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The state diagram from Figure 2 is an example of an extended state machine, in which the complete condition of the system (called the extended state) is the combination of a qualitative aspect—the state variable—and the quantitative aspects—the extended state variables. The obvious advantage of extended state machines is flexibility.
A well-known example implicitly using additive state decomposition is the superposition principle, widely used in physics and engineering. The superposition principle states: For all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.
Even though these two examples of a perfectly localized and a perfectly delocalized state were only shown for the real space wave function and correspondingly for the real space IPR, one could obviously extend the idea to momentum space and even phase space; the IPR then gives some information about the localization in the space at ...
In Everett's terminology, each subsystem state was now correlated with its relative state, since each subsystem must now be considered relative to the other subsystems with which it has interacted. In the example of Schrödinger's cat, after the box is opened, the entangled system is the cat, the poison vial and the observer.
A state on the composite system is a positive element ρ of the dual of C(X × Y), which by the Riesz–Markov theorem corresponds to a regular Borel measure on X × Y. The corresponding reduced state is obtained by projecting the measure ρ to X. Thus the partial trace is the quantum mechanical equivalent of this operation.
Given a pure bipartite quantum state of the composite system, it is possible to obtain a reduced density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for any of the subsystems. If it is non-zero, it indicates the two subsystems are entangled.
A quantum state for an imperfectly isolated system will generally evolve to be entangled with the quantum state for the environment. Consequently, even if the system's initial state is pure, the state at a later time, found by taking the partial trace of the joint system-environment state, will be mixed. This phenomenon of entanglement produced ...
It is an example of a quantum channel, [32] and can be interpreted as expressing how a quantum state changes if a measurement is performed but the result of that measurement is lost. [33] Channels defined by projective measurements can never decrease the von Neumann entropy; they leave the entropy unchanged only if they do not change the ...