Any $2\times2$ matrix can be expressed as the sum and product of the identity, Pauli $X$ and Pauli $Z$ matrices. This is often used to express a singe-qubit state using just Pauli matrices. One can then use tensor products of single-qubit states and take linear combinations to build $n$-qubit states.
If I use the Heisenberg picture and I am interested in an arbitrary single-qubit observable, I can always write it as a sum and product of Pauli $X$ and $Z$ observables because it is just a $2\times2$ matrix.
But can I then write an $n$-qubit observable in terms of these single-qubit observables always by taking tensor products and linear combinations?
