Characterization of Qudit Entanglement Through the Visualization of Spin-Coherent-State Wigner Functions

The purpose of our research is to determine whether or not there is a general relationship between the degree of entanglement and the total amount of negativity in the Wigner function of various combinations of finite-dimensional quantum states. Specifically, by using the Stratonovich-Weyl correspondence we can take the density matrix of a known, finite-dimensional quantum state (hereafter known as a ``qudit'') and generate its corresponding, finite-dimensional Wigner function. This Wigner function reproduces the qudit density matrix through a known volume integral. By doing the same volume integral, but with the absolute value of the Wigner function as the kernel, we get a measure of the total amount of negativity of the Wigner function instead of reproducing the density matrix. Our question is thus, is this ``negative volume'' equivalent to the amount of entanglement in the initial qudit state? Our results for general two-qubit states have confirmed a monotonic relationship between concurrence and this negative volume for specific cases. By analyzing the various Wigner functions of three and more qubits, as well as qubit-qutrit Wigner functions we hope to build a consensus on whether or not the negativity in the Wigner function is a measure of, or witness to, entanglement.