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arXiv:quant-ph/0306074 · Journal of Modern Optics

Supersinglets

Adán Cabello

Published Apr 1, 2003• 59 citations
arXivHTMLPDF

Abstract

Supersinglets |S N (d) 〉 are states of total spin zero of N particles of d levels. Some applications of the |S N (N) 〉 and |S N (2) 〉 states are described. The |S N (N) 〉 states can be used to solve three problems that have no classical solution: The ‘N strangers,’ ‘secret sharing,’ and ‘liar detection’ problems. The |S N (2) 〉 (with N even) states can be used to encode qubits in decoherence-free subspaces.

DOI
10.1080/09500340308234551

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Supersinglets |S N (d) 〉 are states of total spin zero of N particles of d levels. Some applications of the |S N (N) 〉 and |S N (2) 〉 states are described. The |S N (N) 〉 states can be used to solve three problems that have no classical solution: The ‘N strangers,’ ‘secret sharing,’ and ‘liar detection’ problems. The |S N (2) 〉 (with N even) states can be used to encode qubits in decoherence-free subspaces.
Multipartite systems of high-dimensional particles can host forms of quantum entanglement inaccessible to qubit systems. A particularly fascinating class is formed by the fully antisymmetric states, also known as supersinglets, which possess a remarkable symmetry: they are invariant under arbitrary collective local unitaries. Here, we explore their entanglement properties and establish their quantum information applications. Using convex optimisation, we show that central entanglement features of supersinglets are remarkably robust to noise, substantially surpassing the robustness of several paradigmatic entangled states under certain relaxations. We then develop witness methods, based on collective-spin or randomised measurements, that certify both the depth and dimensionality of their entanglement. Finally, we identify supersinglets as a natural resource for high-dimensional gradient sensing and prove that they enable Heisenberg scaling. Together, these results reveal supersinglets as an exceptionally robust and versatile class of multipartite entangled states, combining striking symmetry and strong entanglement with concrete potential for quantum sensing.
Supersinglets are states of spin-zero of d≥3d \ge 3d≥3 particles of ddd levels. They are invariant under unitary transformations of the form U⊗dU^{\otimes d}U⊗d and have applications in metrology, error protection, and communication. They also violate some specific Bell inequalities. However, none any of these applications {\em require} supersinglets nor do any of these Bell inequality violations capture the unique properties of the supersinglets. This leads to two questions. Question 1 is whether there exists a task that can be solved only with supersinglets. Question 2 is whether supersinglets can produce a unique ddd-partite, ddd-dimensional nonlocal signature. We answer both questions affirmatively by presenting a protocol that self-test all supersinglets by producing ddd-partite, ddd-dimensional {\em perfect} quantum strategies for any d≥3d \ge 3d≥3.
We introduce an entanglement distillation (purification) protocol for supersinglet states composed of N qubits. The supersinglet state we target is a total spin zero state with zero spin variance, and has a fully entangled structure involving all qubits. In our distillation protocol, three copies of an initial spin zero state are measured in the local total spin basis such that a higher fidelity supersinglet state is generated upon postselection. The initial state can be prepared using conventional Bell state distillation methods distributed in a way to target the supersinglet symmetries. The protocol uses only local operations and classical communications, and is suitable for long-distance applications such as quantum clock synchronization and cryptography, and avoids a high dimensional Schur transform such that it can be used for tasks such as quantum metrology.