$ and $f\circ f^\dagger \leq 1_T$ , where the dagger of a binary relation in $\mathbf{Rel}$ is just the converse relation. In the same way, we can define a function $F:{\mathcal X}\to \mathcal Y$ between quantum sets as a binary relation that satisfies $F^\dagger \circ F\geq I_{\mathcal X}$ and $F\circ F^\dagger \leq I_{\mathcal Y}$ . We call the category of quantum sets and functions $\mathbf{qSet}$ . We call a function $F:{\mathcal X}\to \mathcal Y$ injective if $F^\dagger \circ F=I_{\mathcal X}$ , surjective if $F\circ F^\dagger =I_{\mathcal Y}$ , and bijective if it is both injective and surjective, in which case $F^\dagger$ is the inverse of $F$ . The injective functions coincide with the monos in $\mathbf{qSet}$ ; the surjective functions coincide with the epis in $\mathbf{qSet}$ , and the bijections are precisely the isomorphisms of $\mathbf{qSet}$ .

An important example of an injective function occurs in case that a quantum set $\mathcal X$ is a subset of a quantum set $\mathcal Y$ . Then we define the canonical injection $J_{\mathcal X}^{\mathcal Y}:{\mathcal X}\to \mathcal Y$ by $J_{\mathcal X}^{\mathcal Y}(X,Y)={\mathbb C}1_X$ if $X=Y$ , and $J_{\mathcal X}^{\mathcal Y}(X,Y)=0$ otherwise, for $X\,\, \propto {\mathcal X}$ and $Y\,\, \propto \mathcal Y$ . When the ambient quantum set $\mathcal Y$ is clear, we often write $J_{\mathcal X}$ instead of $J_{\mathcal X}^{\mathcal Y}$ . Given a relation $R:{\mathcal X}\to \mathcal Y$ between quantum sets, we define the restriction $R|_{\mathcal W}:=\mathcal W\to {\mathcal X}$ of $R$ to a subset $\mathcal W\subseteq {\mathcal X}$ as the quantum relation $R\circ J^{\mathcal X}_{\mathcal W}$ . Likewise, we define the corestriction $R|^{\mathcal Z}:{\mathcal X}\to \mathcal Z$ of $R$ to a subset $\mathcal Z\subseteq \mathcal Y$ to be the quantum relation $(J_{\mathcal Z}^{\mathcal Y})^\dagger \circ R$ . We write $R|_{\mathcal W}^{\mathcal Z}$ for relation $(J_{\mathcal Z}^{\mathcal Y})^\dagger \circ R\circ J_{\mathcal W}^{\mathcal X}$ .

The symmetric monoidal product $\times$ on $\mathbf{qRel}$ restricts to a symmetric monoidal product on $\mathbf{qSet}$ , which we also notate $\times$ , even though this is not the categorical product on $\mathbf{qSet}$ either. We use this notation because this monoidal product on $\mathbf{qSet}$ is the correct quantum generalization of the categorical product $\times$ on $\mathbf{Set}$ . Moreover, $\mathbf{qSet}$ is closed, and $\mathcal Y^{\mathcal X}$ denotes the internal hom between quantum sets $\mathcal X$ and $\mathcal Y$ .

The following theorem, whose content is taken from Kornell (Reference Kornell2020), summarizes the categorical properties of $\mathbf{qSet}$ :

We regard ordinary sets as a special case of quantum sets via the functor $`(\!-\!)$ .

2.5 Modeling physical systems and quantum computations via quantum sets

Finitary physical types are naturally modeled by quantum sets. We can use the atomic quantum set ${\mathcal H}_d$ , whose only atom is a $d$ -dimensional Hilbert space (cf. Definition 2.1.2) to model a qudit. In particular, the qubit can be modeled by the quantum set ${\mathcal H}_2$ . By contrast, the classical bit is modeled by the quantum set $`\{0,1\}$ , which has two one-dimensional atoms. The “memory” of an idealized quantum computer might consist of finitely many qubits, and finitely many bits, so we can model such a quantum computer as a composite physical system.

The symmetric monoidal product $\times$ on $\mathbf{qRel}$ and $\mathbf{qSet}$ generalizes the Cartesian product of ordinary sets. Composite systems consisting of two fully quantum systems, that is, of quantum systems modeled by single Hilbert spaces, are obtained by forming the tensor product of the two Hilbert spaces. Similarly, composite systems consisting of two fully classical systems, that is, of quantum systems modeled by ordinary sets, are obtained by forming the ordinary Cartesian product of the two sets. Thus, our generalized product models composite systems consisting entirely of fully quantum systems, or entirely of fully classical systems. In fact, it is appropriate for modeling mixed quantum/classical systems as well. In particular, an idealized quantum computer possessing $n$ qubits and $m$ bits can be modeled by the quantum set

\begin{equation*} \underbrace {{\mathcal H}_2 \times \cdots \times {\mathcal H}_2}_{n} \times \underbrace {`\{0,1\} \times \cdots \times `\{0,1\}}_{m}.\end{equation*}

This quantum set has $2^m$ atoms, each of dimension $2^n$ .

Other natural type constructors are also easily modeled in quantum sets. Sum types are modeled by disjoint unions of quantum sets, portraying a kind of classical disjunction of potentially quantum systems. The resulting physical system may be in a state of the first type or in a state of the second type, but no superpositions may occur. For example, an idealized quantum computer possessing a single qubit and a single bit is modeled by a quantum set with two atoms, because the computer can be in a configuration where the qubit has value $0$ , and it can be in a configuration where the qubit has value $1$ , but it cannot be in a superposition between two such states. The exponential modality $!$ that permits duplication, is modeled by an operation on quantum sets that extracts just their one-dimensional atoms, so $!{\mathcal X}={\mathcal X}_1$ , where

\begin{equation*}{\mathcal X}_1:=\mathcal Q\{X\in \mathrm{At}({\mathcal X}):\dim (X)=1\}.\end{equation*}

This is essentially an ordinary set, and it is the largest subset that admits a duplication map. Indeed, classical sets all admit duplication maps, but the no-cloning theorem forbids the duplication of higher dimensional atoms. A useful result that characterizes the atoms of ${\mathcal X}_1$ is the following:

We recall that the type systems of many quantum programming languages are based on intuitionistic linear logic. We further note that this logic can be modeled by linear/nonlinear models Benton (Reference Benton1995):

Given such a model, the exponential modality $!\colon \mathbf C\to \mathbf C$ is the comonad induced by the linear/nonlinear adjunction. For instance, the functor $`(\!-\!):\mathbf{Set}\to \mathbf{qSet}$ described in Theorem 2.4.1 has a right adjoint given by ${\mathcal X}\mapsto \mathbf{qSet}(\mathbf 1,-)$ , hence $\mathbf{Set}$ and $\mathbf{qSet}$ form a linear/nonlinear model.Footnote ⁹ By Lemma 2.5.1, the comonad induced by the linear/nonlinear adjunction between $\mathbf{Set}$ and $\mathbf{qSet}$ is precisely the operation described above of extracting the one-dimensional atoms from a quantum set.

Higher types are modeled by quantum function sets, which are more challenging to describe, mainly due to the large automorphism group of the physical qubit. Indeed, the ordinary set $\mathbf{qSet}({\mathcal H}_2, {\mathcal H}_2)$ is in canonical bijection with the rotation group $\mathrm{SO}(3)$ . The quantum function set ${\mathcal H}_2^{{\mathcal H}_2}$ also has atoms of dimension higher than $1$ . In general, the quantum function set from a quantum set $\mathcal X$ to a quantum set $\mathcal Y$ has an atom of dimension $d$ for each unital normal $*$ -homomorphism $\rho \colon \ell ^\infty (\mathcal Y) {\rightarrow } \ell ^\infty ({\mathcal X}) \overline \otimes L(H_d)$ that is irreducible in the sense that $\rho (\ell ^\infty (\mathcal Y))' \cap ({\mathbb C} \overline \otimes L(H_d)) = {\mathbb C}$ , where $\overline \otimes$ denotes the spatial tensor product of von Neumann algebras, and $(\cdot)'$ denotes the commutant.

Example 2.5.4. Pure quantum computation refers to the model of quantum computation in which the evolution of a system is described entirely by quantum gates, typically formalized by unitary operators Huot and Staton (Reference Huot and Staton2019). In this pure setting, a quantum circuit is just the sequential composition of such unitaries. There is no measurement, randomness, or environmental interaction: computation proceeds coherently and reversibly, so quantum gates are automorphisms of fully quantum systems, which consist of qubits. In the category of quantum sets and functions, such an automorphism is formalized by a closely related function between quantum sets. Indeed, each function $F_1\colon {\mathcal H}_d {\rightarrow } {\mathcal H}_d$ , for any positive integer $d$ , is defined by $F_1(H_d, H_d) = {\mathbb C}\cdot u$ , for some unitary operator $u$ . For example, the Hadamard gate is formalized by the function $F_1\colon {\mathcal H}_2 {\rightarrow } {\mathcal H}_2$ defined by

\begin{align*} F_1(H_2, H_2) = {\mathbb C} \cdot \left [ \begin{matrix} 1\;\;\;\; & 1 \\ 1\;\;\;\; & -1 \end{matrix} \right ]. \end{align*}

This function $F_1$ is an automorphism of ${\mathcal H}_2$ in $\mathbf{qSet}$ . The unital normal $*$ -homomorphism $M_2({\mathbb C}) \cong \ell ^\infty ({\mathcal H}_2) {\rightarrow } \ell ^\infty ({\mathcal H}_2) \cong M_2({\mathbb C})$ that corresponds to this function in the sense of Theorem 2.4.1 is simply conjugation by the Hadamard matrix above, appropriately normalized.

