Abstract
We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser–Horne–Shimony–Holt (CHSH) inequality. We make use of the elegant parametrization in the canonical form of these states, proposed by Meill and Meyer (Phys Rev A 96:062310, 2017) based on Majorana geometric representation. The reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state, do not violate the CHSH inequality, and hence, they are CHSH-local. However, when Alice and Bob perform a CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. We have also shown that two different families of three-qubit pure symmetric states, consisting of two and three distinct spinors (qubits), respectively, can be distinguished based on the strength of violation in the conditional CHSH nonlocality test. Furthermore, we identify six of the 46 classes of tight Bell inequalities in the three-party, two-setting, two-outcome, i.e., (3,2,2) scenario (López-Rosa et al. in Phys Rev A 94:062121, 2016). Among the two inequivalent families of three-qubit pure symmetric states, only the states belonging to three distinct spinor class show maximum violations of these six tight Bell inequalities.
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Notes
In the case of a symmetric two-qubit density matrix we have \(T=T^\dag \). Thus, the eigenvalues \( t_1^2,\, t_2^2,\, t_3^2\) of \(T^\dag \, T\) are determined by those of the real symmetric matrix T itself.
Note that \(\langle \mathrm{CHSH}\rangle _{c=-1,\mathrm{opt}}=2\, \sqrt{(t^{c=-1}_1)^2+(t^{c=-1}_2)^2}\) is identically equal to \(\langle \mathrm{CHSH}\rangle _{c=1,\mathrm{opt}}\), when Charlie changes his measurement orientation \({\hat{n}}(\theta ,\phi )\) to \(-{\hat{n}}(\theta ,\phi )={\hat{n}}(\pi -\theta ,\pi +\phi ).\)
The three-qubit states \(\vert \psi ^{(k)}_{\mathrm{ABC}}\rangle , k=2,5,22,26,33,39\) are the ones which exhibit identical pairwise concurrences \(C_{\mathrm{AB}}=C_{\mathrm{BC}}=C_{\mathrm{AC}}\) (see Table VI of Ref. [20]). These states are found to be local unitary equivalent to permutation symmetric states \(\vert \psi ^{(k)}_{\mathrm{sym}}\rangle \).
We determine the explicit form \(\vert \beta \rangle =\cos (\beta /2)\vert 0\rangle +\sin (\beta /2)\vert 1\rangle \) of the constituent qubit states of the three-qubit pure symmetric states \(\vert \psi ^{(k)}_{\mathrm{sym}}\rangle \), \(k=2,\,5,\,22,\,33,\,39\) by solving the Majorana polynomial equation [10] \(b^{(k)}~+~3\,c^{(k)} z^2-a^{(k)}\, z^3 =0,\) where \(z=\tan (\beta /2)\). The Majorana polynomial equation associated with the state \(\vert \psi ^{(26)}_{\mathrm{sym}}\rangle \) reduces to \(z^{-1}\,(-1+\sqrt{3}\,z^2)=0\).
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Acknowledgements
We thank Professor A. K. Rajagopal for going through the manuscript and for making insightful suggestions. KA acknowledges financial support from UGC-RGNF, India. ASH is supported by the Foundation for Polish Science (IRAP Project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth Operational Programme). HSK acknowledges the support of NCN through Grant SHENG (2018/30/Q/ST2/00625). This work was partly done when HSK was at The Institute of Mathematical Sciences, Chennai, India and progressed further at ICTQT, Gdansk, Poland. Sudha and ARU are supported by the Department of Science and Technology(DST), India, through Project No. DST/ICPS/QUST/Theme-2/2019 (Proposal Ref. No. 107).
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Anjali, K., Hejamadi, A.S., Karthik, H.S. et al. Characterizing nonlocality of pure symmetric three-qubit states. Quantum Inf Process 20, 187 (2021). https://doi.org/10.1007/s11128-021-03124-x
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DOI: https://doi.org/10.1007/s11128-021-03124-x