Entanglement

EPR PARADOX AND ENTANGLEMENT

In 1935 Albert Einstein and his two Princeton University colleagues, Boris Podolsky and Nathan Rosen (EPR) proposed a thought experiment to question the completeness of quantum mechanics. Their work led to the famous “EPR Paradox” triggering an avalanche of subsequent research that is continuing even today.

Recall that the Heisenberg’s uncertainty principle applies for non-commuting observables7 like position and momentum: for a subatomic particle whose momentum is well-defined, has a completely ill-defined value of position. Conversely, if the position is fixed, its momentum is uncertain. This is not a matter of an inability to measure position and momentum accurately. The uncertainty principle asserts that the particle can never have well-defined values of both position and momentum at the same time.

Whereas position and momentum are continuous observables, Heisenberg’s uncertainty principle is equally valid for discrete observables, such as spin. The spin of a particle can be resolved along each of the three spatial axes forming a set of mutually non-commuting observables. One can only measure the spin along one axis at a time. For example, a proton with spin up along the x-axis has an undefined spin value along the y and z axes, or in other words, one cannot measure the x and y spin projections of a proton simultaneously. Here we provide David Bohm’s version of the EPR thought experiment that uses spin as an observable.

Consider two entangled particles emitted in what is called a singlet state, in which the total spin must be zero. The spins of the two outgoing particles are linked: if one particle shows spin up, the other will inevitably show spin down.

To visualize this, consider, for example, the decay of a neutral pion at rest emitting a pair of back-to-back photons that can be described by a single wave function. According to the Copenhagen interpretation, the spin of either particle is undefined before any measurement can be made. But suppose the spin of photon 1 is measured along the y-axis. It could be either up or down but suppose the measured value was found to be up along the y-axis. Then photon 2 must have spin pointing down along the y-axis, since the angular momentum must be conserved in the decay process: The total angular momentum of the final two-photon state must be the same as the angular momentum of the initial state of the single neutral pion (neutral pion is a scalar particle).

As long as the two photons remain entangled, then this link survives—even if they are separated across the galaxy. The actual spin direction of either photon cannot be determined prior to the measurement according to the Copenhagen interpretation of quantum mechanics; the only thing that can be said with certainty is that the spins are anti-aligned. Oddly, until they have been measured, both photons are actually spinning up and down at the same time. Their measured state is probabilistic: there is a fifty-fifty chance that, once one of the photons is measured, it will be “fixed” in a state of spinning up or down. Because the two photons are in a single quantum state, they are entangled. As soon as a measurement is made on one (say, photon 1), the wave function of the two-particle system collapses instantaneously. By the law of conservation of angular momentum, the spin of photon 2 immediately gets fixed by pointing to the opposite direction. One knows the spin of photon 2 even without measuring it, even though it could be light years away from photon 1.

Now, if one tries to measure the spin of photon 2, not along the y-axis, but along the x-axis, she or he will find this value to be undefined because according to Heisenberg’s uncertainty principle, one cannot measure the x and y spin projections of a proton simultaneously. But if the spin of photon 1 was not already measured along the y-axis first, measuring the spin of photon 2 along the x-axis wouldn’t have been an issue at all. As if photon 2 “knows” whether the y-spin of photon 1 has already been measured or not! But how do we reconcile the fact that photon 2 “knows” that the y-spin of photon 1 has been measured, even though they are separated by light years and far too little time has passed for information to have travelled to it according to the rules of special relativity?

The same argument can be generalized to any other observable quantity, not just spin. Measurement of the first system will determine the measured value of the same observable quantity of the second system, even though the systems are no longer physically linked (Einstein’s “spooky action at a distance”). Thus, one could imagine the two measurements were taken so far apart in space that special relativity would prohibit any “influence” of one measurement over the other. So what does this all mean?

There are two possible consequences:

  • Quantum mechanics is incomplete and it could be overlooking something. Perhaps a “hidden variables” description of quantum states and properties is possible in which properties are always determinate (possess values) at all times.
  • Accept the postulates of quantum mechanics as a fact of life, in spite of its seemingly uncomfortable coexistence with special relativity. In that case we are forced to accept that a particle can somehow transmit “influence” to another through no known device at an infinite speed, leading to the rejection of the well-established principle of locality—that no influence can travel faster than the speed of light.

EPR favored the first option that quantum mechanics is incomplete and does not completely describe physical reality: “One is thus led to the conclusion that the description of reality as given by a wave function is not complete”. They postulated that the existence of “hidden variables” or some currently unknown properties of the systems, could account for the discrepancy.

Einstein disliked this “essential, irreducible” central role of the observer. Famously, he said “God does not play dice”. Barrow and Tipler (Barrow & Tipler, 1986) point out, “…to a realist like Einstein, who held that a physical reality existed independently of Man the Observer, Bohr’s view was anathema.” Einstein had no idea that it would be possible to test this thought experiment in practice–but that is exactly what came about in the 1980s, as we shall see.

The characteristics of the measurement process before publication of the EPR paper was as follows: the process of measuring the position of an elementary particle such as an election often involves shining light (photons) into it. This action disturbs its location and manifests in uncertainties in its position. EPR successfully challenged this by showing that a measurement can be performed on a particle without disturbing it directly, but by performing a measurement on a distant entangled particle instead.

Quantum entanglement promises a host of exciting applications. In quantum cryptography, entangled particles are used to transmit signals that cannot be compromised without leaving a trace. In quantum computation, entangled quantum states are used to perform computations in parallel, which may allow certain calculations to be performed much more quickly than they ever could be with classical computers.


7. If two operators \hat A and \hat B commute with each other, then \hat A\hat B - \hat B\hat A = 0. For non-commuting operators \hat A\hat B\neq \hat B\hat A.

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