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For example, the Bell state is a pure state, but the reduced density operator of the first qubit is a mixed state. The mixed state implies that not all the information on this first qubit is known. [1] Bell States are either symmetric or antisymmetric with respect to the subsystems. [2]
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 lower bound is obtained by the completely mixed state, represented by the matrix . The purity of a quantum state is conserved under unitary transformations acting on the density matrix in the form †, where U is a unitary matrix.
Density matrix; Scattering theory ... In density operator form, a Bell diagonal state is defined as ... A Bell-diagonal state is separable if all the probabilities ...
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.
This is a pure state with zero entropy, but each spin has maximum entropy when considered individually, because its reduced density matrix is the maximally mixed state. This indicates that it is an entangled state; [ 19 ] the use of entropy as an entanglement measure is discussed further below.
A mixed state is described by a density matrix ρ, that is a non-negative trace-class operator of trace 1 on the tensor product . The partial trace of ρ with respect to the system B, denoted by , is called the reduced state of ρ on system A.
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.