Quantum information science is an area of study based on the idea that information science depends on quantum effects in physics. It includes theoretical issues in computational models as well as more experimental topics in quantum physics including what can and cannot be done with quantum information. The term quantum information theory is sometimes used, but it fails to encompass experimental research in the area.
The following includes in quantum physics:-
Quantum computing, which deals on the one hand with the question how and whether one can build a quantum computer and on the other hand, algorithms that harness its power (see quantum algorithm)
Quantum complexity theory
Quantum cryptography and its generalization, quantum communication
Quantum error correction
Quantum communication complexity
Quantum entanglement, as seen from an information-theoretic point of view
Quantum dense coding
Quantum teleportation is a well-known quantum information processing operation, which can be used to move any arbitrary quantum state from one particle (at one location) to another.
Quantum mechanics is the science of the very small. Quantum mechanics explains the behaviour of matter and its interactions with energy on the scale of atoms and subatomic particles.
By contrast, classical physics only explains matter and energy on a scale familiar to human experience, including the behaviour of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain.[1] Coming to terms with these limitations led to two major revolutions in physics which created a shift in the original scientific paradigm: the theory of relativity and the development of quantum mechanics.[2] This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. These concepts are described in roughly the order in which they were first discovered. For a more complete history of the subject, see History of quantum mechanics.
Light behaves in some respects like particles and in other respects like waves. Matter—particles such as electrons and atoms—exhibits wavelike behaviour too. Some light sources, including neon lights, give off only certain frequencies of light. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its energies, colours, and spectral intensities. Since one never observes half a photon, a single photon is a quantum, or smallest observable amount, of the electromagnetic field. More broadly, quantum mechanics shows that many quantities, such as angular momentum, that appeared to be continuous in the zoomed-out view of classical mechanics, turn out to be (at the small, zoomed-in scale of quantum mechanics) quantized. Angular momentum is required to take on one of a set of discrete allowable values, and since the gap between these values is so minute, the discontinuity is only apparent at the atomic level.
Many aspects of quantum mechanics are counterintuitive and can seem paradoxical, because they describe behaviour quite different from that seen at larger length scales. In the words of quantum physicist Richard Feynman, quantum mechanics deals with "nature as She is – absurd".[3] For example, the uncertainty principle of quantum mechanics means that the more closely one pins down one measurement (such as the position of a particle), the less precise another measurement pertaining to the same particle (such as its momentum) must become.
The first quantum theory: Max Planck and black-body radiation
Main article: Ultraviolet catastrophe
Hot metalwork. The yellow-orange glow is the visible part of the thermal radiation emitted due to the high temperature. Everything else in the picture is glowing with thermal radiation as well, but less brightly and at longer wavelengths than the human eye can detect. A far-infrared camera can observe this radiation.
Thermal radiation is electromagnetic radiation emitted from the surface of an object due to the object's internal energy. If an object is heated sufficiently, it starts to emit light at the red end of the spectrum, as it becomes red hot.
Heating it further causes the colour to change from red to yellow, white, and blue, as light at shorter wavelengths (higher frequencies) begins to be emitted. A perfect emitter is also a perfect absorber: when it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, an ideal thermal emitter is known as a black body, and the radiation it emits is called black-body radiation.
Predictions of the amount of thermal radiation of different frequencies emitted by a body. Correct values predicted by Planck's law (green) contrasted against the classical values of Rayleigh-Jeans law (red) and Wien approximation (blue).
In the late 19th century, thermal radiation had been fairly well characterized experimentally.[note 1] However, classical physics led to the Rayleigh-Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, but strongly disagrees at high frequencies. Physicists searched for a single theory that explained all the experimental results.
The first model that was able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900.[4] He proposed a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number of units of energy at its single characteristic frequency, rather than being able to emit any arbitrary amount of energy. In other words, the energy emitted by an oscillator was quantized.[note 2] The quantum of energy for each oscillator, according to Planck, was proportional to the frequency of the oscillator; the constant of proportionality is now known as the Planck constant. The Planck constant, usually written as h, has the value of 6.63×10−34 J s. So, the energy E of an oscillator of frequency f is given by
E n h f where n 1 2 3 [5]
To change the colour of such a radiating body, it is necessary to change its temperature. Planck's law explains why: increasing the temperature of a body allows it to emit more energy overall, and means that a larger proportion of the energy is towards the violet end of the spectrum.
Planck's law was the first quantum theory in physics, and Planck won the Nobel Prize in 1918 "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".[6] At the time, however, Planck's view was that quantization was purely a heuristic mathematical construct, rather than (as is now believed) a fundamental change in our understanding of the world.
