Two Local Observables are Sufficient to Characterize Maximally Entangled States of N Qubits

Maximally entangled states (MES) represent a valuable resource in quantum information processing. In $N$-qubit systems the MES are $N$-GHZ states, i.e. the collection of $\ket{GHZ_N}=\frac{1}{\sqrt{2}}(\ket{00...0}+\ket{11...1})$ and its local unitary (LU) equivalences. While it is well-known that such states are uniquely stabilized by $N$ commuting observables, in this Letter we consider the minimum number of non-commuting observables needed to characterize an $N$-qubit MES as the unique common eigenstate. Here, we prove that in this general case, any $N$-GHZ state can be uniquely stabilized by only two observables. Thus, for the task of MES certification, only two correlated measurements are required with each party observing the spin of his/her system along one of two directions.

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