# The mechanics of calculating Reduced Density Matrices

1. May 27, 2012

### QITStudent

Hi all. I'm having a little trouble in understanding precisely how to calculate reduced density matrices. No literature I've been able to get my hands on has made it clear how precisely to work out partial matrices.

For example, if we have a bi-partite state for Alice's and Bob's particles:

Phi(ab) = 3-1/2(|bo> + |a1> + |c2>)

Where |b>, |a> and |c> are superpositions of basis states |0>, |1> and |2>. Then we must have density matrix

p(ab) = 1/3 (|bo><bo| + |bo><a1| + |bo><c2| + |a1><bo| + |a1><a1| + |a1><c2| + |c2><bo| + |c2><a1| + |c2><c2|)

So then working out the reduced density matrix means that we must take the partial Trace over Bob's. However, I'm not completely clear about this part.

Does this mean that we take the trace over Bob's matrix and multiply this scalar value by all of Alice's matrix. i.e, the trace here is 3 so Alice's state will be:

p(a) = |b><b| + |b><a| + |b><c| + |a><b| + |a><a| + |a><c| + |c><b| + |c><a| + |c><c|

Or, when taking the partial trace do we eliminate any of Alice's terms that do not have a diagonal term in Bob's matrix. I.e. getting the reduced matrix as

p(a) = |b><b| + |a><a| + |c><c|

The latter seems like it should not be the case, because that would mean every reduced density matrix is diagonal. But I don't seem to be getting the required answers whenever I perform the former method.

Any help would be greatly appreciated,

2. May 27, 2012

### kith

Your useage of the term Bob's matrix is wrong. There is no Bob's matrix before you trace, there is only the whole density matrix. In your case, this is a 9x9 matrix.

Bob's and Alice's density matrices are the reduced density matrices, which you get by taking the partial trace of the whole density matrix with respect to Alice's resp. Bob's state space. These matrices are 3x3.

As far as the result is concerned, the second one is correct. I'm not sure where your misunderstanding comes from, so let's start with the basics. What is the trace of an arbitrary operator ρ using the basis states |ai>?