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Stokes vector and its relationship to Discrete Wigner Functions of multiphoton states

Stokes vectors and Discrete Wigner functions (DWF) provide two alternate ways of representing the polarization state of multiphoton systems. The Stokes vector associated with a n-photon polarization state is unique, and its Minkowski squared norm provides a direct way of quantifying entanglement through n-concurrence. However, the quantification of entanglement from DWF is not straight forward. The DWF associated with a given quantum state is not unique but depends on the way in which basis vectors are assigned to various lines in the phase space. For a Hilbert space of dimension N, there exists N N+1 such possible assignments. While a given DWF corresponds to a unique Stokes vector, the converse is not true. In the present work, we show that, for each particular assignment called a quantum net, there exist a unique Hadamard matrix which relates the Stokes vector to the corresponding DWF. This method provides an elegant and direct method of constructing the DWFs from the Stokes vector for every possible choice of the quantum net. Using these results, we derive the relationship between the Stokes vector of a spin-flipped state and the DWF. Finally, we also present a method to express the Minkowskian squared norm of the Stokes vector directly in terms of the DWF.