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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.
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. is prepared with probability , known as an ensemble. The probability of obtaining projective measurement result when using projectors is given by :99This definition can be motivated by considering a situation where a pure state
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 for some states and coefficients that are non-negative and add up to one. :102 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 :107
where denotes the partial trace over the Hilbert space . This makes the operator
a convenient tool to calculate the probabilities of these local measurements. It is known as the reduced density matrix of 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 for some state .
A pure quantum state is a state that can not be written as a probabilistic mixture, or convex combination, of other quantum states. 73 A density operator represents a pure state if and only if:There are several equivalent characterizations of pure states in the language of density operators. :
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 or , with equal probability, it can be described by the mixed state
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. 81:
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 state for a qubit can be written as a linear combination of the Pauli matrices, which provide a basis for self-adjoint matrices: :126
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.
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 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 which 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 though a linear polarizer there's 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 an 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 :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. 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. :106
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. be an ensemble. Then for any complex matrix such that (a partial isometry), the ensemble defined byThose cannot be distinguished by any measurement. The equivalent ensembles can be completely characterized: let
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.
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. 73:
Moreover, if has spectral resolution
where is the projection operator into the eigenspace corresponding to eigenvalue , 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 , 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.
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. 217 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 . :518
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, 159 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 . :514:
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.
Note that 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
For a more general Hamiltonian, if is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by
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 is the Hamiltonian, and 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 , 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.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, as noted in the introduction.
The formalism of density operators and matrices was introduced in 1927 by John von Neumannand 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.
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.
In quantum physics, a measurement is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
In quantum statistical mechanics, the von Neumann entropy, named after John von Neumann, is the extension of classical Gibbs entropy concepts to the field of quantum mechanics. For a quantum-mechanical system described by a density matrix ρ, the von Neumann entropy is
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.
The Born rule is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.
In quantum mechanics, notably in quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.
The Sudarshan-Glauber P representation is a suggested way of writing down the phase space distribution of a quantum system in the phase space formulation of quantum mechanics. The P representation is the quasiprobability distribution in which observables are expressed in normal order. In quantum optics, this representation, formally equivalent to several other representations, is sometimes championed over alternative representations to describe light in optical phase space, because typical optical observables, such as the particle number operator, are naturally expressed in normal order. It is named after George Sudarshan and Roy J. Glauber, who worked on the topic in 1963. Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the drawback that it is not always positive, and is not a true probability function.
In physics, a superoperator is a linear operator acting on a vector space of linear operators.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.
A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem.
In quantum mechanics, and especially quantum information theory, the purity of a normalized quantum state is a scalar defined as
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement 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 in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
The Maxwell–Bloch equations, also called the optical Bloch equations describe the dynamics of a two-state quantum system interacting with the electromagnetic mode of an optical resonator. They are analogous to the Bloch equations which describe the motion of the nuclear magnetic moment in an electromagnetic field. The equations can be derived either semiclassically or with the field fully quantized when certain approximations are made.
The entropy of entanglement is a measure of the degree of quantum entanglement between two subsystems constituting a two-part composite quantum system. Given a pure bipartite quantum state of the composite system, it is possible to obtain a reduced density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, i.e. the subsystem is in a mixed state, it indicates the two subsystems are entangled.
In quantum information theory and quantum optics, the Schrödinger–HJW theorem is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger, Lane P. Hughston, Richard Jozsa and William Wootters. The result was also found independently by Nicolas Hadjisavvas building upon work by Ed Jaynes, while a significant part of it was likewise independently discovered by N. David Mermin. Thanks to its complicated history, it is also known by various other names such as the GHJW theorem, the HJW theorem, and the purification theorem.