Example 2.5.5. Measurement is a channel from a fully quantum system to a fully classical system and cannot be described by a unitary operator. As a consequence, it is an impure quantum operation, which in $\mathbf{qSet}$ is formalized by a function from an atomic quantum set to a classical quantum set. It follows from the definition of a function between quantum sets that functions from an atomic quantum set ${\mathcal H}_d$ to a classical quantum set $`S$ are in bijective correspondence with projection-valued measurements on ${\mathcal H}_d$ . Explicitly, $F_2(H_d, {\mathbb C}_s) = L(H_d, {\mathbb C}_s) \cdot p_s$ for each $s \in S$ , where $(p_s: s \in S)$ is a projective POVM.Footnote <sup>10</sup> In particular, the standard measurement of a qubit is formalized by a function from ${\mathcal H}_2$ to $`\{1, -1\}$ defined by

\begin{align*} F_2(H_2, {\mathbb C}_s) = \begin{cases} {\mathbb C} \cdot [\begin{matrix} 1\;\;\;\; & 0 \end{matrix} ] & {s = 1}; \\ {\mathbb C} \cdot [\begin{matrix} 0\;\;\;\; & 1 \end{matrix}] & s = {-1}. \end{cases} \end{align*}

This function $F_2$ is an epimorphism in $\mathbf{qSet}$ . Equivalently, it is surjective in the sense that $F_2 \circ F_2^\dagger \geq I$ . The unital normal $*$ -homomorphism ${\mathbb C}^2 \cong \ell ^\infty (\{1, -1\}) {\rightarrow } \ell ^\infty ({\mathcal H}_2) \cong M_2({\mathbb C})$ corresponding to this function in the sense of Theorem 2.4.1 includes ${\mathbb C}^2$ into $M_2({\mathbb C})$ diagonally.

Example 2.5.6. Generally, quantum computations are described by quantum channels, that is, completely positive unital mapsFootnote ¹¹ . Classically, impure computations correspond to side effects, which are modeled by monads. In a similar way, quantum channels can be described by Kleisli morphisms of a monad $\mathcal D$ on $\mathbf{qSet}$ . This monad is induced by the adjunction between the category $\mathbf{WStar}_{\mathrm{HA}}^{\mathrm{op}}$ , which is equivalent to $\mathbf{qSet}$ , and the category $\mathbf{WCPU}_{\mathrm{HA}}^{\mathrm{op}}$ , the opposite of the category of hereditarily atomic von Neumann algebras and normal completely positive unital maps. Here, the left adjoint is given by the canonical embedding of the former category into the latter. Thus, for any two quantum sets $\mathcal X$ and $\mathcal Y$ , we have a natural bijection $\mathbf{qSet}({\mathcal X},{\mathcal D}(\mathcal Y))\cong \mathbf{WCPU}_{\mathrm{HA}}(\ell ^\infty (\mathcal Y),\ell ^\infty ({\mathcal X}))$ , arbitrary quantum channels can be modeled in $\mathbf{qSet}$ .

If we identify the states on $\mathcal X$ with normal completely positive unital functionals on $\ell ^\infty ({\mathcal X})$ , then it follows from Lemma 2.5.1 that

\begin{equation*}\mathrm{States}({\mathcal X})=\mathbf{WCPU}_{\mathrm{HA}}(\ell ^\infty ({\mathcal X}),\mathbb C)=\mathbf{qSet}({\mathbf 1},{\mathcal D}({\mathcal X}))\cong \mathrm{At}({\mathcal D}({\mathcal X})_1),\end{equation*}

hence by the same lemma, we have an bijection

(1) \begin{equation} `\mathrm{States}({\mathcal X})\cong {\mathcal D}({\mathcal X})_1. \end{equation}

The monad $\mathcal D$ on $\mathbf{qSet}$ is a noncommutative version of the (countable) distributions monad $D$ on $\mathbf{Set}$ in the sense that if $S$ is an ordinary set, then there is a bijection $`D(S)\cong {\mathcal D}(`S)_1$ , so essentially, $D(S)$ is the classical part of ${\mathcal D}(`S)$ . This follows by taking ${\mathcal X}=`S$ in Equation (1), and from a standard result in the theory of von Neumann algebras that the predual of the (hereditarily atomic) von Neumann algebra $\ell ^\infty (S)$ is isometrically isomorphic to $\ell ^1(S)$ , the space of functions $f:S\to {\mathbb C}$ such that $\sum _{x\in S}|f(x)|\lt \infty$ . This isomorphism restricts and corestricts to a bijection $\mathbf{WCPU}(\ell ^\infty (S),{\mathbb C})\cong D(S)$ , where we recognize the left-hand side as $\mathrm{States}(`S)$ .

Example 2.5.7. State preparation cannot be formalized directly as a function, in the manner of the Examples 2.5.4 and 2.5.5 Indeed, from the perspective of quantum mechanics, state preparation is a nondeterministic binary channel, but there are no functions at all from $`\{1, -1\}$ to ${\mathcal H}_2$ . However, it is possible to formalize state preparation in $\mathbf{qSet}$ using the monad $\mathcal D$ of Example 2.5.6. In this example, we saw that $\mathcal D$ has the property that for each quantum set $\mathcal X$ , the one-dimensional atoms of ${\mathcal D}({\mathcal X})$ correspond bijectively to the states on $\mathcal X$ , which we may define to be the normal states on $\ell ^\infty ({\mathcal X})$ in the sense of operator algebras. We thus have a monomorphism $`\mathrm{States}({\mathcal X}) \rightarrowtail {\mathcal D}({\mathcal X})$ in $\mathbf{qSet}$ .

A simple experimental procedure, which consists of preparing a qubit, applying the Hadamard gate, and then measuring its value is modeled as the following composition of functions in $\mathbf{qSet}$ :

where the injection is obtained from the bijection $`\mathrm{States}({\mathcal H}_2)\cong {\mathcal D}({\mathcal H}_2)_1$ in Example 2.5.6, and where $F_1$ and $F_2$ are the functions from Examples 2.5.4 and 2.5.5, respectively. This yields a function from $`\mathrm{States}({\mathcal H}_2)$ to ${\mathcal D}(`\{1,-1\})$ , or equivalently a function.

$\mathrm{States}({\mathcal H}_2) {\rightarrow } \mathrm{States}(`\{1,-1\})$ that maps each prepared state to the corresponding probability distribution on outcomes. This equivalence is due to the fact that a function between quantum sets necessarily maps one-dimensional atoms to one-dimensional atoms. Type theoretically, it is related to the fact that any term of type $A$ that has only nonlinear parameters lifts to a term of type $!A$ .

2.6 Quantum posets

Quantum cpo are in particular quantum posets, which are essentially due to Weaver [Weaver (Reference Weaver2012), Definition 2.4] and consist of a pair $({\mathcal X},R)$ consisting of quantum set $\mathcal X$ and a binary relation $R$ on $\mathcal X$ that generalizes orders to the noncommutative setting. We start by reviewing the exact notion of a quantum poset. We note that the notions of partial order, monotone function, etc. introduced here are all direct generalizations of the standard notions of set theory.

A function $F:({\mathcal X},R)\to (\mathcal Y,S)$ between quantum posets is said to be monotone if $F\circ R\leq S\circ F$ . If $R=F^\dagger \circ S\circ F$ , we say that $F$ an order embedding. A surjective order embedding is called an order isomorphism, and is precisely a bijective monotone function whose inverse is also monotone. The category of quantum posets and monotone maps is denoted by $\mathbf{qPOS}$ . The elementary properties of this category are established in Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022):

Here, the monoidal product $({\mathcal X},R)\times (\mathcal Y,S)$ of two quantum posets $({\mathcal X},R)$ and $(\mathcal Y,S)$ is the quantum poset $({\mathcal X}\times \mathcal Y,R\times S)$ . The monoidal product of two monotone functions is the monoidal product of the two functions when regarded as morphisms of $\mathbf{qSet}$ .

Example 2.6.4. Let $\overline {\mathbb N}$ be the set of natural numbers with infinity, ordered in the obvious way by $\sqsubseteq$ . By the above theorem, $`{\sqsubseteq }$ is an order on $`\overline {\mathbb N}=\mathcal Q\{{\mathbb C}_1,{\mathbb C}_2,\ldots ,{\mathbb C}_\infty \}$ whose nonvanishing components are given by $`{\sqsubseteq }({\mathbb C}_r,{\mathbb C}_s)=L({\mathbb C}_r,{\mathbb C}_s)$ if $r\sqsubseteq s$ in $\overline {\mathbb N}$ . Since every atom of $`\overline {{\mathbb N}}$ is one-dimensional, and $L({\mathbb C}_r,{\mathbb C}_s)$ is also one-dimensional, we can represent $`{\sqsubseteq }$ by the following matrix:

\begin{align*} \left ( \begin{matrix} `{\sqsubseteq }({\mathbb C}_\infty , {\mathbb C}_\infty )\;\;\;\; & \cdots\;\;\;\; & `{\sqsubseteq }({\mathbb C}_2, {\mathbb C}_\infty )\;\;\;\; & `{\sqsubseteq }({\mathbb C}_1, {\mathbb C}_\infty ) \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; &\vdots \\ `{\sqsubseteq }({\mathbb C}_\infty , {\mathbb C}_2)\;\;\;\; & \cdots\;\;\;\; & `{\sqsubseteq }({\mathbb C}_2, {\mathbb C}_2)\;\;\;\; &`{\sqsubseteq }({\mathbb C}_1, {\mathbb C}_2) \\ `{\sqsubseteq }({\mathbb C}_\infty , {\mathbb C}_1)\;\;\;\; & \cdots\;\;\;\; & `{\sqsubseteq }({\mathbb C}_2, {\mathbb C}_1)\;\;\;\; & `{\sqsubseteq }({\mathbb C}_1, {\mathbb C}_1) \end{matrix} \right ) : = \left ( \begin{matrix} {\mathbb C}\;\;\;\; & \cdots\;\;\;\; & {\mathbb C}\;\;\;\; & {\mathbb C} \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; &\vdots \\ 0\;\;\;\; & \cdots\;\;\;\; & {\mathbb C}\;\;\;\; & {\mathbb C} \\ 0\;\;\;\; & \cdots\;\;\;\; & 0\;\;\;\; &{\mathbb C} \end{matrix} \right ). \end{align*}

The triangular shape of the matrix of $`{\sqsubseteq }$ reflects that $\sqsubseteq$ is an ordinary order.

