[5.5 The Density Matrix and Mixed States]: Some thoughts

[dhruvbhq]

Describe the problem I think some things can be added in this section for better clarity for first time readers.

Describe the expected solution In 5.5.2 Mixed States, The content states: "In general, a mixed state consisting of an ensemble of n pure states can be expressed in the form of a list of outcome elements:" ... "where each item has a corresponding probability of occurrence given by".

Here, I think it would be good to emphasize that these probabilities of occurrence are classical probabilities involved in a statistical ensemble, and do not refer to the post-measurement probabilities for quantum states. (i.e. that these probabilities are related to classical uncertainty in the state of the quantum system).

Further, I think we can mention that $n$ need not necessarily be the same as the dimension of the underlying Hilbert state.

Maybe @diemilio you would like to share your thoughts on this? :)

[diemilio]

Hi @dhruvbhq. I actually think this is a better place to keep track of issues in PR #514 because that has been closed and it will be difficult for others to find.

So, here is the list of issues/recommendations so far:

1. Tensor product notation in the chapter does not follow qiskit's qubit ordering.

2. The Reduced Density Matrix section may be rewritten to have a better flow and be less overwhelming. It seems that the section starts too "technical", which can be hard to follow for some, but then the example makes things clear. So the idea is to maybe try to start with the example, and then proceed with the formal definitions.

~~3. In 5.5.1 Pure States, the content states: "We understand that if we were to perform a measurement of this state, the outcome will be probabilisitc". It should be: "We understand that if we were to perform a measurement of this state, the outcome will be probabilistic", i.e. there's a typo in "probabilistic".~~

~~4. In 5.5.2 Mixed States, The content states: "In general, a mixed state consisting of an ensemble of n pure states can be expressed in the form of a list of outcome elements:" ... "where each item has a corresponding probability of occurrence given by". Emphasize that these probabilities of occurrence are classical probabilities and that n need not necessarily be the same as the dimension of the underlying Hilbert space.~~

~~5. There's a type right above section 4.1: "In this specific example, ρA and ρb are equal, but this is not always the case". The subscript b should be capitalized --> ρB.~~

[diemilio]

one more for the list:

~~6. Under Trace and Positivity Conditions:~~ ~~2. The matrix must be positive-definite:~~

~~should instead be:~~ ~~2. The matrix must be positive-semidefinite:~~

[shil-m]

Hi @diemilio @dhruvbhq , I think I can help resolve some of the issues pointed out. Can I work on this?

[diemilio]

Hi @shil-m!

Yeah, I will be OK if you take a shot at it. I was planning to do most myself, but I am having a hard time finding time to do it.

Just let me know once you complete them so I can start crossing them out on this list.

Thanks.

[shil-m]

@diemilio In point 3, I am not sure what change are you expecting. I can't see a typo in "probabilistic".

[dhruvbhq]

I can see it is incorrectly written as "probabilisitc" (notice the last 4 characters 🙂 ).

[shil-m]

Thanks @dhruvbhq . It tricked my eyes somehow :)

[shil-m]

3, 4, 5, and 6 resolved in #1166

[diemilio]

Thanks @shil-m. For number 4, you have:

Here, $p_j$ represents classical probability and n can be anything not necessarily equal to the dimension of the underlying Hilbert Space.

How about:

Here, $p_j$ corresponds to the classical probability of the system being in state $|\psi_j \rangle$, and the total number of possible states $n$ does not need to be equal to the dimension of the underlying Hilbert space.

(make sure to include the link to the section where Hilbert spaces are explained: https://qiskit.org/textbook/ch-appendix/linear_algebra.html#Hilbert-Spaces,-Orthonormality,-and-the-Inner-Product)

[shil-m]

@diemilio This looks better indeed. I will change this.

[shil-m]

I forgot to mention, pls take a look #1166