An important consideration in information theory is the comparison of two
systems: probability distributions in the classical context and states (pure or
mixed) in the quantum. For such comparisons, various measures collectively
labeled distance measures have been proposed. We’ll consider some of them
here, to educate ourselves in the concepts involved.
Characterization of Quantum Information 229
11.3.1 Kolmogorov or trace distance
A sort of distance between two probability distributions p(x) and q(x) for
the same random variable X can be defined as
D(p(x), q(x)) =
1
2
X
x
|p(x) − q(x)|. (11.48)
This is similar to a “metric” for determining the distance between points in a
space.
One context in which such a measure is useful is in a dynamic process,
where information X is sent through a (noisy) channel and appears as Y .
We wish to compute the probability of error in the channel by comparing the
two distributions. To do this, we first make a copy of the input and call it
X
0
, and then look at the probability distribution of the pairs (X
0
, X) and
(X
0
, Y ). Let’s compute the trace distance between these two distributions
p(x) = p(X
0
= x, X = x) and q
y
= p(X
0
= x, Y = y):
D(p, q) =
1
2
X
x,y
|p(x) − q(y)|
=
1
2
X
x6=y
p(x) +
1
2
X
x
|p(x) − q(x)|
=
1
2
(p(X
0
6= Y ) + 1 − p(X
0
= Y ))
= p(X 6= Y )
For two quantum states ρ and σ, we can define the Kolmogorov distance
using the trace function
D(ρ, σ) =
1
2
Tr|ρ − σ|. (11.49)
How do we compute this? We will define the mod of a matrix A by
|A| =
√
A
2
=
√
A
†
A,
where the last equality holds if A is Hermitian, which is true of density ma-
trices. We can easily see how this reduces to the classical distance, if we can
diagonalize ρ and σ in the same basis to write
ρ =
X
x
p(x)|xihx|, σ =
X
x
q(x)|xihx|.
Two matrices can be simultaneously diagonalized if and only if they commute.
Then we see that
D(ρ, σ) =
1
2
Tr
X
x
(p(x) − q(x))|xihx|
230 Introduction to Quantum Physics and Information Processing
=
1
2
X
x
|p(x) − q(x)| Tr|(|xihx|)|
=
1
2
X
x
|p(x) − q(x)|
= D(p(x), q(x)).
Example 11.3.1. It may be instructive to visualize the trace distance between
single qubits by a Bloch sphere picture. Let our states be represented by Bloch
vectors ~p and ~q:
ρ =
1
2
( + ~p · ~σ) , σ =
1
2
( + ~q · ~σ) .
The trace distance is then
D(ρ, σ) =
1
4
Tr|(~p − ~q) · ~σ|
The matrix (~p − ~q) · ~σ = ~a.~σ has eigenvalues ±a. So the eigenvalues of |~a.~σ|
are |a| and Tr|~σ| = 2|a|. So we have
D(ρ, σ) =
1
2
(|~p − ~q|)
which is half of the geometric distance between the points ~p and ~q in the
Bloch ball.
The trace distance can be interpreted as follows: if two quantum states
are close in trace distance, then when measurements are performed in those
states, the resulting probability distributions are close in the classical trace
distance.
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