cqm.tex

\chapter{Structures in Process Theories}
\label{chap:cqm}

\chapabstract{In this chapter we introduce the remaining background on categorical structures for process theories. From this background, we define a quantum-like process theory that comes equipped with a quantum-like interpretation through a generalized Born rule. Many of the properties developed here, while still general, are extensions of ideas from quantum theory, e.g. complementarity.  Process theories that possess these properties can be considered quantum-like, and have diagrammatic representations that can be used as more powerful extensions of quantum circuits.}

\section{The Dagger}
\label{sec:dag}
This section introduces concepts that expand categorical diagrams beyond quantum circuits. We see that abstract linear algebra can be introduced by adding a so-called dagger functor.  In the case of \cat{FHilb} this corresponds to the familiar notion of adjoint (complex-transpose).  The addition of the dagger also allows us to take a perspective on quantum-like symmetric monoidal categories (not just \cat{FHilb}). 

\begin{defn}
\label{defn:dagger}
A \textbf{dagger functor} on a category \cat{C} is an involutive contravariant functor $\dagger:\cat{C}\to\cat{C}$ that is the identity on objects. A \textbf{dagger category} is a category equipped with a dagger functor.
\end{defn}
Spelling out this definition, the dagger functor has the following properties for $f,g:\arrs{C}$ with suitable types and $A\in\objs{C}$:
\begin{equation}
\left(f^{\dagger}\right)^{\dagger} = f 
\end{equation}
\begin{equation}
(g\circ f)^{\dagger} = f^{\dagger}\circ g^{\dagger}
\end{equation}
\begin{equation}
\idm{A}^{\dagger} = \idm{A}
\end{equation}
Thus for any $f:A\to B$ in a dagger category, its dagger or \textbf{adjoint} $f^{\dagger}:B\to A$ also exists in that category. We use the shorthand $\dagger$-SMC for a dagger symmetric monoidal category. In $\dagger$-SMCs the dagger and the monoidal product are required to cooperate, i.e. $(f\tensor g)^{\dagger}=f^{\dagger}\tensor g^{\dagger}$.\footnote{In general monoidal dagger categories the structural morphisms of unitors and associators are required to be unitary as Definition~\ref{def:dagdefs}.}

The quantum setting of \cat{FHilb} is a dagger category whose canonical dagger is the adjoint, and generalizing it in the manner of Definition \ref{defn:dagger} allows us to generalize many familiar terms, following Abramsky and Coecke~\cite{abramsky2004categorical}:
\begin{defn}
\label{def:dagdefs}
A morphism $f:A\to B$ in a dagger category is:
\begin{itemize}
\item \textbf{self-adjoint} when $f^{\dagger} = f$;
\item a \textbf{projector} when self-adjoint and $f\circ f = f$
\item an \textbf{isometry} when $\dag{f} \circ f = \idm{A}$;
\item \textbf{unitary} when both $\dag{f} \circ f = \idm{A}$ and $f\circ\dag{f} = \idm{B}$;
\item \textbf{positive} when $f = \dag{g}\circ g$ for some morphism $g:H\to K$.
\end{itemize}
\end{defn}

\noindent These concepts both have meaning as mathematical objects in linear algebra and as information theoretic concepts in a process theory.  For example, a projector is some process where multiple sequential applications have the same effect as a single application in the broadest possible sense.

Further, the dagger can be intuitively extended to an operation on diagrams: it flips the picture upside-down around the horizontal axes.  As a visual aid, morphisms can now be drawn with broken symmetry as follows:
\begin{equation}
\label{eq:daggerPics}
\input{tikz/1_daggerPics.tikz}
\end{equation}

Thus the unitarity condition becomes:
\begin{equation}
\label{eq:unitarityPics}
\input{tikz/1_unitarityPics.tikz}
\end{equation}

The next two definitions, that of scalars and effects, further our understanding of $\dagger$-SMCs with a generalized version of the Born rule that provides a basic notion of measurement.

\begin{defn}
\label{defn:scalar}
A \textbf{scalar} in a SMC is a morphism $a:I\to I$. As these are maps from the empty diagram to the empty diagram they are unsurprisingly represented as:
\begin{equation}
\begin{tikzpicture}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node [whitedot] (0) at (-1, -0) {$a$};
\end{tikzpicture}
\end{equation}
\end{defn}

This categorical view of scalars was made explicit in \cite{abramsky2004categorical}. To gain perspective on why these properly represent scalars, we consider two facts.  Firstly, Kelly~\cite[Prop 6.1]{kelly1980coherence} showed that these scalars form a commutative monoid under categorical composition, as is graphically expressed by:
\begin{equation}
\begin{aligned}
\begin{tikzpicture}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node [whitedot] (0) at (-1, -0) {$a$};
\node [whitedot] (0) at (-1, -2) {$b$};
\end{tikzpicture}
\end{aligned}
\;=\;
\begin{aligned}
\begin{tikzpicture}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node [whitedot] (0) at (-1, -0) {$b$};
\node [whitedot] (0) at (-1, -2) {$a$};
\end{tikzpicture}
\end{aligned}
\end{equation}
\noindent Secondly, in $\cat{FHilb}$ they correspond exactly to the complex numbers, as linear maps~$\mathbb{C}\to\mathbb{C}$.

Recall from Definition~\ref{def:effect}, that, in a $\dagger$-SMC, effects on an object $A$ are morphisms $\bra{\psi}:A\to I$.
The dagger gives a method for assigning an effect to every state and vice versa: just as some preparation $p:I\to A$ prepares system $A$ in that state, the effect $\dag{p}:A\to I$ eliminates the system $A$ by the process $\dag{p}$.  This duality presents generalized \textbf{inner products} as compositions of states and effects that generalize the usual Dirac notation~\cite{abramsky2004categorical}:
\begin{equation}
\left(
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (1) [state] at (0, -0.75) {$\psi$};
\node (2) at (0, 0.75) {};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned}\right)^{\dagger}
=
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (1) at (0, -0.75) {};
\node (2) [state, hflip] at (0, 0.75) {$\psi$};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned}
\qquad\qquad
\langle\phi|\circ|\psi\rangle = \langle\phi|\psi\rangle =
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (1) [state] at (0, -0.75) {$\psi$};
\node (2) [state, hflip] at (0, 0.75) {$\phi$};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned}
\end{equation}

\section{The generalized Born rule}
\label{sec:bornrule}
Measurement in these process theories comes from a statement connecting probabilities and inner products. For a state $X:I\to A$ and an effect $Y:A\to I$ in a $\dagger$-SMC, the \textbf{amplitude} of outcome $Y$ given preparation $X$ is:
\begin{equation}
a = Y\circ X:I\to I.
\end{equation}

%Diagrammatically this is:
%\begin{equation}
%\mbox{Prob}(X|Y) \;=\; 
%\begin{aligned}
%\begin{tikzpicture}[xscale=\tikzxscale, yscale=\tikzyscale]
%\node (1) [state] at (0, -0.5) {$Y$};
%\node (2) [state, hflip] at (0, 0.5) {$X$};
%\draw (1.center) to (2.center);
%\node (3) [state] at (0, 3.5) {$Y^{*}$};
%\node (4) [state, hflip] at (0, 4.5) {$X^{*}$};
%\draw (3.center) to (4.center);
%\end{tikzpicture}
%\end{aligned}
%\;=\;
%\begin{aligned}
%\begin{tikzpicture}[xscale=\tikzxscale, yscale=\tikzyscale]
%\node (1) [state] at (0, -0.5) {$Y$};
%\node (2) [state, hflip] at (0, 0.5) {$X$};
%\draw (1.center) to (2.center);
%\node (3) [state] at (3, -0.5) {$Y^{*}$};
%\node (4) [state, hflip] at (3, 0.5) {$X^{*}$};
%\draw (3.center) to (4.center);
%\end{tikzpicture}
%\end{aligned}
%\end{equation}

