Supporting animation 1 of 3 for Simple procedure for phase-space measurement
and entanglement validation. It has recently been shown that it is
possible to represent the complete quantum state of any system as a phase-space
quasiprobability distribution (Wigner function) [Phys. Rev. Lett. **117**,
180401 (2016)]. Such functions take the form of expectation values of an
observable that has a direct analogy to displaced parity operators. In this
work we give a procedure for the measurement of the Wigner function that should
be applicable to any quantum system. We have applied our procedure to IBM's *Quantum
Experience* five-qubit quantum processor to demonstrate that we can measure
and generate the Wigner functions of two different Bell states as well as the
five-qubit Greenberger–Horne–Zeilinger state. Because Wigner functions for spin
systems are not unique, we define, compare, and contrast two distinct examples.
We show how the use of these Wigner functions leads to an optimal method for
quantum state analysis especially in the situation where specific
characteristic features are of particular interest (such as for spin
Schrödinger cat states). Furthermore we show that this analysis leads to
straightforward, and potentially very efficient, entanglement test and state
characterization methods.

We provide three supporting animations, showing the evolution of the SU(4),
two qubit Wigner function (animations 1 and 2) and the tensor product Wigner
function (animation 3). The Bloch spheres in the top left show four two-dimensional
slices where we show the equal angle slice and then three additional rotations
by pi of the variables given by the second qubit, corresponding to each of the
three coordinate axes. The plots in the top right show the Wigner functions of
the polar angles for each of the spheres plotted against each other for various
fixed values of the azimuthal angles. The bottom right panel shows the Wigner
functions for the individual qubits calculated from their reduced density
matrices. The bottom left panel shows the progress of the simulation through
the algorithm and the entanglement entropy.

In animation_01 we show the Wigner function dynamics for the Deutsch algorithm
for two qubits where the U gate is a C-NOT gate. Note that there is no
entanglement and the maximum value of the Wigner function for the individual
qubits corresponds to the equivalent point on the Bloch sphere.