---
title: "Pauli matrices via Clifford algebra"
author: "Robin K. S. Hankin"
output: html_vignette
bibliography: clifford.bib
vignette: >
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteIndexEntry{Pauli matrices via Clifford algebra}
  %\usepackage[utf8]{inputenc}
---
 

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library("clifford")
library("onion")
library("jordan")
library("emulator")
```

```{r out.width='15%', out.extra='style="float:right; padding:10px"',echo=FALSE}
knitr::include_graphics(system.file("help/figures/clifford.png", package = "clifford"))
knitr::include_graphics(system.file("help/figures/jordan.png", package = "jordan"))
```

To cite the `clifford` package in publications please use
@hankin2022_clifford.  This short document shows how the Pauli
matrices, often used in quantum mechanics, can be calculated using
Clifford algebra as implemented by the `clifford` R package.  The
Pauli matrices are set of three $2\times 2$ matrices with complex
entries.  They represent observables corresponding to measuring spin
along the $x$, $y$, and $z$ axes.  They are also useful when
considering polarized light.  The Pauli matrices have a pleasing
relationship with Jordan algebra [@hankin2023_jordan].  In component
form, they are:

\[
\sigma_0=\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\qquad
\sigma_x=\left(\begin{matrix}0&1\\1&0\end{matrix}\right)\qquad
\sigma_y=\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)\qquad
\sigma_z=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)
\]

We observe that $\sigma_x\sigma_y=i\sigma_z$,
$\sigma_y\sigma_z=i\sigma_x$, and $\sigma_z\sigma_x=i\sigma_y$, and
further that
$\sigma_x^2=\sigma_y^2=\sigma_z^2=-i\sigma_x\sigma_y\sigma_z=\sigma_0$.

The non-identity Pauli matrices [that is,
$\sigma_x,\sigma_y,\sigma_z$] are subject to the following commutation
relations:

\[
\left[\sigma_x,\sigma_y\right]=2i\sigma_z\qquad
\left[\sigma_y,\sigma_z\right]=2i\sigma_x\qquad
\left[\sigma_z,\sigma_x\right]=2i\sigma_y\]

(here, $\left[x,y\right]=xy-yx$).  We also have the following
anticommutation relations:

\[
\left\lbrace\sigma_x,\sigma_y\right\rbrace=2i\sigma_z\qquad
\left\lbrace\sigma_y,\sigma_z\right\rbrace=2i\sigma_x\qquad
\left\lbrace\sigma_z,\sigma_x\right\rbrace=2i\sigma_y\]

(here, $\left\lbrace x,y\right\rbrace=xy+yx$).

Because any $2\times 2$ Hermitian matrix may be expressed as
$A\sigma_0+B\sigma_x+C\sigma_y+D\sigma_z$ for $A,B,C,D\in\mathbb{R}$,
we observe that the anticommutation relations imply that the Pauli
matrices are closed under the Jordan operator $x\circ y=(xy+yx)/2$.
For more details, see the `jordan` package [@hankin2023_jordan] which
implements this operation in a more general context.  The Jordan
multiplication rule is


\[
\sigma_a\sigma_b=\delta_{ab}I_2 + i\epsilon_{abc}\sigma_c
\]

which suggests the following identification:


\begin{aligned}
\sigma_0&\longleftrightarrow 1\\
\sigma_x&\longleftrightarrow e_1\\
\sigma_y&\longleftrightarrow e_2\\
\sigma_z&\longleftrightarrow e_3\\
\end{aligned}

Then we make the _formal_ identifications:

\begin{aligned}
i\sigma_x&\longleftrightarrow e_2e_3\\
i\sigma_y&\longleftrightarrow e_3e_1\\
i\sigma_z&\longleftrightarrow e_1e_2\\
\end{aligned}

and so we recover the Pauli matrix relations from the Clifford
algebra.


# Implementation

Let us start with the Pauli matrices:



\[
\sigma_0=\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\qquad
\sigma_x=\left(\begin{matrix}0&1\\1&0\end{matrix}\right)\qquad
\sigma_y=\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)\qquad
\sigma_z=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)
\]


\[
i\sigma_0=\left(\begin{matrix}i&0\\0&i\end{matrix}\right)\qquad
i\sigma_x=\left(\begin{matrix}0&i\\i&0\end{matrix}\right)\qquad
i\sigma_y=\left(\begin{matrix}0&1\\-1&0\end{matrix}\right)\qquad
i\sigma_z=\left(\begin{matrix}i&0\\0&-i\end{matrix}\right)
\]


Given a general complex matrix

\[
\left(\begin{matrix}
\alpha  +\beta i  &  \gamma+\delta i\\
\epsilon+\zeta i  &  \eta+\theta i  
\end{matrix}\right)
\]

we see that

\begin{eqnarray}
\sigma_0&=(\alpha+\eta)/2\qquad i\sigma_0=(\beta+\theta)/2\\
\sigma_x&=(\gamma+\epsilon)/2\qquad i\sigma_x=(\delta+\xi)/2\\
\sigma_y&=(\gamma-\epsilon)/2\qquad i\sigma_x=(\delta-\xi)/2\\
\sigma_z&=(\alpha-\eta)/2\qquad i\sigma_z=(\beta-\theta)/2\\
\end{eqnarray}

# R implementation



