# Density matrix

A density matrix is a matrix that describes the statistical state of a system in quantum mechanics. The density matrix is especially helpful for dealing with mixed states, which consist of a statistical ensemble of several different quantum systems; the opposite of a mixed state is a pure state. State vectors, also called kets, describe only pure states, whereas a density matrix can describe both pure and mixed states.

Describing a quantum state by its density matrix is a fully general alternative formalism to describing a quantum state by its ket (state vector) or by its statistical ensemble of kets. However, in practice, it is often most convenient to use density matrices for calculations involving mixed states, and to use kets for calculations involving only pure states.

The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics.

Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Examples include a system in thermal equilibrium at a temperature above absolute zero, or a system with an uncertain or randomly varying preparation history (so one does not know which pure state the system is in). Also, if a quantum system has two or more subsystems that are entangled, then each subsystem must be treated as a mixed state even if the complete system is in a pure state;[1] the density matrix is also a crucial tool in quantum decoherence theory.

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. Both matrix and operator are self-adjoint (or Hermitian), positive semi-definite, of trace one, and may be infinite-dimensional.[2]

## History

The formalism of density operators and matrices was introduced by John von Neumann[3] in 1927 and independently, but less systematically by Lev Landau[4][5] and Felix Bloch[6] in 1927 and 1946 respectively.

## Pure and mixed states

In quantum mechanics, the state of a quantum system is represented by a state vector, denoted ${\displaystyle |\psi \rangle }$ (and pronounced ket). A quantum system with a state vector ${\displaystyle |\psi \rangle }$ is called a pure state. However, it is also possible for a system to be in a statistical ensemble of different state vectors: For example, there may be a 50% probability that the state vector is ${\displaystyle |\psi _{1}\rangle }$ and a 50% chance that the state vector is ${\displaystyle |\psi _{2}\rangle }$. This system would be in a mixed state; the density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix.[citation needed]

A mixed state is different from a quantum superposition; the probabilities in a mixed state are classical probabilities (as in the probabilities one learns in classic probability theory / statistics), unlike the quantum probabilities in a quantum superposition. In fact, a quantum superposition of pure states is another pure state, for example ${\displaystyle |\psi \rangle =(|\psi _{1}\rangle +|\psi _{2}\rangle )/{\sqrt {2}}}$. In this case, the coefficients ${\displaystyle 1/{\sqrt {2}}}$ are not probabilities, but rather probability amplitudes.[citation needed]

### Example: light polarization

The incandescent light bulb (1) emits completely random polarized photons (2) with mixed state density matrix
${\displaystyle {\begin{bmatrix}0.5&0\\0&0.5\\\end{bmatrix}}}$

After passing through vertical plane polarizer (3), the remaining photons are all vertically polarized (4) and have pure state density matrix
${\displaystyle {\begin{bmatrix}1&0\\0&0\\\end{bmatrix}}}$

An example of pure and mixed states is light polarization. Photons can have two helicities, corresponding to two orthogonal quantum states, ${\displaystyle |R\rangle }$ (right circular polarization) and ${\displaystyle |L\rangle }$ (left circular polarization). A photon can also be in a superposition state, such as ${\displaystyle (|R\rangle +|L\rangle )/{\sqrt {2}}}$ (vertical polarization) or ${\displaystyle (|R\rangle -|L\rangle )/{\sqrt {2}}}$ (horizontal polarization). More generally, it can be in any state ${\displaystyle \alpha |R\rangle +\beta |L\rangle }$ (with ${\displaystyle |\alpha |^{2}+|\beta |^{2}=1}$), corresponding to linear, circular, or elliptical polarization. If we pass ${\displaystyle (|R\rangle +|L\rangle )/{\sqrt {2}}}$ polarized light through a circular polarizer which allows either only ${\displaystyle |R\rangle }$ polarized light, or only ${\displaystyle |L\rangle }$ polarized light, intensity would be reduced by half in both cases. This may make it seem like half of the photons are in state ${\displaystyle |R\rangle }$ and the other half in state ${\displaystyle |L\rangle }$. But this is not correct: Both ${\displaystyle |R\rangle }$ and ${\displaystyle |L\rangle }$ photons are partly absorbed by a vertical linear polarizer, but the ${\displaystyle (|R\rangle +|L\rangle )/{\sqrt {2}}}$ light will pass through that polarizer with no absorption whatsoever.[citation needed]

