Some properties of the von Neumann entropy immediately follow from the
definition.
Characterization of Quantum Information 225
1. The minimum value of S(ρ), zero, occurs for pure states.
S(ρ) ≥ 0. (11.29)
Thus even though a pure state embodies probabilities of measurement out-
comes, the information carried by it is zero since it represents a definite vector
in Hilbert space.
2. The maximum value of S(ρ) is log d, where d is the dimensionality of
the Hilbert space.
S(ρ) ≤ log d. (11.30)
This occurs for maximally mixed states with each ρ
i
taking the value 1/d.
You will prove this in an exercise.
3. Invariance under unitary transformations:
Under unitary evolution U of the quantum system, the von Neumann entropy
remains unchanged.
S(UρU
†
) = S(ρ). (11.31)
4. Entropy of preparation:
We can think of entropy as a measure of mixedness of the system, or its
departure from purity. When constructing a state ρ out of an ensemble of
pure states |xi with probability p(x), in general we will find that
H(X) ≥ S(ρ). (11.32)
That is, the Shannon (classical) entropy is greater than the von Neumann
entropy. The equality (Equation 11.27) holds when the |xi are mutually or-
thogonal. The interpretation of this result is that when viewed in a basis in
which ρ is not diagonal, we are not in the same basis in which the system was
prepared. Measurement results in such a basis will have probabilities such that
the entropy is more than the von Neumann entropy. The latter is therefore
called the entropy of preparation of the system.
Example 11.2.2. For a state that is 25% |0i and 75% |+i, the Shannon
entropy is
H(X) =
1
4
log 4 +
3
4
log
4
3
= 0.81 bits.
The density matrix is
ρ =
1
4
1 0
0 0
!
+
3
8
1 1
1 1
!
=
1
8
5 3
3 3
!
with eigenvalues
1
/2 ±
1
/4
p
5
/2, so that the von Neumann entropy is
S(ρ) = 0.485 qubits
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