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In quantum physics, a quantum state is a mathematical entity that provides a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
for the outcomes of each possible
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a
ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...
in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
in a
hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...
, the relevant state vectors are identified by the principal quantum number , the
angular momentum quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe th ...
, the magnetic quantum number , and the spin z-component . For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector (\alpha, \beta), with a length of one; that is, with , \alpha, ^2 + , \beta, ^2 = 1, where , \alpha, and , \beta, are the absolute values of \alpha and \beta. A mixed state, in this case, has the structure of a 2 \times 2 matrix that is Hermitian and positive semi-definite, and has trace 1. A more complicated case is given (in
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
) by the singlet state, which exemplifies quantum entanglement: \left, \psi\right\rang = \frac\bigl(\left, \uparrow\downarrow\right\rang - \left, \downarrow\uparrow\right\rang \bigr), which involves superposition of joint spin states for two particles with spin . The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability. A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. Before a particular
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
is performed on a quantum system, the theory gives only a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.


Conceptual description


Pure states

In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, while each observable quantity (such as the energy or momentum of a
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s. On the other hand, a system in a superposition of multiple different eigenstates ''does'' in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as: , \Psi(t)\rangle = \sum_n C_n(t) , \Phi_n\rang. The coefficient which corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by the time evolution operator. The symbols , and \rangle surrounding the \Psi are part of
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
. Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states \Phi_n. A number P_n represents the probability of a randomly selected system being in the state \Phi_n. Unlike the linear combination case each system is in a definite eigenstate. The expectation value _\sigma of an observable is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories. There is no state which is simultaneously an eigenstate for ''all'' observables. For example, we cannot prepare a state such that both the position measurement and the momentum measurement (at the same time ) are known exactly; at least one of them will have a range of possible values. This is the content of the Heisenberg uncertainty relation. Moreover, in contrast to classical mechanics, it is unavoidable that ''performing a measurement on the system generally changes its state''. More precisely: After measuring an observable ''A'', the system will be in an eigenstate of ''A''; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure ''A'' twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences, however, as follows. Consider two
incompatible observables In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physi ...
, and , where corresponds to a measurement earlier in time than . Suppose that the system is in an eigenstate of at the experiment's beginning. If we measure only , all runs of the experiment will yield the same result. If we measure first and then in the same run of the experiment, the system will transfer to an eigenstate of after the first measurement, and we will generally notice that the results of are statistical. Thus: ''Quantum mechanical measurements influence one another'', and the order in which they are performed is important. Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called ''entangled states'', that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties ( Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.


Schrödinger picture vs. Heisenberg picture

One can take the observables to be dependent on time, while the state ''σ'' was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables , .) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the
Schrödinger picture In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which ma ...
. (This approach was taken in the earlier part of the discussion above, with a time-varying state , \Psi(t)\rangle = \sum_n C_n(t) , \Phi_n\rang.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention. Both viewpoints are used in quantum theory. While non-relativistic
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with
Dirac picture In quantum mechanics, the interaction picture (also known as the Dirac picture after Paul Dirac) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state v ...
.


Formalism in quantum physics


Pure states as rays in a complex Hilbert space

Quantum physics is most commonly formulated in terms of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, as follows. Any given system is identified with some finite- or infinite-dimensional
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1. Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space H can be obtained from another vector by multiplying by some non-zero complex number, the two vectors are said to correspond to the same "ray" in H and also to the same point in the projective Hilbert space of H.


