Photon entanglement
Encyclopedia
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.
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:
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.
s. 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 then 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.
Bohr-Einstein debates
The Bohr–Einstein debates were a series of public disputes about quantum mechanics between Albert Einstein and Niels Bohr, who were two of its founders. Their debates are remembered because of their importance to the philosophy of science. An account of them has been written by Bohr in an article...
and is designed to help clarify the discussion of the Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
-Podolsky
Boris Podolsky
Boris Yakovlevich Podolsky , was an American physicist of Russian Jewish descent.-Education:In 1896, Boris Podolsky was born into a poor Jewish family in Taganrog, in what was then the Russian Empire, and he moved to the United States in 1913...
-Rosen
Nathan Rosen
Nathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...
argument in quantum theory
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
which takes place in that article.
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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
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 toapply 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 EinsteinAlbert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
, Podolsky
Boris Podolsky
Boris Yakovlevich Podolsky , was an American physicist of Russian Jewish descent.-Education:In 1896, Boris Podolsky was born into a poor Jewish family in Taganrog, in what was then the Russian Empire, and he moved to the United States in 1913...
and Rosen
Nathan Rosen
Nathan Rosen was an American-Israeli physicist noted for his study on the structure of the hydrogen molecule and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox.-Background:Nathan Rosen was born into a Jewish family in Brooklyn, New York...
(or "EPR") is based:
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.
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 statewhich 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 superpositionQuantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...
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
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
, 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.
Applications
One important area where entanglement can be applied is in computer microchipIntegrated circuit
An integrated circuit or monolithic integrated circuit is an electronic circuit manufactured by the patterned diffusion of trace elements into the surface of a thin substrate of semiconductor material...
s. Normally, the size of a microchip is restricted by the wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
of the photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
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
No-communication theorem
In quantum information theory, a no-communication theorem is a result which gives conditions under which instantaneous transfer of information between two observers is impossible. These results can be applied to understand the so-called paradoxes in quantum mechanics such as the EPR paradox or...
). 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
Covert channel
In computer security, a covert channel is a type of computer security attack that creates a capability to transfer information objects between processes that are not supposed to be allowed to communicate by the computer security policy...
if not already. This is due to it being impossible to eavesdrop
Eavesdropping
Eavesdropping is the act of secretly listening to the private conversation of others without their consent, as defined by Black's Law Dictionary...
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
No cloning theorem
The no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.The state of one system can be...
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 then the previous nonlinear optical crystals such as beta barium borate
Beta Barium Borate
Beta barium borate is a crystal frequently used for frequency mixing and other nonlinear optics applications...
(BBO) or Potassium titanyl phosphate
Potassium titanyl phosphate
Potassium titanyl phosphate or KTP is a nonlinear optical material which is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. The material has a relatively high optical damage threshold , a great optical nonlinearity and excellent...
(KTP), but also produces them on demand as opposed to the one in ten billion being downconverted
Spontaneous parametric down conversion
Spontaneous parametric down-conversion is an important process in quantum optics, used especially as a source of entangled photon pairs, and of single photons.-Basic process:...
into entangled photons within the crystal. The semiconductor is made from gallium arsenide used in optoelectronics
Optoelectronics
Optoelectronics is the study and application of electronic devices that source, detect and control light, usually considered a sub-field of photonics. In this context, light often includes invisible forms of radiation such as gamma rays, X-rays, ultraviolet and infrared, in addition to visible light...
, 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.