Graph states and local unitary transformations beyond local Clifford operations

Abstract

Graph states are quantum states that can be described by a stabilizer formalism and play an important role in quantum information processing. We consider the action of local unitary operations on graph states and hypergraph states. We focus on non-Clifford operations and find for certain transformations a graphical description in terms of weighted hypergraphs. This leads to the indentification of hypergraph states that are locally equivalent to graph states. Moreover, we present a systematic way to construct pairs of graph states which are equivalent under local unitary operations, but not equivalent under local Clifford operations. This generates counterexamples to a conjecture known as LU-LC conjecture. So far, the only counterexamples to this conjecture were found by random search. Our method reproduces the smallest known counterexample as a special case and provides a physical interpretation.

19 pages, 8 figures, v2: final version

References in corpus (7)

• Multi-party entanglement in graph states
• Graphs, Quadratic Forms, and Quantum Codes
• Entanglement and nonclassical properties of hypergraph states
• Encoding Hypergraphs into Quantum States
• Local unitary symmetries of hypergraph states
• Purification to Locally Maximally Entangleable States
• Compact set of invariants characterizing graph states of up to eight qubits

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• Symmetric hypergraph states: Entanglement quantification and robust Bell nonlocality
• Some Ulam's reconstruction problems for quantum states
• Shor-Laflamme distributions of graph states and noise robustness of entanglement
• Engineering holography with stabilizer graph codes
• Generating graph states with a single quantum emitter and the minimum number of fusions
• Non-symmetric GHZ states: weighted hypergraph and controlled-unitary graph representations
• Distinguishing Graph States by the Properties of Their Marginals
• $\chi$-Colorable Graph States: Closed-Form Expressions and Quantum Orthogonal Arrays
• Geometric Graph-Theoretic Aspects of Quantum Stabilizer Codes