## 1 Quantum Information and The Wigner Function
### Entanglement in the Wigner Function
Then the measure for coherence in phase space is \(\mathrm{C}(f_{w})=\min\)\(\mathcal{D}(f_{w},f_{w,\sigma})\), where \(f_{w,\sigma}\in\mathcal{I}\) is introduced from the measure for distance \(\mathcal{D}\)[16; 17]. The latter can be based on the von Neumann entropy or trace distance [16], which can be equivalently defined in both operator and phase spaces. We conclude that incoherent states involve only diagonal elements \(f_{ii}\): From Equation (2) it follows that coherence is directly related to the existence of nondiagonal elements \(f_{ij}\) in the state representation of \(f_{w}\). The above formal mathematical analysis developed in [20] can be directly linked to the negativity of the Wigner function and actually has a transparent physical background in terms of positive and negative particles. Indeed, if we choose \(\hat{A}\) to be the momentum operator \(\hat{A}=\hat{p}\) with a basis \(e^{i\,pj\times h}\), the corresponding Wigner basis becomes  
\[f_{ij}(x,p)=e^{ix(p_{i}-p_{j})/h}\delta\big{(}\big{(}p_{i}+p_{j}\big{)}/2\pm p \big{)}, \tag{4}\]which, due to the exponential prefactor, has the ability to form interference patterns and, in particular, to exhibit negative values. In contrast, for diagonal Wigner basis states, this prefactor becomes unity and (4) reduces to \(\delta(p_{i}\pm p)\). Thus, according to Equation (3), incoherent states \(f_{\alpha\sigma}\) have non-negative values. The associated physical picture is of classical particles with a distribution function \(f(p)=\sum_{i}P_{i}\ \delta(p_{i}\pm p)\). Here each particle \"carries\" a weight given by the probability \(P_{i}\). In this way the Wigner picture reveals that the concept for incoherent states is deeply rooted in classical mechanics. In particular, the evolution of an initial particle distribution, subject to the action of a quadratic potential, is entirely classical: The Wigner potential reduces to a force, which governs particles over classical Newtonian trajectories. We can extend this picture to coherent Wigner states. The latter contain off-diagonal basis states, thus according to Equation (4) the weight of such particles should have complex values. Furthermore, we know that the Wigner function is a real quantity, thus after the cancellation of all imaginary values in the corresponding decomposition of the chosen coherent state, only positive and negative values remain. In this way, we come to two important conclusions:  
* Quantum coherence is characterized by negative values of the Wigner function;
* In a virtual particle picture, coherent states are distinguished from classical states by the existence of particles with a negative weight. These considerations are independent from the complexity of the system: The existence of subsystems is accounted for by a simple increase of the involved phase space coordinates. An alternative approach uses the negativity of the Wigner function to define a parameter which measures the non-classicality of a given state based on the distance from a family of coherent states [22]. The parameter is given by the doubled volume of the integrated (in the whole phase space) negative part of the Wigner function. This definition can be regarded as a projection of the function onto the set of the real numbers between zero and unity. Here, we show that other ways of projection can provide useful information about the quantum evolution. In the next section, we describe how entanglement can be observed in a simple physical problem, that of electron two-slit propagation around an impurity potential in a quantum nanowire (waveguide). The simulation experiment is very similar to the one already used in [23] to analyze coherence effects by comparing classical and quantum density or current density. In contrast, here, we reconstruct the Wigner function in the phase space in order to investigate how the negative excursions, not existing in the classical case, bear information about the involved quantum effects.