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. These arise in quantum mechanics in two different situations: when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of the ensemble of possible preparations; and when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.
Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states (not to be confused with superposed states), such as quantum statistical mechanics, open quantum systems 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 a choice of an orthonormal basis in the underlying space. In practice, the terms density matrix and density operator are often used interchangeably.
Pick a basis with states , 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 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 (which are not necessarily orthogonal) are prepared with probability each.. This is known as an ensemble of pure states. The probability of obtaining projective measurement result when using projectors is given by
which can be proven equal to
Consequently, the density operator, defined as
is a convenient representation for the state of this ensemble. This operator is positive semi-definite, self-adjoint, and has trace one. Conversely, it follows from the spectral theorem that every operator with these properties can be written as for some states and coefficients 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 be a pure entangled state in the composite Hilbert space . The probability of obtaining measurement result when measuring projectors on the Hilbert space alone is given by
which after algebraic manipulation becomes
where denotes the partial trace over the Hilbert space . This makes the operator
a convenient tool to calculate the probabilities of these local measurements. This operator has all the properties of a density operator and is known as the reduced density matrix of on subsystem 1. Conversely, the Schrödinger–HJW theorem implies that all density operators can be written as for some state .
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 can be written as an outer product of a state vector with itself, that is, it is a projection, in particular of rank one. it is idempotent, that is it has purity one, that is,
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 and the other half in , it can be described by the density matrix:
where and 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 with density matrix
Unlike the probabilistic mixture, this superposition can display quantum interference.
Geometrically, the set of density operators is a convex set, and the pure states are the extremal points of that set. The simplest case is that of a two-dimensional Hilbert space, known as a qubit. 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 self-adjoint matrices:
where the real numbers are the coordinates of a point within the unit ball and
Points with 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 and or a superposition of the two: it can be in any state (with ), corresponding to linear, circular, or elliptical polarization. Consider now a vertically polarized photon, described by the state . If we pass it through a circular polarizer that allows either only polarized light, or only polarized light, half of the photons are absorbed in both cases. This may make it seem like half of the photons are in state and the other half in state , but this is not correct: if we pass through a linear polarizer there is no absorption whatsoever, but if we pass either state or 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 (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 a statistical ensemble, e. g. as each photon having either polarization or polarization with probability 1/2. The same behavior would occur if each photon had either vertical polarization or horizontal polarization 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 . 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 be an ensemble. Then for any complex matrix such that (a partial isometry), the ensemble 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 , with states not necessarily orthogonal. Then for all partial isometries we have that
is a purification of , where is an orthogonal basis, and furthermore all purifications of are of this form.
Measurement Let be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states occurs with probability . Then the corresponding density operator equals
The expectation value of the measurement can be calculated by extending from the case of pure states:
where denotes trace. Thus, the familiar expression for pure states is replaced by
for mixed states.
Moreover, if has spectral resolution
where is the projection operator into the eigenspace corresponding to eigenvalue , the post-measurement density operator is given by Translated by K. A. Kirkpatrick as
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 , 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 of a mixture can be expressed in terms of the eigenvalues of or in terms of the trace and logarithm of the density operator . Since is a positive semi-definite operator, it has a spectral decomposition such that , where are orthonormal vectors, , and . Then the entropy of a quantum system with density matrix is
This definition implies that the von Neumann entropy of any pure state is zero. If 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 and the Shannon entropy of the probability distribution :
When the states do not have orthogonal supports, the sum on the right-hand side is strictly greater than the von Neumann entropy of the convex combination .
Given a density operator and a projective measurement as in the previous section, the state 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 , except if . It is however possible for the produced by a generalized measurement, or POVM, to have a lower von Neumann entropy than ..
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 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 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