In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles of states. In practice, the terms density matrix and density operator are often used interchangeably.
Pick a basis with states <math>|0\rangle</math>, <math>|1\rangle</math> in a two-dimensional Hilbert space, then the density operator is represented by the matrix
<math display="block">
(\rho_{ij}) = \left( \begin{matrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11}
\end{matrix} \right)
= \left( \begin{matrix}
p_{0} & \rho_{01} \\
\rho^*_{01} & p_{1}
\end{matrix} \right)
</math>
where the diagonal elements are real numbers that sum to one (also called populations of the two states <math>|0\rangle</math>, <math>|1\rangle</math>).
The off-diagonal elements are complex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that <math>(\rho_{ij})</math> be a positive semi-definite operator, see below.
A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system. This definition can be motivated by considering a situation where some pure states <math>|\psi_j\rangle</math> (which are not necessarily orthogonal) are prepared with probability <math>p_j</math> each. This is known as an ensemble of pure states. The probability of obtaining projective measurement result <math>m</math> when using projectors <math>\Pi_m</math> is given by A density operator represents a pure state if and only if:
- it can be written as an outer product of a state vector <math>|\psi\rangle</math> with itself, that is, <math display="block"> \rho = |\psi \rangle \langle \psi|.</math>
- it is a projection, in particular of rank one.
- it is idempotent, that is <math display="block">\rho = \rho^2.</math>
- it has purity one, that is, <math display="block">\operatorname{tr}(\rho^2) = 1.</math>
It is important to emphasize the difference between a probabilistic mixture (i.e. an ensemble) of quantum states and the superposition of two states. If an ensemble is prepared to have half of its systems in state <math>| \psi_1 \rangle</math> and the other half in <math>| \psi_2 \rangle</math>, it can be described by the density matrix:
: <math>\rho = \frac12\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}, </math>
where <math>| \psi_1 \rangle</math> and <math>| \psi_2 \rangle</math> 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 <math>| \psi \rangle = (| \psi_1 \rangle + | \psi_2 \rangle)/\sqrt{2},</math> with density matrix
: <math>|\psi\rangle\langle\psi| = \frac12\begin{pmatrix} 1 & 1 \\ 1 & 1\end{pmatrix}.</math>
Unlike the probabilistic mixture, this superposition can display quantum interference. Those cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let <math>\{p_j,|\psi_j\rangle\}</math> be an ensemble. Then for any complex matrix <math>U</math> such that <math>U^\dagger U = I</math> (a partial isometry), the ensemble <math>\{q_i,|\varphi_i\rangle\}</math> defined by
: <math>\sqrt{q_i} \left| \varphi_i \right\rangle = \sum_j U_{ij} \sqrt{p_j} \left| \psi_j \right\rangle </math>
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
: <math>\rho = \sum_j p_j |\psi_j \rangle \langle \psi_j| </math>
be the density operator generated by the ensemble <math>\{p_j,|\psi_j\rangle\}</math>, with states <math>|\psi_j\rangle</math> not necessarily orthogonal. Then for all partial isometries <math>U</math> we have that
: <math> |\Psi\rangle = \sum_j \sqrt{p_j} |\psi_j \rangle U |a_j\rangle </math>
is a purification of <math>\rho</math>, where <math>|a_j\rangle</math> is an orthogonal basis, and furthermore all purifications of <math>\rho</math> are of this form.
Measurement
Let <math>A</math> be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states <math>\textstyle |\psi_j\rangle</math> occurs with probability <math>p_j</math>. Then the corresponding density operator equals
: <math>\rho = \sum_j p_j |\psi_j \rangle \langle \psi_j|.</math>
The expectation value of the measurement can be calculated by extending from the case of pure states:
: <math> \langle A \rangle = \sum_j p_j \langle \psi_j|A|\psi_j \rangle = \sum_j p_j \operatorname{tr}\left(|\psi_j \rangle \langle \psi_j|A \right) = \operatorname{tr}\left(\sum_j p_j |\psi_j \rangle \langle \psi_j|A\right) = \operatorname{tr}(\rho A),</math>
where <math>\operatorname{tr}</math> denotes trace. Thus, the familiar expression <math>\langle A\rangle=\langle\psi|A|\psi\rangle</math> for pure states is replaced by
: <math> \langle A \rangle = \operatorname{tr}( \rho A)</math>
for mixed states.
