EPR paradox and Bell inequality
Popular science books or articles on quantum mechanics love the topic of EPR paradox. It relates to a legend of Albert Einstein and his debates with Neils Bohr. As well known in physics history, Einstein was not completely convinced by the quantum mechanics interpretation, and the so-called EPR paradox was proposed as an attempt to prove that quantum mechanics is "incomplete". Nowadays, we are used to quantum mechanics. EPR paradox is no longer a paradox. It is solved by Bell inequality and its various experimental verifications.
Thanks to the simplification made by physicists David Bohm and Yakir Aharanov, we can describe EPR paradox in terms of a spin singlet state of two entangled electrons \begin{equation}|S=0\rangle=\frac{1}{\sqrt{2}}\left[|\uparrow\rangle_A |\downarrow\rangle_B - |\downarrow\rangle_A |\uparrow\rangle_B\right]\,.\tag{1}\end{equation} Remarks:
- The spins of two electrons are alway along the exact opposite directions, with or without measurements.
- Although there are $\uparrow,\,\downarrow$ arrows in (1), there is no predefined up or down direction unless specified by a measurement. For example, we can also write the state (1) as $\left[|\leftarrow\rangle_A |\rightarrow\rangle_B - |\rightarrow\rangle_A |\leftarrow\rangle_B\right]/\sqrt{2}$.
It is time for Alice and Bob's debut. Suppose they are separated in a large distance and each of them owns one electron in the state (1). If Alice performs a measurement with respect to a chosen direction, her electron will randomly collapse to a state of either "spin up" or "spin down" with equal probability. By quantum mechanics, Alice's measurement also forces Bob's electron instantly be in a state of respective "spin down" or "spin up", even though it is far away from Alice's electron. Such “spooky action at a distance” leads to a paradox since it looks like special relativity is violated here.
Einstein's solution to the paradox is the existence of some undiscovered "hidden variable": When the two electrons become entangled in the same place, the hidden variable makes them reach a consensus on how to response to all possible measurements in future when they are separated. For example, in the simplest case that Alice and Bob agrees to only measure on one direction, the hidden variable could generate a random bit for one electron and assign the opposite bit to the other electron. Later once met a measurement, the electron simple reads its own bit and becomes "spin up" if the bit is 1 or "spin down" if the bit is 0. In this way, the two electrons always show the opposite spins as observed in experiments but without any "spooky action at a distance". In general, it is enough to assign one value for each possible measurement direction since the entanglement varnishes after one measurement.
Einstein's solution to the paradox is the existence of some undiscovered "hidden variable": When the two electrons become entangled in the same place, the hidden variable makes them reach a consensus on how to response to all possible measurements in future when they are separated. For example, in the simplest case that Alice and Bob agrees to only measure on one direction, the hidden variable could generate a random bit for one electron and assign the opposite bit to the other electron. Later once met a measurement, the electron simple reads its own bit and becomes "spin up" if the bit is 1 or "spin down" if the bit is 0. In this way, the two electrons always show the opposite spins as observed in experiments but without any "spooky action at a distance". In general, it is enough to assign one value for each possible measurement direction since the entanglement varnishes after one measurement.
Nowadays, we know quantum mechanics is correct and the "spooky action at a distance" does exist. However, special relativity is not violated: special relativity only asserts that information can not travel faster than light, and there is NO information transfer in the "spooky action at a distance". To see why, note that Bob cannot know his electron's state unless he also makes a measurement. Suppose Bob performs a measurement with respect to the same direction as Alice's, and he finds that his electron is in the "spin down" state. Bob cannot distinguish the following two cases:
- Alice performed a measurement on her electron before him. Bob can infer immediately that the outcome of Alice's measurement is the "spin up" without a communication with Alice.
- Alice has not performed any measurement. It is Bob's own measurement that makes his electron collapse to "spin down" state.
