In this example system A represents the system under study and system B

In this example system a represents the system under

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In this example system A represents the system under study and system B represents the environment of A, which we defined to be whatever is dy- namically coupled to A but incompletely instrumented. If, for example, A is a hydrogen atom, then the electromagnetic field inside the vessel containing the atom would form part of B because a hydrogen atom, being comprised of two moving charged particles, is inevitably coupled to the electromagnetic field. If we start with the atom in its first excited state and the electro- magnetic field in its ground state, then atom, field and atom-plus-field are initially all in pure states. After some time the atom-plus-field will evolve into the state | ψ,t ) = a 0 ( t ) | A; 0 )| F; 1 ) + a 1 ( t ) | A; 1 )| F; 0 ) , (6 . 81) where | A; n ) is the n th excited state of the atom, while | F; n ) is the state of the electromagnetic field when it contains n photons of the frequency asso- ciated with transitions between the atom’s ground and first-excited states. In equation (6.81), a 0 ( t ) is the amplitude that the atom has decayed to its ground state while a 1 ( t ) is the amplitude that it is still in its excited state. When neither amplitude vanishes, the atom is entangled with the electro- magnetic field. If we fail to monitor the electromagnetic field, we have to describe the atom by its reduced density operator ρ A = | a 0 | 2 | A; 0 )( A; 0 | + | a 1 | 2 | A; 1 )( A; 1 | . (6 . 82) This density operator indicates that the atom is now in an impure state. In practice a system under study will sooner or later become entangled with its environment, and once it has, we will be obliged to treat the system as one for which we lack complete information. That is, we will have to predict the results of measurements with a non-trivial density operator. The transition of systems in this way from pure states to impure ones is called quantum decoherence . Experimental work directed at realising the possi- bilities offered by quantum computing is very much concerned with arresting the decoherence process by weakening all couplings to the environment.
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6.3 Density operator 127 6.3.2 Shannon entropy Once we recognise that systems are typically in impure states, it’s natural to want to quantify the impurity of a state: for example, if in the definition (6.64) of the density operator, p 3 = 0 . 99999999, then the system is almost certain to be found in the state | 3 ) and predictions made by assuming that the system is in the pure state | 3 ) will not be much in error, while if the largest probability occurring in the sum is 10 20 , the effects of impurity will be enormous. A probability distribution { p i } provides a certain amount of information about the outcome of some investigation. If one probability is close to unity, the information it provides is nearly complete. Conversely, if all the probabil- ities are small, no outcome is particularly likely and the missing information is large. The question we now address is “what is the appropriate measure of the missing information that remains after a probability distribution { p i } has been specified?” Logic dictates that the required measure s ( p 1 ,...,p n
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