\noindent We could then define the operational probability Prob$(X|Y)=|a|^2$. While this clearly makes sense in \cat{FHilb}, where amplitudes are complex numbers, one might ask under what general conditions the scalars $I\to I$ will square to our usual notion of probability. Vicary provides an answer in the following theorem:
\begin{theorem}{\cite[Thm 4.2]{vicary2011completeness}}
In a monoidal dagger-category with simple tensor unit, which has all finite dagger-limits and for which the self-adjoint scalars are Dedekind-complete, the scalars have an involution-preserving embedding into the complex numbers.
\end{theorem}
\noindent For any category satisfying these conditions, this embedding can be used, along with appropriate normalization, to extract real-valued probabilities. Still, even if the scalars do not embed in the complex numbers one can generally handle the needed structure of complex conjugation to obtain a generalized Born rule. This is done using monoidal categories with duals.\footnote{These are sometimes called autonomous categories in the literature \cite{joyal1993braided,selinger2011survey}.}

Duals in a process theory help us define the quantum-like properties of entanglement and conjugation.

\begin{defn}
\label{def:dual}
In a $\dagger$-SMC, a system $A$ has a\footnote{In general, there can be separate left and right duals for an object in any category, where the definition given here corresponds to a left dual. In a \dsmc(in fact in any braided monoidal category with left duals), these are necessarily equal, so we need only speak of one dual~\cite[Prop.~7.2]{joyal1993braided}. } \textbf{dual} system $A^*$ if there exist processes 
\begin{align}
e_A:I\to A^*\otimes A \quad \mbox{and}\quad d_A:A\otimes A^*\to I,
\end{align}
such that the following equations hold:
\begin{align}
\label{eq:snakeeq}
(d_A\otimes \idm{A})\circ(\idm{A}\otimes e_A) = \idm{A} \qquad (\idm{A^*}\otimes d_A)\circ(e_A\otimes \idm{A^*})=\idm{A^*}
\end{align}
\end{defn}

When duals are introduced, we can denote them by placing arrows on the objects:
\begin{equation}
\input{tikz/2_arrowdual.tikz}
\end{equation}

Thus the duality maps, commonly called ``cups" and ``caps", and their ``snake equations"~\eqref{eq:snakeeq} have the following diagrammatic form:
\begin{align}
\input{tikz/2_capcup.tikz}
\end{align}
\begin{equation}
\label{eq:snake}
\input{tikz/2_snake.tikz}
\end{equation} 
\noindent and are well behaved under the dagger functor, i.e.
\begin{equation}
\input{tikz/2_daggerdual.tikz}
\end{equation}

We can compose these caps and cups to define the dual of any process $f:A\to B$ as $f^*:B^*\to A^*$:
\begin{equation}
\label{eq:dualmorph}
\input{tikz/2_dualmorph.tikz}
\end{equation}
This is called the \textbf{upper-star} of $f$.

\begin{example}
\label{ex:bellduals}
To get a better handle on what duals are, we consider the example of \cat{FHilb}. Given some finite dimensional Hilbert space $A\in\objs{FHilb}$, its dual $A^*$ is the usual dual space, i.e. the space of linear functionals $A\to \mathbb{C}$. Note that in this case $A$ is isomorphic to $A^*$, and thus we can canonically map $\bra{i}\mapsto\ket{i}$. Because of this isomorphism, we will often omit the arrows on wires when working in \cat{FHilb}. Using this and given a basis $\{ \ket{i} \}$ for $A$, the cups and the caps are maps:
\begin{align}
e_A &:: 1\mapsto \sum_i\bra{i}\otimes\ket{i}\cong\sum_i\ket{i}\otimes\ket{i}
\\
d_A &:: \sum_i\ket{i}\otimes\bra{i}\cong\sum_i\bra{i}\otimes\bra{i} \mapsto 1
\end{align}
We then immediately recognize $e_A$ as entangled state preparation. In particular, when $A$ is a two-dimensional Hilbert space, $e_A$ is a Bell state preparation of $\ket{\psi_{00}}=\ket{00}+\ket{11}$ and  and $d_A$ is a post-selected Bell measurement for $\ket{\psi_{00}}$. For any process in \cat{FHilb} its dual gives the transpose.
\end{example}

This provides a motivating example for the following, which acts as a generalized conjugation operation:

\begin{defn}
Given a process theory with duals \cat{C}, the \textbf{lower-star} is an involutive map 
\begin{align*}
(-)_*:\arrs{C}&\to\arrs{C} \\
\left(f:A\to B\right) &\mapsto f_*:=(\dag{f})^*=\dag{(f^*)}.
\end{align*}
\end{defn}

The lower-star operation introduces abstract probabilities for the generalized Born rule:
\begin{defn}[Generalized Born Rule]
\label{def:bornrule}
For a state $X:I\to A$ and an effect $Y:A\to I$ in a $\dagger$-SMC, the probability of outcome $Y$ given preparation $X$ is:
\begin{equation}
\mbox{Prob}(Y|X) = (Y\circ X)_*\otimes Y\circ X:I\to I.
\end{equation}
\end{defn}
\noindent It is easy to see that this reduces to the usual Born rule in \cat{FHilb}.

The cup and cap maps also provide a general definition for the trace of a process:
\begin{equation}
\label{eq:trace}
\Tr \left(\begin{pic}
\node (0) at (0,0) {};
\node [morphism, wedge] (1) at (0,1) {$f$};
\node (2) at (0,2) {};
\draw [arrow=0.5] (0) to (1.south);
\draw [arrow=0.5] (1.north) to (2);
\end{pic} \right) = 
\input{tikz/3_trace.tikz},
\end{equation}
whose correspondence with the usual trace is easily shown~\cite{coecke2015generalised}:
\begin{equation}
\mbox{Tr}(f) = \left(\sum_i\bra{ii}\right)\circ\left(\idm{A}\tensor f\right)\circ\left(\sum_j\ket{jj}\right) = \sum_i\langle i | f |i\rangle = \sum_if_{ii}
\end{equation}

\subsection{Quantum-like process theories}

The addition of states and a generalized Born rule (with further details in this section), give our process theories a notion of measurement and a quantum-like character.

\begin{defn}(Quantum-like Process Theories)
\label{thm:QPT}
A $\dagger$-SMC where all objects have duals is called a \textbf{$\dagger$-compact category}. A \textbf{quantum-like process theory} (QPT) is a \dcc where objects are interpreted as systems and morphisms are interpreted as processes.
\end{defn}

Duals and their ``wire-straightening" equations give diagrams in a quantum-like process theory a lot of power to encode topological equivalences.  This is well expressed in the following graphical coherence theorem, whose mathematical details can be traced in Selinger's survey~\cite{selinger2011survey}.

