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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 .
Because Bell states are entangled states, information on the entire system may be known, while withholding information on the individual subsystems. 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]
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.
1. A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2, i.e., /. [2]2. Many entanglement measures have a simple formulas for entangled Bell-diagonal states: [1]
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.
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.
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.
The classical analog of the CNOT gate is a reversible XOR gate. How the CNOT gate can be used (with Hadamard gates) in a computation.. In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-X gate, controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer.