Despite the fundamental importance of quantum entanglement in many-body systems, our understanding is mostly limited to bipartite situations. Indeed, even defining appropriate notions of multipartite entanglement is a significant challenge for general quantum systems. In this work, we initiate the study of multipartite entanglement in a rich, yet tractable class of quantum states called stabilizer tensor networks. We demonstrate that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state. In particular, we show that the average number of Greenberger-Horne-Zeilinger (GHZ) triples that can be extracted from a stabilizer tensor network is small, implying that tripartite entanglement is scarce. This, in turn, restricts the higher-partite entanglement structure of the states. Recent research in quantum gravity found that stabilizer tensor networks reproduce important structural features of the AdS/CFT correspondence, including the Ryu-Takayanagi formula for the entanglement entropy and certain quantum error correction properties. Our results imply a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks, as well as a novel formula for the third moment of random stabilizer states, which we expect to find further applications in quantum information.
We study multi-invariants---a multi-partite generalization of Renyi entropies---of simple stabilizer states, for an arbitrary replica symmetry group. We develop a group theoretic technique for evaluating such invariants, which operates on the replica group and involves counting intersections among its cosets. Our techniques streamline the calculations of multi-invariants for graph states with two GHZ-like nodes and for certain generalizations thereof. The methodology developed here is most suitable when the quantum state is simple but flexibility over the replica group is desired.
We develop a method for visualizing the internal structure of multipartite entanglement in pure stabilizer states. Our algorithm graphically organizes the many-body correlations in a hierarchical structure. This provides a rich taxonomy from which one can simultaneously extract many quantitative features of a state including some traditional quantities such as entanglement depth, k-uniformity and entanglement entropy. Our method also presents an alternative computational tool for extracting the exact entanglement depth and all separable partitions of a stabilizer state. Our construction is gauge invariant and goes beyond traditional entanglement measures by visually revealing how quantum information and entanglement is distributed. We use this tool to analyze the internal structures of prototypical stabilizer states (GHZ state, cluster state, stabilizer error correction codes) and are able to contrast the complexity of highly entangled volume law states generated by random unitary operators and random projective measurements.
We study genuine multi-entropy as a diagnostic of multipartite entanglement in the toric code, which provides a controlled setting for probing multipartite structures in topologically ordered states. Our main question is whether genuine multi-entropy captures information that is not reducible to conventional lower-party entropic data, such as topological entanglement entropy. We first analyze toric-code ground states that admit a stabilizer-state description, where the relevant quantities can be evaluated exactly. We conjecture that, for stabilizer states and
q≥4, the
q-partite genuine multi-entropy at replica index
n<q collapses to a linear combination of multi-entropies involving at most
q−2 parties. We establish this pattern explicitly for
q=4 in the toric code stabilizer sector: for
n=2,3, the genuine multi-entropy is proportional to the tripartite information
I3 and, for the Kitaev-Preskill partition, contains no independent genuine four-partite information beyond that captured by the topological entanglement entropy. At
n=4, however, this reduction breaks down: the genuine multi-entropy is no longer proportional to
I3, but remains a topological invariant of the toric-code stabilizer ground states under Kitaev-Preskill-preserving deformations that do not change whether the numbers of vertex- and plaquette-type junctions coincide. For generic non-stabilizer superpositions within the ground-state manifold and for coherent superpositions of local excitations, the low-
n reduction also fails. These results show that genuine multi-entropy probes multipartite entanglement structure beyond the tripartite information, and hence beyond the topological entanglement entropy in the Kitaev-Preskill partition, whereas for stabilizer states at low replica index it reduces to lower-partite entropic data.
Rényi multi-entropy probes multi-partite structure beyond bipartite spectra, but its behavior in general quantum states and the entanglement structure it captures remain poorly understood. We develop a finite field Gauss sum framework for calculating it in stabilizer states. Within this framework, the replica contraction reduces to a quadratic Gauss sum if and only if the state is locally unitarily equivalent to a stabilizer state. We obtain analytic expressions for the Rényi multi-entropy of all tree graph states and large families of loop graph states. In contrast to their flat bipartite Rényi spectra, stabilizer states generically have nonflat Rényi multi-entropy spectra.
Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.