  We formulate a variant of Fourier restriction for operators in Schatten
classes, where the Fourier-Wigner transform of a bounded operator replaces the
Fourier transform of a function. The Fourier-Wigner transform is closely
related to the group Fourier transform of the Heisenberg group. The first
result shows that Fourier-Wigner restriction for Schatten class operators is
equivalent to the restriction of the symplectic Fourier transform of functions
on phase space. We deduce various Schatten class results for the quantum
Fourier extension operator and answer a conjecture by Mishra and Vemuri
concerning the Weyl transform of measures in the affirmative.
