Seminar: Quantum Theory
30.06.2016
Characterizations of quantum channels and
Neumark’s theorem
Leonardo Araujo
Leon Bollmann
Proposition (Choi-Jamiolkowski isomorphism). Let HA and HB be finite dimensional Hilbert
spaces. Define the map
C : B (B (HA ) , B (HB )) → B(HA ⊗ HB ) ,
T 7→ C(T ) := (1A ⊗ T ) |Ωi hΩ|
where |Ωi is the maximally entangled state in HA ⊗ HA .
Then:
• C is an isomorphism
• T is completely positive ⇔ C(T ) ≥ 0
Theorem (Kraus representation). Let HA and HB be finite dimensional Hilbert spaces and
T : B(HA ) → B(HB ) be a linear map. Then:
T is a quantum channel ⇔ There is a finite set of operators {Bk }k ⊆ B(HA , HB ) such that
X
∀ρ ∈ B(HA )
T (ρ) =
Bk ρ Bk∗
k
X
Bk∗ Bk
= 1A
k
Theorem (Stinespring’s dilation). Let HA and HB be finite dimensional Hilbert spaces and
T : B(HA ) → B(HB ) be a linear map. Then:
T is a quantum channel ⇔ There is a Hilbert space HE and a linear map V : HA ⊗HE → HB ⊗HE
such that
T (ρ) = trE (V (ρ ⊗ σE ) V ∗ )
V V = 1HA ⊗HE
∗
∀ρ ∈ B(HA )
i.e. V is an isometry
where σE can be chosen to be a pure state.
Theorem (Neumark’s dilation). Let {Mi }i=1..n be a POVM acting on B(Cd ), i.e.
and 0 ≤ Mi ∈ B(Cd ). Then:
P
i Mi
= 1d
There is a Hilbertspace Cm , a state σE ∈ B(Cm ) and a projection-valued measurement {Pi }i=1..n
in B(Cd·m ) such that:
tr[Mi ρ] = tr[Pi (ρ ⊗ σE )]
∀ρ ∈ B(Cd )