How to exclude the hidden variable from quantum mechanics? It is Bell inequality that makes this philosophical question at first glance into a testable scientific problem. Instead of rephasing the proof in Bell's seminal paper, here I would like to share a simplified version that I first learned from this youtube video. For this purpose, let me first introduce a device called spin filter which either lets an electron pass through or blocks it. There is a particular axis for a spin filter (for example, the magnetic field of Stern-Gerlach magnet) and one can rotate the spin filter to make its axis point to different orientations. By quantum mechanics, if an electron's spin direction has an angle $\theta$ with the axis orientation, the electron can pass through the spin filter with probability $\cos^2\theta$ while it is blocked with probability $\sin^2\theta$. Now consider the scenario that both Alice and Bob carry a spin filter. Before separation they agree on three particular orientations denoted by $a, b, c$ such that the pairwise angles between $a, b, c$ are all equal to $2\pi / 3$ (the logo of Mercedes-Benz). When separated far away, both Alice and Bob perform the same experiments: randomly select one orientation out of $a, b, c$; rotate spin filter to align its axis with the selected orientation; and finally make the electron pass the spin filter. We then ask the following questions:
What is the conditional probability that Alice and Bob obtain the same results (both electrons pass through or both are blocked) given that they select different orientations?
The answer from quantum mechanics is $1/4$. Suppose Alice completes her experiment first and assume she selects the orientation $a$. Her electron collapses to the state of either "spin up" or "spin down" along $a$, and thus can always pass through her spin filter. Because of Alice's experiment, Bob's electron collapses instantly to the respective state of either "spin down" or "spin up" along $a$. After that, Bob performs his experiment and selects a different orientation of either $b$ or $c$. His electron that has already spin direction along $a$ will pass through his spin filter with probability $\cos^2 2\pi / 3=1/4$.
But by hidden variable explanation, this probability can not be less than $1/3$. As controlled by the hidden variable, we first list out all the eight possible outcomes of whether an electron can pass through a spin filter for each of the three axis orientations:
Assume both Alice and Bob select an orientation among $a, b, c$ with equal probability: (1) For the first case, no matter which orientations Alice and Bob select, both electrons can pass through spin filters. So the probability that they obtain the same result is 1. (2) For the second case, Alice and Bob obtain the same result (both electrons pass through spin filters) only if they select the orientations $a$ and $b$, which has probability 1/3. (3) For the sixth case, Alice and Bob obtain the same result (both electrons are blocked by spin filters) only if they select the orientations $a$ and $b$, which has probability 1/3. ... After going through all the cases listed in the table, we will find that the asked conditional probability is 1 for the first and eighth cases, and 1/3 for the remaining six cases. Therefore, even though we lack the details of hidden variable and thus have no idea on the each case's real occurrence probability controlled by hidden variable, we are able to conclude that the asked probability $\geq$ 1/3.
With no surprises, various experiments on Bell's inequality support quantum mechanics rather than hidden variable. Some people argue that Bell inequality only rules out "local" hidden variables. In another post, I will present a stronger form than Bell inequality: free will theorem.
| a | b | c | |
|---|---|---|---|
| outcome 1 | passed | passed | passed |
| outcome 2 | passed | passed | blocked |
| outcome 3 | passed | blocked | passed |
| outcome 4 | blocked | passed | passed |
| outcome 5 | passed | blocked | blocked |
| outcome 6 | blocked | passed | blocked |
| outcome 7 | blocked | blocked | passed |
| outcome 8 | blocked | blocked | blocked |
Assume both Alice and Bob select an orientation among $a, b, c$ with equal probability: (1) For the first case, no matter which orientations Alice and Bob select, both electrons can pass through spin filters. So the probability that they obtain the same result is 1. (2) For the second case, Alice and Bob obtain the same result (both electrons pass through spin filters) only if they select the orientations $a$ and $b$, which has probability 1/3. (3) For the sixth case, Alice and Bob obtain the same result (both electrons are blocked by spin filters) only if they select the orientations $a$ and $b$, which has probability 1/3. ... After going through all the cases listed in the table, we will find that the asked conditional probability is 1 for the first and eighth cases, and 1/3 for the remaining six cases. Therefore, even though we lack the details of hidden variable and thus have no idea on the each case's real occurrence probability controlled by hidden variable, we are able to conclude that the asked probability $\geq$ 1/3.
With no surprises, various experiments on Bell's inequality support quantum mechanics rather than hidden variable. Some people argue that Bell inequality only rules out "local" hidden variables. In another post, I will present a stronger form than Bell inequality: free will theorem.
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