Bra–ket notation

Calculations in quantum mechanics make frequent use of linear operators, scalar products,
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
s and
Hermitian conjugation In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule :\langle Ax,y \rangle = \langle x,A^*y \rangle, where ...
. In order to make such calculations flow smoothly, and to make it unnecessary (in some contexts) to fully understand the underlying linear algebra,
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
invented a notation to describe quantum states, known as ''
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathem ...
''. Although the details of this are beyond the scope of this article, some consequences of this are: *The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form , \psi\rangle (where the "\psi" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual ''mathematical'' notation, where vectors are usually lower-case Latin letters, and it is clear from the context that they are indeed vectors. *Dirac defined two kinds of vector, ''bra'' and ''ket'', dual to each other. *Each ket , \psi\rangle is uniquely associated with a so-called ''bra'', denoted \langle\psi, , which corresponds to the same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen basis, writing , \psi\rangle as a column vector, \langle\psi, is a row vector; to obtain it just take the transpose and entry-wise complex conjugate of , \psi\rangle. *Scalar products (also called ''brackets'') are written so as to look like a bra and ket next to each other: (The phrase "bra-ket" is supposed to resemble "bracket".)


Spin

The
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
has the same dimension ( M· L· T) as the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
and, at quantum scale, behaves as a ''discrete'' degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the
group representations In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...
of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number that, in units of
Planck's reduced constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equiv ...
, is either an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
(0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin , its spin quantum number always assumes one of the 2''S'' + 1 possible values in the set \ As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2''S''+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete
quantum number In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can ...
variable (for the spin) is added to the usual three continuous variables (for the position in space).


Many-body states and particle statistics

The quantum state of a system of ''N'' particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3
spatial coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
and spin, e.g. , \psi (\mathbf r_1,\, m_1;\; \dots;\; \mathbf r_N,\, m_N)\rangle. Here, the spin variables ''mν'' assume values from the set \ where S_\nu is the spin of ''ν''-th particle. S_\nu = 0 for a particle that does not exhibit spin. The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above ''N''-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all ''N'' particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic). Electrons are fermions with ,
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s (quanta of light) are bosons with (although in the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
they are massless and can't be described with Schrödinger mechanics). When symmetrization or anti-symmetrization is unnecessary, ''N''-particle spaces of states can be obtained simply by
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s of one-particle spaces, to which we will return later.


Basis states of one-particle systems

As with any
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets , \rang, any ket , \psi\rang can be written , \psi \rang = \sum_i c_i , \rangle where are
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. In physical terms, this is described by saying that , \psi\rang has been expressed as a ''quantum superposition'' of the states , \rang. If the basis kets are chosen to be
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
(as is often the case), then c_i = \lang , \psi \rang. One property worth noting is that the ''normalized'' states , \psi\rang are characterized by \lang\psi, \psi\rang = 1, and for orthonormal basis this translates to \sum_i \left , c_i \right , ^2 = 1. Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the , \rang are eigenstates (with
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s ) of an observable, and that observable is measured on the normalized state , \psi\rang, then the probability that the result of the measurement is is . (The normalization condition above mandates that the total sum of probabilities is equal to one.) A particularly important example is the ''position basis'', which is the basis consisting of eigenstates , \mathbf\rang with eigenvalues \mathbf of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket , \psi\rang is associated with a complex-valued function of three-dimensional space \psi(\mathbf) \equiv \lang \mathbf , \psi \rang. This function is called the wave function corresponding to , \psi\rang. Similarly to the discrete case above, the probability ''density'' of the particle being found at position \mathbf is , \psi(\mathbf), ^2 and the normalized states have \int d^3 \mathbf \, , \psi(\mathbf), ^2 = 1. In terms of the continuous set of position basis , \mathbf\rang, the state , \psi \rang is: , \psi \rang = \int d^3 \mathbf \, \psi (\mathbf) , \mathbf\rang .


