Supporting animation 3 of 3 for Simple procedure for phase-space measurement and entanglement validation.
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. In animation_02 we show the Wigner function dynamics for the creation of the four Bell states. Here we see that the Wigner functions for these states are rotations of each other in four dimensional space. In animation_03 we show the tensor product Wigner function version of animation_02 for comparative purposes.