Tensor Rank and Other Multipartite Entanglement Measures of Graph States

Abstract

Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. Prior work has revealed elegant connections between the graph structure of these states and their multipartite entanglement content. 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 ($|R_{2n+1}⟩$) to read $2^n+1\le rank(|R_{2n+1}⟩)\le 3\ast 2^{n-1}$. Next, we show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph. Lastly, we give a simple graph rule for computing the n-tangle ${\tau }_n$.

arXiv comment9 + 5 pages, 5 figures

References in corpus (33)

• Quantum entanglement
• Mixed State Entanglement and Quantum Error Correction
• A computable measure of entanglement
• Distributed Entanglement
• Persistent entanglement in arrays of interacting particles
• Multi-party entanglement in graph states
• Geometric measure of entanglement and applications to bipartite and multipartite quantum states
• Entanglement monotones
• Graph States for Quantum Secret Sharing
• Quantum error-correcting codes associated with graphs
• Graphical description of the action of local Clifford transformations on graph states
• Measure of genuine multipartite entanglement with computable lower bounds
• A potential multipartide entanglement measure
• The Schmidt Measure as a Tool for Quantifying Multi-Particle Entanglement
• Concurrence of mixed multi-partite quantum states
• A rigorous treatment of distillable entanglement
• Quantum Algorithms: Entanglement Enhanced Information Processing
• Convex-roof extended negativity as an entanglement measure for bipartite quantum systems
• Fast simulation of stabilizer circuits using a graph state representation
• Quantum Error Correcting Codes Using Qudit Graph States
• Multipartite entanglement detection from correlation tensors
• Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
• A resource theory of entanglement with a unique multipartite maximally entangled state
• Tripartite entanglement transformations and tensor rank
• Obtain $W$-state from three-qubit $GHZ$-state on rate 1
• Entanglement in eight-qubit graph states
• Mapping graph state orbits under local complementation
• Experimentally accessible lower bounds for genuine multipartite entanglement and coherence measures
• Asymptotic entanglement transformation between W and GHZ states
• Entanglement distillation from Greenberger-Horne-Zeilinger shares
• Tensor rank and entanglement of pure quantum states
• Two-colorable graph states with maximal Schmidt measure
• Tensor surgery and tensor rank

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