Example 2.6.5. Let $\mathcal H$ be the atomic quantum set with single atom $H={\mathbb C}^2$ , and let $V$ be the quantum order from Example 2.6.3, that is, $V(H,H)={\mathbb C} a+{\mathbb C}1$ , where $a = \left [\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\right ]$ and $1$ is the identity on ${\mathbb C}^2$ . Let ${\mathcal X}:=`\overline {\mathbb N}\times {\mathcal H}$ . Then, all atoms of $\mathcal X$ are of the form $X_i : = {\mathbb C}_i \otimes {\mathbb C}^2$ for some $i\in \overline {\mathbb N}$ . If $\sqsubseteq$ denotes the usual order on $\overline {\mathbb N}$ as in the previous example, then the above theorem implies that $S:=`{\sqsubseteq }\times V$ is an order on $`\overline {\mathbb N}\times {\mathcal H}$ . Since each atom $X_i$ of $\mathcal X$ is canonically isomorphic to ${\mathbb C}^2$ , we may depict $S$ as a matrix of matrix subspaces as follows:

\begin{align*} \left ( \begin{matrix} S(X_\infty , X_\infty )\;\;\;\; & \cdots\;\;\;\; & S(X_2, X_\infty )\;\;\;\; & S(X_1, X_\infty ) \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; &\vdots \\ S(X_\infty , X_2)\;\;\;\; & \cdots\;\;\;\; & S(X_2, X_2)\;\;\;\; &S(X_1, X_2) \\ S(X_\infty , X_1)\;\;\;\; & \cdots\;\;\;\; & S(X_2, X_1)\;\;\;\; & S(X_1, X_1) \end{matrix} \right ) : = \left ( \begin{matrix} {\mathbb C} a+{\mathbb C} 1\;\;\;\; & \cdots\;\;\;\; & {\mathbb C} a + {\mathbb C} 1\;\;\;\; & {\mathbb C} a + {\mathbb C} 1 \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; & \vdots \\ 0\;\;\;\; & \cdots\;\;\;\; & {\mathbb C}\;\;\;\; a + {\mathbb C} 1\;\;\;\; & {\mathbb C} a + {\mathbb C} 1 \\ 0\;\;\;\; & \cdots\;\;\;\; & 0\;\;\;\; & {\mathbb C} a\;\;\;\; + {\mathbb C} 1 \\ \end{matrix} \right ). \end{align*}

The following result, from Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022) shows the category $\mathbf{qPOS}$ is enriched over $\mathbf{POS}$ , using a generalization of the pointwise order on functions from an ordinary set into an ordinary poset.

We regard ordinary posets as a special case of quantum posets via the functor $`(\!-\!):\mathbf{POS}\to \mathbf{qPOS}$ .

3.1 Convergence

For a monotonically increasing sequence of functions $K_1 \sqsubseteq K_2 \sqsubseteq \cdots \sqsubseteq K_\infty$ from a quantum set $\mathcal W$ to a quantum poset $\mathcal X$ , we next define the notation $K_n \nearrow K_\infty$ , whose intuitive meaning is that $K_\infty$ is the internal pointwise supremum of the sequence $K_1, K_2, \ldots$ . Example 3.1.3 then shows $K_n \nearrow K_\infty$ is a strictly stronger relation than $K_\infty$ being the external supremum of the sequence $K_1, K_2, \ldots$ among all the functions from $\mathcal W$ to $\mathcal X$ .

In order to show that property (2), $K_n\nearrow K_\infty$ is the appropriate one for the quantum setting, we first prove that the limit of a monotonically ascending sequence is also its supremum, as one would expect from the classical case.

The converse of Lemma 3.1.2 is not true, as the next example shows. That is, there exist a quantum set $\mathcal W$ , a quantum poset $({\mathcal X},R)$ and functions $K_1 \sqsubseteq K_2 \sqsubseteq \cdots \sqsubseteq K_\infty \colon W {\rightarrow } {\mathcal X}$ such that $K_\infty =\bigsqcup _{n\in {\mathbb N}}K_n$ , but not $K_n \nearrow K_\infty$ . Nevertheless, Lemma 3.1.2 implies that $K_\infty =\bigsqcup _{n\in {\mathbb N}}K_n$ is equivalent to $K_n \nearrow K_\infty$ when $({\mathcal X},R)$ is a quantum cpo.

Example 3.1.3. Let ${\mathcal X}=`\overline {\mathbb N}\times {\mathcal H}$ ordered by $S$ as in Example 2.6.5. Later we will see that $({\mathcal X},S)$ is a quantum cpo according to Proposition 3.2.7, Corollaries 6.1.14 and 5.1.5. Here, we modify $S$ slightly in order to obtain a quantum poset for which not every monotonically ascending sequence has a limit. So let $R$ be the quantum order that equals $S$ on all atoms, except we equip the atom $X_\infty ={\mathbb C}_\infty \otimes {\mathbb C}^2$ the flat order, that is, $R(X_\infty ,X_\infty )={\mathbb C} 1_{X_\infty }$ . Since each atom $X_i={\mathbb C}_i\otimes {\mathbb C}^2$ is canonically isomorphic to ${\mathbb C}^2$ , we can depict $R$ as a matrix of matrix subspaces as follows:

\begin{align*} \left ( \begin{matrix} R(X_\infty , X_\infty )\;\;\;\; & \cdots\;\;\;\; & R(X_2, X_\infty )\;\;\;\; & R(X_1, X_\infty ) \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; &\vdots \\ R(X_\infty , X_2)\;\;\;\; & \cdots\;\;\;\; & R(X_2, X_2)\;\;\;\; &R(X_1, X_2) \\ R(X_\infty , X_1)\;\;\;\; & \cdots\;\;\;\; & R(X_2, X_1)\;\;\;\; & R(X_1, X_1) \end{matrix} \right ) : = \left ( \begin{matrix} {\mathbb C} 1\;\;\;\; & \cdots\;\;\;\; & {\mathbb C} a + {\mathbb C} 1\;\;\;\; & {\mathbb C} a + {\mathbb C} 1 \\ \vdots\;\;\;\; & \ddots\;\;\;\; & \vdots\;\;\;\; & \vdots \\ 0\;\;\;\; & \cdots\;\;\;\; & {\mathbb C} a + {\mathbb C} 1\;\;\;\; & {\mathbb C} a + {\mathbb C} 1 \\ 0\;\;\;\; & \cdots\;\;\;\; & 0\;\;\;\; & {\mathbb C} a + {\mathbb C} 1 \\ \end{matrix} \right ) \end{align*}

So, the only difference between $R$ and $S$ is the first row, first column entry.

We show $R$ is an order and that there is a monotonically increasing sequence of functions $K_1 \sqsubseteq K_2 \sqsubseteq \cdots \colon {\mathcal H}\to ({\mathcal X},R)$ with a supremum, $K_\infty$ , that does not satisfy $R \circ K_\infty = \bigwedge _{n \in {\mathbb N}} R \circ K_n$ . Appealing to the fact that the algebra ${\mathbb C}1 + {\mathbb C} a$ is antisymmetric, it is easy to verify that $R$ is an order on $`{\mathbb N} \times {\mathcal H}$ . The “origin” of the matrices is in the lower right corner, to reflect the intuition that higher-indexed atoms $X_i$ correspond to atomic subsets that are higher in the order on $\mathcal X$ .

For each $n \in {\mathbb N}$ we define the function $K_n\colon {\mathcal H} \to {\mathcal X}$ and compute $R \circ K_n$ as follows:

\begin{align*} \left ( \begin{matrix} K_n (H, X_\infty ) \\ \vdots \\ K_n (H, X_{n+1}) \\ K_n (H, X_n) \\ K_n (H, X_{n-1}) \\ \vdots \\ K_n(H, X_2) \\ K_n(H, X_1) \end{matrix} \right ) : = \left ( \begin{matrix} 0 \\ \vdots \\ 0 \\ {\mathbb C} 1 \\ 0\\ \vdots \\ 0 \\ 0 \end{matrix} \right ) \qquad \left ( \begin{matrix} (R \circ K_n) (H, X_\infty ) \\ \vdots \\ (R \circ K_n) (H, X_{n+1}) \\ (R \circ K_n) (H, X_n) \\ (R \circ K_n) (H, X_{n-1}) \\ \vdots \\ (R \circ K_n)(H, X_2) \\ (R \circ K_n)(H, X_1) \end{matrix} \right ) = \left ( \begin{matrix} {\mathbb C} a + {\mathbb C} 1 \\ \vdots \\ {\mathbb C} a + {\mathbb C} 1 \\ {\mathbb C} a + {\mathbb C} 1 \\ 0\\ \vdots \\ 0 \\ 0 \end{matrix} \right ) \end{align*}

We also define $K_\infty \colon {\mathcal H} {\rightarrow } {\mathcal X}$ below (it’s straightforward to show $K_\infty = \sup _{n\in {\mathbb N}} K_n$ ), and we compute $R \circ K_\infty$ as follows:

\begin{align*} \left ( \begin{matrix} K_\infty (H, X_\infty ) \\ \vdots \\ K_\infty (H, X_2) \\ K_\infty (H, X_1) \end{matrix} \right ) : = \left ( \begin{matrix} {\mathbb C} 1 \\ \vdots \\ 0 \\ 0 \end{matrix} \right ) \qquad \left ( \begin{matrix} (R \circ K_\infty ) (H, X_\infty ) \\ \vdots \\ (R \circ K_\infty )(H, X_2) \\ (R \circ K_\infty )(H, X_1) \end{matrix} \right ) = \left ( \begin{matrix} {\mathbb C} 1 \\ \vdots \\ 0 \\ 0 \end{matrix} \right ) \end{align*}

Thus, we find that

\begin{align*} \left ( \begin{matrix} (R \circ K_\infty ) (H, X_\infty ) \\ \vdots \\ (R \circ K_\infty )(H, X_2) \\ (R \circ K_\infty )(H, X_1) \end{matrix} \right ) = \left ( \begin{matrix} {\mathbb C} 1 \\ \vdots \\ 0 \\ 0 \end{matrix} \right ) \neq \left ( \begin{matrix} {\mathbb C} a + {\mathbb C} 1\\ \vdots \\ 0 \\ 0 \end{matrix} \right ) = \bigwedge _{n \in {\mathbb N}} \left ( \begin{matrix} (R \circ K_n) (H, X_\infty ) \\ \vdots \\ (R \circ K_n)(H, X_2) \\ (R \circ K_n)(H, X_1) \end{matrix} \right ). \end{align*}

In short, $R \circ K_\infty \neq \bigwedge _{n \in {\mathbb N}} R \circ K_n$ , so we do not have that $K_n \nearrow K_\infty$ in $\mathbf{qPOS}({\mathcal H},({\mathcal X},R))$ .

