For any particular quantum system, the principle of quantum superposition states the existence of certain relations amongst states, respectively pure with respect to particular distinct quantum state analysers. It is a fundamental principle of quantum mechanics.
Mathematically, it refers to a property of pure state solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of pure state solutions to a particular equation will also be a pure state solution of it. Such solutions are often made to be orthogonal (i.e. the vectors are at right-angles to each other), such as the energy levels of an electron. In other words, the overlap of the states is nullified, and the expectation value of an operator is the expectation value of the operator in the individual states, multiplied by the fraction of the superposition state that is "in" that state (see also eigenstates). Such resolution into orthogonal components is the basis of what is known as "quantum measurement", a concept that is characteristic of quantum physics, inexplicable in classical physics.
Physically, it refers to the separation and reconstitution of different quantum states.
Another example is a quantum logical qubit state, as used in quantum information processing, which is a linear superposition of the "basis states" and . Here is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise is the state that will always convert to 1.
The principle of quantum superposition refers to pure states of a quantum system. One considers a primary beam of quantal entities that passes into a primary beam splitter or quantum analyser that has multiple output channels. A beam has a pure state when every one of its quantal entities passes into one and the same output channel. A primary beam with a pure state is passed into another, different, secondary beam splitter or quantum analyser. Then the emerging quanta are probabilistically in its several output channels. These several emergent, intermediate, beams are respectively pure with respect to the secondary analyser. They are then passed to respective copies of the secondary analyser, arranged so as to bring them together into a single reconstituted beam. That beam is then passed to a copy of the primary analyser. In general, it will be probabilistically split into the several output channels.
The principle of quantum superposition, however, states that there exist special spatial arrangements of the devices, establishing specially suitable flight paths for the quanta. Such special arrangements result in the perfect restoration of the original input beam, with all the quanta emerging from the copy of the original primary analyser again in one and the same output channel. The primary pure state has been restored. It is said to be a superposition of the several intermediate pure states. Beyond the special flight path arrangement that restores the primary pure state, in general the output is split or analysed with definite probabilities into the several output channels. Such a definite probabilistic split is called an interference pattern. Again it is said to be a superposition, a different but definite one, of the several intermediate beams. The output that is found in the interference pattern is not some partial or fractional state such as perhaps classical thinking might expect. No, it is a pure state that is either detected or not detected, with a definite probability. The probabilistic occurrence of such pure states is a principle that is characteristic of quantum physics.
For perfect superposition it is essential that the intermediate beams are mutually coherent. That is to say, they are all physically derived from one and the same output channel of the primary analyser and primary beam. Moreover, for coherence, there must be no factor in the several intermediate beam paths that affects some of the quantal entities differently from others. In other words, each and every one of the quantal entities in the beam must be exposed to one and the same arrangement of flight paths. Otherwise, superposition is imperfect. It is evident that for this scheme to work, the analyser arrangement considered as a whole is unaltered by interchanging its input and output of the primary beam. In a manner of speaking, passage of the quantal entities through the analyser arrangement as a whole is reversible. In contrast to this, the combination of production of the beam and its destruction by a detector is irreversible.
It is, however, not necessary that the input beam to the primary analyser be coherent. The input quantal entities may well be generated independently of one another, and mutually incoherent. The requirement for coherence is only for the several intermediate beams. Dirac indicated this in his famous dictum "Each photon then interferes only with itself. Interference between different photons never occurs." This is what is meant by the Copenhagen doctrine that the wave function refers to the single individual quantal entity. For diffraction of a beam of matter particles, it raises the question "how can one single particle somehow seem to go into several intermediate beams at once?" This question leads to differences of interpretation which are outside the scope of the present account.
The scenario can conveniently be described by a statistical density matrix, with rows and columns respectively labelled by the several output channels of the secondary and primary quantal analysers. The density matrix shows whether the beam is of a pure or of a mixed state. A pure state appears as a single column of non-zero entries with all other columns being zero. For a mixed state, there are non-zero entries in several rows of several columns.
The principle was described by Paul Dirac as follows:
The general principle of superposition of quantum mechanics applies to the states [undisturbed motions] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state...
The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e, either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.
If the output from the initial primary analyser is not confined to just one of its output channels, then the primary input beam is said to be mixed, not in a pure state. If in such a case the primary analyser is omitted from the experimental set-up, then the final output from the final copy of the primary analyser is also in several output channels, and is mixed, not pure. Superposition is still occurring physically, but the clear picture of it just previously described, that would be seen with a pure input, is not apparent. The mixed-state input beam contains many several instances of the quantal entity in many respectively several distinct and physically separable states, which are not related by superposition. Then a statistical description is needed, an elaboration that obviously is not needed when the primary input beam is in a pure state.