\begin{theorem}[Fundamental Theorem of Diagrams~\cite{coecke2015generalised}]
\label{thm:fund}
Two diagrams in a QPT are considered equal if one can be smoothly transformed to another by bending, stretching, or crossing wires, and by moving boxes around. Given any two QPTs $\cat{C}$ and $\cat{D}$ and a functor $F:\cat{C}\to\cat{D}$, for any two diagrams $d$ and $d'$ in $\cat{C}$, if $d=d'$ as diagrams then $F(d)=F(d')$ in $\cat{D}$.
\end{theorem}

To recap, we have so far introduced the structures of the dagger and duals, allowing us to compute measurement outcomes by manipulation of the diagrams alone.  This contrasts with the usual quantum circuit formalism, which provides a representation of a protocol, but whose implementation must ultimately be understood through the matrix mechanics that accompany it. The use of this difference - the operational in operational process theories - can be illustrated with quantum teleportation, which provides a motivating example and was introduced in this form by Abramsky and Coecke~\cite{abramsky2004categorical}.

\begin{example} 
\label{ex:teleport}
We saw in Example \ref{ex:bellduals} that caps and cups represent preparation and post-selected measurement of one of the Bell states $\ket{\psi_{00}}$. The other Bell states can be prepared and/or measured by the application of a unitary to one of the entangled systems:
\begin{equation}
\begin{pic}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node (0) at (-1, -0) {};
\node (1) at (1, -0) {};
\node (2) at (0, -1) {};
\node (r) at (1,3) {};
\node (l) at (-1,3) {};
\node [morphism, wedge] (u) at (-1,1) {$U_i$};
\draw (1.center) to (r);
\draw (u.north) to (l);
\draw [bend right=45, looseness=1.00] (u.south) to (2.center);
\draw [bend right=45, looseness=1.00] (2.center) to (1.center);
\end{pic}
\qquad \qquad
\begin{pic}[xscale={\tikzxscale}, yscale={-1*\tikzyscale}]
\node (0) at (-1, -0) {};
\node (1) at (1, -0) {};
\node (2) at (0, -1) {};
\node (r) at (1,3) {};
\node (l) at (-1,3) {};
\node [morphism, wedge, hflip] (u) at (-1,1) {$U_i$};
\draw (1.center) to (r);
\draw (u.south) to (l);
\draw [bend right=45, looseness=1.00] (u.north) to (2.center);
\draw [bend right=45, looseness=1.00] (2.center) to (1.center);
\end{pic}
\end{equation}

A teleportation protocol can then be represented by the process where Alice and Bob share an entangled system that will be used to teleport from Alice to Bob:
\begin{equation}
\input{tikz/2_teleportation.tikz}
\end{equation}
Here the cup map represents the shared Bell state.  Alice performs a Bell measurement on the source system and her half of the entangled system. Bob then performs a unitary that matches Alice's Bell measurement. We can verify the effect of the protocol using purely diagrammatic equivalences:
\begin{equation}
\label{eq:teleportderive}
\input{tikz/2_teleportderive.tikz}
\end{equation}
This protocol provides a good example where space-like and time-like intuition should be applied carefully. The application of the Bell effect and state connects Alice and Bob's systems by a single wire. Thus, the space-like separated pair of Alice and Bob's unitaries become equivalent to their time-like sequential composition, as show in the first step of~\eqref{eq:teleportderive}. In fact, if one is reading the passage of time vertically in the diagrams, then it may be surprising to note that because
\begin{equation}
\begin{pic}[xscale=0.9*\tikzxscale, yscale=\tikzyscale]
    \node (0) at (-1, -2) {};
    \node (1) at (1, -2) {};
    \node (2) at (0, -3) {};
    \node [style=morphism, wedge, hflip] (3) at (-3, -1) {$U_{i}$};
    \node (4) at (-2, 1) {};
    \node (5) at (-3, -0) {};
    \node (6) at (-1, -0) {};
    \node (7) at (-3, -4) {};
    \node [style=morphism, wedge] (8) at (1, 2) {$U_i$};
    \node (9) at (1, 3) {}; 
    \draw [bend right=45, looseness=1.00] (0.center) to (2.center);
    \draw [in=-90, out=0, looseness=1.00] (2.center) to (1.center);
    \draw [bend left=45, looseness=1.00] (5.center) to (4.center);
    \draw [in=90, out=0, looseness=1.00] (4.center) to (6.center);
    \draw (7.center) to (3.south);
    \draw (3.north) to (5.center);
    \draw (6.center) to (0.center);
    \draw (8.south) to (1.center);
    \draw (8.north) to (9.center);
\end{pic}
\quad=\quad
\begin{pic}[xscale=0.9*\tikzxscale, yscale=\tikzyscale]
    \node (0) at (-1, -2) {};
    \node (1) at (1, -2) {};
    \node (2) at (0, -3) {};
    \node [style=morphism, wedge, hflip] (3) at (-3, -0.7) {$U_{i}$};
    \node (4) at (-2, 1) {};
    \node (5) at (-3, -0) {};
    \node (6) at (-1, -0) {};
    \node (7) at (-3, -4) {};
    \node [style=morphism, wedge] (8) at (1, -1.2) {$U_i$};
    \node (9) at (1, 3) {}; 
    \draw [bend right=45, looseness=1.00] (0.center) to (2.center);
    \draw [in=-90, out=0, looseness=1.00] (2.center) to (1.center);
    \draw [bend left=45, looseness=1.00] (5.center) to (4.center);
    \draw [in=90, out=0, looseness=1.00] (4.center) to (6.center);
    \draw (7.center) to (3.south);
    \draw (3.north) to (5.center);
    \draw (6.center) to (0.center);
    \draw (8.south) to (1.center);
    \draw (8.north) to (9.center);
\end{pic}
\end{equation}  
the time-ordering of Alice and Bob's unitaries appears to not affect the protocol.
In some sense, the connectivity of the diagram means that Alice's unitary appears to happen first in the compositional sequence regardless of the time ordering in the protocol's implementation. It should be noted that this behavior is only present in the post-selected teleportation protocol that we present here.  In general, when some classical communication is required, the time-ordering of course certainly matters. 

This presentation of the teleportation protocol is useful in several ways:
\begin{enumerate}
\item The high level structure of the protocol (the teleportation) is immediately manifest without accompanying calculation.
\item It is obvious how to design equivalent teleportation protocols with multiple parties and Bell measurements in more elaborate arrangements, e.g.
\begin{equation}
\input{tikz/2_weirdteleport.tikz}
\end{equation}
\item By specifying the teleportation protocol at the level of QPTs, we can consider models of teleportation in theories other than quantum theory, i.e. categories other than $\cat{FHilb}$. As an example, the QPT $\cat{FRel}$ where systems are sets and processes are relations also has entangled states established by duals, where the dagger functor is the relational converse. All the protocol specifications given in this section apply equally well in the $\cat{FRel}$ setting, just as they will in any other QPT.
\end{enumerate}
\end{example}

Quantum-like process theories already give us some power above and beyond the usual quantum circuit formalism, and we add more structures to the toolbox in the next sections.  We have glossed over some of the details of how measurement should be represented in such theories, but the introduction of classical structures in the next section allows us to be more exact in Section~\ref{sec:measurements}. 

\section{Classical structures}
\label{sec:observables}

Within a process theory we can construct objects that have the phenomenology of classical information: copying, deleting, etc. These structures become the observables of QPTs that allow us to extract classical information via measurements.

