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2 Elements of Quantum Information Theory
Kraus’ theorem is a beautiful result which characterizes quantum operations with
an elegant notation.
Theorem 2.3 (Kraus). A map E is a quantum operation satisfying Axioms 2.1, 2.2
and 2.3 if and only if it can be represented by a Kraus decomposition:
E(ρ) =
i
K i ρ K
†
i ,
(2.40)
for some set of Kraus operators {K i } such that
i K
†
i K i ≤ 1. Moreover, being d the
dimension of the Hilbert space of the system on which E acts, the number of Kraus
operators is not larger than d
2 .
The proof of theorem 2.3 can be found in [2, 7]. We point out that the Kraus decomposition of a quantum operation is not unique, but all the possible decompositions
are linked by unitary transformations.
Unitaries and measurements are two particular cases of quantum operations with
one Kraus operator each, given by: K = U and K = M m , respectively. However,
while unitaries are trace-preserving operations, Tr[UρU
†
] = Tr[ρU
† U ] = 1, quantum measurements in general are not.
An equivalent description of quantum operations interprets them as the result of the
interaction between the system of interest (S) and an environment (E). Conversely, in
absence of interactions with an environment, the system would evolve according to a
unitary transformation (Postulate 2.2). Suppose that the system and the environment
are initially in a product state, where the environment is described by a pure state
|e 0 and the system’s state is ρ. Note that assuming an initial pure state for the
environment is not restrictive as we did not fix its dimension, thus we could always
take its purification. The composite system (S + E) is closed and evolves according
to a unitary U . Then, the final state of system S reads:
E(ρ) = Tr E
U (ρ ⊗ |e 0 e 0 |)U
†
=
i
e i |U (ρ ⊗ |e 0 e 0 |)U
†
|e i ,
(2.41)
for some orthonormal basis {|e i } of the environment. By comparing (2.41) with
(2.40), we deduce an explicit expression for the Kraus operators:
K i = =e i |U |e 0 .
(2.42)
Since the operators (2.42) satisfy the completeness relation
i K
†
i K i = 1, they
describe trace-preserving quantum operations. Instead, non-trace-preserving quantum operations can be viewed as just described with an additional projective measurement on the combined system, following the unitary U .
2 Elements of Quantum Information Theory
Kraus’ theorem is a beautiful result which characterizes quantum operations with
an elegant notation.
Theorem 2.3 (Kraus). A map E is a quantum operation satisfying Axioms 2.1, 2.2
and 2.3 if and only if it can be represented by a Kraus decomposition:
E(ρ) =
i
K i ρ K
†
i ,
(2.40)
for some set of Kraus operators {K i } such that
i K
†
i K i ≤ 1. Moreover, being d the
dimension of the Hilbert space of the system on which E acts, the number of Kraus
operators is not larger than d
2 .
The proof of theorem 2.3 can be found in [2, 7]. We point out that the Kraus decomposition of a quantum operation is not unique, but all the possible decompositions
are linked by unitary transformations.
Unitaries and measurements are two particular cases of quantum operations with
one Kraus operator each, given by: K = U and K = M m , respectively. However,
while unitaries are trace-preserving operations, Tr[UρU
†
] = Tr[ρU
† U ] = 1, quantum measurements in general are not.
An equivalent description of quantum operations interprets them as the result of the
interaction between the system of interest (S) and an environment (E). Conversely, in
absence of interactions with an environment, the system would evolve according to a
unitary transformation (Postulate 2.2). Suppose that the system and the environment
are initially in a product state, where the environment is described by a pure state
|e 0 and the system’s state is ρ. Note that assuming an initial pure state for the
environment is not restrictive as we did not fix its dimension, thus we could always
take its purification. The composite system (S + E) is closed and evolves according
to a unitary U . Then, the final state of system S reads:
E(ρ) = Tr E
U (ρ ⊗ |e 0 e 0 |)U
†
=
i
e i |U (ρ ⊗ |e 0 e 0 |)U
†
|e i ,
(2.41)
for some orthonormal basis {|e i } of the environment. By comparing (2.41) with
(2.40), we deduce an explicit expression for the Kraus operators:
K i = =e i |U |e 0 .
(2.42)
Since the operators (2.42) satisfy the completeness relation
i K
†
i K i = 1, they
describe trace-preserving quantum operations. Instead, non-trace-preserving quantum operations can be viewed as just described with an additional projective measurement on the combined system, following the unitary U .