Abstract
The geometric measure of entanglement is investigated for permutation symmetric pure states of multipartite qubit systems, in particular the question of maximum entanglement. This is done with the help of the Majorana representation, which maps an n qubit symmetric state to n points on the unit sphere. It is shown how symmetries of the point distribution can be exploited to simplify the calculation of entanglement and also help find the maximally entangled symmetric state. Using a combination of analytical and numerical results, the most entangled symmetric states for up to 12 qubits are explored and discussed. The optimization problem on the sphere presented here is then compared with two classical optimization problems on the S2 sphere, namely Tóth's problem and Thomson's problem, and it is observed that, in general, they are different problems.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Highly entangled quantum states are essential for tasks such as quantum teleportation and quantum computation, but little is known about the maximally entangled states of given Hilbert spaces. These problems are simplified when limiting the search to permutation symmetric states. By means of the 'Majorana representation', a generalization of the Bloch sphere representation of single qubits, any symmetric state of n qubits is unambiguously represented by n points on a sphere. For example, a two-qubit Bell state is represented by an antipodal pair of points.
Main results. We analyze symmetric states of up to 12 qubits in terms of the geometric measure, a well-known entanglement measure with operational interpretations, and find theoretical results such as bounds on the maximal possible entanglement. For states with positive coefficients, the maximally entangled states are numerically determined, and for the general case we present candidates for maximal entanglement. The Majorana representations of these states are generally well spread out, and we compare them to the solutions of classical point distribution problems on the sphere.
Wider implications. Recently, the geometric entanglement of symmetric multi-qubit states has become a very active research field, both for theoretical and practical reasons. These states appear, for example, in many-body physics, and they are currently implemented experimentally. They are useful for quantum information tasks such as leader election, and they may be resources for measurement-based quantum computing (MBQC).
Figure. Two ways of displaying the Majorana representation of the extremely entangled twelve qubit 'icosahedron state'. The dots and crosses in the left-hand diagram correspond to the zeroes and maxima of the right-hand 'overlap function', respectively. The volume of the overlap function is constant for all symmetric states, and the entanglement is determined by the value at the global maximum.