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The density matrix is a representation of a linear operator called the density operator. The density matrix is obtained from the density operator by a choice of an orthonormal basis in the underlying space. [2] In practice, the terms density matrix and density operator are often used interchangeably.
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.
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:
A density operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system. [4] [5] [6] A density operator that is a rank-1 projection is known as a pure quantum state, and all quantum states that are not pure are designated mixed.
The conditions for the evolution of the system density matrix to be described by the master equation are: [2] the evolution of the system density matrix is determined by a one-parameter semigroup; the evolution is "completely positive" (i.e. probabilities are preserved) the system and bath density matrices are initially decoupled
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.
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 ...
This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. [1] The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment.