A density matrix is a quantum operator that describes the statistical state of a quantum system.If we don't know exactly what state a quantum system is in, we can describe it with a density matrix – it's like a surprise box: there could be one toy inside, but we're not sure which one, we only know the probabilities.
The concept of the density operator was introduced by John von Neumann in his work 'Probabilistic Construction of Quantum Mechanics' (1927) and simultaneously by Lev Landau to describe damping in quantum systems. Subsequently, the formalism became the foundation of quantum statistical mechanics (Felix Bloch, Richard Feynman). In the 1960s-70s, with the development of quantum optics, the density matrix became a key tool for describing states of light and atoms. Alain Aspect's experiments testing Bell's inequalities (1980s) used the density matrix to analyze entangled photons.
In Hilbert space, the density matrix is written in a basis as ρ_{mn} = ⟨m|ρ|n⟩. Diagonal elements give populations of states, off-diagonal elements – coherences. For a subsystem obtained by taking the partial trace over the environment, ρ_red = Tr_env(ρ_total). This leads to the von Neumann entropy S = –Tr(ρ ln ρ), which characterizes the degree of mixedness. Quantum tomography allows reconstructing all elements of the density matrix through a series of measurements. The density matrix naturally generalizes to continuous variables (Wigner function) and is used in quantum computing for error correction.
A density matrix is like a recipe with probabilities: for example, there might be carrots or onions in the soup, but until you look, you don't know. In the quantum world, this is how mixed states are described.
| ρ | variable | density — density kilogram per cubic metre |
| m | variable | mass — mass kilogram |
| V | variable | volume — volume cubic metre |
| ρ | variable | Density matrix — density matrix |
| p_i | variable | Probability of state |i⟩ — probability |
| |i⟩ | variable | Eigenstate of the density matrix — quantum state |
| ⟨i| | operator | Dual vector (bra) of |i⟩ — bra operator |
| \text{ASD}(f) | variable | Amplitude spectral density at frequency f — amplitude spectral density per root hertz |
| S(f) | variable | Power spectral density at frequency f — power spectral density per hertz |
| f | variable | Frequency — frequency hertz |
| \sqrt{} | operator | Mathematical operation taking the square root — square root |
| S | variable | von Neumann entropy — entropy joule per kelvin |
| ρ | variable | density matrix of the quantum state — density matrix |
| tr | operator | sums diagonal elements of a matrix — trace operator |
| ln | operator | takes the natural logarithm — natural logarithm |
| N | variable | column density — column density per square metre |
| S_ν | variable | line flux density — spectral flux density joule per square metre per second per he |
| Δν | variable | line width — frequency interval hertz |
| Q | variable | rotational partition function — partition function |
| T_rot | variable | rotational temperature — temperature kelvin |
| A_ul | variable | Einstein A coefficient — einstein a coefficient per second |
| Ω | variable | solid angle of source — solid angle steradian |
| h | variable | Planck constant — planck constant joule second |
| c | variable | speed of light — speed of light metre per second |
| g_u | variable | degeneracy of upper level — degeneracy |
| E_u | variable | energy of upper level — energy joule |
| k | variable | Boltzmann constant — boltzmann constant joule per kelvin |
| n | variable | density (or control parameter) — number density per metre cubed |
| R | variable | measured indicator (response) |
| arg max | operator | argument of the maximum: value of n that maximizes the expression — argmax operator |
| d log R / d log n | operator | logarithmic derivative of R with respect to n, measuring relative sensitivity — logarithmic derivative |
