Understanding Interference of Qubits with Varying Density Matrices

[naima]

Hi PF

I wonder how qubits interfere in interferometers when they are not in pure states. Let us take qubits with density matrix = Id/2. There lay at the center of the Bloch sphere.
Half of them can travel unchanged through the left arm. In the other arm they become $UP_Z$. Then there is interference.
What is the density matrix of the result?
I can write $\rho = Id/2 = (Id + a.\sigma)/4 + (Id - a.\sigma)/4$
which is a mixture of equiprobable pure states whith different "a" vectors. We know that we can add the amplitudes of pure states in interferometers. But is the result independent of the choice of the vector in the decomposition of $\rho$?

 

[Strilanc]

The density matrix of the result is always going to be $\frac{I}{2}$ again. Any single-qubit operation $U$ you apply will have the effect of changing $\rho_1 = \frac{I}{2}$ to

$\rho_2 = U^\dagger \frac{I}{2} U$
$= \frac{I}{2} U^\dagger U$
$= \frac{I}{2} I$ (because $U$ is unitary)
$= \frac{I}{2}$
$= \rho_1$

It's even easier to see this in the Bloch sphere representation: $\rho_1$ corresponds to the point at the center of the sphere, and all single-qubit operations are rotations about the center, and all such rotations leave the center point unaffected.

 

[naima]

Yes but i am talking here about interference between two things.

 

[Strilanc]

You'll have to be clearer about the situation you're describing then. If we're talking about a spin being run through an interferometer then we have a two qubit system, one qubit for the spin and one qubit for the path. When are you talking about operating on the path, and when are you talking about operating on the spin? What operation are you performing on the top arm to "make them become UPz"? What specific beam splitter are you using?

 

[naima]

I think that we can easily avoid all these details of spectrometry:
There is a principle in QM which states that, if a particle may be in two different states it can also be in their superposition.
In my case these two states are represented by the center of the Bloch sphere and the other is the "north pole".
The question is to know what is the density matrix associated to the superposition. I do not think that it is here a two-by-two matrix.
My aim is to understand the rules of superposition with density matrices.

 

[Strilanc]

If your knowledge of a qubit is that there's a 50% chance it's in one mixed state and a 50% chance that it's in another mixed state, you scale each of the corresponding density matrices by half and add them together. Density matrices are like probability distributions in that way.

(Note that this is different from the superposition principle for pure states represented as kets, where you'd instead have to scale by the square root of a half when combining them. Also superposition of pure states doesn't correspond well to a lack of knowledge.)

 

[naima]

I found this https://www.researchgate.net/publication/253981635_Inner_composition_law_of_pure-spin_states by Man'ko.
But i do not understand very well the role of the fiducial projector Po in eq 17.
How does $\rho$ vary when Po varies?