```{r}
s0 <- matrix(c(1,0,0,1),2,2)
sx <- matrix(c(0,1,1,0),2,2)
sy <- matrix(c(0,1i,-1i,0),2,2)
sz <- matrix(c(1,0,0,-1),2,2)
```

Given a general complex matrix `M`, we may coerce this to Clifford
form as follows:

```{r}
matrix_to_clifford <- function(M){
      (Re(M[1,1] + M[2,2]))/2             + 
      (Re(M[1,1] - M[2,2]))/2*e(c(  3  )) +
      (Im(M[1,1] + M[2,2]))/2*e(c(1,2,3)) + 
      (Im(M[1,1] - M[2,2]))/2*e(c(1,2  )) +

      (Re(M[2,1] + M[1,2]))/2*e(c(1    )) + 
      (Re(M[2,1] - M[1,2]))/2*e(c(1,  3)) +
      (Im(M[2,1] + M[1,2]))/2*e(c(  2,3)) + 
      (Im(M[2,1] - M[1,2]))/2*e(c(  2  ))
}
```

and then test it as follows:

```{r}
rmat <- function(...){matrix(rnorm(4),2,2) + 1i*matrix(rnorm(4),2,2)}
M <- rmat()
M
matrix_to_clifford(M)
```

We can now test whether `matrix_to_clifford()` is a group
homomorphism:

```{r}
M1 <- rmat()
M2 <- rmat()

diff <- matrix_to_clifford(M1)*matrix_to_clifford(M2) - matrix_to_clifford(M1 %*% M2)
diff
Mod(diff)
```

We see agreement to numerical precision.  Now we can coerce from a
Clifford to a matrix:

```{r}
`clifford_to_matrix` <- function(C){
   return(
                          const(C)*s0 + getcoeffs(C,list(1))*sx 
  +           getcoeffs(C,list(2))*sy + getcoeffs(C,list(3))*sz
  + getcoeffs(C,list(c(1,2,3)))*1i*s0 + getcoeffs(C,list(c(  2,3)))*1i*sx 
  - getcoeffs(C,list(c(1,  3)))*1i*sy + getcoeffs(C,list(c(1,2  )))*1i*sz
  )
} 
```

```{r}
rc <- function(...){rcliff(100,d=3,g=3)}
C <- 104 + rc()
C
clifford_to_matrix(C)
```

Now test that the two coercion functions are inverses of one another:

```{r}
clifford_to_matrix(matrix_to_clifford(M)) - M 
matrix_to_clifford(clifford_to_matrix(C))- C
```

Now we can establish that  `clifford_to_matrix()` is a homomorphism:

```{r}
C1 <- 222 + rc()
C2 <- 333 + rc()
clifford_to_matrix(C1*C2) - clifford_to_matrix(C1)%*%clifford_to_matrix(C2)
```

## Closure

The reason that Pauli matrices are useful in physics is that they are
closed under the Jordan operation $x\circ y=(xy+yx)/2$, which we will
verify for matrices and their Clifford representation.

```{r}
M1 <- as.1matrix(rchm(1,2))
M2 <- as.1matrix(rchm(1,2))
M1
M2
p1 <- (M1 %*% M2 + M2 %*% M1)/2
p1 - ht(p1)  # zero for Hermitian matrices
```

Above, see how $M_1\circ M_2$ is Hermitian.  Now, in Clifford form:

```{r}
C1 <- matrix_to_clifford(M1)
C2 <- matrix_to_clifford(M2)
p2 <- (C1 * C2 + C2 * C1)/2
p2
```

above, see how the clifford product `p2` is a pure Pauli matrix as its
only nonzero coefficients are those of the scalar and the grade-one blades:

```{r}
grades(p2)
```


## References
