An explicit and uniform construction for coherent states (CS) for all the SU(N) groups is given and, on this basis, their properties are investigated. The CS are parametrized by the dots of a coset space, which is, in this particular case, the projective space CP N-1 and which plays the role of the phase space in the corresponding classical mechanics. The logarithm of the modulus of the CS overlap, being interpreted as assymmetric in the space, gives the Fubini-Study metric in CP N-1 . The classical limit is investigated in terms of operator symbols. h=P -1 (where P is the signature of the representation) plays the role of Planck's constant. The classical limit of the so called star commutator of the symbols generates the Poisson bracket in the corresponding phase space. The CS form an overcompleted system in the representation space and, as quantum states possess a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator.
Uncertainty relations constrain the fluctuations of incompatible observables, but most familiar bounds depend on the quantum state. State-independent uncertainty relations instead ask how much fluctuation remains unavoidable for every quantum state. For observables generated by a continuous symmetry, sharp state-independent bounds are known when the symmetry representation is irreducible. Multipartite collective systems, however, generally appear as reducible tensor-product representations of a symmetry algebra, which raises the question of how to determine their total uncertainty.
We resolve this problem for multipartite quantum systems with a compact semisimple symmetry algebra
g. Exploiting the symmetry structure, we formulate a general framework for state-independent uncertainty based on representation theory. The total variance admits an exact decomposition into intrinsic fluctuations within irreducible sectors and a nonnegative dispersion between sectors. This yields the sharp state-independent bound for the total variance
Δρ2(g) on the multipartite Hilbert space
H minρΔρ2(g)=minλ∈Λ(H)2⟨λ,δ⟩, where
ρ is any density operator on
H,
Λ(H) is the set of highest weights
λ labeling those sectors, and
δ is the Weyl vector. This demonstrates that the ultimate uncertainty is completely controlled by its intrinsic symmetry structure. As an application of the general theory, we confirm our previous conjecture [JPA 59, 155302 (2026)] that the total uncertainty floor of collective spin-
1/2 systems depends only on the parity of the particle number. We further illustrate the framework for multipartite spin-
1 systems, demonstrating that the same symmetry-sector mechanism persists beyond spin-
1/2.
The product ( delta s x ) 2 ( delta s y ) 2 is computed where ( delta s x,y ) 2 =(s x,y 2 )-(s x,y ) 2 ; averaging is performed in the coherent spin-state representation given by Radcliffe (1971). After applying the Holstein- Primakoff transformation s - =(2s) 1 2/a Dagger , s + =(2s) 1 2/a and mu = alpha /(2s) 1 2/, and putting s to infinity one proceeds from Radcliffe space into the Glauber space. After this procedure the product ( delta s x ) 2 ( delta s y ) 2 becomes ( delta x) 2 ( delta p) 2 . Using the Jackiw equation it is shown that the function mod mu =0> is the only one which minimizes the uncertainty product, for every s.
Minimum uncertainty states of the conventional Heisenberg uncertainty relation have been extensively studied and are often regarded as the most classical quantum states from the perspective of uncertainty, providing valuable insight into the nature of quantumness and its potential applications. In this work, we investigate the minimum uncertainty states associated with an information-theoretic refinement of the Heisenberg uncertainty relation in general spin systems. Using two different approaches, the matrix formulation and the Wick symbol representation, we derive explicit expressions for the states that saturate the uncertainty bound. We show that spin coherent states indeed achieve minimum uncertainty, consistent with their conventional identification as the classical states of spin systems. Moreover, we also identify additional classes of minimum uncertainty states beyond the coherent family. Finally, we compare the spin-system results with the previously studied bosonic case and elucidate the origin of the differences between the two settings.
Geometric Quantum Mechanics is a mathematical framework that shows how quantum theory may be expressed in terms of Hamiltonian phase-space dynamics. The states are points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is specified by the Schrödinger equation in this framework. The quest to express the uncertainty principle in geometrical language has recently become the focus of significant research in geometric quantum mechanics. One has demonstrated that the Robertson–Schrödinger uncertainty principle, which is a stronger version of the uncertainty relation, can be defined in terms of symplectic form and Riemannian metric. On the basis of this formulation, we study the dynamical behavior of the uncertainty relation for the spin 1 system in this work. We show that under Hamiltonian flow, the Robertson–Schrödinger uncertainty principles are not invariant. This is because, unlike the symplectic area, the Riemannian metric is not invariant under Hamiltonian flow throughout the evolution process.