\subsection{Monoids and comonoids}

Monoids and comonoids on systems embody our notions of copying and comparing states. As they are particular kinds of processes, we draw them with distinct pictures. For $A\in\objs{C}$, a monoid $(A,*,\mathbbm{1})$ has states of $A$ as elements and the following maps:
\begin{equation}
\input{tikz/1_monoidPics.tikz}
\end{equation}

\begin{defn}
\label{defn:monoid}
In a monoidal category, a \textbf{monoid} is a triple \whitemonoid{A} of an object $A$, a morphism $\tinymult[whitedot] : A \otimes A \to A $ called the multiplication, and a state $\tinyunit[whitedot] : I \to A$ called the unit, satisfying associativity and unitality equations:
\begin{equation}
\label{eq:monoid}
\input{tikz/1_monoid.tikz}
\end{equation}
\end{defn}

\begin{defn}
\label{defn:comonoid}
In a monoidal category, a \textbf{comonoid} is a triple \whitecomonoid{A} of an object $A$, a morphism $\tinycomult[whitedot] : A \to A \otimes A$ called the comultiplication, and an effect $\tinycounit[whitedot] : A \to I$ called the counit, satisfying coassociativity and counitality equations:
\begin{equation}
\label{eq:comonoid}
\input{tikz/1_comonoid.tikz}
\end{equation}
\end{defn}

\noindent We use differently colored dots to represent different monoids on the same object. In a dagger monoidal category every monoid has a corresponding comonoid formed by applying the dagger functor, i.e. $\left(\tinymult[whitedot]\right)^{\dagger}=\tinycomult[whitedot]$ and $\left(\tinyunit[whitedot]\right)^{\dagger}=\tinycounit[whitedot]$\,. When a monoid and comonoid are the same color, we take this to mean that each is the dagger of the other.
\begin{defn}
In a monoidal category with object $A$, a \textbf{monoid homomorphism} $f:\whitemonoid{A}\to\blackmonoid{A}$ is a map $f:A\to A$ such that
\begin{equation}
\label{eq:monhom}
\input{tikz/1_mon_hom.tikz}
\end{equation}
\end{defn}
\noindent A \textbf{comonoid homomorphism} is defined similarly, but with the dagger of the conditions in~\eqref{eq:monhom}.

In the next section we will ask for the comonoid and monoid to interact in various ways.

\subsection{Generalized observables}
\label{sec:abstractobservables}

When monoids and comonoids combine under certain rules, we obtain the structure of classical information on systems in a QPT. We can think of this as a way of embedding classical information into the systems of an arbitrary QPT.

\begin{defn}
\label{def:frobenius}
In a $\dagger$-SMC, the pair of a monoid \whitemonoid{A} and comonoid \whitecomonoid{A} form a \textbf{dagger-Frobenius algebra} ($\dagger$-FA) when the following equation holds:
\begin{equation}
\label{eq:frobenius}
\input{tikz/1_Frobenius.tikz}
\end{equation}
\end{defn}
When \tinymult[whitedot] is commutative, the $\dagger$-FA is commutative (is a $\dagger$-CFA). The co-commutativity of \tinycomult[whitedot] for a $\dagger$-CFA follows~\cite[Thm 3.2.8]{kissinger2012pictures}.

\begin{defn}
\label{def:classicalstruct}
A \textbf{classical structure} (\dotonly{whitedot}) is a dagger-Frobenius algebra \whitefrob{A} satisfying the \textbf{specialness} \eqref{eq:special} and \textbf{symmetry} \eqref{eq:sym} conditions:
\begin{equation}
\label{eq:special}
\begin{aligned}
\begin{tikzpicture}[xscale={1.5*\tikzxscale}, yscale={1.5*\tikzyscale}]
\draw (0,0.25) to (0,1) node [whitedot] {} to [out=\nwangle, in=down] (-0.5,1.5) to [out=up, in=\swangle] (0,2) node [whitedot] {} to (0,2.75);
\draw (0,1) to [out=\neangle, in=down] (0.5,1.5) to [out=up, in=\seangle] (0,2);
\end{tikzpicture}
\end{aligned}
\quad=\quad
  \begin{aligned}
  \begin{tikzpicture}[xscale={1.5*\tikzxscale}, yscale={1.5*\tikzyscale}]
  \draw (-0.5,0) to (-0.5,3);
  \end{tikzpicture}
  \end{aligned}
  \end{equation}
  
  \vspace{-10pt}
  \begin{equation}
  \label{eq:sym}
\begin{aligned}
\begin{tikzpicture}[xscale={1.5*\tikzxscale}, yscale={1.5*\tikzyscale}]
\draw (0,-0.5) node [whitedot] {} to (0,0.5) node [whitedot] {} to [out=\nwangle, in=down] (-0.5,1.0) to [out=up, in=down] (0.5,2);
\draw (0,0.5) to [out=\neangle, in=down] (0.5,1) to [out=up, in=down] (-0.5,2);
\end{tikzpicture}
\end{aligned}
\quad=\quad
\begin{aligned}
\begin{tikzpicture}[xscale={1.5*\tikzxscale}, yscale={1.5*\tikzyscale}]
\draw (0,-0.5) node [whitedot] {} to (0,0.5) node [whitedot] {} to [out=\nwangle, in=down] (-0.5,1.0) to [out=up, in=down] (-0.5,2);
\draw (0,0.5) to [out=\neangle, in=down] (0.5,1) to [out=up, in=down] (0.5,2);
\end{tikzpicture}
\end{aligned}
\end{equation}
\end{defn}

\begin{defn}
\label{def:copyables}
The set of \textbf{classical states} $K_{\circ}$ for a classical structure (\dotonly{whitedot}) are all states $j:I\to A$ such that:
\begin{equation}
\label{eq:copy}
\begin{pic}[xscale=\tikzxscale, yscale=-0.8*\tikzyscale]
\node [state] at (1.25,3) {$j$};
\draw (2,0) to [out=up, in=\seangle] (1.25,1.5);
\draw (0.5,0) to [out=up, in=\swangle] (1.25, 1.5);
\draw (1.25,1.5) to (1.25, 3);
\node [whitedot] at (1.25,1.5) {};
\end{pic}
\quad=\quad
\begin{pic}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (a) [state] at (2.5,0) {$j$};
\node (b) [state] at (0,0) {$j$};
\draw (a) to (2.5,2);
\draw (b) to (0,2);
\end{pic}
\end{equation}
\end{defn}
\noindent These classical states will typically be drawn as states that are the same color as the classical structure to which they correspond. It is also easy to determine the behavior of classical states under composition with the (co)unit:
\begin{equation}
\label{eq:unitscalar}
\begin{pic}[xscale=\tikzxscale, yscale=-0.8*\tikzyscale]
\node [state] at (1.25,3) {$j$};
\draw (1.25,0) to (1.25, 3);
\end{pic}
\;\stackrel{\eqref{eq:comonoid}}{=}\;
\begin{pic}[xscale=\tikzxscale, yscale=-0.8*\tikzyscale]
\node [state] at (1.25,3) {$j$};
\draw (2,0) to [out=up, in=\seangle] (1.25,1.5);
\draw (0.5,0) to [out=up, in=\swangle] (1.25, 1.5);
\draw (1.25,1.5) to (1.25, 3);
\node [whitedot] at (1.25,1.5) {};
\node [whitedot] at (0.5,0) {};
\end{pic}
\;\stackrel{\eqref{eq:copy}}{=}\;
\begin{pic}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (a) [state] at (2.5,0) {$j$};
\node (b) [state] at (0,0) {$j$};
\draw (a) to (2.5,2);
\draw (b) to (0,2);
\node [whitedot] at (0,2) {};
\end{pic}
\quad\qquad\Leftrightarrow\qquad\quad
\begin{pic}[xscale=\tikzxscale, yscale=\tikzyscale]

\node (b) [state] at (0,0) {$j$};

\draw (b) to (0,2);
\node [whitedot] at (0,2) {};
\end{pic}
\;=\;1
\end{equation}
\noindent where $1$ is the unit scalar, i.e. $1\circ s = s = s\circ 1$ for all scalars $s$. In this sense the counit ``erases" classical points.

