Multi-party entanglement in graph states

Abstract

Graph states are multi-particle 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, multi-party quantum communication, and quantum computation within the framework of the one-way quantum computer. We characterize and quantify the genuine multi-particle 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 dimension, 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 isomorphies.

22 pages, 15 figures, 2 tables, typos corrected (e.g. in measurement rules), references added/updated

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• All-versus-nothing proofs with n qubits distributed between m parties
• Exclusivity structures and graph representatives of local complementation orbits
• Hardy-like Quantum Pigeonhole Paradox and the Projected-Coloring Graph State
• Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms
• Entanglement of multi-qubit states representing directed networks and its detection with quantum computing
• Interlace Polynomials: Enumeration, Unimodality, and Connections to Codes
• Distinguishing Phases via Non-Markovian Dynamics of Entanglement in Topological Quantum Codes under Parallel Magnetic Field
• Group structures and representations of graph states
• Theory of time-bin entangled photons from quantum emitters
• Non-Markovian Dynamics of Spin Squeezing
• Classical versus Quantum Graph-based Secret Sharing
• Deriving three-outcome permutationally invariant Bell inequalities
• Generation of photonic tensor network states with Circuit QED
• Scheme for constructing graphs associated with stabilizer quantum codes
• Fidelity-Aware Multipath Routing for Multipartite State Distribution in Quantum Networks
• Quantum Random State Generation with Predefined Entanglement Constraint
• Local unitary symmetries and entanglement invariants
• A Substrate Scheduler for Compiling Arbitrary Fault-tolerant Graph States
• Calibrating quantum gates up to 52 qubits in a superconducting processor
• Reliable Quantum Communications based on Asymmetry in Distillation and Coding
• Scalable Determination of Multipartite Entanglement in Quantum Networks
• Efficient entanglement purification based on noise guessing decoding
• Localizing genuine multiparty entanglement in noisy stabilizer states
• Multiparticle singlet states cannot be maximally entangled for the bipartitions
• Entropic measure and hypergraph states
• Entanglement of particles versus entanglement of fields: independent quantum resources
• Generating and verifying graph states for fault-tolerant topological measurement-based quantum computing in 2D optical lattices
• Deterministic Dense Coding and Faithful Teleportation with Multipartite Graph States
• General scheme for preparation of different topological states on the cluster states
• Almost Equivalent Paradigms of Contextuality
• Permutation-equivariant quantum convolutional neural networks
• Entanglement and its facets in condensed matter systems
• A note on the Bloch representation of absolutely maximally entangled states
• The von Neumann entropy of networks
• Designing locally maximally entangled quantum states with arbitrary local symmetries
• Optimal accessing and non-accessing structures for graph protocols
• Twisted graph states for ancilla-driven quantum computation
• A trace distance-based geometric analysis of the stabilizer polytope for few-qubit systems
• Experimental Detection of Non-local Correlations using a Local Measurement-Based Hierarchy on an NMR Quantum Processor
• Operational symmetries of entangled states
• Many-body quantum resources of graph states
• Reversibility in the Extended Measurement-based Quantum Computation
• Identifying non-k-separability of a class of N-qubit complete graph states using correlation tensors
• Generating graph states with a single quantum emitter and the minimum number of fusions
• On the role of coherence for quantum computational advantage
• Graphical Framework for Non-Gaussian Quantum States
• Hypercubic cluster states in the phase modulated quantum optical frequency comb
• Measurement-based quantum machine learning
• Transforming graph states via Bell state measurements
• Finding maximal quantum resources
• The Min-Entropy of Classical-Quantum Combs for Measurement-Based Applications
• Entanglement entropy of higher rank topological phases
• A solvable model for graph state decoherence dynamics
• Entanglement, Complexity, and Causal Asymmetry in Quantum Theories
• Characterization of multipartite entanglement in terms of local transformations
• Nonclassical Structures within the N-qubit Pauli Group
• Finite-Function-Encoding Quantum States
• The Foliage Partition: An Easy-to-Compute LC-Invariant for Graph States
• Entanglement of weighted graphs uncovers transitions in variable-range interacting models
• Stabilizer states and local realism
• Asymmetric Quantum Secure Multi-Party Computation With Weak Clients Against Dishonest Majority
• Demonstration of teleportation across a quantum network code
• Clifford Manipulations of Stabilizer States: A graphical rule book for Clifford unitaries and measurements on cluster states, and application to photonic quantum computing
• Tensor Rank and Other Multipartite Entanglement Measures of Graph States
• Pseudo-telepathy games and genuine NS k-way nonlocality using graph states
• Learning quantum graph states with product measurements
• Cavity polariton blockade for non-local entangling gates with trapped atoms
