Effectus of Quantum Probability on Relational Structures

A graph $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)\subseteq V(G)\times V(G)$. By definition $E(G)$ is a binary relation on the vertex set $V(G)$. We write $v\sim v^{\prime}$ to denote a pair of adjacent vertices $v,v^{\prime}\in V(G)$, i.e. a pair $(v,v^{\prime})\in V(G)\times V(G)$ in the edge relation $(v,v^{\prime})\in E(G)$. A graph homomorphism $G\to H$ is given by a function $f\colon V(G)\to V(H)$ between vertices preserving edges:

Consider the following game involving a given pair of graphs $G$ and $H$, played by Alice and Bob playing against a Verifier. Their goal is to establish the existence of a graph homomorphism from $G$ to $H$. The game is ‘non-local’ which means that Alice and Bob are not allowed to communicate during the game, however they are allowed to agree on a strategy before the game has started. In each round Verifier sends to Alice and Bob vertices $v_{1},v_{2}\in V(G)$, respectively; in response they produce outputs $w_{1},w_{2}\in V(H)$. They win the round if the following conditions hold:

If there is indeed a graph homomorphism $G\to H$, then Alice and Bob can win any round of the game described above by using such homomorphism as strategy for responding accordingly. Conversely, they can win any round with certainty only when there is a graph homomorphism $G\to H$. A strategy for Alice and Bob in which they win with probability 1 is called a perfect strategy. Hence, the existence of a perfect strategy is equivalent to the existence of a graph homomorphism.

In cases where no classical homomorphism exists, one can use quantum resources in the form of a maximally entangled bipartite state, where Alice and Bob can each perform measurements on their part, to construct perfect strategies. These strategies are called quantum because they use quantum resources.

We write $M_{d}(\mathbb{C})$ for the set of all $d\times d$ matrices with complex entries ($d\geq 1$), and $\mathbbm{1}\in M_{d}(\mathbb{C})$ for the $d\times d$ identity matrix. Let $E\in M_{n}(\mathbb{C})$ and $F\in M_{m}(\mathbb{C})$ be two complex square matrices, $n,m\geq 1$. Their tensor product is the matrix defined as $E\otimes F:=(e_{ij}F)\in M_{nm}(\mathbb{C})$ if $E=(e_{ij})$ with $i,j=1,\dots,n$.

Observe that the definition of quantum perfect strategies forgets the two-person aspect of the game and shared state, leaving a matrix-valued relation as the witness for existence of a quantum perfect strategy. Recall perfect strategies are in bijection with graph homomorphisms. This gives rise to the notion of quantum graph homomorphism. This concept was introduced in [14], as a generalisation of the notion of quantum chromatic number from [4]. Analogous results for constraint systems are proved in [7, 12, 2, 3].

An important further step is taken in [14]: a construction $G\mapsto\mathsf{M}G$ on graphs is introduced, such that the existence of a quantum graph homomorphism from $G$ to $H$ is equivalent to the existence of a graph homomorphism of type $G\to\mathsf{M}H$. This construction is called the measurement graph, and it turns out to be a graded monad on the category of graphs. The Kleisli morphisms of this monad are exactly the quantum homorphism between graphs of [14, 12, 3]. One can show equivalence between these three different notions: quantum homomorphisms, quantum perfect strategies, and certain kind of (classical) homomorphisms between graphs [2].

Monads are used in formal semantics of functional and probabilistic programming languages. Building on the work of Giry [8], and inspired by algebraic methods in program semantics, the study of various ‘probability’ monads has evolved and became part of a new branch of categorical logic called effectus theory. The main goal of effectus theory is to describe the salient aspects of quantum computation and logic using the language of category theory. This description includes probabilistic and classical logic and computation as special cases [6, 11]. Quantum perfect strategies form a category. In this paper we shall see that the category of quantum perfect strategies, or quantum graph homomorphisms, is an effectus (see Theorem 4.1). However, specialisation of Theorem 4.1 to the case where the underlying category is the category simple undirected graphs (which is the context used in [14, 12]) is impossible as this category does not have a terminal object. This is why we introduce the quantum monad in the context of the more general notion of relational structures rather than that of simple graphs.