On the other hand, we claim that $K_\infty =\bigsqcup _{n\in {\mathbb N}}K_n$ , that is, that $K_\infty$ is the supremum of the sequence $K_1 \sqsubseteq K_2 \sqsubseteq \cdots$ in $\mathbf{qPOS}({\mathcal H},({\mathcal X},S))$ in the order defined in Lemma 2.6.6. Indeed, let $F$ be a function such that $K_n \sqsubseteq F$ for each $n \in {\mathbb N}$ . It follows immediately that $F \leq \bigwedge _{n \in {\mathbb N}} R \circ K_n$ :

\begin{align*} \left ( \begin{matrix} F (H, X_\infty ) \\ \vdots \\ F(H, X_2) \\ F(H, X_1) \end{matrix} \right ) \leq \left ( \begin{matrix} {\mathbb C} a +{\mathbb C} 1 \\ \vdots \\ 0 \\ 0 \end{matrix} \right ). \end{align*}

By the definition of a function between quantum sets, $F \circ F^\dagger \leq I_{\mathcal X}$ . Thus, $F(H, X_\infty )$ is a subspace of $2\times 2$ complex matrices such that $F(H, X_\infty )\cdot F(H, X_\infty )^\dagger \leq {\mathbb C} 1$ . It is straightforward to show that this implies that either $F(H, X_\infty )= 0$ or $F(H, X_\infty ) = {\mathbb C} u$ for some unitary matrix $u$ . By the definition of a function between quantum sets, $F^\dagger \circ F \geq I_{\mathcal H}$ , so certainly $F(H, X_\infty ) \neq 0$ . Hence, $F(H, X_\infty ) = {\mathbb C} u$ for some unitary matrix $u$ . Applying [Weaver (Reference Weaver2019), Proposition 3.4], we find that $u$ is a scalar and, therefore, that $F = K_\infty$ . Thus, $K_\infty$ is not only the least upper bound of the sequence $K_1 \sqsubseteq K_2 \sqsubseteq \cdots$ , it is the only upper bound of this sequence.

3.2 Quantum cpos

We now define quantum cpos and the Scott continuous functions between them.

The next result shows that we can replace $\mathcal H$ by any quantum set $\mathcal W$ in Definition 3.2.1, as one would expect, since the atomic quantum sets generate $\mathbf{qSet}$ .

Proof. For each $W\,\, \propto \mathcal W$ , let $J_W:\mathcal Q\{W\}\to \mathcal W$ be the canonical inclusion. Let $n,m \in {\mathbb N}$ such that $n \leq m$ . Then, $F_n\sqsubseteq F_m$ , so $F_n\circ J_W\sqsubseteq F_n\circ J_W$ . Hence, $F_1\circ J_W\sqsubseteq F_2\circ J_W\sqsubseteq \cdots :\mathcal Q\{W\}\to {\mathcal X}$ is a monotonically ascending sequence of functions, and since $\mathcal Q\{W\}$ is atomic, it follows that there is a function $F_W:\mathcal Q\{W\}\to {\mathcal X}$ such that $F_n\circ J_W\nearrow F_W$ , that is, $R\circ F_W=\bigwedge _{n\in {\mathbb N}}R\circ F_n\circ J_W.$

We obtain a function $F_\infty =[F_W:W\,\, \propto \mathcal W]: \mathcal W = \biguplus _{W\,\, \propto \mathcal W}\mathcal Q\{W\}\to {\mathcal X}$ , which satisfies $F_\infty (W,X)=F_W(W,X)$ for each $W\,\, \propto \mathcal W$ and each $X\,\, \propto {\mathcal X}$ . We then find that

\begin{align*} (R\circ F_\infty )(W,X) & = \bigvee _{X'\,\, \propto {\mathcal X}}R(X',X)\cdot F_\infty (W,X')=\bigvee _{X'\,\, \propto {\mathcal X}} R(X',X)\cdot F_W(W,X')\\ & =(R\circ F_W)(W,X)=\bigwedge _{n\in {\mathbb N}}(R\circ F_n\circ J_W)(W,X)=\bigwedge _{n\in {\mathbb N}} (R\circ F_n)(W,X). \end{align*}

Therefore, $R\circ F_\infty =\bigwedge _{n\in {\mathbb N}}R\circ F_n$ , that is, $F_n\nearrow F_\infty$ .

Since left multiplication by the monotone function $F$ is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.4], it follows that the functions $F\circ K_n$ form a monotonically increasing sequence.

Proof. Let $\mathcal H$ be an atomic quantum set, and let $K_1\sqsubseteq K_2\sqsubseteq \cdots \subseteq K_\infty :{\mathcal H}\to {\mathcal X}$ satisfy $K_n\nearrow K_\infty$ , that is, $\bigwedge _{n\in {\mathbb N}}R\circ K_n=R\circ K_\infty .$ Since $F$ is a surjective order embedding by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition 2.7], we have $R=F^\dagger \circ S\circ F$ . Using the surjectivity of $F$ and [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6], we then find that

\begin{align*} \bigwedge _{n\in {\mathbb N}}S\circ F\circ K_n & =F\circ F^\dagger \circ \bigwedge _{n\in {\mathbb N}}S\circ F\circ K_n=F\circ \bigwedge _{n\in {\mathbb N}}F^\dagger \circ S\circ F\circ K_n\\ & =F\circ \bigwedge _{n\in {\mathbb N}}R\circ K_n=F\circ R\circ K_\infty =F\circ F^\dagger \circ S\circ F\circ K_\infty =S\circ F\circ K_\infty . \end{align*}

Therefore, $F\circ K_n\nearrow F\circ K_\infty$ .

Proof. Let $\mathcal H$ be an atomic quantum set and let $K_1\sqsubseteq K_2\sqsubseteq K_3\sqsubseteq \cdots$ be a monotonically ascending sequence in $\mathbf{qSet}({\mathcal H},{\mathcal X})$ . By definition of the order on $\mathbf{qSet}({\mathcal H},{\mathcal X})$ , $R\circ K_1\geq R\circ K_2\geq R\circ K_3\geq \cdots$ is a non-ascending sequence in $\mathbf{qRel}({\mathcal H},{\mathcal X})$ . Fix $X\in {\mathcal X}$ , and let $H$ be the unique atom of $\mathcal H$ . Then, $ (R\circ K_1)(H,X)\geq (R\circ K_2)(H,X)\geq \cdots$ is a non-ascending sequence in $L(H,X)$ , which is finite-dimensional. Thus, there is some $n_X\in {\mathbb N}$ such that $(R\circ K_n)(H,X)=(R\circ K_{n_X})(H,X)$ for all $n\geq n_X$ . Since $\mathcal X$ is finite, we know that $\{n_X:X\,\, \propto {\mathcal X}\}$ has a maximum element, which we call $m$ . Let $K_\infty =K_m$ . Then, for each $X\,\, \propto {\mathcal X}$ and each $n\geq m$ , we have $(R\circ K_n)(H,X)=(R\circ K_\infty )(H,X)$ , and therefore, $R\circ K_n=R\circ K_\infty$ . We conclude that $ \bigwedge _{n\in {\mathbb N}}R\circ K_n=R\circ K_\infty$ , that is, that $K_n\nearrow K_\infty$ . Therefore, $({\mathcal X},R)$ is a quantum cpo.

Now let $(\mathcal Y,S)$ be a quantum poset, and let $F:{\mathcal X}\to \mathcal Y$ be monotone. Let $K_1\sqsubseteq K_2\sqsubseteq \cdots \subseteq K_\infty :{\mathcal H}\to {\mathcal X}$ satisfy $K_n\nearrow K_\infty$ . We have already shown that there is some $m\in {\mathbb N}$ such that $K_n=K_m$ for each $n\geq m$ and that $K_\infty =K_m$ . Since $F$ is monotone, left multiplication with $F$ is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.4], hence $F\circ K_1\sqsubseteq F\circ K_2\sqsubseteq \cdots \sqsubseteq F\circ K_m=F\circ K_{m+1}=\cdots = F\circ K_\infty :{\mathcal H}\to {\mathcal X}$ is a monotonically ascending sequence of functions that stabilizes at $m$ , hence

\begin{equation*}S\circ F\circ K_1\geq S\circ F\circ K_2\geq \cdots \geq S\circ F\circ K_m=S\circ F\circ K_{m+1}=\cdots =S\circ F\circ K_\infty .\end{equation*}

Thus $\bigwedge _{n\in {\mathbb N}} S\circ F\circ K_n=S\circ F\circ K_m=S\circ F\circ K_\infty ,$ so $F\circ K_n\nearrow F\circ K_\infty$ . Therefore, $F$ is Scott continuous.

Let $\mathcal X$ be a quantum poset, let $\mathcal H$ be an atomic quantum set, and let $K_1\sqsubseteq K_2\sqsubseteq \cdots :{\mathcal H}\to {\mathcal X}$ be a monotonically ascending sequence with limit $K_\infty$ . Then,

\begin{equation*} \bigwedge _{n\in {\mathbb N}}R\circ I_{\mathcal X}\circ K_n=\bigwedge _{n\in {\mathbb N}}R\circ K_n=R\circ K_\infty =R\circ I_{\mathcal X}\circ K_\infty ,\end{equation*}

and hence $I_{\mathcal X}:{\mathcal X}\to {\mathcal X}$ is Scott continuous. Together with the previous lemma, this shows that quantum posets and Scott continuous maps form a subcategory $\mathbf{qPOS}_{\mathrm{SC}}$ of $\mathbf{qPOS}$ . By restricting the class of objects to quantum cpos, we obtain the category $\mathbf{qCPO}$ .

3.3 Enrichment over $\mathbf{CPO}$

We show that for quantum cpos $({\mathcal X},R)$ and $(\mathcal Y,S)$ , the subset $\mathbf{qCPO}(({\mathcal X},R), (\mathcal Y,S)) \subseteq \mathbf{qPOS}(({\mathcal X},R), (\mathcal Y,S))$ is a cpo in the induced order. Thus, $\mathbf{qCPO}$ is enriched over $\mathbf{CPO}$ .

