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arXiv:2209.06320 · Physical Review A

Tensor rank and other multipartite entanglement measures of graph states

Louis Schatzki, Linjian Ma, Edgar Solomonik, Eric Chitambar

Published Sep 9, 2024• 2 citations
arXivHTMLPDF

Abstract

Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states ($|{R}_{2n+1}\ensuremath{\rangle}$) to read ${2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the $n$-tangle ${\ensuremath{\tau}}_{n}$.

DOI
10.1103/physreva.110.032409

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Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work revealed elegant connections between the graph structure of these states and their multipartite entanglement. We continue this line of investigation by identifying additional entanglement properties for certain types of graph states. From the perspective of tensor theory, we tighten both upper and lower bounds on the tensor rank of odd ring states (∣R2n+1\ensuremath⟩|{R}_{2n+1}\ensuremath{\rangle}∣R2n+1​\ensuremath⟩) to read 2n+1\ensuremath≤rank(∣R2n+1\ensuremath⟩)\ensuremath≤3\ifmmode×\else\texttimes\fi2n\ensuremath−1{2}^{n}+1\ensuremath{\le}\mathrm{rank}(|{R}_{2n+1}\ensuremath{\rangle})\ensuremath{\le}3\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}2n+1\ensuremath≤rank(∣R2n+1​\ensuremath⟩)\ensuremath≤3\ifmmode×\else\texttimes\fi2n\ensuremath−1. Next we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Finally, we give a simple graph rule for computing the nnn-tangle \ensuremathτn{\ensuremath{\tau}}_{n}\ensuremathτn​.
Graph states are multiparticle entangled states that correspond to mathematical graphs, where the vertices of the graph take the role of quantum spin systems and edges represent Ising interactions. They are many-body spin states of distributed quantum systems that play a significant role in quantum error correction, multiparty quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multiparticle entanglement of such graph states in terms of the Schmidt measure, to which we provide upper and lower bounds in graph theoretical terms. Several examples and classes of graphs will be discussed, where these bounds coincide. These examples include trees, cluster states of different dimensions, graphs that occur in quantum error correction, such as the concatenated [7,1,3]-CSS code, and a graph associated with the quantum Fourier transform in the one-way computer. We also present general transformation rules for graphs when local Pauli measurements are applied, and give criteria for the equivalence of two graphs up to local unitary transformations, employing the stabilizer formalism. For graphs of up to seven vertices we provide complete characterization modulo local unitary transformations and graph isomorphisms.
Graph states are multipartite entangled states that correspond to mathematical graphs, where the vertices of the graph now play the role of quantum multilevel systems and edges represent interactions of the systems. Graph states are the basis of quantum error correction and one-way quantum computer. We systematically study the entanglement of non-binary graph states. Using iterative algorithm and entanglement bounds, we calculate the entanglement of all the ternary graph states up to nine vertices and parts of quaternary and quinary graph states modulo local unitary transformations and graph isomorphisms. The entanglement measure can be the geometric measure, the measure of relative entropy of entanglement or the measure of logarithmic robustness. We classify the graph states according to the entanglement values obtained. The closest product states obtained in the calculations are studied.
Graph states are special multipartite entangled states that have been proven useful in a variety of quantum information tasks. We address the issue of characterizing and quantifying the genuine multipartite entanglement of graph states up to eight qubits. The entanglement measures used are the geometric measure, the relative entropy of entanglement, and the logarithmic robustness, have been proved to be equal for the genuine entanglement of a graph state. We provide upper and lower bounds as well as an iterative algorithm to determine the genuine multipartite entanglement.
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.
Computationally feasible multipartite entanglement measures are needed to advance our understanding of complex quantum systems. An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers, showing several advantages over existing methods, including ease of computation, a deep geometrical interpretation, and applicability to multipartite entanglement. Here, we present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states, based on the purity of fragments of the whole systems. Our analysis includes the application of GEM to a two-mode Gaussian state coupled through a combined beamsplitter and a squeezing transformation. Additionally, we explore 3-mode and 4-mode graph states, where each vertex represents a bosonic mode, and each edge represents a quadratic transformation for various graph topologies. Interestingly, the ratio of the geometric entanglement measures for graph states with different topologies naturally captures properties related to the connectivity of the underlying graphs. Finally, by providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory on Rt×S1\mathbb R_t\times S^1Rt​×S1, going beyond the standard bipartite entanglement entropy approach between different regions of spacetime. The results presented herein suggest how the GEM paves the way for using quantum information-theoretical tools to study the topological properties of the space on which a quantum field theory is defined.
Quantum hypergraph states extend the well-studied class of graph states by taking into account multi-qubit interactions through hyperedges. They provide a powerful framework to represent a family of quantum states with genuine multipartite entanglement. In this review, we provide a compact overview of the formal structure, entanglement characteristics, and operational relevance of hypergraph states in quantum information theory. We begin by introducing their mathematical foundations and generalizations of the stabilizer formalism. A central focus is placed on their entanglement properties, including the classification under local unitary (LU) and stochastic local operations with classical communication (SLOCC), the quantification of multipartite entanglement, and detection techniques via entanglement witnesses. We also explore other nonclassical features of hypergraph states, such as contextuality and genuine multipartite nonlocality, derived from stabilizer-based Bell-type inequalities. Additional attention is given to the role of hypergraph states in error correction, and as a computational resource in measurement-based quantum computation (MBQC), and to their non-stabilizer character - quantified via resource-theoretic measures of quantum magic. Finally we review their generalization to higher dimensions, i.e. to qudits and continuous variables.