We revisit the computation of correlations of spin projections onto unit vectors for spin-1/2 particles in Quantum Mechanics. We then choose one of the Boole inequalities that, as we recall, must be obeyed by collections of sequences of normalized spin-1/2 projections onto unit vectors that belong to, say, the plane that is orthogonal to the classical trajectory or to the three dimensional space. Next we define the concept of being a-p for i-indexed sequences of spin-1/2 particles: it turns out that a sequence of spin-1/2 particles is a-p if and only if the particles behave as if they had indeed been prepared (in the usual sense) to have definite spins +1 or -1 along the axis a, the sequence of signs being pre-determined. The a-p property is formulated in a way abstract enough to let one generalize it and we define the concept of being a-p and b-p. However, using our chosen Boole inequality we prove that no particle sequence can be both a-p and b-p when a and b are not co-linear. We next consider two modifications of Quantum Mechanics: - The first one by assuming both Weak Realism, the weakest form of Realism needed to develop any main theorem in Bell's Theory, and Locality. - The second one by assuming Weak Realism and an hypothesis that turns out to be none else than (relativistic) Causality extended to cover as well the observable values that make sense only if one assumes Weak Realism. In both cases we arrive at the conclusion that in the EPRB setting where successive pairs of spin-1/2 particles are prepared in the so-called singlet state, one of these particles sequences is both a-p and b-p and a and b are the opposite of the vectors respectively used to project the spin of the other particle and to get the Realism-based alternate value for a projection. Thus the usual discussions of Bell's inequalities under such hypotheses do not even make physical sense.
Some new, powerful, and very simple Bell inequalities are derived. They establish that for two spin-
s particles in the singlet state, the quantum-theoretic Einstein-Podolsky-Rosen correlations for spin measurements along all pairs of directions from a set of
N axes are incompatible with local realism for any set of distinct coplanar axes when
N=3, and for any set of distinct axes whatever when
N=4, with the possible exception of sets restricted by one of the constraints $\stackrel{^}{a}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{b}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{c}\ifmmode\pm\else\textpm\fi{}\stackrel{^}{d}=0$. These results hold for any value of the spin.
Given a sequence of pairs (
pi,
pi) of spin-
1/2 particles in the singlet state, assume that Alice measures the normalized projections
ai of the spins of the
pi's along vector $\mathbit{a}$ while Bob measures the normalized projections
bi of the spins of the
pi's along vector $\mathbit{b}$. Then quantum mechanics (QM) lets one evaluate the correlation
⟨a,b⟩ as $\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{\mathbit{a}}\ensuremath{-}{\ensuremath{\theta}}_{\mathbit{b}})$ where ${\ensuremath{\theta}}_{\mathbit{v}}$ is the angle between the vector $\mathbit{v}$ and a reference vector chosen once and for all and in a fixed plane. Assuming classical microscopic realism (CMR) there exist also normalized projection pairs (${a}_{i}^{\ensuremath{'}}$, ${b}_{i}^{\ensuremath{'}}$) of the spins of the pairs (
pi,
pi) along (${\mathbit{a}}^{\ensuremath{'}}$, ${\mathbit{b}}^{\ensuremath{'}}$) so that $⟨{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}⟩=\ensuremath{-}\mathrm{cos}({\ensuremath{\theta}}_{{\mathbit{a}}^{\ensuremath{'}}}\ensuremath{-}{\ensuremath{\theta}}_{{\mathbit{b}}^{\ensuremath{'}}})$. Since all projections are in ${\ensuremath{-}1,1}$, $|⟨c,d⟩+⟨c,e⟩|+|⟨f,d⟩\ensuremath{-}⟨f,e⟩|\ensuremath{\le}2$ for
c,
d,
e, and
f in ${a,b,{a}^{\ensuremath{'}},{b}^{\ensuremath{'}}}$. Assuming locality (the impossibility of any effect of an event on another event when said events are spatially separated) beside QM and CMR, Bell's theory lets one deduce various violations of this inequality at some choices of quadruplets $Q\ensuremath{\equiv}(\mathbit{a},\mathbit{b},{\mathbit{a}}^{\ensuremath{'}},{\mathbit{b}}^{\ensuremath{'}})$. Our main result is the existence of such
Q's where at least one of the above inequalities is violated if one only assumes QM, CMR, and some very mild further hypotheses that only concern the behavior of correlations that appear in these inequalities near special
Q's.
The correlations between local measurement results are characteristic for multi-partite quantum systems. Under the condition of local realism, which assumes that properties of physical systems exist independent of being measured, and that space-like separated actions on the system cannot influence each other, these correlations are constrained by Bell-inequalities. Quantum mechanics predicts correlations which are not explainable under these two assumptions. The resulting violation of Bell inequalities was originally studied for pairs of entangled particles (Einstein et al.,1935). Meanwhile these ideas were extended to three-and more partite quantum states in particular with respect to the different kinds of multi-particle entanglement and their implications on the obtainable correlations (Mermin, 1990). Here we present a method to construct characteristic Bell inequalities for particular four-qubit states. These inequalities are a versatile tool, not only to provide evidence for non-locality, but also for the characterization of the states, with respect to detection of genuine multipartite entanglement and to state discrimination. Exemplarily we apply our approach to the symmetric four qubit Dicke-state (Dicke, 1954) with two excitations. (A symmetric N-qubit Dicke state with M excitations is the equal superposition of N-qubit pure product states having M logical 1 's and (N-M) logical 0's.) Experimentally the state was observed with photons as carrier of the qubits, using polarization encoding.
Bell’s inequality can be traced back to a classical probabilistic formula that is invalid clearly, while spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Bell’s two misunderstandings about Bell’s theorem are pointed out: Firstly, the hidden variable theory Bell’s used, which is a special form of such theories, is regarded as a general one. Secondly, there are two theses obtainable from Bell’s hypotheses; one is compatible with quantum mechanics, the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G.Lochak has revealed the first thesis, and thereby he pointed out that Bell’s inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually without regard to Bell’s inequality.