Disclaimer: I wrote this blog post as a way of teaching myself how Laurens Lootens, et al. derive the Jordan-Wigner MPO using category theory. As such this blog is largely a reproduction of their work with some small changes in notation and a slight elaboration on the use of the module functor. I am by no means rigorous in what follows and if the reader would like a more detailed account of these topics I recommend Lootens, 2021 and Lootens, 2023.

We wish to find a magic tensor object (a so called “MPO Intertwiner”) that can apply the Jordan-Wigner transformation locally.

At first glance this problem seems intractable as under the Jordan-Wigner transformation we take local spin operators and convert them to highly non-local fermionic operators (and vise versa):

\begin{equation} X_i \rightarrow (-1)^{\sum_{j< i} c^{\dagger}_{j}c_{j}} c^{\dagger}_i, Y_i \rightarrow (-1)^{\sum_{j< i} c^{\dagger}_{j}c_{j}} c_i, Z_i \rightarrow 2c^{\dagger}_{i}c_{i} -I \end{equation}

To achieve this ambitous goal we must first step back and find a more generalised way of expressing operators that describe the same physics.

In particular, we want to find a way of constructing multiple operator bases $$\{ O_x \}$$ that satisfy the same operator product expansion (i.e. have the same $$f$$ in the following expression):

\begin{equation} O_x O_y = f^z_{x,y} O_z \end{equation}

To construct these operator bases we turn to category theory. Quite a lot of category theory background is required which I will attempt to summarise below:

### Category Theory

#### Fusion Category

First let us define a fusion category. We can think of fusion categories as a more general form of a group and is defined by the following five things:

1. Isomorphism classes called simple objects $$\alpha, \beta, \gamma, \ldots$$
• It is important to note that these classes are isomorphic to themselves but not to each other (or they would reduce into each other). Therefore, $$Hom(\alpha, \beta) = \varnothing$$ and $$Hom(\alpha, \alpha) = End(\alpha)$$
2. A tensor product rule $$\otimes$$ to define new objects e.g. $$\alpha \otimes \beta$$
3. A monoidal associator $$F: \alpha \otimes (\beta \otimes \gamma) \rightarrow (\alpha \otimes \beta) \otimes \gamma$$
4. The trival charge $$\mathbb{I}$$
5. A set of fusion rules which are isomorphisms between tensor products of simple objects and direct sums of simple objects $$\alpha \otimes \beta \cong \oplus_\gamma N_{\alpha \beta}^{\gamma} \gamma$$

We can represent the fusion rules using tensor network diagrams, and in particular if we pick a basis $$\{ i \}$$ of $$N_{\alpha \beta}^{\gamma}$$ then we can further decompose this into a sum over tensor network diagrams as shown below: Therefore, we can write the monoidal associator $$F$$ using the tensor network representation of fusion rules: As we can only have homomorphisms between the same isomorphism classes we can rewrite this map as: We can also use this description to understand the “pentagon” consitency equation these associators must obey:

\begin{equation} FF \cong FFF \end{equation} #### Module Category

Now let us define a module category $$\mathcal{M}$$ over $$\mathcal{D}$$ by the following three properties:

1. Isomorphism classes of simple objects $$A, B, C, \ldots$$
2. An action $$- \triangleleft -: \mathcal{M} \times \mathcal{D} \rightarrow \mathcal{M}$$, e.g. $$A \triangleleft \alpha \cong B$$
3. A module associator $$^\triangleleft F: A \triangleleft (\alpha \otimes \beta) \rightarrow (A \triangleleft \alpha) \triangleleft \beta$$

Similarly to before we can represent the action using a tensor network diagram: and then we can use this along with the fusion rule representation to draw a tensor network representation of the module associator: As the dual to these vectors is found by simply flipping them upside down, we can multiply the LHS by the dual of the right and find an expression for $$^\triangleleft F$$ in terms of the tensor network representation we have defined. Crucial point: This new object $$^\triangleleft F$$ must obey a “pentagon” consistency condition equation similar to that described for the fusion category and this condition is written as below and only involves the monodial associator $$F$$ so holds for any choice of module $$\mathcal{M}$$: As this equation fully defines how we combine these objects, if we construct an algrebra out of these $$^\triangleleft F$$ then it will obey the same operator product expansion for any choice of $$\mathcal{D}$$-module category $$\mathcal{M}$$. For ease of reference throughout the rest of this post we will refer to the algebra constrcuted out of these $$^\triangleleft F$$ objects over $$\mathcal{D}$$-module category $$\mathcal{M}$$ as the $$(\mathcal{D}, \mathcal{M})$$-bond algebra.