The following theorems by Coecke et al. characterize the relationship between classical structures and observables. 

\begin{theorem}[{\cite[Thm 5.1]{coecke2013new}}]
Symmetric dagger Frobenius algebras in \cat{FHilb} are orthogonal bases.
\end{theorem}
\noindent The additional condition of specialness for classical structures acts as a normalizing condition so that:
\begin{theorem}[{\cite[Sec 6]{coecke2013new}}]
\label{thm:cstructBases}
Classical structures in \cat{FHilb} are orthonormal bases.
\end{theorem}

\noindent \noindent Classical structures are bases in \cat{FHilb} and so are recognized as generalized \textbf{observables} in a QPT. The classical states of classical structures  (Definition~\ref{def:copyables}) form the elements of these bases, more specifically the eigenvectors of the observables, though we should be careful that in categories other than $\cat{FHilb,}$ they do not necessarily have all the familiar properties of bases for Hilbert spaces.  For example, we usually expect to be able to distinguish maps by testing them on basis elements. In general this only holds for certain classical structures:

\begin{defn}
\label{def:enoughclassicalpoints}
A classical structure (\dotonly{graydot}) has \textbf{enough classical states} when for all processes $f,g:A\to B$: 
\begin{align}
f=g \qquad \Leftrightarrow \qquad \left(\forall \ket{j}\in K_{\dotonly{graydot}}:f\circ\ket{j}=g\circ\ket{j}\right)\; 
\end{align}
\end{defn}

\begin{remark}
The correspondence in Theorem \ref{thm:cstructBases} was modified for the infinite dimensional case in \cite{abramsky2012h}, where it still holds. We will only use the finite dimensional case in this thesis as we are concerned only with algorithms that run on computers of finite size.
\end{remark}

\begin{defn}
\label{def:dimension}
In a QPT, the \textbf{dimension} $d(A)$ of an object $A$ equipped with a dagger-Frobenius algebra \whitefrob{A}, is given by the following composite:
\begin{equation}
\label{eq:dim}
\input{tikz/1_dimension.tikz}
\end{equation}
\end{defn}
\noindent
When the algebra is in fact a classical structure,~\eqref{eq:dim} can be simplified to the composition of the unit and counit:
\begin{equation}
\label{eq:defdim}
d(A) \;= \;
\begin{pic}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node (t) [whitedot] at (0,1) {};
\node (b) [whitedot] at (0,-1) {};
\draw (t) to (b.center);
\end{pic}
\end{equation}

\begin{remark}
It is important to note that the dimension of a system does not always equal the number of classical states of the associated classical structure. One setting where this does not hold is \cat{FRel}, where this fact plays an important role in its QPT characterization in Chapter~\ref{chap:mermin}.
\end{remark}

An important result of Coecke and Duncan shows that the rules for classical structures can be summarized in a convenient normal form that is in keeping with the sorts of topological equivalences we are used to for diagrams of a QPT. Let the maps $\tinycomult[whitedot]_n:A\to A^{\otimes n}$ be defined recursively by:
\begin{align}
\tinycomult[whitedot]_0:=\tinycounit[whitedot]
\qquad\qquad
\tinycomult[whitedot]_{n+1}:=\left(\tinycomult[whitedot]_{n}\otimes\idm{A}\right)\circ\tinycomult[whitedot]
\end{align}
Let $\tinymult[whitedot]_n$ be similarly defined.
\begin{theorem}[Spider Theorem {\cite[Thm 6.11]{coecke2011interacting}}]
\label{thm:spider}
Given a classical structure, let \newline$f:A^{\otimes n}\to A^{\otimes m}$  be constructed from $\{\tinymult[whitedot],\tinycomult[whitedot],\tinyunit[whitedot],\tinycounit[whitedot]\}$ such that the diagram is connected. Then $f=\tinycomult[whitedot]_m\circ\tinymult[whitedot]_n$.
\end{theorem}

\noindent This normal form has a neat diagrammatic representation.

\begin{proposition}[{\cite[Thm 6.12]{coecke2011interacting}}]
\label{prop:spider} 
Given a classical structure on $A$, let
$(\whitedot)_n^m$ denote the `$(n,m)$-legged spider':
\begin{equation}
\input{tikz/2_spiderdef.tikz}
\end{equation}
then any process $A^{\otimes n}\to A^{\otimes m}$ built from $\{\tinymult[whitedot], \tinycomult[whitedot], \tinyunit[whitedot],\tinycounit[whitedot]\}$ which has a connected graph is equal to $(\whitedot)^m_n$. Spider
composition is:
\begin{equation}\label{eq:spidercomp}
 \input{tikz/2_spidercomp.tikz}
\end{equation}
\end{proposition}

The spider rule makes it clear that every classical structure on $A$ can be used to make $A$ dual to itself (Definition~\ref{def:dual}) using caps ($\tinycounit[whitedot]\circ\tinymult[whitedot]$) and cups ($\tinycomult[whitedot]\circ\tinyunit[whitedot]$) built from the white dot:
\tikzeq{tikz/2_frobcomp.tikz}
Kissinger provides direct proof of this fact~\cite[Thm 3.2.7]{kissinger2012pictures}.

The upper and lower star operations with respect to this white dot cup and cap corresponds in \cat{FHilb} to transposition and conjugation in the white dot  basis respectively. Further it can be shown that transposition with respect to a classical structure is equivalent to the dagger when applied to its classical states:
\begin{equation}
\label{eq:dagfrob}
\input{tikz/2_dagfrobtranspose.tikz}
\end{equation}
\noindent This is of course not true on general states.

\section{Phases}
\label{sec:phases}
Phases for QPTs were introduced by Coecke and Duncan~\cite{coecke2011interacting}. They decorate classical structures with an abelian group in a particular way.

\begin{defn}
\label{def:phases}
A \textbf{phase state} for a Frobenius algebra $\whitecomonoid{A}$ is a state $\ket{\alpha}$ such that:
\begin{equation}
\label{eqn:zphasestate}
\input{tikz/2_phasestate.tikz}
\end{equation}
Each phase state corresponds to a \textbf{phase} that is a unitary map in the following form:
\begin{equation}
\label{eqn:zphase}
\input{tikz/2_phase.tikz}
\end{equation}
\end{defn}

\begin{proposition}[Phase groups]
Given a $\dagger$-FA $\whitefrob{A}$ in a QPT, its phases form a group under the following addition:
\begin{equation}
\input{tikz/2_phasegroups.tikz}
\end{equation}
with unit $\tinyunit[whitedot]$. When the Frobenius algebra is part of a classical structure, then its phases form an abelian group.
\end{proposition}
\begin{proof}
See~\cite[Sec. 7.4]{coecke2011interacting} for proofs.
\end{proof}

Phases can be added to the normal form from Theorem~\ref{thm:spider} to give a decorated spider rule~\cite[Thm 7.11]{coecke2011interacting}:
\begin{equation}
\label{eq:decspider}
\input{tikz/2_decspider.tikz}
\end{equation}
with composition
\begin{equation}
\input{tikz/2_decspidercomp.tikz}
\end{equation}
This normal form is a powerful simplifying tool and will often be used in the analysis of diagrams of processes in QPTs.