However, unpolarized light (such as the light from an incandescent light bulb) is different from any state like ${\displaystyle \alpha |R\rangle +\beta |L\rangle }$ (linear, circular, or elliptical polarization). Unlike linearly or elliptically polarized light, it passes through a polarizer with 50% intensity loss whatever the orientation of the polarizer; and unlike circularly polarized light, it cannot be made linearly polarized with any wave plate because randomly oriented polarization will emerge from a wave plate with random orientation. Indeed, unpolarized light cannot be described as any state of the form ${\displaystyle \alpha |R\rangle +\beta |L\rangle }$ in a definite sense. However, unpolarized light can be described with ensemble averages, e.g. that each photon is either ${\displaystyle |R\rangle }$ with 50% probability or ${\displaystyle |L\rangle }$ with 50% probability. The same behavior would occur if each photon was either vertically polarized with 50% probability or horizontally polarized with 50% probability.[citation needed]

Therefore, unpolarized light cannot be described by any pure state, but can be described as a statistical ensemble of pure states in at least two ways (the ensemble of half left and half right circularly polarized, or the ensemble of half vertically and half horizontally linearly polarized); these two ensembles are completely indistinguishable experimentally, and therefore they are considered the same mixed state. One of the advantages of the density matrix is that there is just one density matrix for each mixed state, whereas there are many statistical ensembles of pure states for each mixed state. Nevertheless, the density matrix contains all the information necessary to calculate any measurable property of the mixed state.[citation needed]

Where do mixed states come from? To answer that, consider how to generate unpolarized light. One way is to use a system in thermal equilibrium, a statistical mixture of enormous numbers of microstates, each with a certain probability (the Boltzmann factor), switching rapidly from one to the next due to thermal fluctuations. Thermal randomness explains why an incandescent light bulb, for example, emits unpolarized light. A second way to generate unpolarized light is to introduce uncertainty in the preparation of the system, for example, passing it through a birefringent crystal with a rough surface, so that slightly different parts of the beam acquire different polarizations. A third way to generate unpolarized light uses an EPR setup: A radioactive decay can emit two photons traveling in opposite directions, in the quantum state ${\displaystyle (|R,L\rangle +|L,R\rangle )/{\sqrt {2}}}$. The two photons together are in a pure state, but if you only look at one of the photons and ignore the other, the photon behaves just like unpolarized light.[citation needed]

More generally, mixed states commonly arise from a statistical mixture of the starting state (such as in thermal equilibrium), from uncertainty in the preparation procedure (such as slightly different paths that a photon can travel), or from looking at a subsystem entangled with something else.[citation needed]

## Definition

For a finite-dimensional function space, the most general density operator is of the form

${\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|}$

where the coefficients pj are non-negative and add up to one, and ${\displaystyle \textstyle |\psi _{j}\rangle \langle \psi _{j}|}$ is an outer product written in bra-ket notation. This represents a mixed state, with probability pj that the system is in the pure state ${\displaystyle \textstyle |\psi _{j}\rangle }$.[citation needed]

For the above example of unpolarized light, the density operator is

${\displaystyle \rho ={\tfrac {1}{2}}|R\rangle \langle R|+{\tfrac {1}{2}}|L\rangle \langle L|.}$

where ${\displaystyle \textstyle |L\rangle }$ is the left-circularly-polarized photon state and ${\displaystyle \textstyle |R\rangle }$ is the right-circularly-polarized photon state.[citation needed]

### Different statistical ensembles with the same density matrix

An earlier section gave an example of two statistical ensembles of pure states that have the same density operator: unpolarized light can be described as both 50% right-circular-polarized and 50% left-circular-polarized, or 50% horizontally-polarized and 50% vertically-polarized; such equivalent ensembles or mixtures cannot be distinguished by any measurement. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator if and only if there is a Unitary operator U with

${\displaystyle |\psi _{i}'\rangle {\sqrt {p_{i}'}}=\sum _{j}u_{ij}|\psi _{j}\rangle {\sqrt {p_{j}}}~.}$

This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M* = N N* if and only if M = NU for some unitary U. (See square root of a matrix for more details.) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator. However, if the kets making up the mixture are restricted to be a specific orthonormal basis, then the original probabilities pj are uniquely recoverable from that basis, as the eigenvalues of the density matrix.[citation needed]

### Mathematical properties and purity condition

In operator language, a density operator is a positive semidefinite, Hermitian operator of trace 1 acting on the state space.[1] A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ describes a pure state if and only if

${\displaystyle \rho =\rho ^{2}}$,

i.e. the state is idempotent. This is true regardless of whether H is finite-dimensional or not.[citation needed]

Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state;[1] the family of mixed states is a convex set and a state is pure if it is an extremal point of that set.

