
8.1 Introduction
In the last chapter, we encountered the effects of entanglement for the first time. It explained our “third mystery” of the double slit experiment, by eliminating the famous interference fringes.
This chapter will be devoted to building up some of the mathematical machinery needed to quantify and study entanglement. The primary tool is the “Density Matrix”. My goal is to convey an intuitive and visual introduction to them, rather than an exhaustive survey. There are many good resources online that provide a more detailed account. Here are a couple.
http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_9corr.pdf
http://www.cs.sandia.gov/nonconventionalcomputing/docs/Gamble,%20John%20King/1.3043847.pdf
The second one is particularly clear and would be a good supplement to this and the next 2 chapters. My hope is that this page will help a newcomer quickly gain a basic understanding of the topic.

8.2 Quantifying Entanglement
We will focus on systems with 2 particles in 1 dimension because of how well this arrangement lends itself to visualization. The first particle will be called the “system”; the second is the “environment”.
One question we could ask is “how entangled is the system particle with the environment particle?” Let’s compare what entangled vs unentangled states look like to get an idea about how we should proceed. Figure 1 shows an unentangled state.
The “system” particle’s wavefunction has 2 peaks (at 3 and 7), and the environment’s wavefunction has 1 peak (at 2). This state has no entanglement because the total function can be written as a simple product of a system function and an environment function.
(1)
Figure 2 shows the opposite situation. The particles are entangled and the total wavefunction cannot be written as a simple product.
It should be pretty apparent that this wavefunction can be written as a sum of 2 products, like this:
(2)
and that each “lump” takes up about half of the probability. Our “quantity of entanglement”, however we define it, should reflect both situations and generalize to more complicated scenarios.
One avenue that springs to mind is to compare various crossections of the wavefunction. Let’s create 2 planes called and to cut the wavefunction as in Figure 3.
The functions created by cutting with these planes are important enough to deserve their own name and notation. Let’s use the “braket” style for these functions and name them after the plane that defines them like this:
(3)
You can think of these as representing “what the environment particle is doing if the system particle is measured at (or ). Also note that we can use them to calculate the probability density of finding the system particle at
(4)
And that
(5)
because our state is assumed to be normalized.
Now, it should be quite clear that there is less “overlappedness” in the entangled case. For the 2 planes I’ve picked out, there is essentially zero overlap. In other words, the inner product vanishes:
(6)
But there’s no reason to focus on just 2 planes. We need information about the entire state. Why don’t we “tabulate” inner products of functions cut by all pairs of planes and plot the result?
We’ll make a new function like this:
(7)
Here’s what it looks like for both cases:
These “inner product tables” are known as “Density Matrices”. More specifically, they are “Reduced Density Matrices” because they only provide information about particle 1. Particle 2 has been “integrated out”. This might seem to be a disadvantage compared to working directly with the total wavefunction, but in some instances, we can solve for the dynamics of the Reduced Density Matrices directly, without worrying about the entire state.
Going back to our last plots, we can see a clear difference between the 2 Density Matrices. In the entangled case, the “off diagonal” elements are suppressed, but the “diagonal” elements are not. This suggests that the integral over and should differ! It turns out that the integral over the square of the magnitude of is going to be what we’re looking for. This quantity is called the “Purity” of the Density Matrix. Let’s write it down.
(8)
Now what we’ll do is break  into components parallel to and perpendicular to 
(9)
where and are the magnitudes of those components.
Insert this into equation (8) and simplify:
(10)
If the state is unentangled, vanishes, which makes = . This, together with equation (5), implies that the inner integral evaluates to 1, which is going to make the entire expression equal to 1. In other words
(11)
In the entangled case, does not vanish, which means that is generally going to be less than . This will cause the Purity to decrease. Therefore,
(12)
It turns out that the Purity is exactly 1/2 for our entangled state, and would evaluate to 1/N if we had N equal disjoint “lumps” of probability. So, the Purity turns out to be a type of “average overlappedness” of our cross sectional functions, and hence gives a reasonable way to quantify entanglement.
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