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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.
ρ A is sometimes called the reduced density matrix of ρ on subsystem A. Colloquially, we "trace out" or "trace over" system B to obtain the reduced density matrix on A. [48]: 207–212 [51]: 133 [55]: §22.4 For example, the reduced density matrix of A for the entangled state
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.
in which is the reduced density matrix (or its continuous-variable analogue [7]) across the bipartition of the pure state, and it measures how much the complex amplitudes deviate from the constraints required for tensor separability. The faithful nature of the measure admits necessary and sufficient conditions of separability for pure states.
Given a pure bipartite quantum state of the composite system, it is possible to obtain a reduced density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled.
The effect of decoherence on density matrices is essentially the decay or rapid vanishing of the off-diagonal elements of the partial trace of the joint system's density matrix, i.e. the trace, with respect to any environmental basis, of the density matrix of the combined system and its environment.
4 Examples. 5 See also. 6 References. 7 ... and can be reduced to the special form. Since the matrix h is ... The Lindblad-type evolution of the density matrix in the ...