Density matrix
In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent "mixed states". 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. Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information. Definition and motivation. The density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by choice of basis in the underlying space. In practice, the terms "density matrix" and "density operator" are  often used interchangeably. In operator language, a density operator for a system is a positive semi-definite, Hermitian operator of trace one acting on the Hilbert space of the system. This definition can be motivated by considering a situation where a pure state formula_1 is prepared with probability formula_2, known as an "ensemble". The probability of obtaining projective measurement result formula_3 when using projectors formula_4 is given by which makes the density operator, defined as a convenient representation for the state of this ensemble. It is easy to check that this operator is positive semi-definite, Hermitian, and has trace  one. Conversely, it follows from the spectral theorem that every operator with these properties can be written as formula_7 for some states formula_1 and coefficients formula_2 that are non-negative and add up to one. However, this representation will not be unique, as shown by the Schrödinger–HJW theorem. Another motivation for the definition of density operators comes from considering local measurements on entangled states. Let formula_10 be a pure entangled state in the composite Hilbert space formula_11. The probability of obtaining measurement result formula_3 when measuring projectors formula_4 on the Hilbert space formula_14 alone is given by where formula_16  denotes the partial trace over the Hilbert space formula_17. This makes the operator a convenient tool to calculate the probabilities of these local measurements. It is known as the reduced density matrix of formula_10 on subsystem 1. It is easy to check that this operator has all the properties of a density operator. Conversely, the Schrödinger–HJW theorem implies that all density operators can be written as formula_20 for some state formula_21. Pure and mixed states. A pure quantum state is a state that can not be written as a probabilistic mixture, or convex combination, of other quantum states. There  are several equivalent characterizations of pure states in the language of density operators. A density operator represents a pure state if and only if: It is important to emphasize the difference between a probabilistic mixture of quantum states and their superposition. If a physical system is prepared to be either in state formula_26 or formula_27, with equal probability, it can be described by the mixed state where formula_26 and formula_27 are assumed orthogonal and of dimension 2, for simplicity. On the other hand, a quantum superposition of these two states with equal probability amplitudes results in the pure state formula_31  with density matrix Unlike the probabilistic mixture, this superposition can display quantum interference. where the real numbers formula_35 are the coordinates of a point within the unit ball and Points with formula_37 represent pure states, while mixed states are represented by points in the interior. This is known as the Bloch sphere picture of qubit state space. Example: light polarization. An example of pure and mixed states is light polarization. An individual photon can be described as having right or left circular polarization, described by the orthogonal quantum states formula_38 and formula_39 or a superposition of the two: it can  be in any state formula_40 (with formula_41), corresponding to linear, circular, or elliptical polarization. Consider now a vertically polarized photon, described by the state formula_42. If we pass it through a circular polarizer which allows either only formula_38 polarized light, or only formula_39 polarized light, half of the photons are absorbed in both cases. This may make it "seem" like half of the photons are in state formula_38 and the other half in state formula_39, but this is not correct: if we pass formula_47 through a linear polarizer there's no absorption whatsoever, but if we pass either state formula_38  or formula_39 half of the photons are absorbed. Unpolarized light (such as the light from an incandescent light bulb) cannot be described as "any" state of the form formula_40 (linear, circular, or elliptical polarization). Unlike polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and it cannot be made polarized by passing it through any wave plate. However, unpolarized light "can" be described as an statistical ensemble, e. g. as each photon having either formula_38 polarization or formula_39 polarization with probability 1/2. The same behavior would occur if each photon  had either vertical polarization formula_53 or horizontal polarization formula_54 with probability 1/2. These two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. For this example of unpolarized light, the density operator equals There are also other ways to generate unpolarized light: one possibility is to introduce uncertainty in the preparation of the photon, for example, passing it through a birefringent crystal with a rough surface, so that slightly different parts of the light beam acquire different polarizations. Another possibility is using entangled states: a radioactive decay can emit two photons traveling in opposite directions , in the quantum state formula_56. The joint state of the two photons "together" is pure, but the density matrix for each photon individually, found by taking the partial trace of the joint density matrix, is completely mixed. Equivalent ensembles and purifications. A given density operator does not uniquely determine which ensemble of pure states gives rise to it; in general there are infinitely many different ensembles generating the same density matrix. