Fig. 2: Self-testing of the relative phase ϕ(N).
From: An elegant scheme of self-testing for multipartite Bell inequalities
This graphic schematically captures the self-testing statements for N partite Uffink’s quadratic Bell inequalities (Theorem 7), as well as the self-testing statements for the complex-valued Bell expressions corresponding to the mean value of Uffink’s non-Hermitian operator \(\left\langle {\tilde{{{{\mathcal{U}}}}}}_{N}\right\rangle\) (Corollary 7.1). The figure depicts the complex plane on which the complex number \(\left\langle {\tilde{{{{\mathcal{U}}}}}}_{N}\right\rangle =\left\langle {{{{\mathcal{M}}}}}_{N}\right\rangle \pm \iota \left\langle {{{{\mathcal{M}}}}}_{N}^{{\prime} }\right\rangle\) lies, where \(\left\langle {{{{\mathcal{M}}}}}_{N}\right\rangle\) is plotted on the real axis while \(\left\langle {{{{\mathcal{M}}}}}_{N}^{{\prime} }\right\rangle\) is plotted on the imaginary axis. The dark blue arc represents the boundary of the quantum set of correlations \({{{\mathcal{Q}}}}\) characterized by the maximum violation of N partite Uffink’s quadratic Bell inequalities, \({{{{\mathcal{U}}}}}_{N}^{{{{\mathcal{M}}}}}=| \left\langle {\tilde{{{{\mathcal{U}}}}}}_{N}\right\rangle {| }^{2}={2}^{N-1}\). Crucially, for all points on the dark blue arc, Theorem 7 implies that the local observables of each party can be taken to be A(j) = σx and \({A}^{{\prime} {(j)}}={\sigma }_{y}\), and the state to be maximally entangled N partite GHZ state, \(\left\vert GH{Z}_{N}\right\rangle =\frac{1}{\sqrt{2}}({\left\vert 0\right\rangle }^{\otimes N}+{e}^{\iota \phi (N)}{\left\vert 1\right\rangle }^{\otimes N})\), where the relative phase ϕ(N) can be chosen arbitrarily (up to local isometries). However, using Corollary 7.1, even the relative phase can be uniquely identified by the observed value of \(\left\langle {\tilde{{{{\mathcal{U}}}}}}_{N}\right\rangle\) if it lies on this arc. To exemplify this, we plot the specific case of \(\left\langle {\tilde{{{{\mathcal{U}}}}}}_{N}\right\rangle =\left\langle {{{{\mathcal{M}}}}}_{N}\right\rangle \pm \iota \left\langle {{{{\mathcal{M}}}}}_{N}^{{\prime} }\right\rangle ={2}^{\frac{N-2}{2}}{e}^{\iota \frac{\pi }{3}}\) (light blue arrow), which uniquely specifies the relative phase to be \(\phi (N)={{{\rm{arccot}}}}\left(\frac{\left\langle {{{{\mathcal{M}}}}}_{{{{\mathcal{N}}}}}\right\rangle }{\left\langle {{{{\mathcal{M}}}}}_{N}^{{\prime} }\right\rangle }\right)=\frac{\pi }{3}\) when N is odd, and \(\phi (N)={{{\rm{arccot}}}}\left(\frac{\left\langle {{{{\mathcal{M}}}}}_{{{{\mathcal{N}}}}}\right\rangle }{\left\langle {{{{\mathcal{M}}}}}_{N}^{{\prime} }\right\rangle }\right)-\frac{\pi }{4}=\frac{\pi }{12}\) when N is even.
