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This decoherence is also the central phenomenon relating the quantum world to the classical realm. Given a mixed state ρ, which is mathematically represented by a non-negative operator with unit trace on a Hilbert space, and a quantum operation E, which is described by a trace-preserving completely positive linear mapping on quantum states,20 in order to quantify the decoherence caused by the quantum operation, one naturally asks how the output state E(ρ) is diﬀerent from the input state ρ, and furthermore, how the relations with their respective outside environments are changed.
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