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let formula_57 be an ensemble. Then for any complex matrix formula_58 such that formula_59  (a partial isometry), the ensemble formula_60 defined by will give rise to the same density operator, and all equivalent ensembles are of this form. A closely related fact is that a given density operator has infinitely many different purifications, which are pure states that generate the density operator when a partial trace is taken. Let be the density operator generated by the ensemble formula_57, with states formula_1 not necessarily orthogonal. Then for all partial isometries formula_58 we have that is a purification of formula_67, where formula_68 is an orthogonal basis, and furthermore all purifications of formula_67 are of this form . Measurement. Let formula_70 be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states formula_71 occurs with probability formula_2. Then the corresponding density operator equals The expectation value of the measurement can be calculated by extending from the case of pure states: where formula_75 denotes trace. Thus, the familiar expression formula_76 for pure states is replaced by for mixed states. Moreover, if formula_70 has spectral resolution where formula_80 is the projection operator into the eigenspace corresponding to eigenvalue formula_81, the post-measurement density operator is given by when outcome  i is obtained. In the case where the measurement result is not known the ensemble is instead described by If one assumes that the probabilities of measurement outcomes are linear functions of the projectors formula_80, then they must be given by the trace of the projector with a density operator. Gleason's theorem shows that in Hilbert spaces of dimension 3 or larger the assumption of linearity can be replaced with an assumption of non-contextuality. This restriction on the dimension can be removed by assuming non-contextuality for POVMs as well, but this has been criticized as physically unmotivated . Entropy. The von Neumann entropy formula_85 of a mixture can be expressed in terms of the eigenvalues of formula_67 or in terms of the trace and logarithm of the density operator formula_67. Since formula_67 is a positive semi-definite operator, it has a spectral decomposition such that formula_89, where formula_90 are orthonormal vectors, formula_91, and formula_92. Then the entropy of a quantum system with density matrix formula_67 is This definition implies that the von Neumann entropy of any pure state is zero. If formula_95 are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex  combination of these states, is given by the von Neumann entropies of the states formula_95 and the Shannon entropy of the probability distribution formula_98: When the states formula_95 do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination formula_67. Given a density operator formula_67 and a projective measurement as in the previous section, the state formula_103 defined by the convex combination which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred, has a von Neumann entropy larger than  that of formula_67, except if formula_106. It is however possible for the formula_103 produced by a "generalized" measurement, or POVM, to have a lower von Neumann entropy than formula_67. The von Neumann equation for time evolution. Just as the Schrödinger equation describes how pure states evolve in time, the von Neumann equation (also known as the Liouville–von Neumann equation) describes how a density operator evolves in time. The von Neumann equation dictates that where the brackets denote a commutator. This equation only holds when the density operator is taken to be in the Schrödinger picture, even though this equation  seems at first look to emulate the Heisenberg equation of motion in the Heisenberg picture, with a crucial sign difference: where formula_111 is some "Heisenberg picture" operator; but in this picture the density matrix is "not time-dependent", and the relative sign ensures that the time derivative of the expected value formula_112 comes out "the same as in the Schrödinger picture". If the Hamiltonian is time-independent, the von Neumann equation can be easily solved to yield For a more general Hamiltonian, if formula_114 is the wavefunction propagator over some interval, then the time evolution of the density matrix over  that same interval is given by Wigner functions and classical analogies. The density matrix operator may also be realized in phase space. Under the Wigner map, the density matrix transforms into the equivalent Wigner function, The equation for the time evolution of the Wigner function, known as Moyal equation, is then the Wigner-transform of the above von Neumann equation, where formula_118 is the Hamiltonian, and formula_119 is the Moyal bracket, the transform of the quantum commutator. The evolution equation for the Wigner function is then analogous to that of its classical limit, the Liouville equation of classical physics. In  the limit of vanishing Planck's constant formula_120, formula_121 reduces to the classical Liouville probability density function in phase space. Example applications. Density matrices are a basic tool of quantum mechanics, and appear at least occasionally in almost any type of quantum-mechanical calculation. Some specific examples where density matrices are especially helpful and common are as follows: C*-algebraic formulation of states. It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified with elements of an abstract C*-algebra "A" (that is one without  a distinguished representation as an algebra of operators) and states are positive linear functionals on "A". However, by using the GNS construction, we can recover Hilbert spaces which realize "A" as a subalgebra of operators. Geometrically, a pure state on a C*-algebra "A" is a state which is an extreme point of the set of all states on "A". By properties of the GNS construction these states correspond to irreducible representations of "A". The states of the C*-algebra of compact operators "K"("H") correspond exactly to the density operators, and therefore the pure states of "K"("H") are exactly the pure states  in the sense of quantum mechanics. The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures. History. The formalism of density operators and matrices was introduced in 1927 by John von Neumann and independently, but less systematically, by Lev Landau and later in 1946 by Felix Bloch. Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The name density matrix itself relates to its classical  correspondence to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics, which was introduced by Wigner in 1932. In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.