Density matrix

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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In quantum mechanics, a density matrix (or density operator) is a matrix that describes an ensemble^[1] of physical systems as quantum states (even if the ensemble contains only one system). It allows for the calculation of the probabilities of the outcomes of any measurements performed upon the systems of the ensemble 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 ensembles (sometimes ambiguously called mixed states). Mixed ensembles arise in quantum mechanics in two different situations:

1. when the preparation of the systems lead to numerous pure states in the ensemble, and thus one must deal with the statistics of possible preparations, and
2. 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 ensembles, such as quantum statistical mechanics, open quantum systems and quantum information.