Let us now apply the concept of a symmetric monoidal category (SMC) to quantum mechanics in finite dimensions. In fact, the first step is rather straight-forward:

Definition 1. [1] The category FdHilb consists of a symmetric monoidal category (SMC) with finite-dimensional complex Hilbert spaces as objects and linear transformations as arrows. Arrow composition is provided by ordinary matrix multiplication. The monoidal structure is provided by the ordinary matrix tensor product $\otimes$.

Remember how we complained right at the beginning that one reason for Dirac notation being clumsy because it allows for meaningless global phases? Well, FdHilb currently has exactly the same issue. Let’s fix this!

Definition 2. [2] The category $\mathbf{FdHilb}_{wp}$ has the same objects and arrows as FdHilb, however linear maps are subject to the equivalence condition $f\equiv g$ iff there exists $\theta\in\mathbb{R}$ such that $f=e^{i\theta}g$.

So what have we achieved so far? Let’s pick some simple object within $\mathbf{FdHilb}_{wp}$ – for example, the well-known Qubit of type $\mathbb{C}^2$. But what can we do with this? The answer is: Not that much. We can juxtapose multiple Qubits into composite systems, and define composite linear transformations to get from one composite system to another. All of this, we can of course do either formally, using equations, or graphically, using the graphical notation introduced for SMCs.

So how can we get to do more exciting things? Let’s look into the nature of Hilbert spaces and augment $\mathbf{FdHilb}_{wp}$ with appropriate inner structure, starting with general properties and then gradually moving on to finer structure.

What again distinguishes a Hilbert space from an Euclidean space? We faintly remember from our undergrad times that Hilbert spaces admit a scalar product, that, unlike in Euclidean spaces, has to be positive-definite complex. This additional constraint gives rise to the concept of a dual space, which we propose to capture as follows:

Definition 3. [3] A compact closed category is a symmetric monoidal category (SMC) where each object $A$ is assigned a dual object $A^*$ together with a unit map $\eta_A:I\rightarrow A^*\otimes A$ and a counit map $\epsilon_A:A\otimes A^*\rightarrow I$, such that \(\\lambda^{-1}\_A\\circ(\\epsilon\_A\\otimes A)\\circ\\alpha^{-1}\_{A,A^\*,A}\\circ(A\\otimes\\eta\_A)\\circ\\rho\_A=id\_{A}\) and \(\\rho^{-1}\_A\\circ(A^\*\\otimes\\epsilon\_A)\\circ\\alpha\_{A^\*,A,A^\*}\\circ(\\eta\_A\\otimes A^\*)\\circ\\lambda\_A=id\_{A^\*}\)

The consistency conditions on $\eta_A$ and $\epsilon_A$ maybe looking slightly outlandish, but if you have a closer look (or draw them as a little diagram), you will realize that they are just constraining $\eta$ and $\epsilon$ to be appropriate primal/dual space preparators and destructors.

What other properties make Hilbert spaces special? After having introduced structure reflecting positive-definite complex scalar products, we now need to impose an important property on morphisms: unitarity.

Definition 4. [4] A $\mathbf{\dagger}$-symmetric monoidal category ($\dagger$-SMC) is a symmetric monoidal category equipped with an identity-on-objects contravariant endofunctor $(-)^\dagger: \mathbf{C}^{op}\rightarrow\mathbf{C}$, which assigns to each morphism $f:A\rightarrow B$ and adjoint morphism $f^\dagger:B\rightarrow A$, which coherently preserves the monoidal structure, i.e.: $(f\circ g)^\dagger=g^\dagger\circ f^\dagger, \ (f\otimes g)^\dagger=f^\dagger\otimes g^\dagger,\ 1^\dagger_A=1_A$ and $f^{\dagger\dagger}=f$. Further, for the natural isomorphisms $\lambda, \rho,\alpha$ and $\sigma$ of the symmetric monoidal structure, the adjoint and the inverse coincide.

Combining both unitarity for morphism and a positive-definite complex scalar products:

Definition 13. [5] A $\mathbf{\dagger}$-compact closed category is a $\mathbf{\dagger}$-symmetric monoidal category that is also, and such that the following diagram commutes:

$\dagger$-compact closed categories equally admit a diagrammatic calculus:

And, to represent primal and dual spaces and the associated scalar product (we enclose the equivalent Dirac bra-ket notation):

Congratulations, now we have captured the fundamental properties of Hilbert spaces.

The following important theorem allows custom yanking and bending of wires in $\dagger$-compact categories:

Theorem 2. [5, 6, 7] An equation in the symbolic language of a $\dagger$-compact category follows from the axioms of $\dagger$-compact categories if and only if it holds up to isotopy in the graphical language.

The symbolic language of $\dagger$-compact categories allows for insightful representations of quantum phenomena, such as quantum teleportation and entanglement swapping. We will not discuss these examples at this stage, but we will get back to them later.

Taking a step back, we realize that we have not yet considered a quantum mechanical process of fundamental importance: measurement.

References

[1] Coecke, B., and Paquette, E. O. Categories for the practising physicist. arXiv e-print 0905.3010, May 2009.

[2] Duncan, R., and Perdrix, S. Graph states and the necessity of Euler decomposition. arXiv e-print 0902.0500, Feb. 2009.

[3] Selinger, P. Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170 (2007) 139-163.

[4] Coecke, B., and Duncan, R. Interacting quantum observables: Categorical algebra and diagrammatics. arXiv e-print 0906.4725, June 2009. New J. Phys. 13 (2011) 043016

[5] G. M. Kelly and M. L. Laplaza (1980) Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213.

[6] P. Selinger (2005) Dagger compact closed categories and completely positive maps. Electronic Notes in Theoretical Computer Science 170, 139–163. www.mathstat.dal.ca/∼selinger/papers.html#dagger

[7] P. Selinger (2010) Autonomous categories in which $A\simeq A^*$. Extended abstract. In: Proceedings of the 7th International Workshop on Quantum Physics and Logic, May 29-30, Oxford. www.mscs.dal.ca/∼selinger/papers.html#halftwist

[8] Abramsky, S., and Coecke, B. A categorical semantics of quantum
protocols. arXiv e-print quant-ph
/0402130, Feb. 2004. Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004).