Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0709.3307 · Physical Review A

Unitary equivalence between ordinary intelligent states and generalized intelligent states

Hyunchul Nha

Published Nov 29, 2007• 7 citations
arXivHTMLPDF

Abstract

Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables ${A,B}$, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schr\"odinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution $U$ that transforms the operators ${A,B}$ to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator $U$ to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator $U$ corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables ${A,B}$.

DOI
10.1103/physreva.76.053834

Related papers

Generalized intelligent states and squeezing

D. A. Trifonov

The Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.

Journal of Mathematical Physics•May 1, 1994•100 citations

Generalized intelligent states and SU(1,1) and SU(2) squeezing

D. A. Trifonov

A sufficient condition for a state |ψ> to minimize the Robertson-Schrödinger uncertainty relation for two observables A and B is obtained which for A with no discrete spectrum is also a necessary one. Such states, called generalized intelligent states (GIS), exhibit arbitrarily strong squeezing (after Eberly) of A and B. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the Bloch CS are subset of SU(2) GIS.

arXiv:quant-ph/0001028•Jan 10, 2000•3 citations

Robertson intelligent states

D. A. Trifonov

The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.

arXiv:quant-ph/9701018•Journal of Physics A Mathematical and General•Sep 7, 1997•56 citations

Unitary equivalence in generalized uncertainty principle theories

Sebastiano Segreto, Matteo Bruno

We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case where equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.

arXiv:2509.05477•Nuclear Physics B•Jan 3, 2026•1 citation

Generalized uncertainty relations and coherent and squeezed states

D. A. Trifonov

Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.

arXiv:quant-ph/0012072•Journal of the Optical Society of America A•Dec 1, 2000•33 citations

On the Squeezed States fornObservables

D. A. Trifonov

Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.

arXiv:quant-ph/9705001•Physica Scripta•Sep 1, 1998•13 citations
Ordinary intelligent states (OISs) hold equality in the Heisenberg uncertainty relation involving two noncommuting observables A,B{A,B}A,B, whereas generalized intelligent states (GISs) do so in the more generalized uncertainty relation, the Schr"odinger-Robertson inequality. In general, OISs form a subset of GISs. However, if there exists a unitary evolution UUU that transforms the operators A,B{A,B}A,B to a new pair of operators in a rotation form, it is shown that an arbitrary GIS can be generated by applying the rotation operator UUU to a certain OIS. In this sense, the set of OISs is unitarily equivalent to the set of GISs. It is the case, for example, with the su(2) and the su(1,1) algebras which have been extensively studied, particularly in quantum optics. When these algebras are represented by two bosonic operators (nondegenerate case), or by a single bosonic operator (degenerate case), the rotation, or pseudorotation, operator UUU corresponds to phase shift, beam splitting, or parametric amplification, depending on two observables A,B{A,B}A,B.
The Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.
A sufficient condition for a state |ψ> to minimize the Robertson-Schrödinger uncertainty relation for two observables A and B is obtained which for A with no discrete spectrum is also a necessary one. Such states, called generalized intelligent states (GIS), exhibit arbitrarily strong squeezing (after Eberly) of A and B. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the Bloch CS are subset of SU(2) GIS.
The diagonalization of the uncertainty matrix and the minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1,1), su(2), and sp(N,R) algebras. Unlike the group-related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N,R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators (q>0) and of any k power of are positive definite and can be diagonalized by symplectic linear transformations.
We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case where equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.
Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.
Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.