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International Journal of Modern Physics B

UNITARILY INEQUIVALENT CLASSES OF MINIMUM UNCERTAINTY STATES OF SU(1, 1)

Ravinder Rupchand Puri, G. S. Agarwal

Published Jun 30, 1996• 15 citations
Publisher

Abstract

The notion of the minimum uncertainty states (MUS) of a Lie algebra is discussed. That is used to obtain the MUS of SU(1, 1) algebra. Those states fall in two classes. One is the class of the states obtained by the SU(1, 1) group transformation on the MUS of the pair of its noncompact hermitian generators and the other is obtained by that on the MUS of the pair of a compact and a noncomp]act generator.

DOI
10.1142/s0217979296000660

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The notion of the minimum uncertainty states (MUS) of a Lie algebra is discussed. That is used to obtain the MUS of SU(1, 1) algebra. Those states fall in two classes. One is the class of the states obtained by the SU(1, 1) group transformation on the MUS of the pair of its noncompact hermitian generators and the other is obtained by that on the MUS of the pair of a compact and a noncomp]act generator.
We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators ${J}_{x}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b+a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2$, ${J}_{y}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{-}a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2i$, and the total photon number ⟨Na+Nb⟩⟨{N}_{a}+{N}_{b}⟩⟨Na​+Nb​⟩. They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.
We formulate the sum uncertainty relations for compact classical Lie algebras of the type su (n), so (2n), so (2n + 1) and sp (n). Then we present resulting uncertainty relations explicitly for the su (2) ; su (3) and su (4) cases. We numerically verified our bound by choosing a large number of random vectors within an irrep of each of these algebras. We verify the bounds for several irreps. We discuss what type of states saturate su (n) bound for n = 2; 3 and 4, and compare these states with states that saturate more familiar products uncertainty relations.
The coherent states of a Hamiltonian linear in SU(1,1) operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables. The proposed approach is thus based on a physical characteristic of the harmonic-oscillator coherent states which is in contrast with the existing ones which rely on the generalization of the mathematical methods used for constructing the harmonic-oscillator coherent states. The set of states obtained by following the proposed method contains not only the known SU(1,1) coherent states but also a different class of states. \textcopyright{} 1996 The American Physical Society.
Uncertainty relations impose fundamental constraints on quantum observables, yet optimal bounds are needed to determine their incompatibility through the minimum attainable uncertainty. Here we introduce a covariance-matrix formulation from which infinitely many uncertainty relations follow, encompassing Hermitian observables and non-Hermitian operators and recovering the variance sum as a special case. Within this family, a basis-independent uncertainty for simple and semisimple Lie algebras admits an exact solution: its optimal state-independent bound is an explicit pairing of root system with the highest weight of the representation. The minimum is attained by a highest-weight state, replacing a search over quantum states with a calculation in representation theory. A complementary result establishes that uncertainty is constant within each weight space, yielding a conservation law. The same commutation relations can therefore support distinct uncertainty limits, depending on their realization on the state space. By rendering these limits analytically accessible, the framework turns representation theory into a tool for quantifying quantum incompatibility and may help establish the boundary between attainable and impossible tasks in quantum theory.
A class of squeezed states for the su(1,1) algebra is found and expressed by the exponential and Laguerre-polynomial operators acting on the vacuum states. As a special case it is proved that Perelomov's coherent state is a ladder-operator squeezed state and therefore a minimum uncertainty state. The theory is applied to the two-particle Calogero-Sutherland model. We find some squeezed states and compared them with the classical trajectories. The connection with some su(1,1) quantum optical systems (amplitude-squared realization, Holstein-Primakoff realization, the two-mode realization, and a four-mode realization) is also discussed. \textcopyright{} 1996 The American Physical Society.