The quantisation of matter: the Bohr model of the atom:-
By the dawn of the 20th century, evidence required a model of the atom with a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. These properties suggested a model in which the electrons circle around the nucleus like planets orbiting a sun.[note 5] However, it was also known that the atom in this model would be unstable: according to classical theory, orbiting electrons are undergoing centripetal acceleration, and should therefore give off electromagnetic radiation, the loss of energy also causing them to spiral toward the nucleus, colliding with it in a fraction of a second.
A second, related, puzzle was the emission spectrum of atoms. When a gas is heated, it gives off light only at discrete frequencies. For example, the visible light given off by hydrogen consists of four different colours, as shown in the picture below. The intensity of the light at different frequencies is also different. By contrast, white light consists of a continuous emission across the whole range of visible frequencies. By the end of the nineteenth century, a simple rule known as Balmer's formula had been found which showed how the frequencies of the different lines were related to each other, though without explaining why this was, or making any prediction about the intensities. The formula also predicted some additional spectral lines in ultraviolet and infrared light which had not been observed at the time. These lines were later observed experimentally, raising confidence in the value of the formula.
Emission spectrum of hydrogen. When excited, hydrogen gas gives off light in four distinct colours (spectral lines) in the visible spectrum, as well as a number of lines in the infrared and ultraviolet.
The mathematical formula describing hydrogen's emission spectrum.
In 1885 the Swiss mathematician Johann Balmer discovered that each wavelength λ (lambda) in the visible spectrum of hydrogen is related to some integer n by the equation
λ B n2n24 n 3 4 5 6
where B is a constant which Balmer determined to be equal to 364.56 nm.
In 1888 Johannes Rydberg generalized and greatly increased the explanatory utility of Balmer's formula. He predicted that λ is related to two integers n and m according to what is now known as the Rydberg formula:[13]
1λ R 1m21n2
where R is the Rydberg constant, equal to 0.0110 nm−1, and n must be greater than m.
Rydberg's formula accounts for the four visible wavelengths of hydrogen by setting m = 2 and n = 3, 4, 5, 6. It also predicts additional wavelengths in the emission spectrum: for m = 1 and for n > 1, the emission spectrum should contain certain ultraviolet wavelengths, and for m = 3 and n > 3, it should also contain certain infrared wavelengths. Experimental observation of these wavelengths came two decades later: in 1908 Louis Paschen found some of the predicted infrared wavelengths, and in 1914 Theodore Lyman found some of the predicted ultraviolet wavelengths.[13]
Note that both Balmer and Rydberg's formulas involve integers: in modern terms, they imply that some property of the atom is quantised. Understanding exactly what this
property was, and why it was quantised, was a major part
in the development of quantum mechanics, as will be shown in the rest of this article.
In 1913 Niels Bohr proposed a new model of the atom that included quantized electron orbits: electrons still orbit the nucleus much as planets orbit around the sun, but they are only permitted to inhabit certain orbits, not to orbit at any distance.[14] When an atom emitted (or absorbed) energy, the electron did not move in a continuous trajectory from one orbit around the nucleus to another, as might be expected classically. Instead, the electron would jump instantaneously from one orbit to another, giving off the emitted light in the form of a photon.[15] The possible energies of photons given off by each element were determined by the differences in energy between the orbits, and so the emission spectrum for each element would contain a number of lines.[16]
Starting from only one simple assumption about the rule that the orbits must obey, the Bohr model was able to relate the observed spectral lines in the emission spectrum of hydrogen to previously known constants. In Bohr's model the electron simply wasn't allowed to emit energy continuously and crash into the nucleus: once it was in the closest permitted orbit, it was stable forever. Bohr's model didn't explain why the orbits should be quantised in that way, nor was it able to make accurate predictions for atoms with more than one electron, or to explain why some spectral lines are brighter than others.
Although some of the fundamental assumptions of the Bohr model were soon found to be wrong, the key result that the discrete lines in emission spectra are due to some property of the electrons in atoms being quantised is correct. The way that the electrons actually behave is strikingly different from Bohr's atom, and from what we see in the world of our everyday experience; this modern quantum mechanical model of the atom is discussed below.
A more detailed explanation of the Bohr model.
Bohr theorised that the angular momentum, L, of an electron is quantised:
L n h2π n ℏ
where n is an integer and h is the Planck constant. Starting from this assumption, Coulomb's law and the equations of circular motion show that an electron with n units of angular momentum will orbit a proton at a distance r given by
r n2h24π2keme2 ,
where ke is the Coulomb constant, m is the mass of an electron, and e is the charge on an electron. For simplicity this is written as
r n2 a0
where a0, called the Bohr radius, is equal to 0.0529 nm. The Bohr radius is the radius of the smallest allowed orbit.
