Q1L3g.txt

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Using the intuition we have built,
we may now give necessary and sufficient criteria
for quantum error correction.
We employ the operator-sum representation
to denote the channel where errors are occurring.
Namely, Ek rho Ek dagger summed over k, as is usual.
This theorem describes the quantum error correction
criteria.
Let C be a quantum code that is a subspace defined
by the projector, P, which is a sum over the basis states psi
sub l.
Psi sub l will be the codewords of the code, C.
There exists a quantum operation, a recovery and syndrome
measurement operator, R, which corrects
a certain set of errors defined by E on the subspace C,
if and only if the following criteria is satisfied.
Namely, that P E sub k dagger Ej P
is equal to a diagonal matrix times the projector, P.
This middle expression, Ek dagger Ej,
denotes the relationship of the two errors Ek and Ej,
and we may equivalently understand
what this expression tells us by decomposing it
into the off-diagonal and diagonal terms.
The off-diagonal terms talk about Ej and Ek
and say that they must be orthogonal for all errors which
are different from each other, and for all codeword states.
This is known as the orthogonality criteria.
For the diagonal elements, the criteria
specify that Ek dagger Ek should be equal to a constant, d sub
k, for all codeword states.
That is, it must not distinguish codewords.
We call this the non-deformation criterion.
When these algebraic conditions are satisfied,
R exists, and can also be found using
the proof of these conditions.
Let us now prove this.
The intuition starts with the picture
that we have looked at before with quantum error detection.
There is a Hilbert space with a subspace,
C. This codeword space has as its image,
under error operators, some transformed vector space.
For example, E may transform C into a smaller or displaced
version of itself.
Another, different error may also map C into an image.
If that image collides with the image of the first operator, E,
then we cannot distinguish these two images from each other.
Collisions like this are ruled out
by the orthogonality criteria of the quantum error correction
conditions.
Next, if the images are all orthogonal to each other,
and, thus, distinguishable, the second error correction
criteria condition requires that images not be deformed.
That is, an error operator cannot differentially change
one basis state by a length which is different from how
another basis state is changed.
That is the non-deformation criterion.
The recovery operator should thus
distinguish these image subspaces
and unshrink them to recover the original.
How do we do that?
By the polar decomposition, each one of the error operators
may be written as a unitary transform and a shrinkage part.
This is just like writing a complex number
as a magnitude times a phase.
So the phase portion is u.
That's the rotation.
And the magnitude portion is PE dagger EP.
From the error correction criteria,
we know that PE dagger EP is d.
The two parts of the recovery operation
are, first, a syndrome measurement, and then, second,
a rescaling.
For the syndrome operation, we want
to project into the kth image of the codeword space,
C. So that is the rotated version
of the codeword space projector, P, given by U P U dagger.
We may rewrite this using the polar decomposition result,
to obtain an expression which has, explicitly,
the error operator E sub k in it.
All we have done, here, is to substitute in
for Uk P into the expression, such that E sub k appears.
Another version of this expression
gives us a relationship with E sub k dagger.
The reason we want this is evident from the QEC criteria.
Namely, we have PEk dagger Ej P must
be diagonal with deformation constant d sub k.
By virtue of this condition, which I will label with a star,
we have that the P sub k, which are
projectors into the kth image of C,
P sub k are orthogonal to each other.
This is expected.
After all, we required that the images of C
be orthogonal to each other, and P sub k project
into the images of C.
Explicitly, we can find that this condition is
that Pk Pj is proportional to-- and now,
let us take the product of the two expressions just obtained,
and, indeed, we find that this is 0.
Thus, the recovery operator may first
use for syndrome measurement the measurement
of P sub k, which gives k as the syndrome for the error.
Given the syndrome for the error,
we now want to recover the initial state.
This state may be obtained by employing the unitary rotation
operator, which we have as part of that polar decomposition
of E. So the recovery will perform u sub k dagger, which
undoes the rotation caused by E sub k.
The overall recovery operation combines
the syndrome measurement projection P sub
k, which gives a post-measurement density matrix
state rho prime.
And then, this unitary operator u sub k.
For a state, psi, in the code, we
find that uk dagger Pk acting on the erroneous state, Ej acting
on psi-- error j, in other words--
gives us, via the syndrome measurement operation P sub
k, the syndrome, k.
And, via the recovery operation u sub
k, which rotates the state back, ideally, psi.
So let us see if that's true.
We plug-in here the expression for P sub k,
and we find that, again, P can be inserted here,
because the state psi is in the codeword space.
We find that we have an expression with PE dagger
EP, which is just equal to d sub k, by the quantum error
correction criteria.
That is d sub k times delta jk necessary
for the orthogonality criterion.
Thus, we find that the recovery operator acting
on the output of the channel, E acting on the input state, psi,
is given by taking the outer product of the pure state
version we just obtained above.
Rewriting that here, explicitly, as a sum over kj,
with u sub k dagger P sub k Ej acting on psi, outer product
with itself giving us-- substituting in the result
with d sub k and delta jk-- simply,
psi times the sum over all d sub k.
And that sum is equal to 1, by the completeness criteria
for the operator sum representation of the error
channel E. Thus, we find that the output of the recovery
operation is the input state, psi,
and we have proven the validity of the quantum error correction
criteria.