  The physics of the paradigmatic one-dimensional transverse field quantum
Ising model $H= J \sum_{\langle i,j \rangle} \sigma^x_i \sigma^x_j + h \sum_i
\sigma^z_i$ is well-known. Instead, let us imagine "applying" the transverse
field via a transverse Ising coupling of the spins to partner auxiliary spins,
i.e. $H= J_x \sum_{\langle i,j \rangle} \sigma^x_i \sigma^x_j + J_z \sum_i
\sigma^z_i \sigma^z_{\text{partner of }i}$. If each spin of the chain has a
unique auxiliary partner, then the theorem states that the resultant
eigenspectrum is still the same as that of the quantum Ising model with
$\frac{h}{J} = \frac{J_z}{J_x}$ and the degeneracy of the entire spectrum is
$2^{\text{number of auxiliary spins}}$. This follows from the existence of
extensively large and mutually anticommuting sets of conserved quantities for
$H$. We can interpret this as the auxiliary spins remaining paramagnetic down
to zero temperature with an extensive ground state degeneracy. This is lost
upon the loss of the unique partner condition for the full spin chain. Other
cases where such extensive degeneracy survives or gets lost are also discussed.
Thus this theorem forms the basis for an exact statement on the existence of
extensive ground state entropy in general.