\section{Complementarity}
The notion of complementary bases can also be lifted to the general level of classical structures in QPTs~\cite{coecke2011interacting}. This extension is perhaps best presented as emergent from a suitable generalization of mutual unbiasedness.\footnote{Our presentation in this regard is heavily influenced by the approach taken by Coecke et al. in~\cite{coecke2015generalised}.}  This means that for two bases $\{\ket{i}\}$ and $\{\ket{j}\}$ on a $D$-dimensional Hilbert space, $|\langle i|j\rangle|^2 = 1/D$ for all $i,j$. In diagrams we then have:
\tikzeq{tikz/2_mub.tikz}
as the mutual unbiasedness condition, where we have assumed that $D$ is invertible (as all dimensions in \cat{FHilb} are) and used \eqref{eq:defdim} and \eqref{eq:unitscalar} to obtain the right hand form.
\begin{defn}[Complementarity]
\label{def:complementarity}
In a QPT, two classical structures (\dotonly{whitedot}) and (\dotonly{graydot}) on the same object are \textbf{complementary} when the following equation holds:
\begin{equation}
\label{eq:complementarity}
\input{tikz/2_antipodehopf.tikz}
\qquad \mbox{where} \qquad
\input{tikz/2_antipode.tikz}
\end{equation}
where $D$ is the dimension of the object and the map $S:A\to A$ is called the \textbf{antipode}.
\end{defn}
The Equation \eqref{eq:complementarity} is equivalent to mutual unbiasedness on classical structures that have enough classical points by Definition \ref{def:enoughclassicalpoints}.
Then by a quick calculation:
\tikzeq{tikz/2_complemub.tikz}
where $(*)$ is an application of Theorem \ref{thm:fund}, and the last step uses the fact that there are enough classical points. 

Complementarity relates the classical states of classical structures to their phase groups.
\begin{lemma}
\label{lem:phaseunbiased}
If two classical structures in a QPT are complementary, then, up to an idempotent scalar, a state that is a classical point of one is a phase state for the other.
\end{lemma}
\begin{proof}
We prove this using diagrams, following Coecke and Kissinger~\cite{qcs-notes}:
\begin{equation}
\input{tikz/3_phaseUnbiased.tikz}
\end{equation}
\end{proof}
Though the converse of this lemma holds in \cat{FHilb}, i.e. each phase is also a classical point of the complementary structure, the converse does not hold in general.

%\noindent 
%Note that this is not a symmetric condition between the gray and white structures. %However, thanks to the symmetric property of the dagger-Frobenius algebras, %it is equivalent to the following alternative condition:
%\begin{equation}
%\input{tikz/2_complementarity2.tikz}
%\end{equation}
%The daggers of these equations give rise to two further equivalent conditions.

\begin{defn}\label{def:coherence}
In a QPT, two classical structures (\dotonly{whitedot}) and (\dotonly{graydot}) are \textbf{coherent} when the following equations hold:
\begin{equation}
\label{eq:coherence}
 \input{tikz/2_coher.tikz}
\end{equation}
\end{defn}

\begin{proposition}[{\cite[Prop.~3]{coecke2015generalised}}]
\label{prop:cohere}
In \cat{FHilb} if we are given two self-adjoint operators corresponding to complementary classical structures, then we can always construct a pair of coherent classical structures with the same classical points.
\end{proposition}
\noindent This means that in the QPT for quantum computation, we can take complementary classical structures to be coherent without loss of generality.

The terminology for the antipode comes from the fact that complementarity is almost enough to make the classical structures a Hopf algebra using this antipode. Some complementary classical structures do indeed form Hopf algebras and these ones are called strongly complementary. They will be discussed in Section~\ref{sec:strcomplFT} in detail.

\begin{example}
\label{ex:cnot}
Complementary observables allow us to construct a generalized form of the controlled-not gate in any QPT. In \cat{FHilb} we can choose classical structures on the two-dimensional Hilbert space that correspond with the usual Z and X observables:  \begin{align*}
    \tinycomult[whitedot] : &
      \begin{array}{rcl}
        \ket{0} & \mapsto & \ket{00} \\
        \ket{1} & \mapsto & \ket{11}
      \end{array}
&
    \tinycomult[graydot] : &
      \begin{array}{rcl}
        \ket{+} & \mapsto & \ket{++} \\
        \ket{-} & \mapsto & \ket{--}
      \end{array}
\\ \\
    \tinycounit[whitedot] : &
      \begin{array}{rcl}
        \ket{0} & \mapsto & 1 \\
        \ket{1} & \mapsto & 1
      \end{array}
&
    \tinycounit[graydot] : &
      \begin{array}{rcl}
        \ket{+} & \mapsto & 1 \\
        \ket{-} & \mapsto & 1
      \end{array}
\\ \\
K_{\dotonly{whitedot}} &= \{\ket{0},\ket{1}\}
& 
K_{\dotonly{graydot}} &= \{\ket{+},\ket{-}\}
\end{align*}