### Module Functors

In this language dualities like the Jordan-Wigner transformation are local tensors that can convert from a $$(\mathcal{D}, \mathcal{M})$$-bond algebra to an $$(\mathcal{D}, \mathcal{N})$$-bond algebra. The natural candidate to consider is module functors.

A $$\mathcal{D}$$-module functor is described by the following properties:

1. Two module categories $$\mathcal{M},\mathcal{N}$$ over $$\mathcal{D}$$
2. A functor $$G: \mathcal{M} \rightarrow \mathcal{N}$$
3. A module functor associator $$^G F: G(A \triangleleft \alpha) \rightarrow G(A) \triangleleft \alpha$$ for all $$\alpha \in \mathcal{D}, A \in \mathcal{M}$$

Consider the “pentagon” equation, here we label $$^\triangleleft _\mathcal{D}F$$ with the module they are taken over:

\begin{equation} G(A \triangleleft (\alpha\otimes \beta)) = {^G F}\> G(A) \triangleleft (\alpha \otimes \beta) ={^G F^\triangleleft _\mathcal{M}F} \>(G(A) \triangleleft \alpha)\triangleleft \beta \end{equation} \begin{equation} G(A \triangleleft (\alpha\otimes \beta)) = {^\triangleleft _\mathcal{N}F} \>G((A \triangleleft \alpha)\triangleleft \beta) = {^\triangleleft _\mathcal{N}F^G F} \>G(A \triangleleft \alpha)\triangleleft \beta = {^\triangleleft _\mathcal{N}F^G F ^G F} \>(G(A) \triangleleft \alpha)\triangleleft \beta \end{equation} \begin{equation} {^\triangleleft _\mathcal{N}F^G F ^G F} = {^G F^\triangleleft _\mathcal{M}F} \end{equation}

We can represent $$^G F$$ in tensor network form similarly to before: where $$A, B \in \mathcal{M}$$ and $$C, D \in \mathcal{N}$$. We can write the “pentagon” equation as: where $$A, B, C \in \mathcal{M}$$ and $$A’, B’, C’ \in \mathcal{N}$$.

It is clear from this diagram that we can pull $$^G F$$ through $$^\triangleleft_\mathcal{M} F$$ to convert it to $$^\triangleleft_\mathcal{N} F$$, and hence by “pulling $$^G F$$ through” a $$(\mathcal{D}, \mathcal{M})$$-bond algebra we must convert it to a $$(\mathcal{D}, \mathcal{N})$$-bond algebra. Therefore, the Jordan-Wigner transformation is a $$\mathcal{D}$$-module functor associator such that the $$(\mathcal{D}, \mathcal{M})$$-bond algebra has a representation in terms of Pauli operators and the $$(\mathcal{D}, \mathcal{N})$$-bond algebra has a representation in terms of fermionic operators.

### TLDR

You can use category theory to construct operator bases that obey the same operator product expansions by composing “module associator” tensors together, and further one can find a “module functor associator” tensor that converts from one basis to the other if you “pull it through” the “module associator” tensors. It should be possible to find a “module functor associator” tensor that converts from spin operator bases to fermionic operator bases whilst retaining the same operator product expansion a.k.a the Jordan-Wigner transformation.

