## 1 Quantum Information and The Wigner Function
### Entanglement in the Wigner Function  
Coherence and entanglement are based on the superposition principle, however, they represent conceptually different ideas. The former reflects the wave nature of states to interfere: States with a fixed phase relationship superimpose to represent another state of the physical system. Thus, coherence can be represented in single quantum systems and it is always with respect to a particular basis. In turn,entanglement implicates the (tensor product of) states of two correlated systems in such a way, that a destruction of one of the states in the first system results in a change of the states in the second system. Notably, the two concepts are quantitatively equivalent, that is, any nonzero amount of coherence in a system can be converted into an equal amount of entanglement between that system and another initially incoherent one [17]. This means that the two concepts which describe very different physical notions have a common mathematical foundation and thus can encrypt and carry the same information. The resources theory of coherence, which has been developed in terms of Hilbert space operator mechanics, designates a basis \(\left|\,a_{i}\right\rangle\) and identifies a set of states \(\mathcal{I}\) with a label incoherent. These concepts can be reformulated in phase space terms, moreover the entire theory can be formulated in the phase space. There is a variety of measures of coherence in complex systems, based on the correlations between the subsystems [18; 19]. A principal postulate in the development of a phase space quantum mechanics is the establishment of the correspondence between operators and functions. The problem is that different ordering of the position and momentum operators gives rise to different operators corresponding to one and the same observable, which is a function of the phase space coordinates. There are different rules for ordering the operators such as all momentum operators to the left or to the right, or a fully symmetric ordering. The Wigner theory is developed around the latter with the help of the Weyl map \(A(x,p)\leftrightarrow\mathcal{W}(\hat{A})\):  
\[\mathcal{W}\big{(}\hat{A}\big{)}=\int\frac{dsdq}{h}Tr\Big{(}\hat{A}e^{\frac{i }{h}(s\hat{x}+q\hat{p})}\Big{)}e^{-\frac{i}{h}(sx+q\hat{p})} \tag{1}\]  
The map establishes a one-to one correspondence between operators and functions, or more precisely an isomorphism from the algebra of operators \(\hat{A}\) (\(\hat{x}\)\(\hat{p}\)) with a product and a commutator \([,]\) to the algebra of phase space functions \(A(x,p)\) with a non-commutative star \((*)\)-product \(A*B=\mathcal{W}(\hat{A}\)\(\cdot\)\(\hat{B})\) and a Moyal bracket \(i\hbar[A,\,B]_{\mathrm{M}}=\mathcal{W}\big{(}[\hat{A},\,\hat{B}]\big{)}\). Recently, the algebraic notions used to derive the quantifying theory of coherence have been reformulated in terms of the Wigner theory [20]. In particular the nondiagonal eigenvector Wigner function \(f_{ij}(x,p)\) is introduced in correspondence to the eigenvectors \(\left|\,a_{i}\right\rangle\) of the operator \(\hat{A}\) of a given physical observable:  
\[f_{ij}(x,p)=\frac{1}{h}\mathcal{W}\big{(}|a_{i}\rangle\langle a_{j}|\big{)} \leftrightarrow|a_{i}\rangle\langle a_{j}| \tag{2}\]  
The Weyl map allows the transfer of the concepts of the information theory of coherence to the Wigner phase space picture. We may then define coherence in terms of a fixed basis \(\left|\,a_{i}\right\rangle\) in which one has to use (2) for the set of eigenvector Wigner functions \(f_{ij}\) to define the incoherent states as  
\[\hat{\sigma}=\sum_{i}P_{i}|a_{i}\rangle\langle a_{i}|\leftrightarrow f_{w, \sigma}=\sum_{i}P_{i}f_{ii}, \tag{3}\]  
where \(P_{i}\) are probabilities. The set of such states are denoted by \(\mathcal{I}\). Any other state which cannot be presented as in Equation (3) in this way is a coherent state [21].