In 1935, Einstein, Podolsky, and Rosen published a paper in the Physical Review entitled “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” The paper constitutes part of the foundation for Einstein’s now well-known skepticism about the entrenched probabilism of quantum mechanics, which is sometimes summarized by the quote often attributed to him that God does not play dice with the universe.

EPR (as they are now called) do not propose experimental results but rather a thought experiment and a philosophical argument. In a particularly vivid variant of their thought experiment due to Bohm (1951)—now sometimes called the EPR experiment—two particles, an electron and a positron, separate at a point of radioactive decay. According to the experiment, they speed off in separate directions in a way where their total spin is zero. Along a chosen axis each of the particles can take one of two real values ( and ). For the observable property of spin along this axis, I will write for short “.”

According to the formalism of quantum mechanics, systems describing these two particles, individually, can be formulated as Hilbert spaces—that is, complex valued inner product spaces—and the representation of the spin observable along the chosen axis is represented in the formalism by a Hermitian (self-adjoint) operator. The observable values for the operator are then the eigenvalues of such operators (and since they are Hermitian, these values will be real numbers). Corresponding to each eigenvalue, there is an eigenspace. As a fact of linear algebra, there is a projection operator which takes any complex valued vector in the space and projects it onto the eigenspace defined by the th eigenvalue. For an observable , I will write the projection operator as “” and its th eigenvalue as “.” Likewise its projection operator for this eigenvalue will be written “.”

States of the positron system and the electron system are given by complex valued vectors. Part the mystery of quantum mechanical systems is that in general there is no certain answer to experimental questions like “Does the electron have value of spin ½ in the direction given that the state is ?” According to the Born rule, that answer is given by a probability value which is obtained from the projection operator on the eigenspace defined by the eigenvalue . So if the state is , then the probability of observable having value its th eigenvalue is given by the following formula:

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