These classical structures are coherent and complementary and each have $\mathbb{Z}_2$ as their phase groups, which we will write as $\{0,\pi\}$ where addition is modulo $2\pi$.  By Lemma~\ref{lem:phaseunbiased}, we know that the classical points of the $\dotonly{whitedot}$-structure are phases for the $\dotonly{graydot}$-structure. In particular we have:
\begin{equation}
\label{ex:cnottypes}
\begin{pic}
\node [style=state] (0) at (1, -0) {$0$};
\node (1) at (1, 1) {};
\draw (0) to (1);
\end{pic}
\;=\;
\begin{pic}
\node [style=graydot] (0) at (1, -0) {};
\node (1) at (1, 1) {};
\draw (0) to (1);
\end{pic}
\qquad\qquad
\begin{pic}
\node [style=state] (0) at (1, -0) {$1$};
\node (1) at (1, 1) {};
\draw (0) to (1);
\end{pic}
\;=\;
\begin{pic}
\node [style=graydot] (0) at (1, -0) {$\pi$};
\node (1) at (1, 1) {};
\draw (0) to (1);
\end{pic}
\qquad\qquad
\begin{pic}[scale=0.8]
\node [style=graydot] (0) at (1, -0) {$\pi$};
\node (1) at (1, 1) {};
\node (2) at (1, -1) {};
\draw (0) to (1);
\draw (0) to (2);
\end{pic}
\;=\;
X\mbox{-gate}::
\left\{\begin{array}{c}
\ket{0}\mapsto\ket{1} \\
\ket{1}\mapsto\ket{0} \\
\end{array}\right.
\end{equation}
These structures can be used to define the controlled-not $\mbox{CNOT}:H_2\otimes H_2\to H_2\otimes H_2$ whose diagram is:
\begin{align}
\input{tikz/2_cnot.tikz}
\end{align}
\noindent where the right system acts as the control. We can verify that this behaves as usual for qubits using Z (the $\dotonly{whitedot}$-classical structure as their computational basis).
\begin{align}
\begin{pic}[dotpic]
                \node [style=whitedot] (0) at (1, -0) {};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [point] (2) at (1, -1) {$0$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (0);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
                \draw (0) to (5.center);
\end{pic}
\;&\stackrel{\eqref{ex:cnottypes}}{=}\;
\begin{pic}[dotpic]
                \node [style=whitedot] (0) at (1, -0) {};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [graydot] (2) at (1, -1) {};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (0);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
                \draw (0) to (5.center);
\end{pic}
\;\stackrel{\eqref{eq:coherence}}{=}\;
\begin{pic}[dotpic]
                \node [style=graydot] (0) at (0, -0) {};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [graydot] (2) at (1, 0) {};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (5.center);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
\end{pic}
\;\stackrel{\eqref{eq:decspider}}{=}\;
\begin{pic}[dotpic]
                \node [point] (2) at (1, 0) {$0$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (5.center);
                \draw (3.center) to (4.center);
\end{pic}
\\
\begin{pic}[dotpic]
                \node [style=whitedot] (0) at (1, -0) {};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [point] (2) at (1, -1) {$1$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (0);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
                \draw (0) to (5.center);
\end{pic}
\;&\stackrel{\eqref{ex:cnottypes}}{=}\;
\begin{pic}[dotpic]
                \node [style=whitedot] (0) at (1, -0) {};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [graydot] (2) at (1, -1) {$\pi$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1, 1) {};
                \draw (2.north) to (0);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
                \draw (0) to (5.center);
\end{pic}
\,\stackrel{\eqref{eq:coherence}}{=}\,
\begin{pic}[dotpic]
                \node [style=graydot] (0) at (0, -0) {$\pi$};
                \node [style=graydot] (1) at (-1, -0) {};
                \node [graydot] (2) at (1.5, 0) {$\pi$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1.5, 1) {};
                \draw (2.north) to (5.center);
                \draw (0) to (1);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
\end{pic}
\;\stackrel{\eqref{eq:decspider}}{=}\;
\begin{pic}[dotpic]
                \node [style=graydot] (1) at (-1, -0) {$\pi$};
                \node [point] (2) at (1.5, 0) {$1$};
                \node (3) at (-1, -1) {};
                \node (4) at (-1, 1) {};
                \node (5) at (1.5, 1) {};
                \draw (2.north) to (5.center);
                \draw (3.center) to (1);
                \draw (1) to (4.center);
\end{pic}
\end{align}
This shows that on computational basis elements, the control is left unchanged while a $X$-gate (a computational bit flip) is conditionally applied.
\end{example}

This example construction motivates the following definition on any kind of system in any QPT:
\begin{defn}
Given two classical structures (\dotonly{whitedot}) and (\dotonly{graydot}) that are complementary and coherent, the generalized \textbf{controlled-not} is the map:
\begin{equation}
\label{eq:generalizedcnot}
\sqrt{\ud(A)}\,\,
\begin{pic}[xscale=\tikzxscale, yscale=\tikzyscale]
\node (b) [graydot] at (0,0) {};
\node (w) [whitedot] at (1,1) {};
\draw (-0.75,2) to [out=down, in=left] (b.center);
\draw (b.center) to [out=right, in=left] (w.center);
\draw (w.center) to (1,2);
\draw (b.center) to (0,-1);
\draw (w.center) to [out=right, in=up] (1.75,-1);
\end{pic}
\end{equation}
where we have assumed that a scalar equal to the square root of the dimension exists.
\end{defn}
\noindent This scalar is needed to ensure the unitarity of the controlled-not gate, which we discuss in further details in Section \ref{sec:unitaryoracles}.

\section{Enriched QPTs}
\label{sec:enrichedQPTs}
Some QPTs, such as \cat{FHilb}, come equipped with linear structure on their processes. Abstractly these are enriched categories, i.e. categories where hom-sets are replaced with other objects.\footnote{Kelly~\cite{kelly1982basic} can be used as a standard reference for enriched category theory.}

\begin{defn}
\label{def:enrichedcat}
Given a monoidal category $K$, a $K$-enriched category \cat{C} has objects $\objs{C}$ such that:
\begin{itemize}
\item For every pair of $A,B\in\objs{C}$ there is a hom-object:
\begin{align}
\cat{C}(A,B)\in \objs{K}.
\end{align}
\item For all objects $A,B,C\in\objs{C}$ composition is given by a morphism in $\arrs{K}$ of type:
\begin{align}
 \circ_{A,B,C}:\cat{C}(B,C)\otimes\cat{C}(A,B)\to\cat{C}(C,A).
\end{align}
\end{itemize}
\end{defn}

Linear maps between Hilbert spaces themselves form a vector space, so \cat{FHilb} is in fact a $\cat{FVect}$-enriched category. Another way of saying this is that \cat{FHilb} is enriched in \cat{FVect}. As diagrams of a process theory are morphisms in a category, diagrams in an enriched category have additional operations. The \cat{FVect} enrichment of \cat{FHilb}, for example, allows us to sum diagrams, as in the following property. In the following definition \cat{Mon} is the category of monoids and monoid homomorphisms.


\begin{defn}
\label{def:resid}
In a \cat{Mon}-enriched QPT, a set of states $\{\ket{x}\}$ on a system $A$ forms a \textbf{resolution of the identity} when:
\begin{equation}
\label{eq:ResolutionId}
\input{tikz/3_multcharresolutionID.tikz}
\end{equation}
\end{defn}

\noindent where the sum operation comes from a monoid structure of the hom-object \cat{C}(A,A). In \cat{FHilb} more specifically, this addition is inherited  from the vector space enrichment. In fact, in Hilbert spaces all the classical states of any classical structure form a resolution of the identity. Further, in \cat{FHilb}, we are able to link the classical structure maps to classical states in the following ways~\cite{coecke2015generalised}:
\begin{align}
\label{eq:spidersums}
\input{tikz/3_spider_sums.tikz}
\end{align}
Arbitrary spiders can then be written as:
\begin{align}
\input{tikz/3_spider_sums_n_legs.tikz}
\end{align}

\begin{remark}
We emphasize that this connection between classical states and classical structures does not hold in general QPTs, but merely serves to further motivate abstract constructions that generalize from \cat{FHilb}. We make use of this technique in the following section on measurements.
\end{remark}

Gogioso provides more details on enriched categories, especially in regards to the results of Section~\ref{sec:strcomplFT} of this thesis, in~\cite[Sec. 6]{gogioso2015fourier}.

\section{Measurements}
\label{sec:measurements}

We have already introduced a notion of post-selected measurement in Definition~\ref{def:bornrule}. This section uses Selinger's CPM (completely positive map) construction~\cite{selinger2007dagger}, to present measurement as a process that outputs classical information~\cite{coecke2012strong}. In particular, measurements are maps that decohere pure states into mixed states in a certain basis. We need to be more precise about systems and their duals here so will be more consistent about using wires decorated with the proper arrows. 

