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arXiv:2105.12721 · Physical Review A

Entanglement in highly symmetric multipartite quantum states

Adam Burchardt, Jakub Czartowski, Karol Życzkowski

Published Aug 23, 2021• 25 citations
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Abstract

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.

DOI
10.1103/physreva.104.022426

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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.
Recent attention has turned to exploring quantum information within larger Hilbert spaces by utilizing qudits, which offer increased information capacity and potential for robust quantum communications. While the efficient generation of multipartite qudit entanglement is crucial for studying quantum correlations in high-dimensional Hilbert spaces, the increased dimension makes the circuit design challanging, especially when the entanglement is generated by heralding detections. In this work, we demonstrate that the graph picture of linear quantum networks (LQG picture) can provide a simplified method to generate qudit multipartite heralded entanglement of high symmetries. The LQG picture enables the reduction of circuit complexity by directly imposing the state symmetry onto the circuit structure. Leveraging this insight, we propose heralded schemes for generating NNN-partite NNN-level anti-symmetric (singlet) and symmetric (Dicke) states. Our study shed light on the optimal circuit design of high-dimensional entanglement with a systematic graphical strategy.
Building upon the results of R. Augusiak et al. [Phys. Rev. Lett. 115, 030404 (2015)] we develop a general approach to the generation of genuinely entangled multipartite states of any number of parties from genuinely entangled states of a fixed number of parties, in particular, the bipartite entangled ones. In our approach, certain isometries whose output subspaces are either symmetric or genuinely entangled in some multipartite Hilbert spaces are applied to local subsystems of bipartite entangled or multipartite genuinely entangled quantum states. To prove that entanglement of the resulting states is indeed genuine we then introduce criteria allowing us to decide it efficiently. The construction is then exploited to provide examples of multipartite states that are genuinely entangled but not genuinely nonlocal, giving further illustration for the inequivalence between entanglement and nonlocality in the multiparticle scenario. It is also shown how to construct genuinely entangled states which are unsteerable across certain bipartite cuts.
One of the key manifestations of quantum mechanics is the phenomenon of quantum entanglement. While the entanglement of bipartite systems is already well understood, our knowledge of entanglement in multipartite systems is still limited. This dissertation covers various aspects of the quantification of entanglement in multipartite states and the role of symmetry in such systems. Firstly, we establish a connection between the classification of multipartite entanglement and knot theory and investigate the family of states that are resistant to particle loss. Furthermore, we construct several examples of such states using the Majorana representation as well as some combinatorial methods. Secondly, we introduce classes of highly-symmetric but not fully-symmetric states and investigate their entanglement properties. Thirdly, we study the well-established class of Absolutely Maximally Entangled (AME) quantum states. On one hand, we provide construction of new states belonging to this family, for instance, an AME state of 4 subsystems with six levels each, on the other, we tackle the problem of equivalence of such states. Finally, we present a novel approach for the general problem of verification of the equivalence between any pair of arbitrary quantum states based on a single polynomial entanglement measure.
Entanglement in multipartite quantum systems is much more elusive than its bipartite counterpart. In recent past the usefulness of multipartite entangled states in several information theoretic tasks have been demonstrated. Being a resource, the detection of multipartite entanglement is an imperative necessity. Among the different classes of multipartite entangled states the Dicke state has found importance in several tasks due to its permutation symmetric nature. In this work we propose a simple and elegant way of detecting n qubit Dicke states using permutation symmetric Bell operators. We conjecture that maximal expectation value of the operator corresponds to the detection of Dicke states.