Disjoint union is the coproduct in the category $\mathbf{Set}$ of sets and functions. The disjoint union of two sets $X,Y$ is the defined to be the set

One can define two functions $X\overset{\kappa_{1}}{\rightarrow}X+Y\overset{\kappa_{2}}{\leftarrow}Y$, as $\kappa_{1}(x):=(x,1)$ and $\kappa_{2}:=(y,2)$, for all $x\in X$ and $y\in Y$, respectively. The functions $\kappa_{1},\kappa_{2}$ are called coprojections. For any pair of functions $X\overset{p}{\rightarrow}Z\overset{q}{\leftarrow}Y$, the function $[p,q]\colon X+Y\to Z$ called cotupling is given by:

If $f\colon A\to B,~{}g\colon X\to Y$ are functions, then the function $f+g\colon A+X\to B+Y$ is defined as

The empty set $0:=\emptyset$ is the initial object of $\mathbf{Set}$, and any choice of a singleton set $1:=\{\ast\}$ is terminal in $\mathbf{Set}$. The unique function $!_{X}\colon X\to 1$ is given by $x\mapsto\ast$ for each $x\in X$. Hence, the category $\mathbf{Set}$ has finite coproducts $(+,0)$ and a terminal object $1$.

The category $\mathbf{Set}$ is the effectus used for modelling classical (deterministic, Boolean) computations. The following result is well-known (see, e.g. [11, Example 4.7]).

There are many examples of categories that are effectuses (for more, see [6]):

• •

  Topological spaces with continuous maps between them.

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  Rings (with multiplicative identity) and ring homomorphisms.

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  Measurable spaces with measurable functions.

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  $C^{\ast}$-algebras with completely positive unital maps.

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  Extensive categories: $\mathbf{B}$ is extensive if it has finite coproducts and

  $$\mathbf{B}/X\times\mathbf{B}/Y\simeq\mathbf{B}/(X+Y)$$

  for all $X,Y$ objects of $\mathbf{B}$. (Every topos is extensive.)

For any set $X$ and any point $x\in X$, let $\mathbf{1}_{x}\colon X\to\{0,1\}$ denote the indicator function at $x$:

A convex combination of elements of the set $X$ is an expression:

with $n\geq 1$, $x_{i}\in X$, $\lambda_{i}\in[0,1]$, and $\lambda_{1}+\cdots+\lambda_{n}=1$. Let $D(X)$ be the set of all convex combinations of elements of $X$. The elements of the set $D(X)$ are called states or distributions on $X$. The indicator function is a distribution $\mathbf{1}_{x}\in D(X)$, for every $x\in X$. A distribution $\sum_{i}p_{i}\mathbf{1}_{x_{i}}\in D(X)$ can be represented by a function $p\colon X\to[0,1]$ with finitely many non-zero values, satisfying:

Functions like this are called discrete probability measures. If $\mathrm{range}(p)=\{x_{1},\dots,x_{n}\},~{}n\geq 1$, then the assignment $p(x_{i})\mapsto p_{i}$ gives the bijective correspondence between these two equivalent representations of distributions, i.e. as convex combinations or as discrete probability measures.

Distributions on $X$ can be pushforwarded along a function $f\colon X\to Y$ to get distributions on $Y$:

That is, we have a function $D(f)\colon D(X)\to D(Y)$ defined for any $p\in D(X)$ as:

For every set $X$, the function $\eta_{X}\colon X\to D(X)$ is defined as:

The function $\mu_{X}\colon D^{2}(X)\to D(X)$ is given by the expectation-value of evaluation functions $p\mapsto p(x)$ with respect to some distribution of distributions $P\in D^{2}(X)$, i.e.

The usual naturality and commutativity requirements are satisfied by these functions, so there is a $[0,1]$-valued discrete distributions monad $D=(D,\eta,\mu)$ on $\mathbf{Set}$.

The Kleisli category $\mathcal{K}(D)$ of the distribution monad $D$ has sets as objects, and functions of type $X\to D(Y)$ as morphisms of type $X\to Y$ in $\mathcal{K}(D)$. The identity morphism $X\to X$ in $\mathcal{K}(D)$ is given by $\eta_{X}$. We define the Kleisli extension $c_{\ast}\colon D(X)\to D(Y)$ of any Kleisli morphism $c\colon X\to D(Y)$ as $c_{\ast}:=\mu_{Y}\circ D(c)$. That is, for all $p\in D(X)$:

Composition of Kleisli maps $c\colon X\to D(Y)$ and $d\colon Y\to D(Z)$, is given using Klesli extension (to simplify notation) as:

For any set function $f\colon X\to Y$, one can define a Kleisli map:

given by naturality of $\eta$.

The category $\mathcal{K}(D)$ has finite coproducts $(0,+)$ given by the empty set $0:=\emptyset$, and disjoint union $X_{1}+X_{2}$ with coprojections:

The following result is well-known, see e.g. [10, Proposition 6.4].

The following and last result is well-known, see e.g. [11, Example 4.7].