Abstract
A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.
Highlights
The modern boom in quantum engineering and quantum computing has reinvigorated the study of the interplay between classical and quantum physics
According to the basic principles of phase space representation of quantum mechanics, there is a mapping between the operators on the Hilbert space of a finite-dimensional quantum system and the functions on the phase space of its classical mechanical counterpart
Finalizing our derivation of the master equations, it is worth commenting on the particular formulation of the Stratonovich–Weyl correspondence rules that we use in this paper
Summary
The modern boom in quantum engineering and quantum computing has reinvigorated the study of the interplay between classical and quantum physics. According to the basic principles of phase space representation of quantum mechanics, there is a mapping between the operators on the Hilbert space of a finite-dimensional quantum system and the functions on the phase space of its classical mechanical counterpart This mapping can be realized with the aid of the Stratonovich–Weyl operator kernel ∆(ΩN) defined over a phase space. The factor vol(H) denotes the volume of the isotropy group H calculated with the measure induced by a given embedding, H ⊂ SU(N) Summarizing all these commonly accepted views, the kernel satisfying postulates (I)–(IV) and providing the mapping from an element of the space state to the Wigner function (3) will hereafter be referred to as the Stratonovich–Weyl kernel
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