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 atomplusfield are
initially all in pure states. After some time the atomplusfield 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 firstexcited 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 nontrivial 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.
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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 Spring '15
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 Physics, mechanics, The Land, David Skinner, probability amplitudes, James Binney, Physics of Quantum Mechanics