Abstract The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from topological data analysis to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the average distillable entanglement, which we develop as a generalization of the Meyer–Wallach entanglement measure studied by Scott (2004 Phys. Rev. A 69 052330), via the integrated Euler characteristic (IEC) defined by Hamilton and Leditzky. This allows us to connect the distance of an error correcting code to the IEC. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and one-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number β 1 ( ε ) allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.
Abstract The study of multipartite entanglement is much less developed than the bipartite scenario. Recently, a string of results have proposed using tools from topological data analysis to attach topological quantities to multipartite states. However, these quantities are not directly connected to concrete information processing tasks making their interpretations vague. We take the first steps in connecting these abstract topological quantities to operational interpretations of entanglement in two scenarios. The first is we provide a bound on the average distillable entanglement, which we develop as a generalization of the Meyer–Wallach entanglement measure studied by Scott (2004 Phys. Rev. A 69 052330), via the integrated Euler characteristic (IEC) defined by Hamilton and Leditzky. This allows us to connect the distance of an error correcting code to the IEC. The second is we provide a characterization of a class of graph states containing the GHZ state via the birth and death times of the connected components and one-dimensional cycles of the entanglement complex. In other words, the entanglement distance behavior of the first Betti number β 1 ( ε ) allows us to determine if a state is locally equivalent to a GHZ state, potentially providing new verification schemes for highly entangled states.
Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.
Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
We introduce an operational interpretation for pure-state global multipartite entanglement based on quantum estimation. We show that the estimation of the strength of low-noise locally depolarizing channels, as quantified by the regularized quantum Fisher information, is directly related to the Meyer-Wallach multipartite entanglement measure. Using channels that depolarize across different partitions, we obtain related multipartite entanglement measures. We show that this measure is the sum of expectation values of local observables on two copies of the state.
We study the task of localizing multipartite entanglement in pure quantum states onto a subsystem by measuring the remaining systems. To this end, we fix a multipartite entanglement measure and consider two quantities: the multipartite entanglement of assistance (MEA), defined as the entanglement measure averaged over the post-measurement states and maximized over arbitrary measurements; and the localizable multipartite entanglement (LME), defined in the same way but restricted to only local single-system measurements. Both quantities generalize previously considered bipartite entanglement localization measures. In our work we choose the n-tangle, the genuine multipartite entanglement concurrence and the concentratable entanglement (CE) as the underlying seed measure, and discuss the resulting MEA and LME quantities. First, we prove easily computable upper and lower bounds on MEA and LME and establish Lipschitz-continuity for the n-tangle and CE-based LME and MEA. Using these bounds we investigate the typical behavior of entanglement localization by deriving concentration inequalities for the MEA evaluated on Haar-random states and performing numerical studies for small tractable system sizes. We then turn our attention to protocols that transform graph states. We give a simple criterion based on a matrix equation to decided whether states with a specified n-tangle value can be obtained from a given graph state, providing no-go theorems for a broad class of such graph state transformations beyond the usual “local Clifford plus local Pauli measurement” framework. This analysis is generalized to weighted graph states, which provide a realistic error model in current experiments preparing graph state. Our entanglement localization framework certifies the near-optimality of recently discussed local-measurement protocols to transform uniformly weighted line graph states into GHZ states, even when considering arbitrary entangled measurements. Finally, we demonstrate how our MEA and LME quantities can be used to detect critical phenomena such as phase transitions in transversal field Ising models. Since entanglement localization is operationally relevant throughout quantum networking and measurement-based quantum computation, our framework of results based on the MEA and LME has the potential for broad applications in these fields.
Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the W entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.
We demonstrate that multipartite entanglement is able to characterize one-dimensional topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the original and dual lattices. We investigate an exactly-solvable long-range extended spin-21 Ising model with nontrivial Z-symmetry topological order identified equivalently by winding numbers and paired Majorana zero modes at each end. For topologically ordered phases with high winding numbers, we can define a Z topological invariant by the scaling coefficient of the dual quantum Fisher information density. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topologically ordered state has promising applications in robust quantum computation and quantum metrology, and could be generalized to higher-dimensional systems.