If I have a system of $N$ fermions, and a system of $N$ qubits/spin-$1/2$ particles, the Jordan-Wigner transformation allows me to represent the fermionic operators $a_i, a_i^\dagger$ in terms of Pauli matrices $X_i, Y_i, Z_i$ in the following way:
$$ a_i = Z_1 \otimes \cdots \otimes Z_{i-1} \otimes \frac{1}{2}(X_i – iY_i) \quad a_i^\dagger = Z_1 \otimes \cdots \otimes Z_{i-1} \otimes \frac{1}{2}(X_i + iY_i).$$
One can also invert this transformation. My intuition here is that this suggests that there is some sort of "isomorphism" between fermionic systems and qubit systems – usually the existence of the Jordan-Wigner transformation together with the fact that the fermionic and qubit Hilbert spaces are isomorphic is used to justify this in a non-rigorous way. But, it is not immediately clear what the precise relationship is. My question is: does the existance of the Jordan-Wigner transformation (or, others like the Parity or Bravyi-Kitaev transformations) rely on the existence of an isomorphism between some fermionic and qubit operator algebra? If so, what is this isomorphism?
I'm not sure how to answer this since the fermionic algebra is an associative $*-$algebra, whereas the qubit algebra $su(2^N)$ is a non-associative Lie algebra. But maybe the connection is between a representation of $su(2^N)$ or the UEA of $su(2^N)$ and the fermionic algebra?
Best Answer
I don't think any fancy technology is required to make Jordan-Wigner rigorous. At the end of the day, the fermionic operators $a_i, a_i^{\dagger}$ of $N$ modes and the Pauli operators $X_i$, $Y_i$, and $Z_i$ on $N$ qubits are both operators on finite dimensional Hilbert spaces over $\mathbb{C}$. You're therefore completely justified in thinking of each of the operators as $2^N \times 2^N$ matrices with complex entries. Associating occupation number states of the fermions with spin up/down states of the qubits in the "natural" way, the Jordan-Wigner transformation is literally just a matrix equality.