The EPR Paradox and the Quest for Completeness

In 1935, Albert Ein­stein, Boris Podol­sky, and Nathan Rosen (EPR) pub­lished a paper titled "Can Quan­tum-Me­chan­i­cal De­scrip­tion of Phys­i­cal Re­al­i­ty Be Con­sid­ered Com­plete?". They argued that quan­tum me­chan­ics was in­com­plete.

Local Re­al­ism

The EPR ar­gu­ment was based on two fun­da­men­tal as­sump­tions, col­lec­tive­ly known as local re­al­ism:

• Re­al­ism: Phys­i­cal prop­er­ties of ob­jects exist in­de­pen­dent­ly of whether they are ob­served. An elec­tron has a def­i­nite spin in a given di­rec­tion even if we don't mea­sure it.
• Lo­cal­i­ty: The result of a mea­sure­ment on one par­ti­cle cannot in­stan­ta­neous­ly in­flu­ence the result of a mea­sure­ment on an­oth­er par­ti­cle that is spa­tial­ly sep­a­rat­ed from it. In­for­ma­tion cannot travel faster than the speed of light.

EPR con­sid­ered an en­tan­gled pair. If we mea­sure the spin of par­ti­cle A along the z-axis and find it to be spin-up, lo­cal­i­ty im­plies this mea­sure­ment cannot affect par­ti­cle B. Re­al­ism im­plies that par­ti­cle B must have had a def­i­nite spin-down prop­er­ty along the z-axis all along, even before A was mea­sured. Since quan­tum me­chan­ics doesn't assign such a def­i­nite prop­er­ty before mea­sure­ment, EPR con­clud­ed it was in­com­plete.

Hidden Vari­ables

To com­plete quan­tum me­chan­ics, EPR and others sug­gest­ed the ex­is­tence of hidden vari­ables. These would be ad­di­tion­al pa­ra­me­ters, not yet known or de­scribed by quan­tum theory, that de­ter­mine the out­come of in­di­vid­ual mea­sure­ments. If we knew these hidden vari­ables, the prob­a­bilis­tic nature of quan­tum me­chan­ics would dis­ap­pear, and out­comes would be de­ter­min­is­tic.

For ex­am­ple, for an en­tan­gled pair, a hidden vari­able $\lambda$ (lambda) might pre­de­ter­mine the out­come of any spin mea­sure­ment on either par­ti­cle. The ap­par­ent ran­dom­ness would be due to our ig­no­rance of $\lambda$.