Partial Trace in Quantum Mechanics?

Assumptions:

This article assumes the reader has basic knowledge of the following concepts:

Joint probability distribution
Density matrix definition
Tensor product and tensor product identities
Hilbert spaces

Introduction and Motivation to Partial Trace:

Often in information theory, we have a composite system. That is, we have a system of the form:

\begin{equation}\rho_a \otimes \rho_b\end{equation}

and we want to know about system A only. This goal is the motivation for a partial trace in quantum mechanics. It is the idea of abstracting away the density matrix of system B such that we have statistical information on all of system A regardless of B. The above can represent anything from two qubits to a system, its surroundings, and much more. Before we get into the math and definition behind a partial trace, I want to draw connections between our system and a joint probability distribution.

Imagine the joint probability distribution seen below:

A joint distribution is a probability distribution over the values X and Y.

The density matrix has an abundance of statistical information woven into it. Furthermore, we can think of the tensor product shown above as a joint distribution of the composite system we are representing. In our distribution, if we want just the probability of getting a specific x value, how will we do so? The way we get the value is through the following:

$$ P_{x_1}= \sum_i P(y_i, x_1) $$

The reason is that we can have our x with any y value. This is also what a partial trace is trying to accomplish. We are only interested in one of the density matrices, so we can do something similar to what we did to the Y distribution in our above example.

Trace and Partial Trace Definition:

Now, when we calculate the trace of a tensor, we add up all of the diagonal values. When figuring out the partial trace, we do the same thing, but only for one density matrix. This means we need something along the following:

Of course, our orthonormal basis is whatever basis it needs to be(The definition of the partial trace has independence of basis). So, mathematically, how do we represent this operation? We can do so through the following:

A and B are both density matrices

This representation is the same as the first representation through the following identity:

\begin{equation} (A \otimes B)(C \otimes D)=(AC \otimes BD) \end{equation}

The trick is putting the identity matrices in the right place to keep the state of the rest of the composite system. This definition can be generalized to any number of systems it wants to, in the sense that we can take the partial trace of a system comprised of N individual systems.

Examples of Partial Trace on Composite Systems:

For example, we want to take the partial trace of system A and system B. This can be done via the following:

Example of Partial Trace on the composite system

Another example is a composite system:

Example of 3 particle system partial trace

Here we applied the definition to each element in the sum.

This will end up equaling:

In conclusion, partial trace in quantum mechanics is a rich but tricky topic. Leave a comment with any questions, and I will get back to you with a detailed answer or references elsewhere. Stay curious!