It follows from the spectral theorem for compact self-adjoint operators that every mixed state is a countable convex combination of pure states; this representation is not unique. Furthermore, a theorem of Andrew Gleason states that certain functions defined on the family of projections and taking values in [0,1] (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See quantum logic for more details.[citation needed]

## Measurement

Let A be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states ${\displaystyle \textstyle |\psi _{j}\rangle }$ occurs with probability pj. Then the corresponding density operator is:

${\displaystyle \rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|.}$

The expectation value of the measurement can be calculated by extending from the case of pure states (see Measurement in quantum mechanics):

${\displaystyle \langle A\rangle =\sum _{j}p_{j}\langle \psi _{j}|A|\psi _{j}\rangle =\sum _{j}p_{j}\operatorname {tr} \left(|\psi _{j}\rangle \langle \psi _{j}|A\right)=\sum _{j}\operatorname {tr} \left(p_{j}|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} \left(\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|A\right)=\operatorname {tr} (\rho A),}$

where ${\displaystyle \operatorname {tr} }$ denotes trace. Thus, the familiar expression ${\displaystyle \langle A\rangle =\langle \psi |A|\psi \rangle }$ for pure states is replaced by

${\displaystyle \langle A\rangle =\operatorname {tr} (A\rho )}$

for mixed states.

Moreover, if A has spectral resolution

${\displaystyle A=\sum _{i}a_{i}|a_{i}\rangle \langle a_{i}|=\sum _{i}a_{i}P_{i},}$

where ${\displaystyle P_{i}=|a_{i}\rangle \langle a_{i}|}$, the corresponding density operator after the measurement is given by:

${\displaystyle \;\rho '=\sum _{i}P_{i}\rho P_{i}.}$

Note that the above density operator describes the full ensemble after measurement; the sub-ensemble for which the measurement result was the particular value ai is described by the different density operator

${\displaystyle \rho _{i}'={\frac {P_{i}\rho P_{i}}{\operatorname {tr} [\rho P_{i}]}}.}$

This is true assuming that ${\displaystyle \textstyle |a_{i}\rangle }$ is the only eigenket (up to phase) with eigenvalue ai; more generally, Pi in this expression would be replaced by the projection operator into the eigenspace corresponding to eigenvalue ai.

More generally, suppose ${\displaystyle \Phi }$ is a function that associates to each observable A a number ${\displaystyle \Phi (A)}$, which we may think of as the "expectation value" of A. If ${\displaystyle \Phi }$ satisfies some natural properties (such as giving positive values on positive operators), then there is a unique density matrix ${\displaystyle \rho }$ such that

${\displaystyle \Phi (A)=\operatorname {tr} (\rho A)}$

for all A.[1] That is to say, any reasonable "family of expectation values" is representable by a density matrix; this observation suggests that density matrices are the most general notion of a quantum state.

## Entropy

The von Neumann entropy ${\displaystyle S}$ of a mixture can be expressed in terms of the eigenvalues of ${\displaystyle \rho }$ or in terms of the trace and logarithm of the density operator ${\displaystyle \rho }$. Since ${\displaystyle \rho }$ is a positive semi-definite operator, it has a spectral decomposition such that ${\displaystyle \rho =\textstyle \sum _{i}\lambda _{i}|\varphi _{i}\rangle \langle \varphi _{i}|}$, where ${\displaystyle |\varphi _{i}\rangle }$ are orthonormal vectors, ${\displaystyle \lambda _{i}>0}$, and ${\displaystyle \textstyle \sum \lambda _{i}=1}$. Then the entropy of a quantum system with density matrix ${\displaystyle \rho }$ is

${\displaystyle S=-\sum _{i}\lambda _{i}\ln \lambda _{i}=-\operatorname {tr} (\rho \ln \rho ).}$

This entropy can increase, but never decrease, with a projective measurement. However, generalised measurements can decrease entropy;[7][8] the entropy of a pure state is zero, while that of a proper mixture is always greater than zero. Therefore, a pure state may be converted into a mixture by a measurement, but a proper mixture can never be converted into a pure state, thus the act of measurement induces a fundamental irreversible change on the density matrix; this is analogous to the "collapse" of the state vector, or wavefunction collapse. Perhaps counterintuitively, the measurement actually decreases information by erasing quantum interference in the composite system, see quantum entanglement, einselection, and quantum decoherence.

