2.4 Composite Systems and Entanglement
19
Indeed, we have that:
Tr E [|ψ AE ψ AE |] =
d
i, j=1
λ i λ j |λ i λ j | Tr[|e i e j |]
=
d
i=1
λ i |λ i λ i | = ρ A ,
(2.39)
as claimed.
Note that all purifications |ψ AE of ρ A are related by unitaries on E.
2.5 Quantum Operations
A quantum operation E, also called quantum channel, provides the most general
description of a physical process acting on a system in state ρ. The final state of
the system, after the process occurs, is given by E(ρ) up to some normalization
factor. Both the unitary evolution of a closed system (Postulate 2.2) and quantum
measurements (Postulate 2.3) are examples of quantum operations.
Quantum operations are defined by the following three axiomatic properties, based
on physical grounds.
Axiom 2.1 The probability that the process represented by E occurs is given by
Tr[E(ρ)] ∈ [0, 1], when ρ is the initial state.
Axiom 2.2 The map E is convex-linear on the set of density operators, i.e.
E
i
p i ρ i
=
i
p i E(ρ i ).
Axiom 2.3 The map E is completely positive (CP). That is, E(ρ) is a positive operator for every input state ρ. Additionally, for every composite state ρ AB on H A ⊗ H B ,
the operator (1 A ⊗ E)(ρ AB ) is positive on H A ⊗ H B .
The axioms are chosen such that quantum operations map density operators to
density operators. Axiom 2.1 includes quantum measurements (where each outcome
occurs with a certain probability) as a possible quantum operation. The normalized
state after the process in this case reads E(ρ)/ Tr[E(ρ)]. The second axiom states a
desirable property, namely that if a system is in one of the states {ρ i } with distribution
{ p i }, after applying E it will be in one of the states {E(ρ i )} with the same probability
distribution. Finally, Axiom 2.3 ensures that the output of a quantum operation is
still a density operator, even when it acts on a subsystem of a composite system. Note
that this requirement is non-trivial, as there are maps which are positive but not CP.
19
Indeed, we have that:
Tr E [|ψ AE ψ AE |] =
d
i, j=1
λ i λ j |λ i λ j | Tr[|e i e j |]
=
d
i=1
λ i |λ i λ i | = ρ A ,
(2.39)
as claimed.
Note that all purifications |ψ AE of ρ A are related by unitaries on E.
2.5 Quantum Operations
A quantum operation E, also called quantum channel, provides the most general
description of a physical process acting on a system in state ρ. The final state of
the system, after the process occurs, is given by E(ρ) up to some normalization
factor. Both the unitary evolution of a closed system (Postulate 2.2) and quantum
measurements (Postulate 2.3) are examples of quantum operations.
Quantum operations are defined by the following three axiomatic properties, based
on physical grounds.
Axiom 2.1 The probability that the process represented by E occurs is given by
Tr[E(ρ)] ∈ [0, 1], when ρ is the initial state.
Axiom 2.2 The map E is convex-linear on the set of density operators, i.e.
E
i
p i ρ i
=
i
p i E(ρ i ).
Axiom 2.3 The map E is completely positive (CP). That is, E(ρ) is a positive operator for every input state ρ. Additionally, for every composite state ρ AB on H A ⊗ H B ,
the operator (1 A ⊗ E)(ρ AB ) is positive on H A ⊗ H B .
The axioms are chosen such that quantum operations map density operators to
density operators. Axiom 2.1 includes quantum measurements (where each outcome
occurs with a certain probability) as a possible quantum operation. The normalized
state after the process in this case reads E(ρ)/ Tr[E(ρ)]. The second axiom states a
desirable property, namely that if a system is in one of the states {ρ i } with distribution
{ p i }, after applying E it will be in one of the states {E(ρ i )} with the same probability
distribution. Finally, Axiom 2.3 ensures that the output of a quantum operation is
still a density operator, even when it acts on a subsystem of a composite system. Note
that this requirement is non-trivial, as there are maps which are positive but not CP.