Q4L14c.txt

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We now show that quantum entanglement
can be thought of as a physical resource
in the sense of this claim.
I claim that all entangled bipartite pure states
are asymptotically equivalent.
Different forms of entanglement are fungible.
We may prove this claim by showing
there is a gold standard for entanglement.
We will use the state 0, 0 plus 1, 1.
Specifically, there are two parts to the argument.
First is entanglement concentration.
There must exist a mechanism to take
n copies of a partially-entangled state, psi,
and turn them into a smaller number
of gold standard states, which we will call phi-- capital phi.
In particular, we will show that concentration
produces n times E of psi minus delta of these states' phi.
The second part of the argument is entanglement dilution.
We start with n times E of psi plus delta
copies of the maximum entangled state
and obtain n copies of the less entangled state.
These protocols will work with epsilon delta parameters
greater than zero and in the asymptotic limit
of sufficiently large n.
Epsilon here is the probability of error of the protocol.
Let us first look at concentration.
Without loss of generality, we may take the state psi
as being a bipartite state with coefficients 1 minus P
square root 0, 0, and square root of P 1, 1.
This is the form provided by the Schmidt decomposition.
Recall that the entanglement of the state psi
is given by the von Neumann entropy of the reduced density
matrix of half of this state.
We find this is nothing more than the binary entropy of P,
which we have seen several times.
What we need from concentration is
to be able to take n copies of the state psi
into approximately E of psi copies
of the maximum entangled state.
Delta here will be the fee that is
paid to be able to create these EPR pairs.
Since the protocol uses n copies of psi,
let us explicitly write out this tensor product of n copies.
We may express this tensor product using
a binomial expansion where each of the usual binomial
coefficients multiplies the term xx.
Here, the absolute value of x denotes the number of 1s
in the binary string x.
Let's rewrite using the number of 1s
as w-- giving us a slightly simpler expression
where we may take into account that there are multiple values
of x with the same weight, w.
The superposition of all of the states with the same weight, w,
is useful to define.
Here, we give that the name s sub w and normalize it properly
by the number of possibilities-- that is, the square root of n
choose w.
Each s sub w state is actually a maximally-entangled state
in a Hilbert space of dimension n
choose w meaning equivalently log n choose w ebits.
What we have, though, is not a single s sub w state
but, rather, with psi to the n, a superposition over a variety
of s sub w states-- each with an amplitude with a form which
has a binomial coefficient, as shown here.
This superposition of maximally-entangled states
is quite useful, despite the different sizes.
And that is because we may obtain one of them
simply by following this procedure.
Alice and Bob-- A and B-- both independently
measure w, the Hamming weight of the strings that they have.
This collapses the superposition into a single s sub
w-- requiring no communication between the two parties.
The probability of a given weight
is n choose w, P to the w, 1 minus P to the n minus w.
And this is approximately Gaussian for large n.
The mean value of w obtained is n times P,
and the variance is n P times 1 minus P.
The average number of entangled pairs obtained
is, therefore, the log of n choose n P, which
is approximately the binary entropy of P.
And this is nothing more than n times E, the entanglement.
To be a little more precise, we don't get exactly nE
but, rather, nE minus some overhead, which goes something
like the width of the variance-- that is, square root of n.
And this determines our overhead phi cost-- delta.
Choosing delta to go as 1 over square root of n
thus allows us to obtain a final protocol which
succeeds to produce approximately n
times E minus delta EPR pairs with high probability.
This completes the argument for part one of the proof.
The second part of the proof requires
that we go in the opposite direction
to dilute a given number of EPR pairs--
specifically, n times E plus delta of these EPR pairs
into n copies of psi.
We want to turn ebits into arbitrary bipartite entangled
states.
A really neat way to do this uses teleportation.
One party-- say, Alice-- starts out
by generating n copies of the state psi
locally in her own apparatus.
Remember, this is a bipartite state.
She takes half of this state-- n qubits-- and compresses
it to produce using Schumacher compression,
for example, an output which has approximately nE qubits.
Here, delta is the overhead required for the compression.
Separately, Bob-- shown here in purple--
produces n perfect EPR pairs, perhaps
with a little overhead delta.
Bob, who holds this entangled state,
then sends half of those qubits--
that is, approximately nE qubits over to Alice.
Alice performs a Bell basis measurement
between the compressed state and this half of the EPR pairs,
sends the classical measurement result to Bob,
who fixes up the other half of the EPR pairs
using the normal teleportation protocol.
And remember, there are nE of these qubits.
Decompresses with the reverse of the Schumacher compression,
obtaining a state with n qubits.
All together then what Bob and Alice have
is with high probability n copies of the state psi A/B.
For this protocol, the fixed parameter is delta.
The overhead and epsilon, the error,
are determined by the Schumacher compression and decompression
steps.
This completes part two of the proof,
and so now we have the full proof-- providing a foundation
for conceptually understanding how
entangled bipartite pure states are a physical resource, which
are fungible and which may hold quantifiable intrinsic value.