Photon Entanglement - Entanglement

Entanglement

Let us recall, for purposes of exposition, that a quantum system is described, at every instant, by a vector state which, according to the theory, represents the maximum amount of information that it is possible to have with respect to it. To simplify discussion, let's take the example of the state of polarization of a photon and associate with it the vector state . What does this information tell us about the properties of photons? The knowledge of the vector state, in fact, provides us exclusively with information on the possible results of measurements which we decide to carry out on the system. For example in the case just referred to we know that if we were to apply a test for vertical polarization to the photon whose state is, it would have a probability of 1/2 of passing and 1/2 of failing. But the theory, which usually provides only probabilistic information on the results of hypothetical measurements, can, with reference to particular tests, assign the value 1 or 0 to the probability of obtaining specific results. So, in the case we are considering, the theory tells us that the photon has a probability of 1 of passing through a filter polarized at 45°, and a probability of 0 of passing through a filter polarized at 135°. In this case, and with precise and exclusive reference to the observables (polarization at 45° and 135°) for which we know a priori the results of measurement, we can assert that the photon possesses the property in question: it is polarized at 45° or "possesses the property which guarantees that it will pass with certainty a test at 45°." This is an important distinction with the situation in classical mechanics: in classical physics, any system always possesses precise values for all of the observables that we can conceive, but in quantum physics, a single system will indeed possess some property, but, with reference to other properties, we can do no better than make probabilistic prediction about the results of possible measurements, of and when they are actually executed. In a certain sense the theory teaches us that a system must not have too many properties and that, in particular, some are incompatible with others. So, for example, a photon that is "polarized at 45°" does not possess any definite property relative to vertical or horizontal polarization. This is important for understanding one of the fundamental assumptions on which the argument of Einstein, Podolsky and Rosen (or "EPR") is based:

(R)If, without disturbing a system in any way, it is possible to predict with certainty the result of the measurement of an observable of the system, then there exists an element of reality associated with the observable in question; the system "objectively possesses" the relative property.

Now consider the following situation: two photons are emitted by a source S and are propagated in two opposite directions. At a certain instant, one of them can be found in the region A, to the right of the source and the other in the region B symmetric to A with respect to S (figure G).

We can call the photon at the right 1 and assume that it possesses a vertical polarization. This can be indicated as the vector state . Analogously, suppose that the photon on the left, indicated as 2, has a horizontal polarization, so that it is described by the vector state . The entire system is described as the state

which corresponds to the single quantum state which asserts that "(one photon is in A with vertical polarization) and (one photon is in B with horizontal polarization)".

This state is called "factored" because it is, technically, the product of the two photons. Its properties are rather obvious and are illustrated in Figure G. For example if, given the state, we carry out a test of vertical polarization on the photon on the right and a test of horizontal polarization on the photon to the left, we know that both of them will pass with certainty. Similarly, if we carry out (center of figure) a test of horizontal polarization on both of the photons, the one on the right will certainly fail, while the one on the left will certainly pass. Lastly, consider the more general case in which the photon to the right is passed through a filter polarized at 45°. In this case, the photon 2 will pass through 1/2 of the time and end up polarized at 45°, and it will fail to pass the other 1/2. The photon on the left has not been tested and therefore remains horizontally polarized.

The state is actually a superposition of states, however, and must be rewritten as follows:

Substituting this into the expression for the state, we have:

According to this formula, a measure of polarization at 45° in A can result, with equal probability, in the photon 1 passing the test, in which case the system will be represented, according to wave packet reduction, as follows:

An important case is the one in which the photons have the same initial polarization:

or

In order to understand entanglement, consider again the two photons discussed above and observe that that states and are both possible states of the system. But, if this is the case, then it follows that the superposition of the two states:

is also a possible state of the system of two photons. What are the properties of this system?

It's immediately clear that each of the two photons does not possess the property of being polarized vertically or horizontally, since the probability of passing, for example, a test of vertical polarization on the part of photon 1 is characterized by the coefficient of the state in which it has this polarization and the square of this coefficient is one half. Therefore if one carries out this test, photon 1 will pass about half of the time in an unpredictable manner. The same reasoning applies to the horizontal test and for the other photon.

Suppose we are now interested in measuring the polarizations at 45° and 135°. We must express the state of vertical and horizontal polarizations as the superpositions of states of polarization at 45° and 135°. Substituting the appropriate expressions into the preceding formula and carrying out the explicit calculations, we have:

The result is the superposition of the states of two photons polarized at 45° and of two polarized at 135°. The two new orthogonal directions have taken the place of the vertical and horizontal of the preceding expressions. This implies, of course, that every photon has a probability of 1/2 to pass a test of this type exactly as it has to pass the tests for vertical and horizontal polarization. If one were to continue and calculate the results for other possible measures of polarizations along arbitrary directions in the plane, it would eventually be noted that this result is generalizable as follows:

In words, this means that the state always has the same form regardless of the directions chosen: it is the superposition of two states, in the first of which both of the photons are polarized in the chosen direction n, and in the second of which both of the photons are polarized on the orthogonal direction .

Now, suppose that an observer decides to carry out a measurement of the polarization of photon 1 along an arbitrarily chosen direction n. If the photon passes the test, then according to the principle of wave packet reduction, we have:

and the final state is factored. Spontaneously, photon 2, which had no property of polarization before the measurement, has acquired a precise property as a result of the measurement of photon 1! This is entanglement.