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 |GHZN = 1 √ 2 (|00 · · · 0 + |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. PACS numbers: 03.65. Ud,03.65.Ta, 03.67.Dd From both a theoretical and practical perspective, maximally entangled states (MES) play an important role in quantum information science. While there may be different ways to consider some state more entangled than another, one can work from an axiomatic perspective to define "maximally" entangled states in the multipartite setting. This is the approach taken by Gisin and Bechmann-Pasquinucci who identify N -GHZ states as maximally entangled in N -qubit systems [1] . Their justification primarily comes from observing these states to maximally violate the Bell-Klyshko inequalities, a generalization of the Bell inequalities to more than two parties. Chen advanced the work of [1] by proving N -GHZ states to be the unique family of states which demonstrate such a maximal violation [2] . Hence, it becomes appropriate to regard N -GHZ states as the maximally entangled multiqubit states. At the same time, MES have been recognized as key ingredients in quantum information processing (QIP). The pioneering bipartite tasks of quantum key distribution (QKD) [3, 4] (|00 + |11 ) to achieve their powerful non-classical effects. Multipartite generalizations of these procedures have been developed [8] [9] [10] [11] , as well as novel schemes such as quantum secret sharing [8, 12] , which like their bipartite ancestors, involve manipulations and measurements on MES. The general attraction of MES for information processing is dual since they not only allow for complete correlation between measurements on subsystems, but their purity also ensures these correlations to exist exclusively within the system, i.e. no external eavesdropper can be correlated with any of the subsystems. Since the use of MES is critical to the success of the aforementioned QIP schemes, it is important for the parties to verify that they indeed are encoding their information in MES and not other types of states. One method of doing this is to prepare sufficiently more MES than needed for the given QIP task. From this larger population, a random subset of states is checked to be MES, and if this inspection passes, the remaining states are certified to also be MES with arbitrarily high probability. The task of verifying channel security then reduces to whether N parties can determine if some collection of mutually shared states are all MES. In the bipartite case, Ekert first proposed using Bell inequalities to ascertain whether two parties hold EPR states [4] . While Bell inequalities involve the expectation values of four different observables, Bennett et al. later observed that only two local observables were necessary to detect the possession of EPR pairs [13] . Specifically, the state |Ψ + is the unique +1 eigenstate of the local spin measurements σ X ⊗ σ X and σ Z ⊗ σ Z , where σ X and σ Z are Pauli matrices. Consequently, repeating these measurements on some sample of states can detect the presence of a potential eavesdropper and ensure the protocol's overall safety. Using stabilizer formalism, this idea can be generalized to check the safety of multipartite MES. The set of commuting product Pauli operators having |GHZ N as the unique common +1 eigenstate forms an Abelian group. Letting {P i } k i=1 denote a minimal set of generators for this group and I the identity, the projector onto their common +1 eigenspace is given by
This property is suitably generalized to any value of N. Yan et al. REF deal with maximally entangled states of N qubits, an interesting property of N-GHZ states.
Two Local Observables are Sufficient to Characterize Maximally Entangled States of N Qubits
Physical Review A 83, 022319 (2011)
Physics