Pauli Magic ✨

A small header-only library that can help you with traces of tensor products that contain Pauli matrices. This is based on previous work by McUrbn for his Bachelor thesis. I merely implemented his ideas.

Background

The Pauli matrices σ_x, σ_y, σ_z are traceless 2×2-matrices that are defined such that σ_iσ_i=𝟙. Additionally there are the commutation relations [σ_aσ_b]=2ⅈε_abcσ_c and the anti-commutation relations {σ_aσ_b} = 2δ_ab𝟙. It follows directly, that for a≠b, σ_aσ_b=ⅈε_abcσ_c. For more information see the corresponding Wikipedia article.

For the product of two tensor products (a₁⊗b₁) and (a₂⊗b₂) it holds (this is very much a simplification, for more detail check out this Wikipedia article on the "Tensor product of algebras ") that (a₁⊗b₁)(a₂⊗b₂) = (a₁a₂⊗b₁b₂). Also for traces of tensor products it holds that Tr(a⊗b) = Tr(a)Tr(b).

This program uses said relations to simplify tensor products containing Pauli matrices and ultimately calculates traces of those.

Usage

Here's a minimal program that calculates the trace of the product of four tensor products:

#include <iostream>

#include "paulimagic/tensorproduct.h"

int main() {
  std::cout << (TensorProduct<4>{PM::X} * TensorProduct<4>{PM::Y} *
                TensorProduct<4>{PM::Z, PM::I, PM::iI} *
                TensorProduct<4>{PM::I, PM::nI}).trace() << '\n';
}

Compile it with the flags -O3 -Wall -Werror -pedantic -std=c++11 -I/path/to/cpp-paulimagic. It should print out the result (16,0) representing the complex number 16+0ⅈ. We can indeed verify this result by hand:

(σ_x⊗𝟙⊗𝟙⊗𝟙)(σ_y⊗𝟙⊗𝟙⊗𝟙)(σ_z⊗𝟙⊗ⅈ𝟙⊗𝟙)(𝟙⊗-𝟙⊗𝟙⊗𝟙) = (σ_xσ_yσ_z⊗-𝟙⊗ⅈ𝟙⊗𝟙) = (ⅈ𝟙⊗-𝟙⊗ⅈ𝟙⊗𝟙)

Tr(ⅈ𝟙⊗-𝟙⊗ⅈ𝟙⊗𝟙) = Tr(ⅈ𝟙)Tr(-𝟙)Tr(ⅈ𝟙)Tr(𝟙) = 2i × (-2) × 2i × 2 = 16