Interpreting symplectic linear transformations in a two-qubit phase space

For the continuous Wigner function and for the simplest discrete Wigner function in odd-prime-power dimensions, permuting the values of the Wigner function in accordance with a symplectic linear transformation is equivalent to performing a certain unitary transformation on the state. That is, performing this unitary transformation is simply a matter of moving Wigner-function values around in phase space. This simple correspondence does not hold for dimensions that are powers of two. Here we show, though, that a generalized version of this correspondence does hold for the case of a two-qubit phase space. The phase space we consider is the two-dimensional vector space over the four-element field. In this case, a symplectic linear permutation of the points of phase space, together with a certain reinterpretation of the Wigner function, is equivalent to a unitary transformation.