Density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble[1] of physical systems as quantum states (even if the ensemble contains only one system).
It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble using the Born rule.
in a two-dimensional Hilbert space, then the density operator is represented by the matrix
where the diagonal elements are real numbers that sum to one (also called populations of the two states
The off-diagonal elements are complex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that
It is easy to check that this operator is positive semi-definite, self-adjoint, and has trace one.
Another motivation for the definition of density operators comes from considering local measurements on entangled states.
[5] There are several equivalent characterizations of pure states in the language of density operators.
with density matrix Unlike the probabilistic mixture, this superposition can display quantum interference.
An arbitrary mixed state for a qubit can be written as a linear combination of the Pauli matrices, which together with the identity matrix provide a basis for
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[9]: 75 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.
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.
defined by will give rise to the same density operator, and all equivalent ensembles are of this form.
Then the corresponding density operator equals The expectation value of the measurement can be calculated by extending from the case of pure states: where
[15] This restriction on the dimension can be removed by assuming non-contextuality for POVMs as well,[16][17] but this has been criticized as physically unmotivated.
is This definition implies that the von Neumann entropy of any pure state is zero.
do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination
defined by the convex combination which can be interpreted as the state produced by performing the measurement but not recording which outcome occurred,[10]: 159 has a von Neumann entropy larger than that of
produced by a generalized measurement, or POVM, to have a lower von Neumann entropy than
The von Neumann equation dictates that[20][21][22] where the brackets denote a commutator.
reduces to the classical Liouville probability density function in phase space.
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: It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable.
[27][28] 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 that realize A as a subalgebra of operators.
By properties of the GNS construction these states correspond to irreducible representations of A.
The formalism of density operators and matrices was introduced in 1927 by John von Neumann[29] and independently, but less systematically, by Lev Landau[30] and later in 1946 by Felix Bloch.
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 Eugene Wigner in 1932.
[3] In contrast, the motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector.