A subsystem of a larger system can be turned from a mixed to a pure state, but only by increasing the von Neumann entropy elsewhere in the system; this is analogous to how the entropy of an object can be lowered by putting it in a refrigerator: The air outside the refrigerator's heat exchanger warms up, gaining even more entropy than was lost by the object in the refrigerator. See second law of thermodynamics. See Entropy in thermodynamics and information theory.

## Systems and subsystems

Another motivation for considering density matrices comes from consideration of systems and their subsystems. Suppose we have two quantum systems, described by Hilbert spaces ${\displaystyle {\mathcal {H}}_{1}}$ and ${\displaystyle {\mathcal {H}}_{2}}$. The composite system is then the tensor product ${\displaystyle {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}$ of the two Hilbert spaces. Suppose now that the composite system is in a pure state ${\displaystyle \psi \in {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}$. If ${\displaystyle \psi }$ happens to have the special form ${\displaystyle \psi =\psi _{1}\otimes \psi _{2}}$, then we may reasonably say that the state of the first subsystem is ${\displaystyle \psi _{1}}$. In this case, we say that the two systems are not entangled. In general, however, ${\displaystyle \psi }$ will not decompose as a single tensor product of vectors in ${\displaystyle {\mathcal {H}}_{1}}$ and ${\displaystyle {\mathcal {H}}_{2}}$. If ${\displaystyle \psi }$ cannot be decomposed as a single tensor product of states in the component systems, we say that the two systems are entangled. In that case, there is no reasonable way to associate a pure state ${\displaystyle \psi _{1}\in {\mathcal {H}}_{1}}$ to the state ${\displaystyle \psi \in {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}$.[1]

If, for example, we have a wave function ${\displaystyle \psi (x_{1},x_{2})}$ describing the state of two particles, there is no natural way to construct a wave function (i.e., pure state) ${\displaystyle \psi _{1}(x_{1})}$ that describes the states of the first particle—unless ${\displaystyle \psi (x_{1},x_{2})}$ happens to be a product of a function ${\displaystyle \psi _{1}(x_{1})}$ and a function ${\displaystyle \psi _{2}(x_{2})}$.

The upshot of the preceding discussion is that even if the total system is in a pure state, the various subsystems that make it up will typically be in mixed states. Thus, the use of density matrices is unavoidable.

On the other hand, whether the composite system is in a pure state or a mixed state, we can perfectly well construct a density matrix that describes the state of ${\displaystyle {\mathcal {H}}_{1}}$. Denote the density matrix of the composite system of two systems by ${\displaystyle \rho }$. Then the state of, say, ${\displaystyle {\mathcal {H}}_{1}}$, is described by a reduced density operator, given by taking the "partial trace" of ${\displaystyle \rho }$ over ${\displaystyle {\mathcal {H}}_{2}}$.[1]

If the state of ${\displaystyle {\mathcal {H}}_{1}\otimes {\mathcal {H}}_{2}}$ happens to be a density matrix of the special form ${\displaystyle \rho =\rho _{1}\otimes \rho _{2}}$ where ${\displaystyle \rho _{1}}$ and ${\displaystyle \rho _{2}}$ are density matrices on ${\displaystyle {\mathcal {H}}_{1}}$ and ${\displaystyle {\mathcal {H}}_{2}}$, then the partial trace of ${\displaystyle \rho }$ with respect to ${\displaystyle {\mathcal {H}}_{2}}$ is just ${\displaystyle \rho _{1}}$. A typical ${\displaystyle \rho }$ will not be of this form, however.

## 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 (in fact, the two equations are equivalent, in the sense that either can be derived from the other.) The von Neumann equation dictates that[9][10]

${\displaystyle i\hbar {\frac {\partial \rho }{\partial t}}=[H,\rho ]~,}$

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:

${\displaystyle i\hbar {\frac {dA^{(H)}}{dt}}=-[H,A^{(H)}]~,}$

where ${\displaystyle A^{(H)}(t)}$ 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 ${\displaystyle \langle A\rangle }$ comes out the same as in the Schrödinger picture.[1]

Taking the density operator to be in the Schrödinger picture makes sense, since it is composed of 'Schrödinger' kets and bras evolved in time, as per the Schrödinger picture. If the Hamiltonian is time-independent, this differential equation can be easily solved to yield

${\displaystyle \rho (t)=e^{-iHt/\hbar }\rho (0)e^{iHt/\hbar }.}$

For a more general Hamiltonian, if ${\displaystyle G(t)}$ is the wavefunction propagator over some interval, then the time evolution of the density matrix over that same interval is given by

${\displaystyle \rho (t)=G(t)\rho (0)G(t)^{\dagger }.}$

However,[11] the density matrix contains both classical and quantum-mechanical probabilities it is necessary to account for changes in both in the presence of external influences.

