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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 quantum Fisher information is the largest function that is convex and that equals four times the variance for pure states. That is, it equals four times the convex roof of the variance [ 14 ] [ 15 ]
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a density operator; Such quantum state is known as a mixed state. The density operator of a mixed state is a trace class , nonnegative ( positive semi-definite ) self-adjoint operator ρ normalized to be of trace 1.
The time evolution of the state of a two qubit implementation of the Mølmer-Sørensen fast gate over two gate times from the ions starting in the ground state. States are labelled first by the qubit of each ion and then by the phonon occupancy of the motional mode. The grey circle represents a probability of 1/2 of that state.
The IPR basically takes the full information about a quantum system (the wave function; for a -dimensional Hilbert space one would have to store values, the components of the wave function) and compresses it into one single number that then only contains some information about the localization properties of the state.
Quantum state tomography is a process by which, given a set of data representing the results of quantum measurements, a quantum state consistent with those measurement results is computed. [50] It is named by analogy with tomography , the reconstruction of three-dimensional images from slices taken through them, as in a CT scan .