Editing Density matrix (section)

== 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:
* [[Statistical mechanics]] uses density matrices, most prominently to express the idea that a system is prepared at a nonzero temperature. Constructing a density matrix using a [[canonical ensemble]] gives a result of the form <math>\rho = \exp(-\beta H)/Z(\beta)</math>, where <math>\beta</math> is the inverse temperature <math>(k_{\rm B} T)^{-1}</math> and <math>H</math> is the system's Hamiltonian. The normalization condition that the trace of <math>\rho</math> be equal to 1 defines the [[Partition function (statistical mechanics)|partition function]] to be <math>Z(\beta) = \mathrm{tr} \exp(-\beta H)</math>. If the number of particles involved in the system is itself not certain, then a [[grand canonical ensemble]] can be applied, where the states summed over to make the density matrix are drawn from a [[Fock space]].<ref name=":1">{{cite book|first=Mehran |last=Kardar |author-link=Mehran Kardar |title=Statistical Physics of Particles |title-link=Statistical Physics of Particles |year=2007 |publisher=[[Cambridge University Press]] |isbn=978-0-521-87342-0 |oclc=860391091}}</ref>{{Rp|174}}
* [[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.<ref name=Schlosshauer>{{cite journal|first=M. |last=Schlosshauer |title=Quantum Decoherence |journal=Physics Reports |volume=831 |year=2019 |pages=1–57 |arxiv=1911.06282 |doi=10.1016/j.physrep.2019.10.001 |bibcode=2019PhR...831....1S|s2cid=208006050 }}</ref>
* Similarly, in [[quantum computation]], [[quantum information theory]], [[open quantum system]]s, and other fields where state preparation is noisy and decoherence can occur, density matrices are frequently used. Noise is often modelled via a [[quantum depolarizing channel|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.<ref name="granade2016">{{Cite journal|last1=Granade|first1=Christopher|last2=Combes|first2=Joshua|last3=Cory|first3=D. G.|date=2016-01-01|title=Practical Bayesian tomography|journal=New Journal of Physics|language=en|volume=18|issue=3|pages=033024|arxiv=1509.03770|doi=10.1088/1367-2630/18/3/033024|issn=1367-2630|bibcode=2016NJPh...18c3024G|s2cid=88521187}}</ref><ref>{{cite journal |last1=Ardila |first1=Luis |last2=Heyl |first2=Markus |last3=Eckardt |first3=André |title=Measuring the Single-Particle Density Matrix for Fermions and Hard-Core Bosons in an Optical Lattice |journal=Physical Review Letters |date=28 December 2018 |volume=121 |issue=260401 |pages=6 |doi=10.1103/PhysRevLett.121.260401|pmid=30636128 |bibcode=2018PhRvL.121z0401P |arxiv=1806.08171 |s2cid=51684413 }}</ref>
* 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 [[electronic correlation|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).<ref>{{cite book |title=Quantum theory of solids |first=Charles |last=Kittel |date=1963 |publisher=Wiley |location=New York |pages=101 |url=https://archive.org/details/quantumtheoryofs00kitt/page/100/mode/2up}}</ref>