Reassessment of Leggett Inequality

Abstract

Leggett formulated an inequality that seems to generalize the Bell theorem to non-local hidden variable theories. Leggett inequality is violated by quantum mechanics, as was confirmed by experiment. However, a careful analysis reveals that the theory applies to a class of local theory. Contrary to what happens in the derivation of Bell inequality, it is not necessary to make the hypothesis of outcome independence to derive the Leggett inequality.

Appendix

Equation (9) is our starting point. Let us study the term

$$ R(\mathbf{a},\mathbf{b})\equiv \int dF \vert \mathbf{u}\cdot \mathbf{a}-\mathbf{v}\cdot\mathbf{b} \vert . $$

(16)

The value of the function inside the absolute value is independent of the choice of coordinate system, so its functional dependence on the coordinates must change with the coordinate system. The same is true for the function F. In Ref. [28] a fixed coordinate system was used, so that the functional dependence of F on its parameters is given. We indicate this fixed function in spherical coordinates as . It is also assumed that and , i.e. the polarizations of the detectors vary in the XY plane of the fixed coordinate system. We call p the unit vector along the Z axis in the coordinate system . The vectors u and v can be written as u=sinθ _u (cosϕ _u ,sinϕ _u ,0)+cosθ _u (0,0,1) and v=sinθ _v (cosϕ _v ,sinϕ _v ,0)+cosθ _v (0,0,1), with θ _u ∈[0,π] and θ _v ∈[0,π] the angles with the fixed Z axis. After defining ξ≡(ϕ _a +ϕ _b )/2 and ϕ≡ϕ _a −ϕ _b , it is immediate to check that Eq. (16) can be rewritten

(17)

with N(≥0) and δ defined by

(18)

(19)

By averaging Eq. (9) over the possible values of ξ, i.e. rotating a and b in the fixed XY plane while keeping their relative direction unchanged, we obtain

$$ -1+L_{\mathbf{p}}(\phi) \le C_{\mathbf{p}}(\mathbf{a}\cdot\mathbf{b}) \le 1-R_{\mathbf{p}}(\phi), $$

(20)

where

(21)

(22)

and

(23)

(24)

with S _±≡sinθ _u ±sinθ _v , α=ϕ _u −ϕ _v −ϕ. Since N depends only on ϕ _u −ϕ _v , θ _u , and θ _v , it is convenient to change variables to χ≡ϕ _u −ϕ _v and ψ=(ϕ _u +ϕ _v )/2. The two inner integrals in Eq. (21) then become

$$ \int_{-\pi}^{\pi} d\phi_u d \phi_v (\ldots) = \int_{-2\pi}^{2\pi} d\chi \int _{-\pi+|\chi/2|}^{\pi-|\chi/2|}d\psi(\ldots). $$

(25)

After defining the marginal distribution

Eq. (21) can be rewritten

(26)

Thus, the correlator satisfies

$$ -1+L_{\mathbf{p}}(\phi)\le C_{\mathbf{p}}(\phi) \le1-R_{\mathbf{p}}(\phi ). $$

(27)

By letting the angle between a and b vary while keeping both vectors in the XY plane, and summing the inequalities, it can be proved, following Ref. [42], that

$$ \bigl \vert C_{\mathbf{p}}(\phi)+C_{\mathbf{p}}\bigl( \phi'\bigr)\bigr \vert \le 2-\frac{2\sqrt{2}}{\pi}J \sin{\biggl \vert \frac{\phi-\phi'}{2}\biggr \vert }, $$

(28)

with

(29)

and another marginal distribution.

Now there is a subtle point to be made: After restoring ϕ _u and ϕ _v , i.e., writing , Eq. (29) can be cast in invariant form, so that⁵

(30)

Now, if we consider another unit vector p′ and let a and b vary in the plane orthogonal to p′, letting the angle between them take the same values ϕ and ϕ′, the inequality reads

(31)

By summing Eqs. (30) and (31), and noticing that [see Eqs. (37)–(43) of Ref. [42] for a proof]

$$\sqrt{2-(\mathbf{p}\cdot\mathbf{u})^2-(\mathbf{p}\cdot \mathbf{v})^2}+ \sqrt{2-\bigl(\mathbf{p'}\cdot\mathbf{u} \bigr)^2-\bigl(\mathbf{p'}\cdot\mathbf{v} \bigr)^2} \ge\sqrt{2}, $$

when p⋅p′=0, we arrive at an inequality independent on the knowledge of the measure dF, namely

$$ \bigl \vert C_{\mathbf{p}}(\phi)+C_{\mathbf{p}}\bigl( \phi'\bigr)\bigr \vert + \bigl \vert C_{\mathbf{p'}}( \phi)+C_{\mathbf{p'}}\bigl(\phi'\bigr)\bigr \vert \le 4- \frac{4}{\pi}\sin{\biggl \vert \frac{\phi-\phi'}{2}\biggr \vert } $$

(32)

for p⋅p′=0, which is Leggett inequality.