24
2 Elements of Quantum Information Theory
distribution governing the outcomes of X is thus {
1
2
,
1
2 n+1 , . . . ,
1
2 n+1 }. The Shannon
entropy of X is given by: H (X ) =
n
2
+ 1. Conversely the minimum compression
length of X , for a single realization of X , is
ε
compr (X ) ≈ n for a sufficiently small ε
[12]. This example shows that the Shannon entropy may arbitrarily deviate from the
operational quantity it represents in non-asymptotic or non-i.i.d. scenarios.
2.7 Min- and Max-Entropy
In this Section we introduce two additional entropy measures that play a fundamental
role in the security of quantum cryptographic schemes. Moreover, the smoothed
versions of such entropies can be regarded as generalizations of the Shannon and
von Neumann entropies to non-i.i.d. and non-asymptotic scenarios.
Definition 2.8 (Min-entropy [12, 13]) Let ρ AB be a bipartite density operator. The
min-entropy of A conditioned on B of the state ρ AB is defined as:
H min (A|B) ρ = − log min
σ B
{Tr(σ B ) : σ B ≥ 0, (1 A ⊗ σ B ) − ρ AB ≥ 0}.
(2.54)
Definition 2.9 (Max-entropy [12, 13]) Let ρ AB be a bipartite density operator and
let ρ ABC be a purification of ρ AB . The max-entropy of A conditioned on B of the
state ρ AB is defined as:
H max (A|B) ρ = −H min (A|C) ρ .
(2.55)
When the system B is trivial, i.e. one-dimensional, the conditioning is omitted and
the entropies are represented by H min (A) and H max (A). For instance, the min-entropy
of a quantum state ρ A reduces to: H (A) ρ = − log λ max (ρ A ), where λ max (ρ A ) is the
largest eigenvalue of ρ A , i.e the largest probability value in the probability distribution defined by the eigenvalues of ρ A . Min- and max-entropy are also defined on a
probability distribution P X by evaluating them on the state: ρ X =
x P X (x)|xx |,
where {|x} is an orthonormal basis.
In general, the entropies are related as follows to the von Neumann entropy of a
bipartite density operator ρ AB [14]:
H min (A|B) ≤ H (A|B) ≤ H max (A|B).
(2.56)
2 Elements of Quantum Information Theory
distribution governing the outcomes of X is thus {
1
2
,
1
2 n+1 , . . . ,
1
2 n+1 }. The Shannon
entropy of X is given by: H (X ) =
n
2
+ 1. Conversely the minimum compression
length of X , for a single realization of X , is
ε
compr (X ) ≈ n for a sufficiently small ε
[12]. This example shows that the Shannon entropy may arbitrarily deviate from the
operational quantity it represents in non-asymptotic or non-i.i.d. scenarios.
2.7 Min- and Max-Entropy
In this Section we introduce two additional entropy measures that play a fundamental
role in the security of quantum cryptographic schemes. Moreover, the smoothed
versions of such entropies can be regarded as generalizations of the Shannon and
von Neumann entropies to non-i.i.d. and non-asymptotic scenarios.
Definition 2.8 (Min-entropy [12, 13]) Let ρ AB be a bipartite density operator. The
min-entropy of A conditioned on B of the state ρ AB is defined as:
H min (A|B) ρ = − log min
σ B
{Tr(σ B ) : σ B ≥ 0, (1 A ⊗ σ B ) − ρ AB ≥ 0}.
(2.54)
Definition 2.9 (Max-entropy [12, 13]) Let ρ AB be a bipartite density operator and
let ρ ABC be a purification of ρ AB . The max-entropy of A conditioned on B of the
state ρ AB is defined as:
H max (A|B) ρ = −H min (A|C) ρ .
(2.55)
When the system B is trivial, i.e. one-dimensional, the conditioning is omitted and
the entropies are represented by H min (A) and H max (A). For instance, the min-entropy
of a quantum state ρ A reduces to: H (A) ρ = − log λ max (ρ A ), where λ max (ρ A ) is the
largest eigenvalue of ρ A , i.e the largest probability value in the probability distribution defined by the eigenvalues of ρ A . Min- and max-entropy are also defined on a
probability distribution P X by evaluating them on the state: ρ X =
x P X (x)|xx |,
where {|x} is an orthonormal basis.
In general, the entropies are related as follows to the von Neumann entropy of a
bipartite density operator ρ AB [14]:
H min (A|B) ≤ H (A|B) ≤ H max (A|B).
(2.56)