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Chinese Physics Letters

Rotationally Invariant Proof of Bell＇s Theorem Without Inequalities for 4k-Dimensional （k = 1, 2,...） Systems

梁林梅, 李承祖

Published Jan 1, 2004• 0 citations
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Abstract

We present rotationally invariant proof of Bell's theorem without inequalities for spin-(2N+1)/2system (N = 2k-1,k = 1, 2, 3, 4...) with only two particles, which is a generalization of Cabello's study [Phys. Rev. A 67 (2003)032107].

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In this Comment we show that Cabello's proof of Bell's theorem without inequalities [Phys. Rev. Lett. 91, 230403 (2003)] does not exhibit two of the three "remarkable properties" which the proof is claimed to possess. More precisely it is our purpose to show that property (c) of Cabello's proof, stating that ``local observables can be measured by means of tests on individual qubits" is definitely false since it has been obtained by a wrong use of the basic rules of the quantum formalism, while property (b), stating that ``distant local setups do not need to be aligned since the require perfect correlations are achieved for any local rotation of the setups", is also wrong due to the failure of validity of property (c).

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Bell’s inequality fornspin- s particles

Adán Cabello

The Mermin-Klyshko inequality for n spin-$\frac{1}{2}$ particles and two dichotomic observables is generalized to n spin-$s$ particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any $s.$ For a fixed $n,$ the magnitude of the violation is constant for any $s,$ which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-$s$ particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed $s,$ the violation grows exponentially with $n,$ which provides a generalization to any s of Mermin's conclusion for n $\mathrm{spin}\ensuremath{-} \frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].

arXiv:quant-ph/0202126•Physical Review A•Jun 5, 2002•27 citations
The singlet state of two spin-32\frac{3}{2}23​ particles allows a proof of Bell's theorem without inequalities with two distinguishing features: any local observable can be regarded as an Einstein-Podolsky-Rosen element of reality, and the contradiction with local realism occurs not only for some specific local observables but for any rotation whereof.
In this Comment we show that Cabello's proof of Bell's theorem without inequalities [Phys. Rev. Lett. 91, 230403 (2003)] does not exhibit two of the three "remarkable properties" which the proof is claimed to possess. More precisely it is our purpose to show that property (c) of Cabello's proof, stating that ``local observables can be measured by means of tests on individual qubits" is definitely false since it has been obtained by a wrong use of the basic rules of the quantum formalism, while property (b), stating that ``distant local setups do not need to be aligned since the require perfect correlations are achieved for any local rotation of the setups", is also wrong due to the failure of validity of property (c).
The Mermin-Klyshko inequality for n spin-12\frac{1}{2}21​ particles and two dichotomic observables is generalized to n spin-sss particles and two maximal observables. It is shown that some multiparty multilevel Greenberger-Horne-Zeilinger states [A. Cabello, Phys. Rev. A 63, 022104 (2001)] maximally violate this inequality for any s.s.s. For a fixed n,n,n, the magnitude of the violation is constant for any s,s,s, which provides a simple demonstration and generalizes the conclusion reached by Gisin and Peres for two spin-sss particles in the singlet state [Phys. Lett. A 162, 15 (1992)]. For a fixed s,s,s, the violation grows exponentially with n,n,n, which provides a generalization to any s of Mermin's conclusion for n $\mathrm{spin}\ensuremath{-} \frac{1}{2}$ particles [Phys. Rev. Lett. 65, 1838 (1990)].