An extended phase space x – p – X – P is introduced for one-mode systems, where x – p is the position–momentum plane and X – P is the quantum correlations plane. The Wigner functions W ( x , p ) and the Weyl functions ( X , P ) for arbitrary operators are studied in this space. Creation and annihilation operators for Wigner and Weyl functions are introduced and are used to perform displacements and squeezing in the x – p – X – P extended phase space. The formalism can be used for the construction of quantum states with desirable correlation properties.
The Wigner function
W(x,p) is related to the Weyl function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{W}(X,P)$ through a two-dimensional Fourier transform. Several properties for these functions are proved which lead naturally to an extended phase space $x\ensuremath{-}p\ensuremath{-}X\ensuremath{-}P.$ Uncertainty relations for $\ensuremath{\delta}x\ensuremath{\delta}P$ and for $\ensuremath{\delta}p\ensuremath{\delta}X$ are studied. They reduce to the usual uncertainty relation in the special case of pure states, but are different for mixed states. Thermal density matrices obey these uncertainty relations as equalities.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.
Various phase-space distributions, from the celebrated Wigner function, to the Husimi
Q function and the Glauber–Sudarshan
P distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the
s -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter
s , we interpret the
s -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the
s -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the
s -ordered phase-space distribution. In this connection, the parameter
−s/2 (rather than the original
s ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter
s , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the
s -ordered phase-space distribution is derived.
Phase space methods, particularly Wigner functions, provide intuitive tools for representing and analyzing quantum states. We focus on systems with SU(2) dynamical symmetry, which naturally describes spin and a wide range of two-mode quantum models. We present a unified phase space framework tailored to these systems, highlighting its broad applicability in quantum optics, metrology, and information. After reviewing the core SU(2) phase-space formalism, we apply it to states designed for optimal quantum sensing, where their nonclassical features are clearly revealed in the Wigner representation. We then extend the approach to systems with an indefinite number of excitations, introducing a generalized framework that captures correlations across multiple SU(2)-invariant subspaces. These results offer practical tools for understanding both theoretical and experimental developments in quantum science.
Quantum systems with finite Hilbert space where position x and momentum \np take values in Z(d) (integers modulo d) are considered. Symplectic tranformations \nS(2¿,Z(p)) in ¿-partite finite quantum systems are studied and \nconstructed explicitly. Examples of applying such simple method is given \nfor the case of bi-partite and tri-partite systems. The quantum correlations \nbetween the sub-systems after applying these transformations are discussed \nand quantified using various methods. An extended phase-space x¿p¿X¿P \nwhere X, P ¿ Z(d) are position increment and momentum increment, is introduced. \nIn this phase space the extended Wigner and Weyl functions are \ndefined and their marginal properties are studied. The fourth order interference \nin the extended phase space is studied and verified using the extended \nWigner function. It is seen that for both pure and mixed states the fourth \norder interference can be obtained.