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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 .
The right hand side shows the resulting 2-by-2 reduced density matrix . In linear algebra and functional analysis , the partial trace is a generalization of the trace . Whereas the trace is a scalar -valued function on operators, the partial trace is an operator -valued function.
A graphical intuition of purity may be gained by looking at the relation between the density matrix and the Bloch sphere, = (+), where is the vector representing the quantum state (on or inside the sphere), and = (,,) is the vector of the Pauli matrices.
A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations within the bath are lost extremely quickly, and amounts to replacing ρ ( t ′ ) → ρ ( t ) {\displaystyle \rho (t ...
Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g(x, p) with the Wigner function. The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is Moyal's evolution equation for the Wigner function:
The density matrix provides a fully general tool that can incorporate both quantum uncertainties (present even if the state of the system were completely known) and classical uncertainties (due to a lack of knowledge) in a unified manner. Any physical observable X in quantum mechanics can be written as an operator, ^.
An example of this function would be [1] = ... The microcanonical ensemble is defined by taking the limit of the density matrix as the energy width goes to zero ...
The fidelity between two quantum states and , expressed as density matrices, is commonly defined as: [1] [2] (,) = ().The square roots in this expression are well-defined because both and are positive semidefinite matrices, and the square root of a positive semidefinite matrix is defined via the spectral theorem.