Q4L14b.txt

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Let us define quantum entanglement
and how entanglement may be quantitatively measured.
For a pure state, psi AB, a composite state of two systems,
A and B, we say that psi is entangled if and only
if there does not exist a pure state phi of A and chi of B,
such that psi of A and B is the tensor product of phi A
and chi B. Entanglement is more than just a binary
state, however.
There are measures which quantify
how entangled a given state is.
The most important of these is the entropy.
Let rho sub A be the trace over B of the bipartite AB system.
The entanglement of psi AB is defined as the Von Neumann
entropy of this reduced density matrix on A
or the reduced density matrix on B.
This is called The Entanglement with a capital E
because of how important this measure is.
For example, consider the bipartite state 0 0 plus 1 1.
The density matrix of this state is a familiar quantity
which has four one half values in the corners.
The reduced density matrix is the identity matrix
divided by 2.
And thus, the entropy is one bit.
And the entanglement is, therefore, one E bit.
In fact, we will see this defines an E bit.
Another useful measure is the Schmidt number.
This is a number of non-zero coefficients
which appear in the Schmidt decomposition
of a bipartite pure state.
More explicitly, for psi AB, this
means we take the Schmidt decomposition, which
is a sum over square root lambda times K sub A K sub
B, where the square root of lambda sub K
is the Schmidt coefficient.
For example, consider two E bits of the form 0 0 plus 1 1.
Let's expand this, keeping A in blue and B in orange.
When we expand this product, we end up
with the A and B labels all mixed up.
And this is a very inconvenient representation.
So let us collect all the A terms together and all the B
terms together.
That gives us 0 0 0 0 plus 0 1 0 1 plus 1 0 1 0 plus 1 1 1 1.
And this is conveniently represented
by taking x to be an integer from 0 to 3.
And it's a sum of xx.
The Schmidt number for this state is 4.
And in fact, this representation--
a sum over x of xx-- is a general and very useful
representation for maximally entangled states.