John Bell and His Theorem

In 1964, John Stew­art Bell, a physi­cist at CERN, took the EPR ar­gu­ment fur­ther. He showed that the as­sump­tions of local re­al­ism (or, more pre­cise­ly, local hidden vari­able the­o­ries) lead to testable pre­dic­tions that differ from those of quan­tum me­chan­ics.

Bell's As­sump­tions

Bell con­sid­ered an ex­per­i­ment with a source emit­ting pairs of en­tan­gled par­ti­cles. Two ob­servers, Alice and Bob, are spa­tial­ly sep­a­rat­ed. Alice mea­sures a prop­er­ty of her par­ti­cle (e.g., spin along axis $\mathbf{a}$), and Bob mea­sures a prop­er­ty of his par­ti­cle (e.g., spin along axis $\mathbf{b}$). Alice can choose her mea­sure­ment set­ting $\mathbf{a}$ from a set of pos­si­bil­i­ties, and Bob can choose his set­ting $\mathbf{b}$.

Bell's the­o­rem relies on the fol­low­ing as­sump­tions for a local hidden vari­able theory:

1. Re­al­ism: The mea­sure­ment out­comes $A(\mathbf{a}, \lambda)$ and $B(\mathbf{b}, \lambda)$ are de­ter­mined by the set­tings $\mathbf{a}$, $\mathbf{b}$ and the hidden vari­able(s) $\lambda$. The out­comes are def­i­nite values (e.g., $+1$ or $-1$).
2. Lo­cal­i­ty: Alice's out­come $A$ cannot depend on Bob's set­ting $\mathbf{b}$, and Bob's out­come $B$ cannot depend on Alice's set­ting $\mathbf{a}$. So, $A = A(\mathbf{a}, \lambda)$ and $B = B(\mathbf{b}, \lambda)$.
3. Sta­tis­ti­cal In­de­pen­dence (or Free Will/Mea­sure­ment In­de­pen­dence): The choice of mea­sure­ment set­tings $(\mathbf{a}, \mathbf{b})$ is in­de­pen­dent of the hidden vari­ables $\lambda$. The dis­tri­b­u­tion of hidden vari­ables $\rho(\lambda)$ is fixed and does not depend on the set­tings chosen by the ex­per­i­menters.

Bell's In­equal­i­ty

Bell de­rived an in­equal­i­ty that must be sat­is­fied by any local hidden vari­able theory. There are sev­er­al forms of Bell's in­equal­i­ty. One of the most famous and ex­per­i­men­tal­ly con­ve­nient is the CHSH in­equal­i­ty, named after John Clauser, Michael Horne, Abner Shi­mo­ny, and Richard Holt, who de­rived it in 1969.

Let Alice choose be­tween two set­tings $\mathbf{a}$ and $\mathbf{a}'$, and Bob choose be­tween $\mathbf{b}$ and $\mathbf{b}'$. The out­comes are $A = \pm 1$ and $B = \pm 1$.

The ex­pec­ta­tion value of the prod­uct of their out­comes for set­tings $\mathbf{a}$ and $\mathbf{b}$ is:

Where:

• $A(\mathbf{a}, \lambda)$ is Alice's out­come for set­ting $\mathbf{a}$ and hidden vari­able $\lambda$.

• $B(\mathbf{b}, \lambda)$ is Bob's out­come for set­ting $\mathbf{b}$ and hidden vari­able $\lambda$.

• $\rho(\lambda)$ (rho) is the prob­a­bil­i­ty dis­tri­b­u­tion of the hidden vari­ables $\lambda$.

The CHSH in­equal­i­ty states:

Let $S$ be the CHSH quan­ti­ty:

Then the CHSH in­equal­i­ty is $|S| \le 2$.

De­riva­tion Sketch of CHSH In­equal­i­ty (Sim­pli­fied):

Assume local hidden vari­ables. For each par­ti­cle pair (char­ac­ter­ized by $\lambda$), the out­comes $A_1 = A(\mathbf{a}, \lambda)$, $A_2 = A(\mathbf{a}', \lambda)$, $B_1 = B(\mathbf{b}, \lambda)$, $B_2 = B(\mathbf{b}', \lambda)$ are all $\pm 1$. Con­sid­er the quan­ti­ty $A_1 B_1 - A_1 B_2 + A_2 B_1 + A_2 B_2 = A_1(B_1 - B_2) + A_2(B_1 + B_2)$. Since $B_1, B_2 = \pm 1$, either $B_1 - B_2 = 0$ (if $B_1=B_2$) or $B_1 + B_2 = 0$ (if $B_1 \neq B_2$).

• If $B_1 = B_2$, then $A_1(B_1 - B_2) + A_2(B_1 + B_2) = A_2(2B_1) = \pm 2$.
• If $B_1 = -B_2$, then $A_1(B_1 - B_2) + A_2(B_1 + B_2) = A_1(2B_1) = \pm 2$.

So, for any spe­cif­ic $\lambda$, the ex­pres­sion $A(\mathbf{a}, \lambda)B(\mathbf{b}, \lambda) - A(\mathbf{a}, \lambda)B(\mathbf{b}', \lambda) + A(\mathbf{a}', \lambda)B(\mathbf{b}, \lambda) + A(\mathbf{a}', \lambda)B(\mathbf{b}', \lambda)$ must be equal to $\pm 2$.

Taking the av­er­age over all $\lambda$ (i.e., the ex­pec­ta­tion value):

This gives $|S| \le 2$.

Quan­tum Me­chan­i­cal Pre­dic­tion

Quan­tum me­chan­ics pre­dicts cor­re­la­tions that can vi­o­late this in­equal­i­ty. For en­tan­gled spin-$1/2$ par­ti­cles in the sin­glet state, the ex­pec­ta­tion value of the prod­uct of spin mea­sure­ments along di­rec­tions $\mathbf{a}$ and $\mathbf{b}$ is given by:

Where $\theta_{ab}$ (theta-sub-ab) is the angle be­tween the mea­sure­ment di­rec­tions $\mathbf{a}$ and $\mathbf{b}$.

Let's choose spe­cif­ic angles for the CHSH in­equal­i­ty:

• Alice's set­ting $\mathbf{a}$ (angle $0^\circ$)

• Alice's set­ting $\mathbf{a}'$ (angle $90^\circ$)

• Bob's set­ting $\mathbf{b}$ (angle $45^\circ$)

• Bob's set­ting $\mathbf{b}'$ (angle $135^\circ$)

Then:

• $E(\mathbf{a}, \mathbf{b}) = -\cos(45^\circ) = -1/\sqrt{2}$

• $E(\mathbf{a}, \mathbf{b}') = -\cos(135^\circ) = -(-\cos(45^\circ)) = +1/\sqrt{2}$

• $E(\mathbf{a}', \mathbf{b}) = -\cos(45^\circ) = -1/\sqrt{2}$ (angle be­tween $90^\circ$ and $45^\circ$ is $45^\circ$)

• $E(\mathbf{a}', \mathbf{b}') = -\cos(45^\circ) = -1/\sqrt{2}$ (angle be­tween $90^\circ$ and $135^\circ$ is $45^\circ$)

Plug­ging these into the CHSH quan­ti­ty $S$:

So, $|S_{QM}| = 2\sqrt{2} \approx 2.828$.

Since $2\sqrt{2} > 2$, quan­tum me­chan­ics pre­dicts a vi­o­la­tion of the CHSH in­equal­i­ty. This means that quan­tum me­chan­ics is in­com­pat­i­ble with local hidden vari­able the­o­ries.