Geometry of spin coherent states

Abstract

Spin states of maximal projection along some direction in space are called (spin) coherent, and are, in many aspects, the "most classical" available. For any spin $s$, the spin coherent states form a 2-sphere in the projective Hilbert space $\mathbb{P}$ of the system. We address several questions regarding that sphere, in particular its possible intersections with complex lines. We also find that, like Dali's iconic clocks, it extends in all possible directions in $\mathbb{P}$. We give a simple expression for the Majorana constellation of the linear combination of two coherent states, and use Mason's theorem to give a lower bound on the number of distinct stars of a linear combination of two arbitrary spin-$s$ states. Finally, we plot the image of the spin coherent sphere, assuming light in $\mathbb{P}$ propagates along Fubini-Study geodesics. We argue that, apart from their intrinsic geometric interest, such questions translate into statements experimentalists might find useful.

Added an explanatory comment on eq. (68) and several new references. 16 pages, 5 figures

References in corpus (12)

• The geometric measure of entanglement for symmetric states
• Multiqubit symmetric states with high geometric entanglement
• The maximally entangled symmetric state in terms of the geometric measure
• Extremal quantum states and their Majorana constellations
• Tensor Representation of Spin States
• Quantum metrology at the limit with extremal Majorana constellations
• Multiqubit symmetric states with maximally mixed one-qubit reductions
• Anticoherence of spin states with point group symmetries
• Quantumness of spin-1 states
• Geometric descriptions of entangled states by auxiliaries varieties
• Anticoherence measures for pure spin states
• Entanglement classification with matrix product states

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• Platonic dynamical decoupling sequences for interacting spin systems
• Maximum entanglement of mixed symmetric states under unitary transformations
• When geometric phases turn topological
• Stellar Representation of Grassmannians
• Curves in quantum state space, geometric phases, and the brachistophase
• Boson Sampling for Generalized Bosons
• Phase characterization of spinor Bose-Einstein condensates: a Majorana stellar representation approach
• Speed excess and total acceleration: a kinematical approach to entanglement
• Stellar representation of extremal Wigner-negative spin states
• Mixed-state geometric phases of coherent and squeezed spin states
• Entangling power of symmetric multiqubit systems: a geometrical approach
• Thermal effects on the spin domain phases of high spin-f Bose-Einstein condensates with rotational symmetries
• Non-Gaussianity from superselection rules
• Polynomial deformations of $sl(2)$ and unified algebraic framework for solutions of a class of spin models
• Universality in fidelity-based quantum metrology
• Spin coherence scale: operator-ordering sensitivity beyond the Heisenberg-Weyl group