Q4L14a.txt

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Space, time, and energy are fundamental physical resources.
Does quantum information theory introduce
additional such resources?
For example, we have seen how noisy, classical channels
may provide correlation between two signals, x and y.
Shared classical randomness is known
to be useful for cryptography.
Might noisy quantum channels also be resources?
Could entangled quantum states be resources?
Well let us see what entanglement may be useful for.
We have seen that teleportation may
be utilized to turn one entangled pair,
called an ebit, plus two classical bits
into the ability to communicate one quantum bit.
Such conversion is not possible without the entanglement.
Another well known protocol is super dense coding,
by which an ebit and one qubit are used to communicate
two classical bits.
Entangled quantum states are known
to also be useful for clock synchronization, distributed
quantum computation, and cryptography
such as quantum key distribution.
Entanglement may also sometimes replace
classical, shared randomness.
So given that entanglement is useful,
the problem is, is entanglement a resource?
And by this I mean, a mathematical, formal definition
of resource.
For example, here are two entangled states, 0 0 plus 1 1,
versus 0.9 square root 0 0 plus 0.1 square root 1 1.
How are these the same or different?
After all, one can't have different kinds of resources
for different states.
For entanglement to be a resource,
there must be a way to convert between states
of different qualities of entanglement.
One strict criterion for a definition of a resource
is to say that A and B are equivalent if there
is a mechanism to convert from A to B and from B to A.
Does this apply to entangled states?
In fact yes, to some degree.
A state psi and another state phi,
which are by bipartite pure states,
are equivalent under local operations
and classical communication, that is,
operations which do not create entanglement, if and only
if psi majorizes phi, and phi also majorizes psi.
Here, psi and phi are the Schmidt decomposition
coefficients of the pure bipartite states.
Equivalently, the eigenvalues of the reduced density matrices
of psi and phi for one side or another must be the same.
This is a fine criterion, but it is clearly too strict,
because it means that we would have many different categories
of entanglement.
A better approach utilizes a broader concept
for what a resource is.
This is asymptotic equivalence.
For example, currency, such as dollars and the English pound,
are considered equivalent because they
may be converted to each other at a constant rate with a fixed
fee.
We capture this idea mathematically
with the following definition, saying
that A and B are asymptotically equivalent if there exists
a conversion rate that is a ratio, which we will call r,
such that for some epsilon and delta greater than 0,
there exists a size n such that when the amount being converted
is larger than capital N, we have a procedure, which
takes n times r plus delta copies of a and returns n
copies of b.
And n copies of b may be converted
to n times r minus delta copies of A.
These conversions should be possible
with error less than epsilon.
Think of r as being the exchange rate
and delta as being the fee.
This model works for many resources.
Does it work for entanglement?
Do entangled states have a well-defined notion
of asymptotic equivalence?
Let us find out.