Introduction to Quantum Physics and Information Processing

Quantum Error Correction 207

|ei|1i −→



|e

3

iXP

1

+ |e

4

iP

1



|1i. (10.20b)

The error acting on |ψi can be written as

|ei|ψi −→

h



|e

1

i + |e

2

iX



P

0

+



|e

3

iX + |e

4

i



P

1

i

|ψi. (10.21)

Now the projection operators can be written in terms of the Pauli matrices:

Z = |0ih0| − |1ih1|; = |0ih0| + |1ih1|;

=⇒ P

0

=

+ Z

2

; P

1

=

− Z

2

. (10.22)

Also, using XZ = iY , we get

|ei|ψi −→



|E

1

i + |E

2

i

ˆ

X + |E

3

i

ˆ

Y + |E

4

i

ˆ

Z



|ψi, (10.23)

where we have appropriately regrouped the environment states |e

i

i to obtain

the new environment states |E

i

i. (We do not care about the exact form of

these states since we are not going to observe them.) We thus see that the

generic error can be expressed as a linear ...