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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.
A related concept is the quantum metrological gain, which for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher information for the same Hamiltonian for separable states
The purity of a normalized quantum state satisfies , [1] where is the dimension of the Hilbert space upon which the state is defined. The upper bound is obtained by tr ( ρ ) = 1 {\displaystyle \operatorname {tr} (\rho )=1\,} and tr ( ρ 2 ) ≤ tr ( ρ ) {\displaystyle \operatorname {tr} (\rho ^{2})\leq \operatorname {tr} (\rho ...
When a Dicke states has been prepared in an experiment, it is important to verify that the state has been prepared with a good quality. Apart from obtaining the fidelity, a usual goal is to show that the quantum state was highly entangled. If for a quantum state the fidelity with respect to W-states
These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a ...
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] ‖ ‖ = | |.
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states are complex numbers, quantized, limited by uncertainty relations, [1]: 159 and only provide a probability distribution for the outcomes for a system. These constraints alter the nature of ...