Transformations of Stabilizer States in Quantum Networks

Abstract

Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to $3$ parties [Bravyi et al., J. Math. Phys. {\bf 47}, 062106~(2006)], we show that for $4$ and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for $4$ parties up to $10$ qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to $2$, which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.

37 pages, 7 figures, final version

References in corpus (8)

• The Quantum Internet
• Entanglement detection
• Multi-party entanglement in graph states
• Bell nonlocality in networks
• Classification problems for system of forms and linear mappings
• Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques
• Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild
• Homology of Generic Stabilizer States

Cited by in corpus (8)

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• Quantum Internet: from Medium Access Control to Entanglement Access Control
• Localizing genuine multiparty entanglement in noisy stabilizer states
• Multipartite Greenberger-Horne-Zeilinger Entanglement in Monitored Random Clifford Circuits
• Symmetry-resolved genuine multi-entropy: Haar random and graph states
• Frustration graph formalism for qudit observables
• Qudit Noisy Stabilizer Formalism
• Entanglement witnesses for stabilizer states and subspaces beyond qubits