Abstract
We construct multipartite graph states whose dimension is the power of a prime number. This is realized by the finite field, as well as the generalized controlled-NOT quantum circuit acting on two qudits. We propose the standard form of graph states up to local unitary transformations and particle permutations. The form greatly simplifies the classification of graph states as we illustrate up to five qudits. We also show that some graph states are multipartite maximally entangled states in the sense that any bipartition of the system produces a bipartite maximally entangled state. We further prove that 4-partite maximally entangled states exist when the dimension is an odd number at least three or a multiple of four.
Highlights
Maximal entanglement is the key ingredient in quantum teleportation, computing and the violation of Bell inequality
We propose the dual graph state of the standard form in (94), and show that they are equivalent under local unitary transformation in Theorem 2
Our main task is to find out the maximally entangled state by the quantum circuit composed of generalized CNOT gates
Summary
We construct the generalized CNOT gates by two one-qudit operations A(am) and D(am). They are mathematically realized by the known finite field and the commutation relations. Before investigating the properties of the generated states, let us first calculate the basic commutation relations for related unitary transformations widely used throughout the paper. The proof of these relations will be given in the end of this subsection.
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