• Local Invariance of Divergence-based Quantum Information Measures
• Measurement-Based Entanglement Distillation and Constant-Rate Quantum Repeaters over Arbitrary Distances
• Entanglement in Quantum Systems Based on Directed Graphs
• Storing Small Photonic Cluster States in a Dephasing Environment
• An Extensible Quantum Network Simulator Built on ns-3: Q2NS Design and Evaluation
• Graph state extraction from two-dimensional cluster states
• Superconducting qubits in the millions: the potential and limitations of modularity
• Exact Entanglement Dynamics of Two Spins in Finite Baths
• Entanglement bound for multipartite pure states based on local measurements
• All genuinely entangled stabilizer subspaces are multipartite fully nonlocal
• Atom-mediated deterministic generation and stitching of photonic graph states
• Distinguishing Graph States by the Properties of Their Marginals
• Localizing multipartite entanglement with local and global measurements
• The Study of Entangled States in Quantum Computation and Quantum Information Science
• Twin GHZ-states behave differently
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• From Graph States to Two-Graph States
• Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations
• Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
• Efficient and high-fidelity entanglement in cavity QED without high cooperativity
• Quantum Theory from Principles, Quantum Software from Diagrams
• Measurement-based quantum computing with qudit stabilizer states
• Minimising the number of edges in LC-equivalent graph states
• Multipartite separability of density matrices of graphs
• Generation of frequency-bin-encoded dual-rail cluster states via time-frequency multiplexing of microwave photonic qubits
• Maximal Entanglement: Applications in Quantum Information and Particle Physics
• The genuinely multipartite nonlocality of graph states is model-dependent
• Local density matrices of many-body states in the constant weight subspaces
• Tree-size complexity of multiqubit states
• Symmetry-resolved genuine multi-entropy: Haar random and graph states
• Three-Qubit State Preparation: Classification and Explicit Circuits
• Quantum Zero-error Capacity
• Small quantum networks in the qudit stabilizer formalism
• Robustness of Bell Violation of Graph States to Qubit Loss
• Entanglement of Assistance as a measure of multiparty entanglement
• Entanglement of stabilizer codewords
• Efficient bounds on quantum communication rates via their reduced variants
• Continuous-Variable Multiplexed Quantum Repeater Networks
• Quantum Optimal Control for Pure-State Preparation Using One Initial State
• Geometric Graph-Theoretic Aspects of Quantum Stabilizer Codes
• Gauge and diffeomorphism invariance from quantum information principles
• Optical Quantum Computing
• Hybrid Reward-Driven Reinforcement Learning for Efficient Quantum Circuit Synthesis
• Sharing classical secrets with continuous-variable entanglement: Composable security and network coding advantage
• Long-ranged gates in quantum computation architectures with limited connectivity
• Unifying communication paradigms in measurement-based delegated quantum computing
• Optimal Verification of the Bell State and Greenberger-Horne-Zeilinger States in Untrusted Quantum Networks
• Determinism and Computational Power of Real Measurement-based Quantum Computation
• Z-States Algebra for a Tunable Multi-Party Entanglement-Distillation Protocol
• Fluxon-based generation of graph states in Josephson qubits
• Generalized Type II Fusion of Cluster States
• Multipartite Entanglement Distribution in Quantum Networks using Subgraph Complementations
• Fully multi-qubit entangled states
• Threetangle in the XY-model class with a non-integrable field background
• Topology-Dependent Enhancement of Entanglement Extraction in Repeater Graph States
• Quantum states supported by matroids
• Exploring Bell nonlocality of quantum networks with stabilizing and logical operators
• Efficient tomography of microwave photonic cluster states
• Quantum Godwin's Law
• $\chi$-Colorable Graph States: Closed-Form Expressions and Quantum Orthogonal Arrays
• Detecting Entanglement by State Preparation and Local Measurements
• The Virtual Quantum Device (VQD): A tool for detailed emulation of quantum computers
• A Lie Theoretic Framework for Controlling Open Quantum Systems
• A streamlined demonstration that stabilizer circuits simulation reduces to Boolean linear algebra
• Entanglement, Flow and Classical Simulatability in Measurement Based Quantum Computation
• On the relation between a graph code and a graph state
• Reducing depth and measurement weights in Pauli-based computation
• Logic Functions and Quantum Error Correcting Codes
• Entanglement witnesses for stabilizer states and subspaces beyond qubits
• Automated discovery of heralded ballistic graph state generators for fusion-based photonic quantum computation
• Self-Testing Graph States Permitting Bounded Classical Communication
• Quadratic Form Expansions for Unitaries
• Studies of properties of bipartite graphs with quantum programming
• Evaluating Pauli errors on cluster states by weighted distances
• Blind quantum computing with different qudit resource state architectures
• One-Way Quantum Computer Simulation
• Quantum Information Processing with Continuous Variables and Atomic Ensembles
• Scalable and fault-tolerant preparation of encoded k-uniform states
• Access structure in graphs in high dimension and application to secret sharing