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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 ]
A pure quantum state is a state that can not be written as a probabilistic mixture, or convex combination, of other quantum states. [5] There are several equivalent characterizations of pure states in the language of density operators. [9]: 73 A density operator represents a pure state if and only if:
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] ‖ ‖ = | |.
For mixed states, the concurrence can be defined by convex roof extension. [ 3 ] For the tangle, there is monogamy of entanglement , [ 9 ] [ 10 ] that is, the tangle of a qubit with the rest of the system cannot ever exceed the sum of the tangles of qubit pairs which it is part of.
Graph states are useful in quantum error-correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models. A graph state is a particular case of a stabilizer state as well as a 2-uniform hypergraph state, a generalization where the edges have ...
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 .