: <math>\rho_i' = \frac{P_i \rho P_i}{\operatorname{tr}\left[\rho P_i\right]}</math>
when outcome i is obtained. In the case where the measurement result is not known the ensemble is instead described by
: <math>\; \rho ' = \sum_i P_i \rho P_i.</math>
If one assumes that the probabilities of measurement outcomes are linear functions of the projectors <math>P_i</math>, 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 <math>S</math> of a mixture can be expressed in terms of the eigenvalues of <math>\rho</math> or in terms of the trace and logarithm of the density operator <math>\rho</math>. Since <math>\rho</math> is a positive semi-definite operator, it has a spectral decomposition such that <math>\rho = \textstyle\sum_i \lambda_i |\varphi_i\rangle \langle\varphi_i|</math>, where <math>|\varphi_i\rangle</math> are orthonormal vectors, <math>\lambda_i \ge 0</math>, and <math>\textstyle \sum \lambda_i = 1</math>. Then the entropy of a quantum system with density matrix <math>\rho</math> is
: <math>S = -\sum_i \lambda_i \ln\lambda_i = -\operatorname{tr}(\rho \ln\rho).</math>
This definition implies that the von Neumann entropy of any pure state is zero. If <math>\rho_i</math> are states that have support on orthogonal subspaces, then the von Neumann entropy of a convex combination of these states,
: <math>\rho = \sum_i p_i \rho_i,</math>
is given by the von Neumann entropies of the states <math>\rho_i</math> and the Shannon entropy of the probability distribution <math>p_i</math>:
: <math>S(\rho) = H(p_i) + \sum_i p_i S(\rho_i).</math>
When the states <math>\rho_i</math> do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination <math>\rho</math>. has a von Neumann entropy larger than that of <math>\rho</math>, except if <math>\rho = \rho'</math>. It is however possible for the <math>\rho'</math> produced by a generalized measurement, or POVM, to have a lower von Neumann entropy than <math>\rho</math>.
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
: <math> i \hbar \frac{d}{dt} \rho = [H, \rho]~, </math>
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:
: <math> i \hbar \frac{d}{dt} A_\text{H} = -[H_\text{H}, A_\text{H}]~,</math>
where <math>A_\text{H}(t)</math> 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 <math>\langle A \rangle</math> comes out the same as in the Schrödinger picture.
- Quantum decoherence theory typically involves non-isolated quantum systems developing entanglement with other systems, including measurement apparatuses. Density matrices make it much easier to describe the process and calculate its consequences. Quantum decoherence explains why a system interacting with an environment transitions from being a pure state, exhibiting superpositions, to a mixed state, an incoherent combination of classical alternatives. This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them.
- Similarly, in quantum computation, quantum information theory, open quantum systems, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used. Noise is often modelled via a depolarizing channel or an amplitude damping channel. Quantum tomography is a process by which, given a set of data representing the results of quantum measurements, a density matrix consistent with those measurement results is computed.
- When analyzing a system with many electrons, such as an atom or molecule, an imperfect but useful first approximation is to treat the electrons as uncorrelated or each having an independent single-particle wavefunction. This is the usual starting point when building the Slater determinant in the Hartree–Fock method. If there are <math>N</math> electrons filling the <math>N</math> single-particle wavefunctions <math>|\psi_i\rangle</math> and if only single-particle observables are considered, then their expectation values for the <math>N</math>-electron system can be computed using the density matrix <math display="inline">\sum_{i=1}^N |\psi_i\rangle \langle \psi_i|</math> (the one-particle density matrix of the <math>N</math>-electron system).
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 that realize A as a subalgebra of operators.
Geometrically, a pure state on a C*-algebra A is a state that 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
This formalism of the 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 a matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements. The term density was introduced by Dirac in 1931 when he used von Neumann's operator to calculate electron density clouds.
Nowadays the term "density matrix" obtained a significance of its own, and corresponds to a classical phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics, which was introduced by Eugene Wigner in 1932.