# Quantum superposition

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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.

For example, a physically observable manifestation of superposition is interference peaks from an electron wave in a double-slit experiment.

Another example is a quantum logical qubit state, as used in quantum information processing, which is a linear superposition of the "basis states" ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$. Here ${\displaystyle |0\rangle }$ is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise ${\displaystyle |1\rangle }$ is the state that will always convert to 1.

## Concept

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.[1] 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."[2] 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.[3]

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'.

## Theory

### Examples

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 = c1f1 + c2f2 + c3f3 would also be a solution, in which each Template:Mvar is a coefficient. The Schroedinger equation is linear, so quantum mechanics follows this.

For example, consider an electron with two possible configurations, up and down. This describes the physical system of a qubit.

${\displaystyle c_{1}\mid \uparrow \rangle +c_{2}\mid \downarrow \rangle }$

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.

${\displaystyle p_{\text{up}}=\mid c_{1}\mid ^{2}}$
${\displaystyle p_{\text{down}}=\mid c_{2}\mid ^{2}}$
${\displaystyle p_{\text{up or down}}=p_{\text{up}}+p_{\text{down}}=1}$

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.

${\displaystyle |\psi \rangle ={3 \over 5}i|\uparrow \rangle +{4 \over 5}|\downarrow \rangle .}$

In this, the probability for up is ${\displaystyle \left|\;{\frac {3i}{5}}\;\right|^{2}={\frac {9}{25}}}$. The probability for down is ${\displaystyle \left|\;{\frac {4}{5}}\;\right|^{2}={\frac {16}{25}}}$. Note that ${\displaystyle {\frac {9}{25}}+{\frac {16}{25}}=1}$.

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 ${\displaystyle \alpha }$

${\displaystyle |\psi \rangle \approx \alpha |\psi \rangle }$

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 ${\displaystyle \psi }$ turns into a mixture of A′ and B′ with the same coefficients as A and B.

For example, if we have the following

${\displaystyle \mid \uparrow \rangle \to \mid \downarrow \rangle }$
${\displaystyle \mid \downarrow \rangle \to {\frac {3i}{5}}\mid \uparrow \rangle +{\frac {4}{5}}\mid \downarrow \rangle }$

Then after those 10 seconds our state will change to

${\displaystyle c_{1}\mid \uparrow \rangle +c_{2}\mid \downarrow \rangle \to c_{1}\left(\mid \downarrow \rangle \right)+c_{2}\left({\frac {3i}{5}}\mid \uparrow \rangle +{\frac {4}{5}}\mid \downarrow \rangle \right)}$

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 Template:Mvar. These are written:

${\displaystyle |x\rangle }$

The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:

${\displaystyle \sum _{x}\psi (x)|x\rangle }$

This sum is defined only if the index Template:Mvar is discrete. If the index is over ${\displaystyle \mathbb {R} }$, then the sum replaced by an integral. The quantity ${\displaystyle \psi (x)}$ 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:

${\displaystyle \sum _{x}\psi _{+}(x)|x,\uparrow \rangle +\psi _{-}(x)|x,\downarrow \rangle \,}$

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 ${\displaystyle |x,y\rangle }$. The most general state is a superposition of the possibilities:

${\displaystyle \sum _{xy}A(x,y)|x,y\rangle \,}$

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:

${\displaystyle P(x,y)=P_{x}(x)P_{y}(y)\,}$