32
2 Elements of Quantum Information Theory
The above homogeneous system is solvable if the two equations are linearly dependent:
det
1 − λ i −−ψ|φ
ψ|φ −(1 + λ i )
= 0,
(2.85)
which is verified for λ i = ±
1 − |ψ|φ|
2 . By substituting the expression for λ i in
(2.81) we get the following result.
The trace distance between two pure states |ψ and |φ is a function of their
overlap (inner product):
T (ψ, φ) =
1 − |ψ|φ|
2
.
(2.86)
Equation (2.79) combined with (2.86) indicate that the higher the overlap
between two pure states, the less distinguishable they are, as expected.
The notion of distance between quantum states that is commonly used in defining
the ε-smooth min- and max-entropy is the purified distance.
Definition 2.15 (Purified distance [16]) The purified distance between two positive
operators ρ and τ is given by:
P(ρ, τ ) =
1 − F(ρ, τ ) 2 ,
(2.87)
where F(ρ, τ ) is the generalized fidelity:
F(ρ, τ ) =
√ τ
√
ρ
+
(1 − Tr ρ)(1 − Tr σ ).
(2.88)
An important property of the purified distance, which is not satisfied by the trace distance, is that if two states are separated by a distance P(ρ, τ ), there exist purifications
of ρ and τ with the same purified distance.
References
1. Kaye, P., Laflamme, R., & Mosca, M. (2007). An introduction to quantum computing. Oxford
University Press.
2. Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information (10th
Anniversary ed.). Cambridge University Press.
3. Rossetti, C. (2011). Rudimenti di meccanica quantistica. Levrotto & Bella.