Abstract We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.
Abstract We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.
Although variance, as one of the most fundamental and ubiquitous quantities in quantifying uncertainty, has been widely used in both classical and quantum physics, there are still new applications awaiting exploration. In this work, by interchanging the roles of the state variable and the observable variable, i.e., by formally regarding any state as an observable (which is rational because any state is a priori a Hermitian operator ) and considering the average variance of this state (now in the position of an observable ) in all spin coherent states, we introduce a quantifier of spin nonclassicality with respect to a resolution of identity induced by spin coherent states. This quantifier is easy to compute and it admits various operational interpretations, such as the purity deficit, the Tsallis 2-entropy deficit, and the squared norm deficit between the Wigner function and the Husimi function. We reveal several intuitive properties of this quantifier, connect it to the phase-space distribution uncertainty, and illustrate it with some prototypical examples. Various extensions are further indicated.
Two basic ingredients in the quantum formalism are the concepts of states and observables, and apart from the average value (which includes the Born probability as a paramountly important case), the variance is the most fundamental and prominent quantity arising from the coupling of a state and an observable. Given the simplicity and ubiquity of the variance, it seems surprising that we can still exploit the variance to gain conceptually different insight into the quantum substratum. The purpose of the present work is to advocate the following idea: By inputting a general quantum state as the observable in the variance and incorporating a resolution of identity induced by coherent states, we obtain a mathematically simple and physically intuitive method for quantifying nonclassicality. We reveal its fundamental properties and highlight its connection with phase-space quantification of nonclassicality. The idea further suggests a whole family of appealing quantifiers of nonclassicality involving deep mathematical subtlety and an intriguing conjecture with physical significance. Some prototypical examples are worked out to illustrate the concept. This approach of regarding states as observables presents an opportunity for investigating quantum features from an alternative perspective of uncertainty.
The idea of construction of the nonlinear coherent states based on the hypergeometric- type operators associated to the Weyl-Heisenberg group [J:P hys:A 45(2012) 095304], are generalized to the similar states for the arbitrary Lie group SU(1, 1). By using of a discrete, unitary and irreducible representation of the Lie algebra su(1, 1) wide range of generalized nonlinear coherent states(GNCS) have been introduced, which admit a resolution of the identity through positive definite measures on the complex plane. We have shown that realization of these states for different values of the deformation pa- rameters r = 1 and 2 lead to the well-known Klauder-Perelomov and Barut-Girardello coherent states associated to the Irreps of the Lie algebra su(1, 1), respectively. It is worth to mention that, like the canonical coherent states, GNCS possess the temporal stability property. Finally, studying some statistical characters implies that they have indeed nonclassical features such as squeezing, anti-bunching effect and sub-Poissonian statistics, too.
Using the non-negativity of Gram determinants of arbitrary order, we derive higher-order uncertainty relations for the symmetric uncertainty matrices of corresponding order n > 2 to n Hermitean operators (n = 2 is the usual case). The special cases of third-order and fourth-order uncertainty relations are considered in detail. The obtained third-order uncertainty relations are applied to the Lie groups SU(1,1) with three Hermitean basis operators (K 1,K 2,K 0) and SU(2) with three Hermitean basis operators (J 1,J 2,J 3) where, in particular, the group-coherent states of Perelomov type and of Barut–Girardello type for SU(1,1) and the spin or atomic coherent states for SU(2) are investigated. The uncertainty relations for the determinant of the third-order uncertainty matrix are satisfied with the equality sign for coherent states and this determinant becomes vanishing for the Perelomov type of coherent states for SU(1,1) and SU(2). As an example of the application of fourth-order uncertainty relations, we consider the canonical operators (Q 1,P 1,Q 2,P 2) of two boson modes and the corresponding uncertainty matrix formed by the operators of the corresponding mean deviations, taking into account the correlations between the two modes. In two mathematical appendices, we prove the non-negativity of the determinant of correlation matrices of arbitrary order and clarify the principal structure of higher-order uncertainty relations.
In this paper, after a brief review on the entangled coherent state, a general form of the entangled nonlinear coherent state has been introduced based on the nonlinear coherent states approach. By choosing particular f-deformation functions, theoretical schemes for generation of two distinct classes of such states associated with SU(1,1) and SU(2) groups are presented. The entanglement characteristic of the corresponding entangled coherent states has been studied by evaluating the concurrence. In the continuation, other nonclassical behaviors of the produced states have been analyzed by investigating sub-Poissonian photon statistics, the oscillatory photon number distribution, one- and two-mode quadrature squeezing, and sum and difference squeezing. Accordingly, in addition to their entanglement property, other nonclassicality features of the introduced states are well established in detail.
Generalized coherent states associated with SU(1,1) Lie algebra are reviewed. A state is called intelligent if it satisfies the strict equality in the Heisenberg uncertainty relation. The eigenvalue problem satisfied by intelligent states (IS) is solved. The IS associated with SU(1,1) Lie algebra are investigated. We have constructed some realizations for our results of IS, and some applications are discussed. Some nonclassical properties such as Glauber second-order correlation function, photon number distribution and squeezing are investigated.