Another important measure for comparing probability distributions is the
fidelity, which is easily extended to quantum states. This is variously defined
in different texts, but we will stick to a simple operational definition here:
F(p(x), q(x)) =
X
x
p
p(x)q(x). (11.50)
The square root is used so that we have F(p(x), p(x)) = 1. This definition
is compatible with the inner product of two vectors with components {p(x)}
and {q(x)}
Characterization of Quantum Information 231
In the quantum case, the fidelity between a pure state |ψi and a state |φi
is the inner product:
F(ψ, φ) = hφ|ψi (11.51)
|F|
2
can also be thought of as the probability of confusing the state |ψi with
|φi in an experimental situation. Another way of looking at it is that if the
state |ψi is sent through a communication protocol, the probability that the
end state |φi is the same as the input state is (the mod-square of) the fidelity
of the process. The fidelity is minimum, 0, if the two states are orthogonal, and
maximum, 1, if the two states are identical. Classically, these are the only two
situations that could possibly arise. But in the quantum world, there exists
a continuity of states connecting the two possibilities, and this distinguishes
quantum information from classical.
One can extend this definition to mixed states as well: for states ρ and σ,
F(ρ, σ) = Tr(
√
ρσ). (11.52)
Example 11.3.2. If we have a pure state |ψi and a mixed state ρ, we can
calculate the fidelity as
F(|ψi, ρ) = Tr(
p
|ψihψ|ρ)
= Tr(
p
hψ|ρ|ψi)
=
p
hψ|ρ|ψi (11.53)
Example 11.3.3. If two density matrices ρ and σ commute then they can be
diagonalized in the same basis and the fidelity can be calculated as
F(ρ, σ) = Tr
s
X
x
(p(x)q(x))|xihx|
= Tr
X
x
p
p(x)q(x)|xihx|
=
X
x
p
p(x)q(x)|xihx| = F(p(x), q(x)). (11.54)
Fidelity is not a distance, but can be used to define one between density
operators, the so-called Bures distance
D
B
=
√
2 − 2F, (11.55)
which is a metric on the space of states.
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