# Photon entanglement

This brief explanation of photon entanglement is a supplement to the article Bohr-Einstein debates and is designed to help clarify the discussion of the Einstein-Podolsky-Rosen argument in quantum theory which takes place in that article.

## Entanglement

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. To simplify discussion, let's take the example of the state of polarization of a photon and associate with it the vector state $\left|45\right\rang$. 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 $\left|45\right\rang$, 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.
File:Phots.PNG
The behaviour, during various processes of measurement, of two distant photons as discussed in the text

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 2 and assume that it possesses a vertical polarization. This can be indicated as the vector state $\left|2,V\right\rang$. Analogously, suppose that the photon on the left, indicated as 1, has a horizontal polarization, so that it is described by the vector state $\left|1,H\right\rang$. The entire system is described as the state $\left|\Psi\right\rang = \left|2,V\right\rang \left|1,H\right\rang$

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 $\left|\Psi\right\rang = \left|2,V\right\rang \left|1,H\right\rang$, 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 $\left|1,H\right\rang$ is actually a superposition of states, however, and must be rewritten as follows[why?]: $\left|1,H\right\rang = {1 \over \sqrt{2}} \left|1,45\right\rang + {1 \over \sqrt{2}} \left|1,135\right\rang$

Note that we define the angle of polarization with respect to the vertical axis (pointing upwards), where positive angles represent clockwise rotation. Substituting this into the expression for the state $\left|\Psi\right\rang$, we have: $\left|\Psi\right\rang = \left|1,H\right\rang \left|2,V\right\rang$ $= {1 \over \sqrt{2}} \left|1,45\right\rang \left|2,V\right\rang + {1 \over \sqrt{2}} \left|1,135\right\rang \left|2,V\right\rang$

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: $\left|\Psi\right\rang = \left|1,H\right\rang \left|2,V\right\rang$ $= {1 \over \sqrt{2}} \left|1,45\right\rang \left|2,V\right\rang + {1 \over \sqrt{2}} \left|1,135\right\rang \left|2,V\right\rang$ $\Rightarrow \left|1,45\right\rang \left|2,V\right\rang$

An important case is the one in which the photons have the same initial polarization: $\left|\Theta\right\rang = \left|1,V\right\rang \left|2,V\right\rang$

or $\left|\Lambda\right\rang = \left|1,H\right\rang \left|2,H\right\rang$

In order to understand entanglement, consider again the two photons discussed above and observe that the states $\left|\Theta\right\rang$ and $\left|\Lambda\right\rang$ are both possible states of the system. But, if this is the case, then it follows that the superposition of the two states: $\left|\Psi\right\rang = {1 \over \sqrt{2}}[\left|\Theta\right\rang + \left|\Lambda\right\rang]$ $= {1 \over \sqrt{2}} \left|1,V\right\rang \left|2,V\right\rang + {1 \over \sqrt{2}} \left|1,H\right\rang \left|2,H\right\rang$

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: $\left|\Psi\right\rang = {1 \over \sqrt{2}}{1 \over \sqrt{2}} [\left|1,45\right\rang + \left|1,135\right\rang {1 \over \sqrt{2}} [\left|2,45\right\rang + \left|2,135\right\rang] + {1 \over \sqrt{2}} [\left|1,45\right\rang - \left|1,135\right\rang]{1 \over \sqrt{2}} [\left|2,45\right\rang - \left|2,135\right\rang]$ $= {1 \over 2}(\sqrt{2}){[\left|1,45\right\rang \left|2,45\right\rang + \left|1,45\right\rang \left|2,135\right\rang + \left|1,135\right\rang \left|2,45\right\rang + \left|1,135\right\rang \left|2,135\right\rang + \left|1,45\right\rang \left|2,45\right\rang - \left|1,45\right\rang \left|2,135\right\rang - \left|1,135\right\rang \left|2,45\right\rang + \left|1,135\right\rang \left|2,135\right\rang}$ $= {1 \over \sqrt{2}}{\left|1,45\right\rang \left|2,45\right\rang + \left|1,135\right\rang \left|2,135\right\rang}$

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: $\left|\Psi\right\rang = {1 \over \sqrt{2}} \left|1,n\right\rang \left|2,n\right\rang + {1 \over \sqrt{2}} \left|1,n\bot\right\rang \left|2,n\bot\right\rang$

In words, this means that the state $\left|\Psi\right\rang$ 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 $n\bot$.

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: $\left|\Psi\right\rang = {1 \over \sqrt{2}}\left|1,n\right\rang \left|2,n\right\rang + {1 \over \sqrt{2}}\left|1,n\bot\right\rang \left|2,n\bot\right\rang$ $\Rightarrow \left|1,n\right\rang \left|2,n\right\rang$

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.

## Applications

An important area where entanglement can be applied is in computer microchips. Normally, the size of a microchip is restricted by the wavelength of the photon carving the chip, being able to carve at one-half of the wavelength in accordance with the Rayleigh criterion. However, entangled photons can be separated and then rejoined together, and since they have exactly the same position the constructive interference doubles the energy so that it can carve as low as 1/4 of the original wavelength and thus make microelectronic devices half the size of what was previously possible. Entangling more than one photon can lead to even greater energies, hitting 1/6 and theoretically even 1/8 the original wavelength.

Instantaneous communication by means of quantum entanglement is actually impossible because neither side can manipulate the state of the entangled particles, they can only measure it (see No-communication theorem). This fact means that if you measure one particle you cannot infer anything meaningful about the observers measuring the other particle, except you know what state they will measure, or have already measured. Thus causality is preserved.

Photon entanglement may soon be used as a Covert channel if not already. This is due to it being impossible to eavesdrop on the channel, at least for now. Although it may be possible to entangle additional photons and thus observe the communication or tamper with it in the future, this would most likely require physical access to the photons. See the No cloning theorem for additional information.

It may soon be possible to mass-produce entangled photons since scientists have discovered a way to produce these photons using a simple semi-conductor. This approach is not only simpler than the previous nonlinear optical crystals such as beta barium borate (BBO) or Potassium titanyl phosphate (KTP), but also produces them on demand as opposed to the one in ten billion being downconverted into entangled photons within the crystal. The semiconductor is made from gallium arsenide used in optoelectronics, and dots made from indium arsenide mere nanometers in size; this compound is convenient since it self-organizes into dots. Currently it has to be produced at low temperatures to produce infrared light, but some companies predict that it can be produced at room temperature soon.