Symmetric Multiqudit States: Stars, Entanglement, Rotosensors

Abstract

A constellation of $N=d-1$ Majorana stars represents an arbitrary pure quantum state of dimension $d$ or a permutation-symmetric state of a system consisting of $n$ qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of $k$ subsystems with $d$ levels each. For $d\ge 3$, such states are equivalent, as far as rotations are concerned, to a collection of various spin states, with definite relative complex weights. Following Majorana's lead, we introduce a multiconstellation, consisting of the Majorana constellations of the above spin states, augmented by an auxiliary, "spectator" constellation, encoding the complex weights. Examples of stellar representations of symmetric states of four qutrits, and two spin-$3/2$ systems, are presented. We revisit the Hermite and Murnaghan isomorphisms, which relate multipartite states of various spins, number of parties, and even symmetries. We show how the tools introduced can be used to analyze multipartite entanglement and to identify optimal quantum rotosensors, i.e., pure states which are maximally sensitive to rotations around a specified axis, or averaged over all axes.

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Cited by in corpus (9)

• Entanglement in highly symmetric multipartite quantum states
• Interconversion of $W$ and Greenberger-Horne-Zeilinger states for Ising-coupled qubits with transverse global control
• Generating a 4-photon Tetrahedron State: Towards Simultaneous Super-sensitivity to Non-commuting Rotations
• Quantum Polarimetry
• Quantum metrology of rotations with mixed spin states
• Platonic dynamical decoupling sequences for interacting spin systems
• Phase characterization of spinor Bose-Einstein condensates: a Majorana stellar representation approach
• Minimal-noise estimation of noncommuting rotations of a spin
• Entangling power of symmetric multiqubit systems: a geometrical approach