The EPR argument

Intuition

In 1935 Einstein, Podolsky and Rosen argued that quantum mechanics is incomplete. In Bohm’s version a pair in the singlet flies apart. Measuring $A$ along $z$ predicts $B$ ’s $z$ result with certainty, without touching $B$ ; measuring $A$ along $x$ would have predicted $B$ ’s $x$ result with certainty instead. If whatever can be predicted with certainty without disturbing a system is an element of reality, and if $A$ ’s choice cannot reach $B$ far away, then $B$ must carry definite values of both $\hat{S}_{z}$ and $\hat{S}_{x}$ — and quantum mechanics has no state with both. So, they concluded, the theory is incomplete, and further variables should complete it. For thirty years this seemed a matter of taste. Bell turned it into an experiment.

Two sealed envelopes with a card each are posted to two cities: opening one tells you the other card at once, and nobody thinks the opening sent anything. EPR argued that entangled pairs must be like that, every answer written in advance. The next lessons show the envelopes cannot hold answers that agree with quantum mechanics.

Bohm’s form of the EPR experiment: a source sends the two spins of a singlet to $A$ and $B$ , where each is measured along an axis chosen there, $\mathbf{a}$ and $\mathbf{b}$ .

The singlet has the same form along every axis $\mathbf{n}$ :

\ket{\Psi^{-}}=\tfrac{1}{\sqrt2}\bigl(\ket{\uparrow_{\mathbf{n}}\downarrow_{\mathbf{n}}}-\ket{\downarrow_{\mathbf{n}}\uparrow_{\mathbf{n}}}\bigr)\quad\text{for every}\ \mathbf{n}

The two assumptions and the conclusion

Reality: if a value can be predicted with certainty without disturbing the system, it is an element of reality, there before any measurement.
Locality: what is chosen or done at $A$ cannot change what exists at $B$ far away.
With the singlet, a measurement at $A$ along any axis predicts $B$ along that axis with certainty, so $B$ would carry values of and at once.

The singlet is anticorrelated along every axis

\bigl\langle(\mathbf{n}\cdot\vec{\sigma})\otimes(\mathbf{n}\cdot\vec{\sigma})\bigr\rangle_{\Psi^{-}}=-1\quad\text{for every}\ \mathbf{n}

Check that each component of the total spin annihilates the singlet. Then the spin of $A$ along any axis acts on the singlet as minus the spin of $B$ along that axis, and the product of the two acts as $-1$ : the results are opposite every time.

Proof steps

\bigl(\sigma_{i}\otimes\hat{1}+\hat{1}\otimes\sigma_{i}\bigr)\ket{\Psi^{-}}=0\quad(i=x,y,z)

For $x$ : $\ket{01}-\ket{10}$ becomes $\ket{11}-\ket{00}+\ket{00}-\ket{11}=0$ ; and are alike.

\bigl(\mathbf{n}\cdot\vec{\sigma}\otimes\hat{1}\bigr)\ket{\Psi^{-}}=-\bigl(\hat{1}\otimes\mathbf{n}\cdot\vec{\sigma}\bigr)\ket{\Psi^{-}}

Combine the three components with the weights of $\mathbf{n}$ .

\begin{aligned}\bigl(\mathbf{n}\cdot\vec{\sigma}\otimes\mathbf{n}\cdot\vec{\sigma}\bigr)\ket{\Psi^{-}}&=-\bigl(\hat{1}\otimes(\mathbf{n}\cdot\vec{\sigma})^{2}\bigr)\ket{\Psi^{-}}\\ &=-\ket{\Psi^{-}}\end{aligned}

Move $A$ ’s factor onto $B$ , where $(\mathbf{n}\cdot\vec{\sigma})^{2}=\hat{1}$ .

\bigl\langle(\mathbf{n}\cdot\vec{\sigma})_{A}(\mathbf{n}\cdot\vec{\sigma})_{B}\bigr\rangle=-1

The product of the two $\pm1$ results is $-1$ every time: always opposite.

Applications

Practice

Two Assumptions

EPR assumed reality — a value that can be predicted with certainty, without disturbing the system, exists before it is measured — and locality — a choice at $A$ cannot change what exists at $B$ .

Try it

On which two assumptions does the EPR argument rest?