4. A particular class of quantum channels [12 points; 4 each].

Define the qubit-to-qubit channels XA and XB as

where

Ao

Во

=

noMRIENTERED ON WIR

XA(P) = APA + A₁A₁

XB(P) = BopB + B₁B₁,

0

[6]

[A]

and

0

[₁]

State your answer and justify it.

and

Co =

A₁

=

for parameters p, q = [0, 1].

(a) Confirm that A and A₁ are valid Kraus operators by calculating

AA0 + A₁ A₁.

B₁ =

[8 VT-P]

(b) Describe in words what the channel XA does in the case where p = 0. Describe in

words what the channel XA does in the case where p = 1.

(c) Consider the channel obtained by first applying XA and then applying XB. Denote

this composed channel as XBO XA, where (XBO XA) (P) = XB(XA(P)). From the

definition, the composed channel XBO XA has Kraus operators AoBo: AB₁, A₁B0,

and A₁B₁. Write out the entries of these four matrices. Is XBOXA the same channel

as Xc, with Kraus operators

and C₁

(5)

pq

- [8 VT-P4] ?

G=

(6)

(7)

(9)