Fortschr. Phys. 51, No. 4–5, 531 – 538 (2003) / DOI 10.1002/prop.200310071 On Bell’s theorem for N-qubits Marek ˙ Zukowski 1, ∗ and Caslav Brukner 2 1 Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gda´ nski, 80-952 Gda´ nsk, Poland 2 Institut f¨ ur Experimentalphysik, Universit¨ at Wien, Boltzmanngasse 5, 1090 Wien, Austria Received 29 July 2002, accepted 5 August 2002 Published online 30 April 2003 WepresentasinglegeneralBellinequalitythatsummarizesallpossibleBelltypeconstrainsonthecorrelation function for the N-particle system (i.e., it is a necessary and sufficient condition for the existence of local realistic models for the correlation function). It is applicable in the case when measurements on each particle can be chosen between two arbitrary dichotomic observables. We also obtain a necessary and sufficient condition for an arbitrary N-qubit mixed state to violate this general Bell inequality. 1 Introduction Bell type inequalities [1–5] are bounds on certain combinations of statistical correlations for measurements on multi-particle systems which can be understood within a realistic picture based on local properties of each individual particle. In such a picture particles carry properties prior to and independent of observation that are performed on the other particle(s) at space-like separation. Quantum mechanics predicts violation of the inequalities even if the individual particles are separated. This is known as Bell’s theorem in quantum mechanics. However, the problem which states violate Bell type inequalities has been solved only for pure states [6,7] and for the mixed states only in the most simple case of two-qubit systems [8]. Here we present a generalized single Bell inequality that is a necessary and sufficient condition for local realistic description of a correlation function for the N -particle system, for the case where the measurements oneachparticlearechosenbetweentwodichotomicobservables[9].Thederivationwilluseaspecificfeature of dichotomic measurements. Namely this is the simplest possible (yes/no) measurement. It turns out that in this simplest case one can formulate the problem of existence of a local realistic description, for the full set of values of the correlation function involved in a given Bell-GHZ experiment, in such a way so that the probabilistic distribution of the predetermined hidden results acquires the form of an orthonormal basis expansion. This unique feature of the dichotomic observables, for the case when each local observer can choose between two of them, allows for the proof of the general inequality to go through. From the general inequality one can obtain, for example, Mermin-Klyshko [4,5] inequalities as a partic- ular case. We also formulate the condition for an arbitrary N -qubit mixed state to violate the general Bell inequality. The presentation will be given for N =3. The general proof for arbitrary N follows exactly the same line of thought. 2 General Bell inequality for correlation functions In order to make our reasoning more clear, we shall employ here the well known fact that local realistic theories can always be put in the form of deterministic local hidden variable theories. In a hidden variable ∗ Corresponding author E-mail: fizmz@univ.gda.pl c  2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0015-8208/03/4–505-0531 $ 17.50+.50/0