The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.)

Is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

The only non-negative Wigner functions for pure states are the Gaussian ones. Quadratic Hamiltonians (like the simple Harmonic oscillator) preserve the Gaussianity of pure states, so the Wigner function never becomes partially negative after starting out non-negative, or vice versa.

On the other hand, non-quadratic Hamiltonians do not have this property. So just about any non-quadratic Hamiltonian you pull out of a hat will in general destroy and create strict positivity. However, there really isn't anything profound going on, because you shouldn't be taking a probability interpretation of the Wigner function either way; it's called a quasiprobability distribution for a reason.

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.)

Is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

The only non-negative Wigner functions for pure states are mixtures of pure Gaussian wavepackets. Quadratic Hamiltonians (like the simple Harmonic oscillator) preserve the Gaussianity of pure states, so the Wigner function never becomes partially negative after starting out non-negative, or vice versa.

On the other hand, non-quadratic Hamiltonians do not have this property. So just about any non-quadratic Hamiltonian you pull out of a hat will in general destroy and create strict positivity. However, there really isn't anything profound going on, because you shouldn't be taking a probability interpretation of the Wigner function either way; it's called a quasiprobability distribution for a reason.

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.)

Is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

Yes.

The only non-negative Wigner functions for pure states are the Gaussian ones. Quadratic Hamiltonians (like the simple Harmonic oscillator) preserve the Gaussianity of pure states, so the Wigner function never becomes partially negative after starting out non-negative, or vice versa. On the other hand, non-quadratic Hamiltonians do not have this property.

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.)

Is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

The only non-negative Wigner functions for pure states are the Gaussian ones. Quadratic Hamiltonians (like the simple Harmonic oscillator) preserve the Gaussianity of pure states, so the Wigner function never becomes partially negative after starting out non-negative, or vice versa.

On the other hand, non-quadratic Hamiltonians do not have this property. So just about any non-quadratic Hamiltonian you pull out of a hat will in general destroy and create strict positivity. However, there really isn't anything profound going on, because you shouldn't be taking a probability interpretation of the Wigner function either way; it's called a quasiprobability distribution for a reason.