Proof. By the definition of limit (Definition 3.1.1),

(2) \begin{equation} S\circ F_\infty =\bigwedge _{n\in {\mathbb N}} S\circ F_n. \end{equation}

Assume that each $F_n$ is monotone, i.e., that $S\circ F_n\geq F_n\circ R$ . Then,

\begin{align*} S\circ F_\infty & =\bigwedge _{n\in {\mathbb N}}S\circ S\circ F_n\geq \bigwedge _{n\in {\mathbb N}}S\circ F_n\circ R\geq \left (\bigwedge _{n\in {\mathbb N}}S\circ F_n\right )\circ R=S\circ F_\infty \circ R\geq F_\infty \circ R, \end{align*}

where the first equality follows by combining Equation (2) with the definition of an order; that definition is also used in the last inequality. We use [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma A.1] in the second inequality.

Now, assume that each $F_n$ is Scott continuous. Let $\mathcal H$ be an atomic quantum set, and let $K_1\sqsubseteq K_2\sqsubseteq \cdots :{\mathcal H}\to {\mathcal X}$ be a monotonically ascending sequence of functions with limit $K_\infty$ . Then, for each $n \in {\mathbb N}$ , we have that $F_n\circ K_m\nearrow F_n\circ K_\infty$ , that is,

(3) \begin{equation} \bigwedge _{m\in {\mathbb N}}S\circ F_n\circ K_m=S\circ F_n\circ K_\infty . \end{equation}

Using [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6] implicitly, we find that $S\circ F_\infty \circ K_\infty = \bigwedge _{n\in {\mathbb N}}S\circ F_n\circ K_\infty = \bigwedge _{n,m\in {\mathbb N}} S\circ F_n\circ K_m =\bigwedge _{m\in {\mathbb N}}S\circ F_\infty \circ K_m,$ where we use Equation (2) in the first and last equalities and Equation (3) in the middle equality.

Proof. Since right multiplication is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.2], it follows that $G_1\circ F\sqsubseteq G_2\circ F\sqsubseteq \cdots :{\mathcal X}\to \mathcal Z$ is a monotonically ascending sequence of functions. Then,

\begin{equation*} \bigwedge _{n\in {\mathbb N}}T\circ G_n\circ F=\left (\bigwedge _{n\in {\mathbb N}}T\circ G_n\right )\circ F=T\circ G_\infty \circ F,\end{equation*}

where the first equality is by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6], and the second equality since $G_n\nearrow G_\infty$ . We conclude that $G_n\circ F\nearrow G_\infty \circ F$ .

Proof. Fix $X \,\, \propto {\mathcal X}$ , and let $J_X:\mathcal Q\{X\}\to {\mathcal X}$ be the inclusion function. By Lemma 3.3.3, $F_n \circ J_X \nearrow F_\infty \circ J_X$ . Since $\mathcal Q\{X\}$ is atomic, it follows from the Scott continuity of $G$ that $G\circ F_n\circ J_X\nearrow G\circ F_\infty \circ J_X$ . We now reason that

\begin{align*} & \left (\bigwedge _{n\in {\mathbb N}}T\circ G\circ F_n\right ) \circ J_X = \bigwedge _{n\in {\mathbb N}}T\circ G\circ F_n \circ J_X = T \circ G \circ F_\infty \circ J_X, \end{align*}

where the second equality follows by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6]. We vary $X \,\, \propto {\mathcal X}$ to conclude that $\bigwedge _{n\in {\mathbb N}}T\circ G\circ F_n = T \circ G \circ F_\infty$ , i.e., that $G \circ F_n \nearrow G \circ F_\infty$ , as desired.

3.4 Completeness

We next show that the category $\mathbf{qCPO}$ is complete. For denotational semantics, completeness on its own has no special meaning. Later, we use completeness of $\mathbf{qCPO}$ to show that $\mathbf{qCPO}$ is also cocomplete, which in turn will be used to construct a category of quantum cpos that is $\mathbf{CPO}$ -algebraically compact, a property that is fundamental for modeling recursive types. Eventually, we will show that $\mathbf{qCPO}$ is symmetric monoidal closed. In combination with completeness and cocompleteness, this implies that $\mathbf{qCPO}$ is a cosmos, which is a category that has suitable properties for enriched category theory.

Proof. Consider a diagram of shape $A$ , consisting of objects $({\mathcal X}_\alpha ,R_\alpha )\in \mathbf{qCPO}$ and with Scott continuous maps $F_f:{\mathcal X}_\alpha \to {\mathcal X}_\beta$ for each morphism $f:\alpha \to \beta$ in $A$ . We can regard this as a diagram in $\mathbf{qPOS}$ . The latter category is complete by Theorem 5.5 Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), which also states that the limit of the diagram in $\mathbf{qPOS}$ is given by $({\mathcal X},R)$ , where $\mathcal X$ is the limit of the diagram $({\mathcal X}_\alpha )_{\alpha \in A}$ in $\mathbf{qSet}$ , and where $R=\bigwedge _{\alpha \in A}F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha$ , where $F_\alpha :{\mathcal X}\to {\mathcal X}_\alpha$ are the limiting functions of the diagram in $\mathbf{qSet}$ . Theorem 5.5 Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022) also assures that these limiting functions $F_\alpha$ are monotone, so they are also limiting functions in $\mathbf{qPOS}$ .

We first show that $({\mathcal X},R)$ is a quantum cpo. For a fixed atomic quantum set $\mathcal H$ let $K_1\sqsubseteq K_2$ $\sqsubseteq \ldots :{\mathcal H}\to {\mathcal X}$ be a monotonically ascending sequence of functions. Since left multiplication is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.4], we find for each $\alpha \in A$ that $F_\alpha \circ K_1\sqsubseteq F_\alpha \circ K_2\sqsubseteq \ldots :{\mathcal H}\to {\mathcal X}_\alpha$ is a monotonically ascending sequence of functions, too.

We first claim that if there exists a function $K_\infty :{\mathcal H}\to {\mathcal X}$ such that

(4) \begin{equation} F_\alpha \circ K_n\nearrow F_\alpha \circ K_\infty \end{equation}

for each $\alpha \in A$ , then $K_n\nearrow K_\infty$ . Equation (4) means that $ R_\alpha \circ F_\alpha \circ K_\infty =\bigwedge _{n\in {\mathbb N}}R_\alpha \circ F_\alpha \circ K_n$ ; hence, we find that

\begin{align*} R\circ K_\infty & = \left (\bigwedge _{\alpha \in A}F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha \right )\circ K_\infty =\bigwedge _{\alpha \in A}F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha \circ K_\infty \\ & = \bigwedge _{\alpha \in A}F_\alpha ^\dagger \circ \bigwedge _{n\in {\mathbb N}}R_\alpha \circ F_\alpha \circ K_n=\bigwedge _{\alpha \in A} \bigwedge _{n\in {\mathbb N}}F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha \circ K_n\\ & = \bigwedge _{n\in {\mathbb N}}\bigwedge _{\alpha \in A} F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha \circ K_n= \bigwedge _{n\in {\mathbb N}}\left (\bigwedge _{\alpha \in A} F_\alpha ^\dagger \circ R_\alpha \circ F_\alpha \right )\circ K_n = \bigwedge _{n\in {\mathbb N}}R\circ K_n, \end{align*}

where we use [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6] in the second, the third and the penultimate equalities. Therefore, $K_n\nearrow K_\infty$ , as claimed.

Next we construct a function $K_\infty$ that satisfies (4). Since $({\mathcal X}_\alpha ,R_\alpha )$ is by assumption a quantum cpo for each $\alpha \in A$ , there exists a function $G_\alpha :{\mathcal H}\to {\mathcal X}_\alpha$ such that $F_\alpha \circ K_n\nearrow G_\alpha$ . Consider a morphism $f:\alpha \to \beta$ in $A$ . By the definition of $F_\alpha$ and $F_\beta$ being limiting maps, we have $ F_f\circ F_\alpha =F_\beta .$ By the Scott continuity of $F_f$ , we have $ F_\beta \circ K_n=F_f\circ F_\alpha \circ K_n\nearrow F_f\circ G_\alpha .$ It now follows from Lemma 3.1.2 that $G_\beta =F_{f}\circ G_\alpha$ . Thus, the functions $G_\alpha :{\mathcal H}\to {\mathcal X}_\alpha$ form a cone, and since the functions $F_\alpha :{\mathcal X}\to {\mathcal X}_\alpha$ form the limiting cone, there must be a monotone function $K_\infty :{\mathcal H}\to {\mathcal X}$ such that $G_\alpha =F_\alpha \circ K_\infty$ . Since $F_\alpha \circ K_n\nearrow G_\alpha$ , it follows that Equation (4) holds for each $\alpha \in A$ , hence we conclude that $K_n\nearrow K_\infty$ . Therefore, $({\mathcal X},R)$ is a quantum cpo.

We have to show that the limiting functions $F_\alpha$ are Scott continuous. Let $\mathcal H$ be an atomic quantum set, and let $K_1\sqsubseteq K_2\sqsubseteq \cdots :{\mathcal H}\to {\mathcal X}$ be a monotonically ascending sequence. Then, $K_n\nearrow K_\infty '$ for some $K'_\infty :{\mathcal H}\to {\mathcal X}$ since $\mathcal X$ is a quantum cpo. Now, as before, construct some $K_\infty$ such that (4) holds for each $\alpha \in A$ , and note that $K_n\nearrow K_\infty$ . Then Lemma 3.1.2 implies $K_\infty =K_\infty '$ . Thus, Equation (4) expresses that $F_\alpha$ is Scott continuous. We have shown that the vertex $({\mathcal X},R)$ of the limiting cone is a quantum cpo and that the limiting functions $F_\alpha$ are all Scott continuous; hence, this cone is entirely in $\mathbf{qCPO}$ . It remains to show that it is also limiting in $\mathbf{qCPO}$ .

