Pauli matrices — simplify expressions without printing out the raw matrix

Question

Squaring a Pauli matrix results in the identity matrix.

These bits of documentation (weakly, to a Mathematica-newbie like me) imply that some algebraic identities that link the Pauli matrices together are built into Mathematica:

• http://mathworld.wolfram.com/notebooks/LinearAlgebra/PauliMatrices.nb
• (other basic Pauli Matrix documentation on the Wolfram pages)

I enter in Mathematica:

PauliMatrix[2]

FullSimplify[PauliMatrix[1] . PauliMatrix[1] . PauliMatrix[2]]

Is there a way of getting the second expression simplified to just PauliMatrix[2] because PauliMatrix[1] . PauliMatrix[1] is the identity matrix?

Mathematica simply prints out the resulting matrices for comparison rather than sticking to the algebra.

I see that someone else is trying to manipulate the Pauli matrices algebraically (for a similar purpose): http://homepage.cem.itesm.mx/lgomez/quantum/v7pauli.pdf (page 2. Setup Pauli Algebra)

... so I imagine that Mathematica doesn't have the behaviour built in?

Answer 1

I had actually answered this in a different thread that was eventually closed. Since that answer fits perfectly for this question, I'll use it here:

I will use $\sigma$ as an abbreviation for PauliMatrix.

The goal is to get back a result in terms of the symbolic matrices $\sigma$. Fortunately, this can always be done because the Pauli matrices when combined with the unit matrix form a basis of the vector space of two-dimensional matrices. This means that for a given matrix a, the equation Solve[{x1, x2, x3, x4}.PauliMatrix[{0, 1, 2, 3}] == a] has one and only one solution {x1, x2, x3, x4}. These coefficients can be used to form a linear expression using the symbolic matrices $\sigma$ in the end.

So here is a function pauliReduce which takes an expression involving the symbolic σ[i] (where i = 0, 1, 2, 3), and returns a simplified result in terms of the same symbols, and potentially the unit matrix which I called $\hat{1}$ to distinguish it from the number 1. The unit matrix is also given by σ[0].

Clear[pauliReduce]
pauliReduce[a_] := 
 Module[{x, symbolicPauliIndices, expression}, 
  x = Array[\[FormalX], 4];
  symbolicPauliIndices = 
   DeleteDuplicates[Cases[a, σ[i_Symbol] :> i, Infinity]];
  expression = {OverHat[1], σ[1], σ[2], σ[
      3]}.(x /. 
      First[Solve[
        x.PauliMatrix[{0, 1, 2, 3}] == a /. σ[i_] :> 
          Sum[KroneckerDelta[i, k] PauliMatrix[k], {k, 0, 3}], x]]);
  FullSimplify[expression, 
   Assumptions -> 
    Map[# ∈ Integers && 1 <= # <= 3 &, 
     symbolicPauliIndices]]]

Here are some relations that can now be proved:

pauliReduce[σ[1].σ[2]]

$\mathbb{i}$ σ$[3]$

pauliReduce[σ[1].σ[2] - σ[2].σ[1]]

2 $\mathbb{i}$ σ$[3]$

pauliReduce[σ[1].σ[1]]

$\hat{1}$

pauliReduce[σ[1].σ[2] + σ[2].σ[1]]

0

pauliReduce[MatrixExp[α σ[1]]]

Cosh[α] $\hat{1}$ + Sinh[α] σ$[1]$

pauliReduce[σ[i].σ[j] + σ[j].σ[i]]

pauliReduce[σ[1].σ[2].σ[3]]

$\mathbb{i} \hat{1}$

pauliReduce[σ[1].σ[1].σ[2]]

σ$[2]$

Answer 2

Try the following:

SetAttributes[simplifyPM, HoldFirst]

simplifyPM[expression_] := Module[
  {intermediate},
  intermediate = HoldForm[expression] //. PauliMatrix[a_].PauliMatrix[a_] :> IdentityMatrix[Dimensions[PauliMatrix[a]]];
  intermediate /. IdentityMatrix[_].m_?MatrixQ :> m
 ]

You can then use it as follows:

results = simplifyPM[PauliMatrix[1].PauliMatrix[1].PauliMatrix[2]]
(* Out: PauliMatrix[2] *)

The results are in a held form. If you want to obtain their actual value, you can release the hold:

ReleaseHold@results

(* Out: {{0, -I}, {I, 0}} *)

The function also works on multiple substitutions:

simplifyPM[PauliMatrix[4].PauliMatrix[4].PauliMatrix[3].PauliMatrix[3]]
ReleaseHold[%]

(* Out: 
IdentityMatrix[Dimensions[PauliMatrix[3]]] 
{{1, 0}, {0, 1}}
*)

Since Pauli matrices are always 2x2 matrices, you could also simplify the code above by using IdentityMatrix[2] in place of IdentityMatrix[Dimensions[PauliMatrix[a]]] in the definitions, but I thought I would leave the current expression in as a more general approach that may come in handy for cases in which the size of the matrix is not known a priori.