## "Quantum Liouville", Moyal's equation

The density matrix operator may also be realized in phase space. Under the Wigner map, the density matrix transforms into the equivalent Wigner function,

${\displaystyle W(x,p){\stackrel {\mathrm {def} }{=}}{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{*}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy.}$

The equation for the time evolution of the Wigner function is then the Wigner-transform of the above von Neumann equation,

${\displaystyle {\frac {\partial W(q,p,t)}{\partial t}}=-\{\{W(q,p,t),H(q,p)\}\},}$

where H(q, p) 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 ħ, W(q, p, t) reduces to the classical Liouville probability density function in phase space.

The classical Liouville equation can be solved using the method of characteristics for partial differential equations, the characteristic equations being Hamilton's equations; the Moyal equation in quantum mechanics similarly admits formal solutions in terms of quantum characteristics, predicated on the ∗−product of phase space, although, in actual practice, solution-seeking follows different methods.

## 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:

• 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.
• Similarly, in quantum computation, quantum information theory, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used.
• 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 N electrons filling the N single-particle wavefunctions ${\displaystyle |\psi _{i}\rangle }$, then the collection of N electrons together can be characterized by a density matrix ${\displaystyle \sum _{i=1}^{N}|\psi _{i}\rangle \langle \psi _{i}|}$.

## 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.[13][14] 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.

## Notes and references

1. Hall, Brian C. (2013). "Systems and Subsystems, Multiple Particles". Quantum Theory for Mathematicians. Graduate Texts in Mathematics. 267. pp. 419–440. doi:10.1007/978-1-4614-7116-5_19. ISBN 978-1-4614-7115-8.
2. ^ Fano, U. (1957). "Description of States in Quantum Mechanics by Density Matrix and Operator Techniques". Reviews of Modern Physics. 29 (1): 74–93. Bibcode:1957RvMP...29...74F. doi:10.1103/RevModPhys.29.74.
3. ^ von Neumann, John (1927), "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik", Göttinger Nachrichten, 1: 245–272
4. ^ "The Damping Problem in Wave Mechanics". Collected Papers of L.D. Landau. 1965. pp. 8–18. doi:10.1016/B978-0-08-010586-4.50007-9. ISBN 9780080105864.
5. ^ Schlüter, Michael; Sham, Lu Jeu (1982). "Density functional theory". Physics Today. 35 (2): 36–43. Bibcode:1982PhT....35b..36S. doi:10.1063/1.2914933.
6. ^ Fano, Ugo (1995). "Density matrices as polarization vectors". Rendiconti Lincei. 6 (2): 123–130. doi:10.1007/BF03001661.
7. ^ Nielsen, Michael; Chuang, Isaac (2000), Quantum Computation and Quantum Information, Cambridge University Press, ISBN 978-0-521-63503-5. Chapter 11: Entropy and information, Theorem 11.9, "Projective measurements cannot decrease entropy".
8. ^ Everett, Hugh (1973), "The Theory of the Universal Wavefunction (1956) Appendix I. "Monotone decrease of information for stochastic processes"", The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press, pp. 128–129, ISBN 978-0-691-08131-1
9. ^ Breuer, Heinz; Petruccione, Francesco (2002), The theory of open quantum systems, p. 110, ISBN 978-0-19-852063-4
10. ^ Schwabl, Franz (2002), Statistical mechanics, p. 16, ISBN 978-3-540-43163-3
11. ^ Grandy, W. T (2003). "Time Evolution in Macroscopic Systems. I: Equations of Motion". Foundations of Physics. 34: 1–20. arXiv:cond-mat/0303290. Bibcode:2004FoPh...34....1G. doi:10.1023/B:FOOP.0000012007.06843.ed.
12. ^ Ardila, Luis; Heyl, Markus; Eckardt, André (28 December 2018). "Measuring the Single-Particle Density Matrix for Fermions and Hard-Core Bosons in an Optical Lattice". Physical Review Letters. 121 (260401): 6. arXiv:1806.08171. Bibcode:2018PhRvL.121z0401P. doi:10.1103/PhysRevLett.121.260401. PMID 30636128.
13. ^ See appendix, Mackey, George Whitelaw (1963), Mathematical Foundations of Quantum Mechanics, Dover Books on Mathematics, New York: Dover Publications, ISBN 978-0-486-43517-6
14. ^ Emch, Gerard G. (1972), Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, ISBN 978-0-471-23900-0