Let $(\mathcal Y,S)$ be a quantum poset, and let $C_\alpha :\mathcal Y\to {\mathcal X}_\alpha$ be Scott continuous maps that form a cone. Since $({\mathcal X},R)$ is the limit of the $({\mathcal X}_\alpha ,R_\alpha )$ in $\mathbf{qPOS}$ , there exists a unique monotone map $M:\mathcal Y\to {\mathcal X}$ such that $F_\alpha \circ M=C_\alpha$ for each $\alpha \in A$ . We claim that $M$ is Scott continuous: Indeed, let $\mathcal H$ be an atomic quantum set, let $E_1\sqsubseteq E_2\sqsubseteq \cdots :{\mathcal H}\to \mathcal Y$ be a monotonically ascending sequence of functions, and assume that there exists a function $E_\infty :{\mathcal H}\to \mathcal Y$ such that $E_n\nearrow E_\infty$ . Note that $E_\infty$ exists if we assume that $(\mathcal Y,S)$ is a quantum cpo. Let $K_n=M\circ E_n$ for each $n\in \{1,2,\ldots ,\infty \}$ . Since $M$ is monotone, left multiplication with $M$ is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.4], hence the sequence $K_1\sqsubseteq K_2\sqsubseteq \cdots$ is also monotonically ascending. Since the $C_\alpha$ are Scott continuous, we have $C_\alpha \circ E_n\nearrow C_\alpha \circ E_\infty$ . So, for each $\alpha \in A$ we obtain

\begin{equation*} F_\alpha \circ K_n=F_\alpha \circ M\circ E_n=C_\alpha \circ E_n\nearrow C_\alpha \circ E_\infty =F_\alpha \circ M\circ E_\infty =F_\alpha \circ K_\infty ,\end{equation*}

which shows the $K_n$ satisfy condition (4) for each $\alpha \in A$ ; i.e., we obtain $K_n\nearrow K_\infty$ . But this exactly expresses that $M\circ E_n\nearrow M\circ E_\infty$ , so $M$ is Scott continuous. If $\mathcal Y$ is a quantum cpo, it follows that the Scott continuous functions $F_\alpha :{\mathcal X}\to {\mathcal X}_\alpha$ form a limiting cone in $\mathbf{qCPO}$ , so $({\mathcal X},R)$ is the limit of the diagram.

Choosing $(\mathcal Y,S)$ to be a quantum cpo shows that $\mathbf{qCPO}$ is complete. Since we allowed $(\mathcal Y,S)$ to be an arbitrary quantum poset, it follows that the embedding of $\mathbf{qCPO}$ into $\mathbf{qPOS}_{\mathrm{SC}}$ preserves all limits.

4.1 Coproducts

The main technical challenge of showing the existence of coproducts in $\mathbf{qCPO}$ is the case that a monotone ascending sequence $K_1 \sqsubseteq K_2 \sqsubseteq \cdots \colon {\mathcal H}\to {\mathcal X}_i\uplus {\mathcal X}_2$ of functions from an atomic quantum set to a coproduct of quantum posets - intuitively a family of monotone ascending sequences in ${\mathcal X}_1 \uplus {\mathcal X}_2$ indexed by $\mathcal H$ - have non-zero range in both summands.

We briefly recall the construction of coproducts of quantum sets and of quantum posets. The coproduct $\biguplus _{\alpha \in A}{\mathcal X}_\alpha$ of a family $({\mathcal X}_\alpha )_{\alpha \in A}$ of quantum sets coincides in $\mathbf{qSet}$ and $\mathbf{qRel}$ and is defined as follows. If all quantum sets in the family are pairwise disjoint, their coproduct is simply the union $\bigcup _{\alpha \in A}{\mathcal X}_\alpha :=\mathcal Q\bigcup _{\alpha \in A}\mathrm{At}({\mathcal X}_\alpha )$ , otherwise the coproduct is given by the disjoint union $\bigcup _{\alpha \in A}{\mathcal X}_\alpha \times `\{\alpha \}$ . So, just as for ordinary sets, for most results involving coproducts we can assume that the family over which the coproduct is formed consists of pairwise disjoint quantum sets. For each $\beta \in A$ , we denote the canonical injection ${\mathcal X}_\beta \to \biguplus _{\alpha \in A}{\mathcal X}_\alpha$ by $J_\beta$ .

If each quantum set in the family $({\mathcal X}_\alpha )_{\alpha \in A}$ is equipped with an order $R_\alpha$ , then the coproduct is $\biguplus _{\alpha \in A}({\mathcal X}_\alpha ,R_\alpha ):=({\mathcal X},R)$ , where ${\mathcal X}=\biguplus _{\alpha \in A}{\mathcal X}_\alpha$ in $\mathbf{qSet}$ , and $R=\biguplus _{\alpha \in A}R_\alpha$ in $\mathbf{qRel}$ , that is, the unique relation on $\mathcal X$ such that $R\circ J_\alpha =R_\alpha \circ J_\alpha$ for each $\alpha \in A$ [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition 6.2].

We have a canonical one-to-one correspondence between the decompositions of $\mathcal H$ and orthogonal direct sum decompositions $H = \bigoplus _{\alpha \in A} X_\alpha$ . The nonzero subspaces $X_\alpha$ are pair-wise distinct, so $\bigcup _{\alpha \in A} \mathcal Q\{X_\alpha \} = \biguplus _{\alpha \in A} \mathcal Q\{X_\alpha \}$ .

Proof. For each $\alpha \in A$ , let $\mathrm{proj}_{X_\alpha }:H\to H$ be the projection operator whose range is the subspace $X_\alpha$ of $H$ . Clearly $\mathrm{proj}_{X_\alpha }\mathrm{proj}_{X_\beta }^\dagger = 0$ for distinct $\alpha , \beta \in A$ . We now calculate that for all $\alpha , \beta \in A$ ,

\begin{align*} (D\circ D^\dagger )(X_\beta ,{X_\alpha }) & = \bigvee _{H\,\, \propto {\mathcal H}}D(H,X_\alpha )D^\dagger ({X_\beta },H)=D(H, X_\alpha )D^\dagger (X_\beta , H) =D(H, X_\alpha )D(H,X_\beta )^\dagger \\ & = {\mathbb C} \mathrm{proj}_{X_\alpha } \mathrm{proj}_{X_\beta }^\dagger =\delta _{\alpha ,\beta }{\mathbb C} \mathrm{proj}_{X_\alpha }\mathrm{proj}_{X_\alpha }^\dagger = \delta _{\alpha ,\beta }{\mathbb C} 1_{X_\alpha }=I_{{\mathcal X}}(X_\beta ,X_\alpha ). \end{align*}

In general, a decomposition $D:{\mathcal H}\to {\mathcal X}$ of $\mathcal H$ is not injective. In fact, $D$ is injective precisely when it is bijective, in which case $\mathcal X$ also is atomic. In this case, it necessarily follows that the unique atoms of $\mathcal X$ and $\mathcal H$ have the same dimension.

Proof. Fix $\alpha \in A$ , and let $J_\alpha$ be the inclusion of $\mathcal Y_\alpha$ into the disjoint union $\biguplus _{\alpha '} \mathcal Y_{\alpha '}$ . We have that $J_\alpha ^\dagger \circ F \circ F^\dagger \circ J_\alpha \leq J_\alpha ^\dagger \circ J_\alpha = I_{\mathcal Y_\alpha }$ , and thus, $J_\alpha ^\dagger \circ F$ is a partial function. Applying [Kornell (Reference Kornell2020), Proposition C.6], we obtain a subspace $X_\alpha \leq H$ and a function $F_\alpha \colon \mathcal Q\{X_\alpha \} \to \mathcal Y_\alpha$ such that $F_\alpha \circ K_\alpha =J_\alpha ^\dagger \circ F$ , where $K_\alpha$ is the partial function from $\mathcal H$ to ${\mathcal X}_\alpha = \mathcal Q\{X_\alpha \}$ defined by $K_\alpha (H, X_\alpha ) = {\mathbb C} \cdot \mathrm{proj}_{X_\alpha }^\dagger$ .

If we vary $\alpha \in A$ , we obtain families $\{X_\alpha \,\, \propto {\mathcal X}_\alpha \}_{\alpha \in A}$ , $\{K_\alpha \}_{\alpha \in A}$ and $\{F_\alpha \}_{\alpha \in A}$ . Now, let $D = $ $[K_\alpha ^\dagger : \alpha \in A]^\dagger$ , where the square-bracket notation refers to the coproduct in $\mathbf{qRel}$ . We compute that

\begin{align*} \left (\biguplus _\alpha F_\alpha \right ) \circ D &= \left (\biguplus _\alpha F_\alpha \right ) \circ [K_\alpha ^\dagger : \alpha \in A]^\dagger = \left ( [K_\alpha ^\dagger : \alpha \in A] \circ \left (\biguplus _\alpha F_\alpha ^\dagger \right )\right )^\dagger \\ & = [K_\alpha ^\dagger \circ F_\alpha ^\dagger : \alpha \in A]^\dagger = [(F_\alpha \circ K_\alpha )^\dagger : \alpha \in A]^\dagger = [(J_\alpha ^\dagger \circ F)^\dagger : \alpha \in A]^\dagger \\ & = [F^\dagger \circ J_\alpha : \alpha \in A]^\dagger = (F^\dagger )^\dagger = F. \end{align*}

Thus, $(\biguplus _\alpha F_\alpha ) \circ D = F$ .

To show that $D$ is a decomposition, we show that it is a function. For distinct $\alpha , \beta \in A$ , we apply [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1(a)] to find that

\begin{align*} K_\alpha \circ K_\beta ^\dagger & \leq F_\alpha ^\dagger \circ F_\alpha \circ K_\alpha \circ K_\beta ^\dagger \circ F_\beta ^\dagger \circ F_\beta = F_\alpha ^\dagger \circ J_\alpha ^\dagger \circ F \circ F^\dagger \circ J_\beta \circ F_\beta \\ & \leq F_\alpha ^\dagger \circ J_\alpha ^\dagger \circ J_\beta \circ F_\beta = F_\alpha ^\dagger \circ \bot \circ F_\beta = \bot , \end{align*}

so $K_\alpha \circ K_\beta ^\dagger = \bot$ . It follows that

\begin{align*} D \circ D^\dagger & = \bigvee _{\alpha , \beta \in A} J_\alpha \circ J_\alpha ^\dagger \circ D \circ D^\dagger \circ J_\beta \circ J_\beta ^\dagger = \bigvee _{\alpha , \beta \in A} J_\alpha \circ (D^\dagger \circ J_\beta )^\dagger \circ ( D^\dagger \circ J_\beta ) \circ J_\beta ^\dagger \\ & = \bigvee _{\alpha , \beta \in A} J_\alpha \circ K_\alpha \circ K_\beta ^\dagger \circ J_\beta ^\dagger = \bigvee _{\alpha \in A} J_\alpha \circ K_\alpha \circ K_\alpha ^\dagger \circ J_\alpha ^\dagger \leq \bigvee _{\alpha \in A} J_\alpha \circ J_\alpha ^\dagger = I_{(\biguplus _\alpha \mathcal Y_\alpha )}. \end{align*}

Thus, $D$ is a partial function.

