18
2 Elements of Quantum Information Theory
The Schmidt decomposition allows us to immediately compute the reduced state
of the two subsystems according to Definition 2.5:
ρ A =
R
k=1
λ
2
k |α k α k | ; ρ B =
R
k=1
λ
2
k |β k β k |
(2.36)
We observe that ρ A and ρ B are written in their spectral decomposition and have the
same eigenvalues! Since many properties in quantum information are determined by
the eigenvalues of a state (e.g., the von Neumann entropy, see Sect. 2.6), they will
be the same for the two subsystems of a composite quantum system in a pure state.
Moreover, if the Schmidt rank is R = 1, the global state is separable and the two
subsystems are pure. Otherwise, for R > 1 the global state is entangled and the two
subsystems are mixed. In this case, we can completely determine the (pure) state of
the combined system (2.33), but we lack information when we focus on its single
constituents (2.36)—the reduced states are mixed. This bizarre fact is one of the
hallmarks of entanglement.
The missing information on system A is represented by its classical randomness (2.36), which is correlated to system B as visualized in (2.33). Only a global
description of systems A and B, provided by the pure state (2.33), presents no classical randomness and hence cannot be correlated with any other system. Therefore,
everything that might possibly be correlated with system A is contained in system
B.
This fact is widely used in quantum cryptography. Here, a group of honest parties holds a quantum system A. One then assumes the worst-case scenario where
the eavesdropper, Eve, holds the quantum system E that contains all the possible
correlations with A, i.e. the composite system AE is in a pure state. We say that Eve
holds the purifying system, which can be identified as follows.
Proposition 2.1 (Purification) Let ρ A on H A be the state of a quantum system A.
Then there exists an auxiliary system E with state space H E and a pure state |ψ AE ∈
H A ⊗ H E , called a purification of ρ A , such that:
Tr E [|ψ AE ψ AE |] = ρ A .
(2.37)
Proof Consider the spectral decomposition of ρ A :
d
i=1 λ i |λ i λ i | and a Hilbert
space H E of the same dimension d of H A , with orthonormal basis {|e i }. The purification of ρ A is given by:
|ψ AE =
d
i=1
λ i |λ i ⊗ |e i .
(2.38)
2 Elements of Quantum Information Theory
The Schmidt decomposition allows us to immediately compute the reduced state
of the two subsystems according to Definition 2.5:
ρ A =
R
k=1
λ
2
k |α k α k | ; ρ B =
R
k=1
λ
2
k |β k β k |
(2.36)
We observe that ρ A and ρ B are written in their spectral decomposition and have the
same eigenvalues! Since many properties in quantum information are determined by
the eigenvalues of a state (e.g., the von Neumann entropy, see Sect. 2.6), they will
be the same for the two subsystems of a composite quantum system in a pure state.
Moreover, if the Schmidt rank is R = 1, the global state is separable and the two
subsystems are pure. Otherwise, for R > 1 the global state is entangled and the two
subsystems are mixed. In this case, we can completely determine the (pure) state of
the combined system (2.33), but we lack information when we focus on its single
constituents (2.36)—the reduced states are mixed. This bizarre fact is one of the
hallmarks of entanglement.
The missing information on system A is represented by its classical randomness (2.36), which is correlated to system B as visualized in (2.33). Only a global
description of systems A and B, provided by the pure state (2.33), presents no classical randomness and hence cannot be correlated with any other system. Therefore,
everything that might possibly be correlated with system A is contained in system
B.
This fact is widely used in quantum cryptography. Here, a group of honest parties holds a quantum system A. One then assumes the worst-case scenario where
the eavesdropper, Eve, holds the quantum system E that contains all the possible
correlations with A, i.e. the composite system AE is in a pure state. We say that Eve
holds the purifying system, which can be identified as follows.
Proposition 2.1 (Purification) Let ρ A on H A be the state of a quantum system A.
Then there exists an auxiliary system E with state space H E and a pure state |ψ AE ∈
H A ⊗ H E , called a purification of ρ A , such that:
Tr E [|ψ AE ψ AE |] = ρ A .
(2.37)
Proof Consider the spectral decomposition of ρ A :
d
i=1 λ i |λ i λ i | and a Hilbert
space H E of the same dimension d of H A , with orthonormal basis {|e i }. The purification of ρ A is given by:
|ψ AE =
d
i=1
λ i |λ i ⊗ |e i .
(2.38)