# Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product

Given a fermionic Hamiltonian in a matrix form, we can write it as a sum over Kronecker products of Pauli matrices using the Hilbert-Schmidt inner product. However if the same Hamiltonian is given in a operator form, we can use the Jordan-Wigner transformation to write it as a sum over Kronecker products of Pauli operators.

How can one show that both the methods will give the same result, or if that is not the case then how does one show that the two results are related in some way?

Any material that discusses this is also appreciated.

First let us consider spin-5/2 particles (they are still fermions). The fermionic Hamiltonian in this case will be $$6^n \times 6^n$$ matrix for $$n$$ particles, so for one spin-5/2 particle we have a 6x6 matrix which cannot be decomposed as a Kronecker product of 2x2 Pauli matrices.
• That does clear up a few things. What I am understanding from this is that if the fermionic Hamiltonian matrix is of the size $2^n \times 2^n$ then the Hilbert-Schmidt inner product is one of the possible methods to express the Hamiltonian as a Paulinomial. Is that the correct interpretation? Jul 27, 2021 at 1:06