## Back to Jordan-Wigner

Now we have established all the nitty-gritty of the category theory required, we can get back to considering how this applies in particular to the Jordan-Wigner transformation. In order to convert our cateogry theory abstract nonsense into a more set theoretic understanding, we need to introduce a Hilbert space for our bond algebras to act on. We make the following choice of Hilbert space as we are interested in operators acting upon local states:

\begin{equation} \mathcal{H} = \bigoplus_{\{A\}}\bigoplus_{\{\alpha\}} \bigotimes_i \mathcal{V}_{i+\frac{1}{2}} \end{equation} where \begin{equation} \mathcal{V}_{i+\frac{1}{2}} = Hom_\mathcal{M}(A_i \triangleleft \alpha_{i+\frac{1}{2}}, A_{i+1}), \alpha_{j+\frac{1}{2}} \in \mathcal{D}, A_j \in \mathcal{M} \end{equation}

This space is of the correct form to be acted upon by the bond algebra as the indicies on the outside of $$^\triangleleft F$$ are labels for basis vectors in the space $$Hom_\mathcal{M}(A \triangleleft \alpha,B), \alpha \in \mathcal{D}, A, B \in \mathcal{M}$$

First we need to show for $$\mathcal{D} = sVec$$ and $$\mathcal{M} = sVec$$ this hilbert space represents the spin basis hilbert space, whilst for $$\mathcal{D} = sVec$$ and $$\mathcal{M} = sVec/\{\mathbb{1} = \psi\}$$ this hilbert space represents the fermionic hilbert space.

$$sVec$$ is a braided fusion category equipped with:

1. Two simple isomorphism categories $$\mathbb{1}, \psi$$
2. Fusion rules $$\mathbb{1} \otimes \mathbb{1} \cong \psi \otimes \psi \cong \mathbb{1}, \>\>\mathbb{1} \otimes \psi \cong \psi \otimes \mathbb{1} \cong \psi$$
3. A field $$\mathbb{C}^{1|1} = \mathbb{C}^{1|0} \oplus \mathbb{C}^{0|1}$$ with $$\mathbb{1} \cong \mathbb{C}^{1|0}$$ and $$\psi \cong \mathbb{C}^{0|1}$$
4. Braiding rule $$\alpha \otimes \beta \cong (-1)^{|\alpha||\beta|} \beta \otimes \alpha$$ where $$|\alpha| = 0$$ if $$\alpha \in \mathbb{C^{1|0}}$$ and $$|\alpha| = 1$$ if $$\alpha \in \mathbb{C^{0|1}}$$

We need to further specify the module action in the two choices:

• $$\mathcal{M} = sVec$$ - we have $$A \triangleleft \alpha \cong A \otimes \alpha$$
• $$\mathcal{M} = sVec/\{\mathbb{1} = \psi\}$$ - we have $$\mathbb{1} \triangleleft \mathbb{1} \cong \mathbb{C}^{1|0} \times \mathbb{1}$$, $$\mathbb{1} \triangleleft \psi \cong \mathbb{C}^{0|1} \times \mathbb{1}$$

Consider $$\mathcal{V}_{i+\frac{1}{2}}$$ for $$\mathcal{D} = sVec$$ and $$\mathcal{M} = sVec$$. In this instance, we have

\begin{equation} \mathcal{V}_{i+\frac{1}{2}} = \begin{cases}End(A_{i+1}) & \text{if } A_{i}\otimes \alpha_{i+\frac{1}{2}} \cong A_{i+1}\\ \varnothing & \text{if } A_{i}\otimes \alpha_{i+\frac{1}{2}}\ncong A_{i+1}\\ \end{cases} \end{equation}

Therefore, the non-vanishing parts of this hilbert space are completely specified by the choices of modules $$\{A_i\}$$, as $$A_{i}\otimes \alpha_{i+\frac{1}{2}} \cong A_{i+1}$$ if only satsfied by a unique $$\alpha_{i+\frac{1}{2}}$$ for a given $$A_i, A_{i+1}\in sVec$$. Therefore, the Hilbert space is given by:

\begin{equation} \mathcal{H} = \bigotimes_i End(\mathbb{1}) \oplus End(\psi)= \bigotimes_i \mathbb{C}^{1|0}\oplus \mathbb{C}^{1|0} \cong \bigotimes_i \mathbb{C}\oplus \mathbb{C} \cong \bigotimes_i \alpha \ket{0} + \beta \ket{1}, \alpha, \beta \in \mathbb{C} \end{equation}

Therefore, this hilbert space is equivalent to the hilbert space over the spin basis (or qubit computational basis).

