Testing

---EPR Paradox---

Before diving into entanglement in the context of quantum computing and information, it would be prudent to discuss the now-famous EPR paradox. It was first proposed in a paper by Einstein, Polosky, and Rosen in 1935. The original paper does include a fairly simple mathematical explanation of the paradox---it is, however, not really necessary as the thought experiment is quite easily understood conceptually with (mostly) words. A slightly simplified version of the experiment will be given here.

Suppose a neutral pi meson, which has no spin, is at rest. It then decays into an electron and a positron, necessarily going in opposite directions. The wave function can now be written as

{Equation|$\left\vert \Psi _{-}\right\rangle ={\frac {1}{\sqrt {2}}}(\left\vert \uparrow \downarrow \right\rangle -\left\vert \downarrow \uparrow \right\rangle ).$|4.1}

As can be seen, we now have a system of two particles that have a correlated spin---one being up and the other being down---with an equal probability for each configuration being the outcome of a measurement. The system is said to be entangled, as a measurement on one will guarantee that the other particle is in the correlated state. In other words, it cannot be written as $\left\vert \psi _{1}\right\rangle \left\vert \psi _{2}\right\rangle$.

Now what is the significance here? It all depends on what interpretation of quantum mechanics is being used. The orthodox position says that the wave function is the complete representation of the system. When the measurement occurs, the wave function collapses, changing the system.

But, in the context of EPR, how can this be? Imagine the entangled electron-positron pair are at opposite ends of the galaxy when one of them is measured. The conservation of angular momentum says that the other particle all of the way on the other side of the galaxy must instantly be the opposite spin as the measured particle. EPR argued that this is a violation of locality, which says an effect cannot travel faster than the speed of light. If the very action of measurement on one particle is what caused the other particle to realize the opposite spin, then locality has been violated. Therefore, the measurement could not have caused the collapse of the wave function.

EPR concluded that this proves that quantum mechanics is incomplete---that the wave function is missing some information. There was no "spooky-action-at-a-distance," there must be some underlying property that is absent from the wave function. Einstein rejected the notion that a measurement caused this quasi-mystical collapse of the wave function---the particles do not care if they are being watched or not.

---Bell's Theorem---

The peculiarities of the EPR paradox were convincing enough to drive many to examine possible "hidden variable theories." The basic idea is that there exists a quantity, often denoted by $\lambda$, that must be included in the wave function to completely describe the system. J.S. Bell very elegantly showed in 1964 that this is not the case, using the very thought experiment (although slightly modified) that EPR proposed.

Suppose we have another pion at rest about to decay with detectors oriented equidistant and on opposite sides, ready to measure the spin of the electron and positron. Further suppose that, unlike the previous scenario, these detectors can be rotated in order to detect the spin in the direction of unit vectors ${\vec {a}}$ and ${\vec {b}}$ for the electron and positron respectively.

When the electron and positron pair strikes the detectors, a spin up ($-1$) or spin down ($-1$) is registered. The product of the results is then examined. If they are oriented parallel, where ${\vec {a}}={\vec {b}}$, then the result will be -1. If anti-parallel, the result is then +1. The averages are, obviously,

{Equation| ${\begin{aligned}P({\vec {a}},{\vec {a}})&=-1,\\P({\vec {a}},-{\vec {a}})&=+1.\end{aligned}}$|4.2}

Quantum mechanics tells us that for arbitrary vectors,

{Equation| $P({\vec {a}},{\vec {b}})=-{\vec {a}}\bullet {\vec {b}}.$|4.3}

We can now introduce the hidden variable, $\lambda$. This can represent any possible amount of variables that complete the description of the system and allow for locality. We then define some functions, $A({\vec {a}},\lambda )$ and $B({\vec {b}},\lambda ),$ that will give the results for the measurement (either +1 or -1) for the electron and positron respectively.

The locality assumption tells us that the orientation of one detector will not affect the outcome of the measurement of the other detector; one can imagine a scenario where the orientation is chosen at a time too late for any information to be transfered slower than light. It must also be true that, when the detectors are parallel, the results must be

{Equation| $A({\vec {a}},\lambda )=-B({\vec {a}},\lambda )$|4.4}

due to the conservation of angular momentum. Let us also define a probability density, $\rho (\lambda ),$ for the hidden variable. Since we know nothing of $\lambda$, this can be anything as long as it is nonnegative and normalizable ($\int \rho (\lambda )d(\lambda )=1$). We can now look at the product of the measurements,

{Equation| $P({\vec {a}},{\vec {b}})=\int \rho (\lambda )A({\vec {a}},\lambda )B({\vec {b}},\lambda )d\lambda .$|4.5}

We know from Eq.(4.4) that this can be rewritten:

{Equation| $P({\vec {a}},{\vec {b}})=-\int \rho (\lambda )A({\vec {a}},\lambda )A({\vec {b}},\lambda )d\lambda .$|4.6}

Now for the clever part. Introducing another unit vector, ${\vec {c}}$, and noting that $[A({\vec {b}},\lambda )]^{2}=1,$

{Equation| ${\begin{aligned}P({\vec {a}},{\vec {b}})-P({\vec {a}},{\vec {c}})&=-\int \rho (\lambda )[A({\vec {a}},\lambda )A({\vec {b}},\lambda )-A({\vec {a}},\lambda )A({\vec {c}},\lambda )]d\lambda \\&=-\int \rho (\lambda )[1-A({\vec {b}},\lambda )A({\vec {c}},\lambda )]A({\vec {a}},\lambda )A({\vec {b}},\lambda )d\lambda \end{aligned}}$|4.7}

Recognizing some inequalities,

{Equation| ${\begin{aligned}-1\leq A({\vec {a}},\lambda )A({\vec {b}},\lambda )\leq 1,\\\rho (\lambda )[1-A({\vec {b}},\lambda )A({\vec {c}},\lambda )]\geq 0,\end{aligned}}$|4.8}

we get to a remarkable result,

{Equation| ${\begin{aligned}|P({\vec {a}},{\vec {b}})-P({\vec {a}},{\vec {c}})|&\leq \int \rho (\lambda )[1-A({\vec {b}},\lambda )A({\vec {c}},\lambda )]d\lambda \\&\leq 1+P({\vec {b}},{\vec {c}}).\end{aligned}}$|4.9}

The last form is known as the Bell inequality. This inequality is true for any local hidden variable theory.

What does this mean? Let us define ${\vec {a}}$ and ${\vec {b}}$ to be orthoganal and ${\vec {c}}$ to make a $45^{\circ }$ angle with both of them. Using quantum mechanics (Equation(4.3)),

${\begin{aligned}P({\vec {a}},{\vec {b}})=0\\P({\vec {a}},{\vec {c}})=P({\vec {b}},{\vec {c}})=-{\frac {1}{\sqrt {2}}}.\end{aligned}}$

Inserting the values into the Bell inequality (Equation (4.9)),

${\begin{aligned}\left|-{\frac {1}{\sqrt {2}}}\right|&\leq 1+\left(-{\frac {1}{\sqrt {2}}}\right)\\{\frac {1}{\sqrt {2}}}&\leq {\frac {{\sqrt {2}}-1}{\sqrt {2}}}\\1&\leq {\sqrt {2}}-1\end{aligned}}$

Since ${\sqrt {2}}-1\approx .41,$ the inequality is violated!

This means that quantum mechanics is incompatible with any local hidden variable theory. The EPR paradox had stronger implications than the authors realized; if local realism is held, then quantum mechanics is incorrect. This has been repeatedly disproved experimentally. Thus no local hidden variable theory can resolve the "spooky-action-at-a-distance."