The equation $(\biguplus _\alpha F_\alpha ) \circ D = F$ in the category $\mathbf{qPar}$ of quantum sets and partial functions is equivalent to the equation $D^\star \circ (\biguplus _\alpha F_\alpha )^\star = F^\star$ in the category $\mathbf{WStar}_{\mathrm{HA}}$ of hereditarily atomic von Neumann algebras and normal $*$ -homomorphisms (cf. Theorem 2.4.1). The normal $*$ -homomorphism $F^\star$ is unital, because $F$ is a function. Then, $D^\star \circ (\biguplus _\alpha F_\alpha )^\star = F^\star$ implies the normal $*$ -homomorphism $D^\star$ is also unital, so $D$ is also a function.

It follows that $D$ is a decomposition: By a short direct computation, we have that $D(H, X_\alpha ) = K_\alpha (H, X_\alpha ) = {\mathbb C} \cdot \mathrm{proj}_{X_\alpha }^\dagger$ for all $\alpha \in A$ . For all distinct $\alpha , \beta \in A$ , if $X_\alpha$ and $X_\beta$ are both nonzero, then they must be orthogonal anyway, because

\begin{equation*}{\mathbb C} \cdot \mathrm{proj}_{X_\alpha } \cdot \mathrm{proj}_{X_\beta }^\dagger = D(H,X_\alpha )^\dagger \cdot D(H, X_\beta ) \leq I_{(\biguplus _{\alpha \in A} {\mathcal X}_\alpha )}(X_\alpha , X_\beta ) = 0.\end{equation*}

Similarly, $1_H \in I_{\mathcal H}(H, H) = \bigvee _{\alpha \in A} D(H, X_\alpha ) \cdot D(H, X_\alpha ) = \bigvee _{\alpha \in A} {\mathbb C} \cdot \mathrm{proj}_{X_\alpha }^\dagger \cdot \mathrm{proj}_{X_\alpha }$ , so $1_H = $ $\sum _{\alpha \in A}\mathrm{proj}_{X_\alpha }^\dagger \cdot \mathrm{proj}_{X_\alpha }$ , implying that $\bigvee _{\alpha \in A} X_\alpha = H$ .

Lemma 4.1.4. Let $\{(\mathcal Y_\alpha ,S_\alpha )\}_{\alpha \in A}$ be an indexed family of quantum posets, and let $\mathcal Y=\biguplus _{\alpha \in A}\mathcal Y_\alpha$ be the coproduct of $\{\mathcal Y_\alpha \}_{\alpha \in A}$ in $\mathbf{qSet}$ . Let $\mathcal H$ be an atomic quantum set, let $S$ be an order on $\mathcal Y$ , and let $F^1,F^2:{\mathcal H}\to \mathcal Y$ be functions such that $F^1\sqsubseteq F^2$ . For each $i \in \{1,2\}$ , let $D_i:{\mathcal H}\to \biguplus _{\alpha \in A}\mathcal Q\{X^i_\alpha \}$ be a decomposition of $\mathcal H$ and let $\{F^i_\alpha :\mathcal Q\{X^i_\alpha \}\to \mathcal Y_\alpha \}_{\alpha \in A}$ be an indexed family of functions such that

\begin{equation*} F^i=\left (\biguplus _{\alpha \in A}F^i_\alpha \right )\circ D_i,\end{equation*}

as in Lemma 4.1.3 .

Let $\beta \in A$ . If $J_\beta \circ S_\beta =S\circ J_\beta ,$ where $J_\beta \colon \mathcal Y_\beta \to \mathcal Y$ is the canonical injection, then

1. (1) $X_\beta ^1\perp X_\gamma ^2$ for each $\gamma \in A$ distinct from $\beta$ and

2. (2) $X^1_\beta = X^2_\beta$ implies $F_\beta ^1\sqsubseteq F_{\beta }^2$ .

Proof. The condition $F^1\sqsubseteq F^2$ is equivalent to $F^2\circ (F^1)^\dagger \leq S$ . Hence, we obtain

\begin{equation*} \left (\biguplus _{\alpha \in A} F^2_\alpha \right )\circ D_2\circ D_1^\dagger \circ \left (\biguplus _{\alpha \in A}F^1_\alpha \right )^\dagger \leq S, \end{equation*}

and hence

\begin{align*} D_2\circ D_1^\dagger & \leq \left (\biguplus _{\alpha \in A} F_\alpha ^2\right )^\dagger \circ \left (\biguplus _{\alpha \in A} F_\alpha ^2\right )\circ D_2\circ D_1^\dagger \circ \left (\biguplus _{\alpha \in A}F^1_\alpha \right )^\dagger \circ \left (\biguplus _{\alpha \in A}F^1_\alpha \right )\\ & \leq \left (\biguplus _{\alpha \in A} F_\alpha ^2\right )^\dagger \circ S \circ \left (\biguplus _{\alpha \in A}F^1_\alpha \right ). \end{align*}

For $i \in \{1,2\}$ and $\beta \in A$ , let $J_\beta ^i:\mathcal Q\{X_\beta ^i\}\hookrightarrow \biguplus _{\alpha \in A}\mathcal Q\{X^i_\alpha \}$ be inclusion. Let $\beta ,\gamma \in A$ be distinct. Then,

\begin{align*} (J_\gamma ^2)^\dagger \circ D_2\circ D_1^\dagger \circ J_\beta ^1 & \leq (J_\gamma ^2)^\dagger \circ \left (\biguplus _{\alpha \in A} F_\alpha ^2\right )^\dagger \circ S \circ \left (\biguplus _{\alpha \in A}F^1_\alpha \right )\circ J^1_\beta =(F_\gamma ^2)^\dagger \circ J_\gamma ^\dagger \circ S\circ J_\beta \circ F_\beta ^1\\ &= (F_\gamma ^2)^\dagger \circ J_\gamma ^\dagger \circ J_\beta \circ S_\beta \circ F_\beta ^1= \bot , \end{align*}

since $J_\gamma ^\dagger \circ J_\beta =\bot$ by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1(a)]. Applying [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma A.5], we now infer that $((J^1_\beta )^\dagger \circ D_1)(H,X_\beta )=D_1(H,X_\beta )={\mathbb C} \cdot \mathrm{proj}_{X^1_{\beta }}$ and similarly that $((J_\gamma ^2)^\dagger \circ D_2)(H,X^2_\gamma )={\mathbb C}\cdot \mathrm{proj}_{X^2_\gamma }.$ Thus, for distinct $\beta ,\gamma \in A$ , if $X_\beta ^1$ and $X_\gamma ^2$ are nonzero, we find that

\begin{align*} 0 & = \bot (X^1_\beta , X^2_\gamma ) = [(J_\gamma ^2)^\dagger \circ D_2\circ D_1^\dagger \circ J^1_\beta ](X_\beta ^1,X^2_\gamma )=[((J_\gamma ^2)^\dagger \circ D_2)\circ ( (J^1_\beta )^\dagger \circ D_1)^\dagger ](X^1_\beta ,X^2_\gamma )\\ & = \bigvee _{H\,\, \propto {\mathcal H}}((J_\gamma ^2)^\dagger \circ D_2)(H,X_\gamma ^2)\cdot ((J_\beta ^1)^\dagger \circ D_1)^\dagger (X^1_\beta ,H)=((J_\gamma ^2)^\dagger \circ D_2)(H,X_\gamma ^2)\cdot ((J^1_\beta )^\dagger \circ D_1)(H,X^1_\beta )^\dagger \\ & = ({\mathbb C} \cdot \mathrm{proj}_{X_\gamma ^2}) \cdot ({\mathbb C} \cdot \mathrm{proj}_{X_\beta ^1})^\dagger = {\mathbb C} \cdot \mathrm{proj}_{X_\gamma ^2} \cdot \mathrm{proj}_{X_\beta ^1}^\dagger , \end{align*}

which implies that $X_\gamma ^2$ and $X_\beta ^1$ are orthogonal subspaces of $H$ .

For (2), assume $X_1^\beta =X_2^\beta$ . For each $i \in \{1,2\}$ , the decomposition $D_i$ is surjective by Lemma 4.1.2, and thus, we have that

\begin{align*} S_\beta \circ F_\beta ^i & = J_\beta ^\dagger \circ J_\beta \circ S_\beta \circ F_\beta ^i=J_\beta ^\dagger \circ S\circ J_\beta \circ F^i_\beta = J_\beta ^\dagger \circ S\circ \left (\biguplus _{\alpha \in A}F_\alpha ^i\right )\circ J_\beta ^i\\ & =J_\beta ^\dagger \circ S\circ \left (\biguplus _{\alpha \in A}F_\alpha ^i\right )\circ D_i \circ D_i^\dagger \circ J_\beta ^i =J_\beta ^\dagger \circ S\circ F^i\circ D_i^\dagger \circ J_\beta ^i. \end{align*}

As above, the partial functions $(J^1_\beta )^\dagger \circ D_1$ and $(J^2_\beta )^\dagger \circ D_1$ are completely determined by $X^1_\beta$ and $X^2_\beta$ , respectively, and hence we have that $(J^1_\beta )^\dagger \circ D_1 = (J^2_\beta )^\dagger \circ D_2$ . Recalling that $F^1\sqsubseteq F^2$ is equivalent to $S\circ F^2\leq S\circ F^1$ , we calculate

\begin{align*} S_\beta \circ F^2_\beta & = J_\beta \circ S\circ F^2\circ D_2^\dagger \circ J^2_\beta = J_\beta \circ S\circ F^2\circ D_1^\dagger \circ J^1_\beta \\ & \leq J_\beta \circ S\circ F^1\circ D_1^\dagger \circ J^1_\beta =S_\beta \circ F_\beta ^1. \end{align*}

Therefore, $F^1_\beta \sqsubseteq F^2_\beta$ , as claimed.

