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arXiv:2502.18574 · Quantum

Dicke subsystems are entangled

Szilárd Szalay, Péter Nyári

Published Nov 6, 2025• 2 citations
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Abstract

We show that all reduced states of nonproduct symmetric Dicke states of arbitrary number of qudits are genuinely multipartite entangled, and of nonpositive partial transpose with respect to any subsystem.
DOI
10.22331/q-2025-11-06-1908

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The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterized by specifying how entangled its subsystems are. A generally mixed subsystem of m qudits is obtained by tracing over the other $N\ensuremath{-}m$ qudits. We examine the entanglement in the space of mixed states of m qudits. We show that for a typical pure state of N qudits, its subsystems smaller than N/3N/3N/3 qudits will have a positive partial transpose and hence are separable or bound entangled. Additionally, our numerical results show that the probability of finding entangled subsystems smaller than N/3N/3N/3 falls exponentially in the dimension of the Hilbert space. The bulk of pure state Hilbert space thus consists of highly entangled states with multipartite entanglement encompassing at least a third of the qudits in the pure state.
A construction of genuinely entangled multipartite quantum states based on the group theory is presented. Analyzed states resemble the Dicke states, whereas the interactions occur only between specific subsystems related by the action of the selected group. The states constructed by this technique exhibit desired symmetry properties and form a natural resource for less-symmetric quantum informational tasks. We propose quantum circuits efficiently generating such states, which in general have smaller complexity than the circuits necessary to create fully symmetric entangled states. Exemplary five-qubit quantum states are simulated on quantum computers.
Dicke states represent a class of multipartite entangled states that can be generated experimentally with many applications in quantum information. We propose a method to experimentally detect genuine multipartite entanglement in the vicinity of arbitrary Dicke states. The detection scheme can be used to experimentally quantify the entanglement depth of many-body systems and is easy to implement as it requires measurement of only three collective spin operators. The detection criterion is strong as it heralds multipartite entanglement even in cases where the state fidelity goes down exponentially with the number of qubits.
Links of rings offer an intuitive picture of multipartite entanglement. Cutting one ring can free some of the others, just as losing a particle, i.e. tracing it out, can disentangle some of the remaining ones. This analogy classifies multipartite entanglement according to which groups of particles stay entangled when others are lost, a question at the heart of robust quantum networks. However, pure states realizing a given link class were known only in special cases. Here, we close this gap. Recasting link polynomials as hypergraphs, we give an explicit formula for pure qudit states and prove that, for every link class, their entanglement under any pattern of particle loss matches the link exactly. The same construction also controls genuine multipartite entanglement, determining exactly which of the groups that stay entangled are genuinely multipartite entangled, and all of them can be made so. As a consequence, we prove the conjecture that mmm-resistant pure qudit states, which stay entangled after the loss of any mmm particles but become fully separable after the loss of m+1m+1m+1, exist for any number of particles and any mmm.