Q4L14d.txt

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Entanglement concentration, also known
as entanglement purification, has a deep relationship
with quantum error correction.
Recall that we have seen that dilution can be performed
using teleportation, and in that case,
teleportation served as a noiseless channel.
Concentration can use teleportation
as a noisy channel.
Recall the goal of concentration is
to create nE EPR pairs between Alice and Bob.
The first step in this protocol is for Alice
to create these EPR pairs herself locally.
She then takes half of each of these EPR pairs
and encodes them into an n qubit code.
Meanwhile, Bob has n copies of the state psi,
which we wish to concentrate.
He sends half of each of these bipartite states to Alice.
Alice performs a Bell basis measurement between the two n
qubits that she holds.
Alice sends those measurement results
to Bob, who uses them to complete the teleportation
protocol.
But the teleportation is noisy because it
used states psi instead of perfectly entangled EPR pairs
like it should have.
So Bob sends his n qubits through an error correction
code decoder, resulting in nE qubits
and all together therefore Alice and Bob
hold an entangled pair with nE ebits.
This protocol requires a quantum error correction code,
which encodes nE ebits with n physical qubits
and has a certain distance d, which I
will let you work out yourself.
If the code is a CSS code, it turns out
this is equivalent to coding in a random CSS code,
as is shown in the security proof
of quantum key distribution.
Alternatively, suppose that instead of constructing
an entanglement concentration protocol,
you were given such a protocol.
Call it an entanglement purification protocol, EPP.
Suppose then, we wanted to do something which normally you
do with error correction codes.
That is, to send from Alice a state psi
to Bob over a noisy channel.
This may be accomplished by using
teleportation and entanglement purification as follows.
Alice starts out with a state psi
but also creates some EPR pairs.
She sends half of each of the EPR pairs
over the noisy channel.
Let the number of EPR pairs be n and nE be the number of qubits
in the state psi.
The output of the noisy channel goes to Bob,
shown here in purple.
After receiving this state, Alice and Bob
then perform the entanglement purification protocol
using whatever classical communication
and local operations is specified in the protocol.
No qubits are transmitted between Alice and Bob
during this portion of the protocol.
The output of the entanglement purification protocol
is nE EPR pairs.
Alice then performs a Bell basis measurement between psi
and these nE ebits and sends the measurement results to Bob.
Bob uses these to complete the teleportation protocol, finally
obtaining an nE qubit state, which
is a good approximation of the original state psi
that Alice transmitted.
This shows how entanglement purification protocols
can serve in place of quantum error correction codes
using teleportation.
In fact, this approach can provide advantages
over error correction.
Entanglement purification protocols
can provide communication over noisy channels,
even when quantum error correction codes fail.
This is the case, for example, for the depolarization channel.
This single qubit channel transmits the state
unchanged with probability p but otherwise replaces the qubit
with one of the Pauli states.
When p is less than 3 over 4, it is
known that the capacity of the channel
is 0 as proven, for example, by the quantum singleton bound.
But the entanglement purification protocol
gives a non-0 rate, and that is because the channel output
may still have distillable entanglement for p
less than 3/4.
It turns out that the reason for this difference
is because the entanglement purification protocol may
employ two-way communication in contrast
to the one-way communication traditionally employed in quantum error correction.