Proof. Let $\{(\mathcal Y_\alpha ,S_\alpha )\}_{\alpha \in A}$ be an indexed family of quantum cpos, and let $(\mathcal Y,S)$ be its coproduct $(\mathcal Y,S)$ in $\mathbf{qPOS}$ (cf. [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition 6.2]). Hence,

(5) \begin{equation} J_\alpha \circ S_\alpha =S\circ J_\alpha \end{equation}

for each $\alpha \in A$ . Let ${\mathcal H} = \mathcal Q\{H\}$ be an atomic quantum set, and let $K^1\sqsubseteq K^2\sqsubseteq \cdots :{\mathcal H}\to \mathcal Y$ be a monotonically ascending sequence of functions. By Lemma 4.1.3, for each $n\in {\mathbb N}$ , there exist a decomposition $D_n:{\mathcal H}\to \biguplus _{\alpha \in A}\mathcal Q\{X^n_\alpha \}$ and an indexed family $\{K^n_\alpha : \mathcal Q\{X^n_\alpha \} \to \mathcal Y_\alpha \}_{\alpha \in A}$ of functions such that $K^n=\left (\biguplus _{\alpha \in A}K_\alpha ^n\right )\circ D_n$ .

We show that the decompositions $D_n$ are all the same, as follows: Since Equation (5) holds for each $\alpha \in A$ , we can apply Lemma 4.1.4(1) to conclude that for each $n\in {\mathbb N}$ , we have $X^1_\alpha \perp X^{n}_\beta$ for each $\alpha \neq \beta$ in $A$ . Since $H=\bigoplus _{\alpha \in A}X_\alpha ^n$ by the definition of a decomposition, it follows that $X_\alpha ^1=X_{\alpha }^{n}$ for each $\alpha \in A$ . Hence $D_1=D_{n}$ for each $n\in {\mathbb N}$ . Let $D=D_1 = D_2 = \ldots$ , and for each $\alpha \in A$ , let $X_\alpha = X_\alpha ^1 = X_\alpha ^2 = \cdots$ .

We now apply Lemma 4.1.4(2) to conclude that $K_\alpha ^1\sqsubseteq K_\alpha ^2\sqsubseteq \cdots$ for each $\alpha \in A$ . If $X_\alpha$ is zero-dimensional, then $\mathcal Q\{X_\alpha \} = `\emptyset$ is initial in $\mathbf{qSet}$ ; hence $K^1_\alpha =K^n_\alpha$ for each $n\in {\mathbb N}$ . Taking $K^\infty _\alpha =K^1_\alpha$ , we automatically have $K_\alpha ^n\nearrow K_\alpha ^\infty$ . If $X_\alpha$ is not zero-dimensional, then the fact that $\mathcal Y_\alpha$ is a quantum cpo ensures the existence of a function $K^\infty _\alpha :{\mathcal X}_\alpha \to \mathcal Y_\alpha$ such that $K^n_\alpha \nearrow K^\infty _\alpha$ . In either case,

(6) \begin{equation} S_\alpha \circ K_\alpha ^\infty =\bigwedge _{n\in {\mathbb N}}S_\alpha \circ K_\alpha ^n. \end{equation}

Let $K^\infty =\left (\biguplus _{\alpha \in A}K_\alpha ^\infty \right )\circ D$ . We show that $K^n\nearrow K^\infty$ :

\begin{align*} S\circ K^\infty & = \left (\biguplus _{\alpha \in A}S_\alpha ^\infty \right )\circ \left (\biguplus _{\alpha \in A}K_\alpha ^\infty \right )\circ D =\left (\biguplus _{\alpha \in A}(S_\alpha \circ K_\alpha ^\infty )\right )\circ D=\left (\biguplus _{\alpha \in A}\left (\bigwedge _{n\in {\mathbb N}}(S_\alpha \circ K_\alpha ^n)\right )\right )\circ D\\ & =\left (\bigwedge _{n\in {\mathbb N}}\left (\biguplus _{\alpha \in A}(S_\alpha \circ K_\alpha ^n)\right )\right )\circ D=\bigwedge _{n\in {\mathbb N}}\left (\biguplus _{\alpha \in A}(S_\alpha \circ K_\alpha ^n)\circ D\right )\\ & =\bigwedge _{n\in {\mathbb N}}\left (\left (\biguplus _{\alpha \in A}S_\alpha \right )\circ \left (\biguplus _{\alpha \in A}K_\alpha ^n\right )\circ D\right )=\bigwedge _{n\in {\mathbb N}}S\circ K^n, \end{align*}

where the fourth equality follows from [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1], and the fifth equality follows from [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6]. Therefore, $\mathcal Y$ is a quantum cpo.

Next, we show that the canonical injections $J_\beta :\mathcal Y_\beta \to \mathcal Y$ are Scott continuous for each $\beta \in A$ . So, let $\mathcal H$ be an atomic quantum set, and let $E_1\sqsubseteq E_2\sqsubseteq \cdots :{\mathcal H}\to \mathcal Y_\beta$ be a monotonically ascending sequence of functions with limit $E_\infty$ . Let $K^i=J_\beta \circ E_i$ for each $i\in {\mathbb N}\cup \{\infty \}$ . Since the $J_\alpha$ are monotone by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition 6.2], left multiplication with $J_\alpha$ is monotone [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 4.4], and it follows that $K^1\sqsubseteq K^2\sqsubseteq \cdots \sqsubseteq$ is a monotonically ascending sequence. We need to show that $K^n\nearrow K^\infty$ . For each $\alpha \in A$ , we calculate that

\begin{align*} J_\alpha ^\dagger \circ \bigwedge _{n\in {\mathbb N}} S\circ K^n & =\bigwedge _{n\in {\mathbb N}}J_\alpha ^\dagger \circ S\circ K^n=\bigwedge _{n\in {\mathbb N}}J_\alpha ^\dagger \circ S\circ J_\beta \circ E_n=\bigwedge _{n\in {\mathbb N}}J_\alpha ^\dagger \circ J_\beta \circ S_\beta \circ E_n\\ & =\Delta _{\alpha ,\beta } \cdot \bigwedge _{n\in {\mathbb N}}S_\beta \circ E_n=\Delta _{\alpha ,\beta }\cdot (S_\beta \circ E_\infty ) =J_\alpha ^\dagger \circ J_\beta \circ S_\beta \circ E_\infty \\ &= J_\alpha ^\dagger \circ S\circ J_\beta \circ E_\infty =J_\alpha ^\dagger \circ S\circ K_\infty , \end{align*}

where we have used [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6] for the first and the penultimate equality, [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition 6.2] for the third equality, and [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1] for the fourth and sixth equalities. We use $E_n\nearrow E_\infty$ for the fifth equality. It now follows by [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1] that $\bigwedge _{n\in {\mathbb N}}S\circ K^n=S\circ K^\infty$ , i.e., that $K^n\nearrow K^\infty$ . Therefore, each of the canonical injections $J_\beta$ , for $\beta \in A$ , is Scott continuous.

Finally, let $(\mathcal Z,T)$ be a quantum cpo, and for each $\alpha \in A$ , let $F_\alpha :\mathcal Y_\alpha \to \mathcal Z$ be a Scott continuous function. We need to show that $[F_\alpha :\alpha \in A]:\mathcal Y\to \mathcal Z$ is Scott continuous, where the notation refers to the universal property of the coproduct. Hence, let $\mathcal H$ be an atomic quantum set and let $K^1\sqsubseteq K^2\sqsubseteq \cdots :{\mathcal H}\to \mathcal Y$ be a monotonically ascending sequence of functions with limit $K^\infty$ . As before, we have a decomposition $D:{\mathcal H}\to \biguplus _{\alpha \in A}\mathcal Q\{X_\alpha \}$ and an indexed family of functions $\{K^i_\alpha :\{X_\alpha \}\to \mathcal Y_\alpha \}_{\alpha \in A}$ such that $K^i=\left (\biguplus _{\alpha \in A}K^i_\alpha \right )\circ D$ , for each index $i \in {\mathbb N} \cup \{\infty \}$ , and $K^n_\alpha \nearrow K^\infty _\alpha$ . Since each function $F_\alpha$ is Scott continuous, we also have that $F_\alpha \circ K_\alpha ^n\nearrow F_\alpha \circ K_\alpha ^\infty$ . We show that $\left [F_\alpha :\alpha \in A\right ]\circ K^n\nearrow \left [F_\alpha :\alpha \in A\right ]\circ K^\infty$ :

\begin{align*} \bigwedge _{n\in {\mathbb N}}T\circ [F_\alpha :\ &\alpha \in A]\circ K^n = \bigwedge _{n\in {\mathbb N}}T\circ [F_\alpha :\alpha \in A]\circ \left (\biguplus _{\alpha \in A}K^n_\alpha \right )\circ D \\ & = \bigwedge _{n\in {\mathbb N}}[T\circ F_\alpha \circ K_\alpha ^n:\alpha \in A]\circ D = \left (\bigwedge _{n\in {\mathbb N}} [T\circ F_\alpha \circ K_\alpha ^n:\alpha \in A]\right )\circ D\\ & = \left [\bigwedge _{n\in {\mathbb N}}T\circ F_\alpha \circ K_\alpha ^n:\alpha \in A\right ]\circ D = \left [T\circ F_\alpha \circ K_\alpha ^\infty :\alpha \in A\right ]\circ D\\ & = T\circ \left [F_\alpha \in A\right ]\circ \left (\biguplus _{\alpha \in A}K_\alpha ^\infty \right )\circ D = T\circ [F_\alpha :\alpha \in A]\circ K^\infty , \end{align*}

where the third equality follows from [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Proposition A.6], the fourth from [Kornell et al. (Reference Kornell, Lindenhovius and Mislove2022), Lemma 6.1], and the fifth equality is $F_\alpha \circ K_\alpha ^n\nearrow F_\alpha \circ K_\alpha ^\infty$ . Therefore, the monotone function $[F_\alpha :\alpha \in A]:\mathcal Y\to \mathcal Z$ , whose existence and uniqueness is guaranteed by the universal property of the coproduct in $\mathbf{qPOS}$ , is Scott continuous. All together, we find that $(\mathcal Y,S)$ is also its coproduct of the family $\{(\mathcal Y_\alpha , S_\alpha )\}_{\alpha \in A}$ in $\mathbf{qCPO}$ .