Consider $$\mathcal{V}_{i+\frac{1}{2}}$$ for $$\mathcal{D} = sVec$$ and $$\mathcal{M} = sVec/\{\mathbb{1} = \psi\}$$. In this instance, we have

\begin{equation} \mathcal{V}_{i+\frac{1}{2}} = \begin{cases}Hom_{sVec/\{\mathbb{1} = \psi\}}( \mathbb{C}^{1|0} \times \mathbb{1}, \mathbb{1}) & \text{if } \alpha_{i+\frac{1}{2}} \cong \mathbb{1}\\ Hom_{sVec/\{\mathbb{1} = \psi\}}( \mathbb{C}^{0|1} \times \mathbb{1}, \mathbb{1}) & \text{if } \alpha_{i+\frac{1}{2}} \cong \mathbb{\psi}\\ \end{cases} \end{equation}

Therefore,

\begin{equation} \mathcal{H} = \bigotimes_i Hom_{sVec/\{\mathbb{1} = \psi\}}( \mathbb{C}^{1|0} \times \mathbb{1}, \mathbb{1}) \oplus Hom_{sVec/\{\mathbb{1} = \psi\}}( \mathbb{C}^{0|1} \times \mathbb{1}, \mathbb{1}) \end{equation} \begin{equation} \mathcal{H} = \bigotimes_i \mathbb{C}^{1|0} \oplus \mathbb{C}^{0|1} \cong \bigotimes_i \alpha \ket{\varnothing} + \beta c^{\dagger}_i \ket{\varnothing}, \alpha, \beta \in \mathbb{C} \end{equation}

where $$c^{\dagger}_i$$ is the fermionic creation operator. Therefore, this hilbert space is equivalent to the fermionic basis.

Therefore, the $$(sVec, sVec)$$-bond algebra is made up of qubit operators that act on the computational basis states (e.g. Pauli gates etc.), whereas the $$(sVec, sVec/\{\mathbb{1} = \psi\})$$-bond algebra is made up of fermionic operators that act on the fermionic basis (e.g. creation/annihilation operators). If we can define a module functor between $$sVec$$ and $$sVec/\{\mathbb{1} = \psi\}$$ then we can find a module functor associator which we have proved allows us to convert from one bond algebra to the other.

We can choose the forgetful module functor $$G: A \rightarrow \mathbb{1}$$ for $$A \in sVec$$. This functor has two basis vectors: $$Hom(\psi, \mathbb{1}) \cong \mathbb{C}^{0|1}$$ and $$Hom(\mathbb{1}, \mathbb{1})\cong \mathbb{C}^{1|0}$$. Let $$N_i(A) = (c_i^{\dagger})^{n(A)} \ket{\varnothing}$$ where

\begin{equation} n(A) = \begin{cases}1 & \text{if } A = \psi\\ 0 & \text{if } A = \mathbb{1}\end{cases} \end{equation}

As $$N(\mathbb{1}) \cong \mathbb{C}^{1|0}$$ and $$N(\psi) \cong \mathbb{C}^{0|1}$$, $$N(A)$$ is the basis for this module functor. Similarly $$N(\alpha)$$ is the basis for $$Hom(\mathbb{1} \triangleleft \alpha, \mathbb{1})$$ with $$\alpha \in sVec$$. More simply $$\alpha$$ is the basis for $$Hom(A \triangleleft \alpha, B)$$ with $$\alpha, A, B \in sVec$$. With this in mind let us consider the module functor associator: as we know there is only one unique $$B$$ that $$A \triangleleft \alpha$$ maps to we can write: As this module functor must satisfy the “pentagon” equation when we “pull it through” the $$(sVec, sVec/\{\mathbb{1} = \psi\})$$-bond algebra it will transform to the $$(sVec, sVec)$$-bond algebra and vise versa. Therefore, this is a local MPO that transforms fermionic operators to and from spin operators. Therefore, our Jordan-Wigner MPO is given by: 