The energy of the electron[note 6] can also be calculated, and is given by
E kee22a0 1n2 .
Thus Bohr's assumption that angular momentum is quantised means that an electron can only inhabit certain orbits around the nucleus, and that it can have only certain energies. A consequence of these constraints is that the electron will not crash into the nucleus: it cannot continuously emit energy, and it cannot come closer to the nucleus than a0 (the Bohr radius).
An electron loses energy by jumping instantaneously from its original orbit to a lower orbit; the extra energy is emitted in the form of a photon. Conversely, an electron that absorbs a photon gains energy, hence it jumps to an orbit that is farther from the nucleus.
Each photon from glowing atomic hydrogen is due to an electron moving from a higher orbit, with radius rn, to a lower orbit, rm. The energy Eγ of this photon is the difference in the energies En and Em of the electron:
Eγ En Em kee22a0 1m21n2
Since Planck's equation shows that the photon's energy is related to its wavelength by Eγ = hc/λ, the wavelengths of light that can be emitted are given by
1λ kee22a0hc 1m21n2
This equation has the same form as the Rydberg formula, and predicts that the constant R should be given by
R kee22a0hc
Therefore, the Bohr model of the atom can predict the emission spectrum of hydrogen in terms of fundamental constants.[note 7] However, it was not able to make accurate predictions for multi-electron atoms, or to explain why some spectral lines are brighter than others.
Wave-particle duality Edit
Main article: Wave-particle duality
Louis de Broglie in 1929. De Broglie won the Nobel Prize in Physics for his prediction that matter acts as a wave, made in his 1924 PhD thesis.
Just as light has both wave-like and particle-like properties, matter also has wave-like properties.[17]
Matter behaving as a wave was first demonstrated experimentally for electrons: a beam of electrons can exhibit diffraction, just like a beam of light or a water wave.[note 8] Similar wave-like phenomena were later shown for atoms and even small molecules.
The wavelength, λ, associated with any object is related to its momentum, p, through the Planck constant, h:[18][19]
p hλ
The relationship, called the de Broglie hypothesis, holds for all types of matter: all matter exhibits properties of both particles and waves.
The concept of wave–particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behaviour of quantum-scale objects, either photons or matter. Wave–particle duality is an example of the principle of complementarity in quantum physics.[20][21][22][23][24] An elegant example of wave–particle duality, the double slit experiment, is discussed in the section below.
The double-slit experiment Edit
Main article: Double-slit experiment
The diffraction pattern produced when light is shone through one slit (top) and the interference pattern produced by two slits (bottom). The much more complex pattern from two slits, with its small-scale interference fringes, demonstrates the wave-like propagation of light.
In the double-slit experiment, as originally performed by Thomas Young and Augustin Fresnel in 1827, a beam of light is directed through two narrow, closely spaced slits, producing an interference pattern of light and dark bands on a screen. If one of the slits is covered up, one might naively expect that the intensity of the fringes due to interference would be halved everywhere. In fact, a much simpler pattern is seen, a simple diffraction pattern. Closing one slit results in a much simpler pattern diametrically opposite the open slit. Exactly the same behaviour can be demonstrated in water waves, and so the double-slit experiment was seen as a demonstration of the wave nature of light.
Play media
The double slit experiment for a classical particle, a wave, and a quantum particle demonstrating wave-particle duality
The double-slit experiment has also been performed using electrons, atoms, and even molecules, and the same type of interference pattern is seen. Thus it has been demonstrated that all matter possesses both particle and wave characteristics.
Even if the source intensity is turned down, so that only one particle (e.g. photon or electron) is passing through the apparatus at a time, the same interference pattern develops over time. The quantum particle acts as a wave when passing through the double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave in an experiment to measure its wave-like properties, and like a particle in an experiment to measure its particle-like properties. The point on the detector screen where any individual particle shows up will be the result of a random process. However, the distribution pattern of many individual particles will mimic the diffraction pattern produced by waves.
Application to the Bohr model Edit
De Broglie expanded the Bohr model of the atom by showing that an electron in orbit around a nucleus could be thought of as having wave-like properties. In particular, an electron will be observed only in situations that permit a standing wave around a nucleus. An example of a standing wave is a violin string, which is fixed at both ends and can be made to vibrate. The waves created by a stringed instrument appear to oscillate in place, moving from crest to trough in an up-and-down motion. The wavelength of a standing wave is re