Using caps and cups, we can construct the unique ``name" of a process $\rho$ as:
\begin{equation}
\label{eq:cj}
\input{tikz/2_mapstateduality.tikz}
\end{equation}
This is the usual Choi-Jamiolkowski  isomorphism for map-state duality in quantum information. The ``doubling" that occurs here, provides a natural way to represent measurements as completely positive maps. Suppose we wish to measure a system with respect to a classical structure (\dotonly{graydot}),
whose classical states form an orthonormal basis $\{ \ket{x_i} \}$. The probability
of getting the $i$-th measurement outcome is computed using the Born
rule: 
\begin{equation}
 \mbox{Prob}(i, \rho) = \mbox{Tr}(\ketbra{x_i}{x_i} \rho) 
\end{equation}

We can write this probability distribution as a vector in the basis $\{ \ket{x_i} \}$. That is, a vector whose $i$-th entry is the probability of the $i$-th outcome:
\begin{equation}
M_{\dotonly{graydot}}(\rho) = \sum 
\mbox{Tr}(\ketbra{x_i}{x_i} \rho) \ket{x_i}
\end{equation}

So, $M$ defines a linear map from density matrices to probability distributions. Expanding this graphically, we have:
\begin{equation}
\sum_i \mbox{Tr}\left( \input{tikz/2_Prho.tikz} \right) 
\begin{pic}
\node [style={gray point}] (m) at (0, 1.25) {$i$};
\draw [diredge] (m) to (0,2.25);
\end{pic}
\;\stackrel{\eqref{eq:trace}}{=}\;
\input{tikz/2_measurementderivation2.tikz}
\end{equation}

\begin{defn}[{\cite{coecke2012strong}}]
\label{def:meas2}
  For a classical structure \graycomonoid{A}, a measurement in that classical structure is defined as
  the following map: 
  \begin{equation}
\input{tikz/2_measurement.tikz}
  \end{equation}
\end{defn}

While the exact Born rule derivation does not apply in all QPTs (as traces and the linear structures are different in general), we can still consider Definition~\ref{def:meas2} as the abstract version of a QPT measurement.

\begin{example}
We will illustrate this measurement using an example on a qubit. Take the $Z$ and $X$ classical structures to be defined as they were in Example~\ref{ex:cnot}. Here $Z$ is the computational basis and the two classical points of $X$ are $\begin{pic}[scale=0.5]
\node (o) at (0,1) {};
\node (t) [point, fill=gray, scale=0.9] at (0,0) {$0$};
\draw (o) to (t);
\end{pic}=\ket{+}$ and
$\begin{pic}[scale=0.5]
\node (o) at (0,1) {};
\node (t) [point, fill=gray, scale=0.9] at (0,0) {$1$};
\draw (o) to (t);
\end{pic}=\ket{-}$. Thus a measurement of the state $\ket{0}$ in the $X$ (gray) basis is $m_{\dotonly{graydot}}\circ\ket{0}\bra{0}$, which is diagrammatically:
\begin{equation}
\input{tikz/3_measExample.tikz}
\end{equation}
This gives the expected result that a measurement of $\ket{0}$ in the $X$ basis is a mixed state of $\ket{+}$ and $\ket{-}$. 
\end{example}

We especially make use of this measurement presentation in Chapter~\ref{chap:mermin}, to describe Mermin non-locality tests.

\section{Summary of QPTs}

This chapter has introduced the main framework that is used in this thesis. This framework performs two functions:
\begin{enumerate}
\item our framework lifts the structure of quantum computation into the general setting of quantum-like process theories. This allows the study of protocols like teleportation, state transfer~\cite{cqm-notes}, quantum secret sharing (Section~\ref{section_QSS}), quantum bit commitment~\cite{katriel-commitment}, Mermin non-locality tests (Chapter~\ref{chap:mermin}), and other protocols (Chapters~\ref{chap:qalg} and~\ref{chap:qDisCo}) to be handled at an abstract level and studied in different QPTs.
\item The categorical diagrams that accompany QPTs present a more powerful quantum circuit language. This is one that includes bases, classical information, complementarity, and phases explicitly and gives rules for manipulating these structures within the diagrams themselves. We present a summarizing table of these structures at the end of this chapter.
\end{enumerate}
Both the generalization and diagrammatic formalism provide powerful tools throughout the results in this thesis.

%\newpage

\begin{figure}[b]
\caption{A summary of the diagrammatic elements for QPTs above and beyond quantum circuit diagrams.}\vspace{-10pt}
{\renewcommand{\arraystretch}{2}\small
\begin{tabulary}{\linewidth}{|p{7cm}|c|}\hline
The \textbf{dagger} $\dagger:\cat{C}\to\cat{C}$ generalizes the adjoint and vertically flips the whole diagram. See Definition \ref{defn:dagger}.
& \input{tikz/1_daggerPics.tikz} \\\hline
\textbf{Scalars} $s:I\to I$. These float freely in diagrams. See Definition \ref{defn:scalar}.
& \begin{pic}[xscale={\tikzxscale}, yscale={\tikzyscale}]
\node [whitedot] (0) at (0, -1) {$s$};
\node [none] (0) at (0, -0) {};
\end{pic} \\\hline
\textbf{States} and \textbf{effects}. Definition~\ref{def:effect}
& \left(
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=0.5*\tikzyscale]
\node (1) [state] at (0, -0.75) {$\psi$};
\node (2) at (0, 0.75) {};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned}\right)^{\dagger}
=
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=0.5*\tikzyscale]
\node (1) at (0, -0.75) {};
\node (2) [state, hflip] at (0, 0.75) {$\psi$};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned} \\\hline
\textbf{Post-selected measurement}.\newline Section~\ref{sec:bornrule}.
& \langle\phi|\psi\rangle =
\begin{aligned}
\begin{tikzpicture}[xscale=\tikzxscale, yscale=0.5*\tikzyscale]
\node (1) [state,scale=0.8] at (0, -0.75) {$\psi$};
\node (2) [state, scale=0.8, hflip] at (0, 0.75) {$\phi$};
\draw (1.center) to (2.center);
\end{tikzpicture}
\end{aligned} \\\hline
\textbf{Bell states} and measurements, i.e. cups and caps. Definition~\ref{def:dual}.
& \input{tikz/2_capcup.tikz} \\\hline
\textbf{Duals}. See \eqref{eq:dualmorph}.
& \input{tikz/2_dualmorph.tikz} \\\hline
\textbf{Classical structures} i.e. generalized observables (Def.~\ref{def:classicalstruct}). Their \textbf{classical states} (Def. \ref{def:copyables}) act as ``basis elements". 
& \input{tikz/2_spiderdef.tikz} \\\hline
\textbf{Dimension} of a system. Def.~\ref{def:dimension}
& d(A) \;= \;
\begin{pic}[xscale={\tikzxscale}, yscale={0.5*\tikzyscale}]
\node (t) [whitedot] at (0,1) {};
\node (b) [whitedot] at (0,-1) {};
\draw (t) to (b.center);
\end{pic} \\\hline
\textbf{Phases}: A group of states for each classical structure. Section~\ref{sec:phases}.
& \input{tikz/2_decspidercomp.tikz} \\\hline
\textbf{Complementarity} between classical structures. Definition~\ref{def:complementarity}.
& \input{tikz/2_antipodehopf.tikz} \\\hline
\textbf{Measurement} by a classical structure. Definition~\ref{def:meas2}.
& \input{tikz/2_measurement.tikz} \\\hline
\end{tabulary}
}

\end{figure}