An isolated instance of a quantal entity, considered without respect to any quantum state analyser, is thereby seen without relation with possible superposed quantum states. Such an instance is simply what it is in itself, neither in a pure state nor in a mixed state. Without a quantum analyser, purity, mixture, and superposition are undefined. For classical thinking, a quantum analyser is not part of the conceptual scenario, and no question of quantum superposition arises. A classically thinking observer sees no quantum superposed state, because quantum superposition does not arise in his thinking. As remarked above, for a classical thinker, the fundamental mystery is 'how can a quantum analyser exist and define a quantum pure state?', not 'how can a certain relation hold between pure quantum states?' In contrast, that quantum analysers exist and define quantum states is essential in Niels Bohr's 'postulate of the quantum'.
For an equation describing a physical phenomenon, the superposition principle states that a combination of solutions to a linear equation is also a solution of it. When this is true the equation is said to obey the superposition principle. Thus if state vectors f1, f2 and f3 each solve the linear equation on ψ, then ψ = c1 f1 + c2 f2 + c3 f3 would also be a solution, in which each c is a coefficient. The Schroedinger equation is linear, so quantum mechanics follows this.
is the most general state. But these coefficients dictate probabilities for the system to be in either configuration. The probability for a specified configuration is given by the square of the absolute value of the coefficient. So the probabilities should add up to 1. The electron is in one of those two states for sure.
Continuing with this example: If a particle can be in state up and down, it can also be in a state where it is an amount 3i/5 in up and an amount 4/5 in down.
In this, the probability for up is . The probability for down is . Note that .
In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned. Either of these describe the same state for any nonzero
The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition turns into a mixture of A′ and B′ with the same coefficients as A and B.
For example, if we have the following
Then after those 10 seconds our state will change to
So far there have just been 2 configurations, but there can be infinitely many.
In illustration, a particle can have any position, so that there are different configurations which have any value of the position x. These are written:
The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:
This sum is defined only if the index x is discrete. If the index is over , then the sum replaced by an integral. The quantity is called the wavefunction of the particle.
If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:
The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.
A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written . The most general state is a superposition of the possibilities:
The description of the two particles is much larger than the description of one particle—it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random variables are correlated. If two particles are uncorrelated, the probability distribution for their joint position P(x, y) is a product of the probability of finding one at one position and the other at the other position:
In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces the word correlation, but the analogy[which?] is exact. A disentangled wave function has the form:
while an entangled wavefunction does not have this form.
|This section requires expansion. (July 2012)|
Analogy with probability
In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.
For example, if we have a probability distribution for where a particle is, it is described by the "state"
Where is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.
The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.
Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.
In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:
where the probabilities are positive numbers. Rescaling x,y,z so that
The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.
In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:
rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high-dimensional sphere
A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.
The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:
So that the infinite list of amplitudes completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes add up to one:
For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities , which give the probability of any position. The quantities that describe how they change in time are the transition probabilities , which gives the probability that, starting at x, the particle ends up at y after time t. The total probability of ending up at y is given by the sum over all the possibilities
The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:
So that the total probability will be preserved, K is what is called a stochastic matrix.
When no time passes, nothing changes: for zero elapsed time , the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.
where is the time derivative of the K matrix:
The equation for the probabilities is a differential equation which is sometimes called the master equation:
The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:
One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle which has a constant rate of random walking. In this case is zero unless y is either x+1,x, or x−1, when y is x+1 or x−1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of must be −2c. So the probabilities obey the discretized diffusion equation:
which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:
Which is the diffusion equation.
Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:
Since the sum of the absolute squares of the amplitudes must be constant, must be unitary:
or, in matrix notation,
The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:
The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:
which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy.
For a particle which has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for is the time differential equation:
In the case that left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, , the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.
which is the right choice of phase to take the continuum limit. When c is very large and psi is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:
If there is an additional term in the H matrix which is an extra phase rotation which varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:
These equations describe the motion of a single particle in non-relativistic quantum mechanics.
Quantum mechanics in imaginary time
The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step , the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
where the sum extends over all paths with the property that and . The analogous expression in quantum mechanics is the path integral.
A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution has the property:
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:
In the new coordinates, the R matrix is rescaled as follows:
and H is symmetric
This matrix H defines a quantum mechanical system:
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:
and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.
For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.
Experiments and applications
- A "cat state" has been achieved with photons.
