Transformations of Stabilizer States in Quantum Networks

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