Superposition of pure states

As mentioned above, quantum states may be superposed. If , \alpha\rangle and , \beta\rangle are two kets corresponding to quantum states, the ket c_\alpha, \alpha\rang + c_\beta, \beta\rang is a different quantum state (possibly not normalized). Note that both the amplitudes and phases ( arguments) of c_\alpha and c_\beta will influence the resulting quantum state. In other words, for example, even though , \psi\rang and e^, \psi\rang (for real ) correspond to the same physical quantum state, they are ''not interchangeable'', since , \phi\rang + , \psi\rang and , \phi\rang + e^ , \psi\rang will ''not'' correspond to the same physical state for all choices of , \phi\rang. However, , \phi\rang+, \psi\rang and e^(, \phi\rang+, \psi\rang) ''will'' correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important. One practical example of superposition is the double-slit experiment, in which superposition leads to quantum interference. The
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
state is a superposition of two different states, one corresponding to the photon travel through the left slit, and the other corresponding to travel through the right slit. The relative phase of those two states depends on the difference of the distances from the two slits. Depending on that phase, the interference is constructive at some locations and destructive in others, creating the interference pattern. We may say that superposed states are in ''coherent superposition'', by analogy with
coherence Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a deriv ...
in other wave phenomena. Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. The resulting superposition ends up oscillating back and forth between two different states.


Mixed states

A ''pure quantum state'' is a state which can be described by a single ket vector, as described above. A ''mixed quantum state'' is a statistical ensemble of pure states (see
quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
). Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations; and second, when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state. Mixed states inevitably arise from pure states when, for a composite quantum system H_1 \otimes H_2 with an entangled state on it, the part H_2 is inaccessible to the observer. The state of the part H_1 is expressed then as the partial trace over H_2. A mixed state ''cannot'' be described with a single ket vector. Instead, it is described by its associated ''density matrix'' (or ''density operator''), usually denoted ''ρ''. Note that density matrices can describe both mixed ''and'' pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space H can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system H \otimes K for a sufficiently large Hilbert space K. The density matrix describing a mixed state is defined to be an operator of the form \rho = \sum_s p_s , \psi_s \rangle \langle \psi_s , where p_s is the fraction of the ensemble in each pure state , \psi_s\rangle. The density matrix can be thought of as a way of using the one-particle
formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scien ...
to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in. A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ''ρ''2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state. The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average ( expectation value) of a measurement corresponding to an observable is given by \langle A \rangle = \sum_s p_s \langle \psi_s , A , \psi_s \rangle = \sum_s \sum_i p_s a_i , \langle \alpha_i , \psi_s \rangle , ^2 = \operatorname(\rho A) where , \alpha_i\rangle and a_i are eigenkets and eigenvalues, respectively, for the operator , and "" denotes trace. It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets , \psi_s\rangle of the pure states, and the other being a statistical (said ''incoherent'') average with the probabilities of those states. According to Eugene Wigner, the concept of mixture was put forward by Lev Landau. English translation reprinted in: p.8–18


Mathematical generalizations

States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.


See also

*
Atomic electron transition Atomic electron transition is a change (or jump) of an electron from one energy level to another within an atom or artificial atom. It appears discontinuous as the electron "jumps" from one quantized energy level to another, typically in a fe ...
* Bloch sphere *
Greenberger–Horne–Zeilinger state In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, qubits, or qudits). It was first ...
* Ground state * Introduction to quantum mechanics * No-cloning theorem *
Orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
*
PBR theorem The PBR theorem is a no-go theorem in quantum foundations due to Matthew Pusey, Jonathan Barrett, and Terry Rudolph (for whom the theorem is named) in 2012. It has particular significance for how one may interpret the nature of the quantum state ...
* Quantum harmonic oscillator * Quantum logic gate *
State vector reduction In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observa ...
, for historical reasons called a ''wave function collapse'' * Stationary state *
W state The W state is an entangled quantum state of three qubits which in the bra-ket notation has the following shape : , \mathrm\rangle = \frac(, 001\rangle + , 010\rangle + , 100\rangle) and which is remarkable for representing a specific type of ...


Notes


References


Further reading

The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics. For a discussion of conceptual aspects and a comparison with classical states, see: * For a more detailed coverage of mathematical aspects, see: * In particular, see Sec. 2.3. For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes fo
Physics 219
at Caltech. For a discussion of geometric aspects see: *

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