- A beryllium ion has been trapped in a superposed state.
- A double slit experiment has been performed with molecules as large as buckyballs.
- An experiment involving a superconducting quantum interference device ("SQUID") has been linked to theme of the "cat state" thought experiment.
- By use of very low temperatures, very fine experimental arrangements were made to protect in near isolation and preserve the coherence of intermediate states, for a duration of time, between preparation and detection, of SQUID currents. Such a SQUID current is a coherent physical assembly of perhaps billions of electrons. Because of its coherence, such an assembly may be regarded as exhibiting "collective states" of a macroscopic quantal entity. For the principle of superposition, after it is prepared but before it is detected, it may be regarded as exhibiting an intermediate state. It is not a single-particle state such as is often considered in discussions of interference, for example by Dirac in his famous dictum stated above. Morever, though the 'intermediate' state may be loosely regarded as such, it has not been produced as an output of a secondary quantum analyser that was fed a pure state from a primary analyser, and so this is not an example of superposition as strictly and narrowly defined.
- Nevertheless, after preparation, but before measurement, such a SQUID state may be regarded in a manner of speaking as a "pure" state that is a superposition of a clockwise and an anti-clockwise current state. In a SQUID, collective electron states can be physically prepared in near isolation, at very low temperatures, so as to result in protected coherent intermediate states. Remarkable here is that there are found two well-separated respectively self-coherent collective states that exhibit such metastability. The crowd of electrons tunnels back and forth between the clockwise and the anti-clockwise states, as opposed to forming a single intermediate state in which there is no definite collective sense of current flow.
- In contrast, for actual real cats, such well-separated metastable collective states states do not exist and consequently cannot be physically prepared. Schrödinger's point was that classical thinking does not in general anticipate such physically distinct and separate metastable quantum states. In classical thinking, distinct quantum states even of single atoms can indeed be regarded as metastable, and are remarkable and unexpected. In the days when Schrödinger raised his argumentative example, no one had imagined the invention of SQUIDs that exhibit such states on a macroscopic scale. The present-day physicist here pays close attention to the requirement mentioned above, that the intermediate states must be carefully physically shielded to protect them from any factor that affects some of the independent quantal entities (in this case collective not single particle) differently from others. Contrary to this requirement, the living cat breathes. This destroys intermediate state coherence, and so the conditions required for exhibition of the principle of superposition are not fulfilled.
- A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non vibrating states. The resonator comprises about 10 trillion atoms.
- An experiment involving a flu virus has been proposed.
- Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.
Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the Huygens principle.
For any physical property in quantum mechanics, there is a list of all the states where that property has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical property. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.
Write each state with the value q of the physical quantity as a vector in some basis , a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix
where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.
Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.
It is natural to ask why ordinary everyday "real" (macroscopic, Newtonian) objects and events do not seem empirically to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded even as "mysterious", for example by Richard Feynman. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted the dissonance between quantum mechanics and Newtonian physics, where only one configuration occurs, although a configuration for a particle in Newtonian physics specifies both position and momentum.
The explanation is as follows. It is a logical truism that a single detection of a quantal entity, observed alone, empirically considered, is not an example of a relation of several states. For the several states are not empirically defined when the quantal entity is observed alone. It would therefore be nonsense to try to say that it, a single state, observed alone, empirically shows superposition. Superpositon is a relation of several states that are empirically defined only when several intermediate beams are empirically verified to be present. Actual empirical observation of superposition requires that the several intermediate beams should be actually observed, in several distinct experimental set-ups. Without the several experiments, talk of superposition is mere theoretical speculation, not empirical observation. The superposed state is, as defined above, also a pure state, with respect to the primary analyser. It is classically inexplicable how a quantum analyser can have several pure states as outputs. That is Feynman's "mystery".
Quantum superposition is exhibited in fact in many directly observable phenomena, such as interference peaks from an electron wave in a double-slit experiment. Superposition persists at all scales, provided that coherence is shielded from disruption by intermittent external factors. This is a reason for differences of opinion, as between the Copenhagen or other interpretations.
The Heisenberg uncertainty principle states that for any given instant of time, the position and velocity of an electron or other subatomic particle cannot both be exactly determined.
If the operators corresponding to two observables do not commute, they have no simultaneous eigenstates and they obey the uncertainty principle. A state where one observable has a definite value corresponds to a superposition of many states for the other observable.
- Mach-Zehnder interferometer
- Penrose Interpretation
- Pure qubit state
- Quantum computation
- Schrödinger's cat
- Wave packet
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- Dirac, P.A.M. (1930/1958), p. 9.
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