Linear Algebra: Orthogonality of Hermitean Projectors

In summary, the problem is to show that P_i P_j = 0 whenever i \ne j. The attempt is to first write P_i = \textbf{1} - \sum_{i \ne m}{P_m} and P_j = \textbf{1} - \sum_{j \ne n}{P_n}. Then try to create some identities to see if that would make things more clear: P_i P_j = P_i^\dagger P_j^\dagger = (P_j P_i)^\dagger and P_i P_j = P_i^2 P_j
  • #1
Martin Muñoz
5
0
I'm studying for my Quantum Computing exam. It's at 2 PM EST today. If anyone can give me a nudge in the right direction before then that would be excellent!

Problem:

Assume the operators [tex]P_i[/tex] satisfy:
  • [tex]\textbf{1} = \sum_i{P_i}[/tex]
  • [tex]P_i^{\dagger} = P_i[/tex]
  • [tex]P_j^2 = P_j[/tex].
Show that [tex]P_i P_j = 0[/tex] whenever [tex] i \ne j[/tex].

Attempt:

This seemed really obvious to me intuitively but I've been struggling with a proof.

First I wrote [tex]P_i = \textbf{1} - \sum_{i \ne m}{P_m}[/tex] and [tex]P_j = \textbf{1} - \sum_{j \ne n}{P_n}[/tex]. I then tried to apply the first to the second, but it got messy and I couldn't get anywhere.

Then I tried to create some identities to see if that would make things more clear:

[tex]P_i P_j = P_i^\dagger P_j^\dagger = (P_j P_i)^\dagger[/tex]
and
[tex]P_i P_j = P_i^2 P_j^2 = P_i P_i P_j P_j = P_i (P_j P_i)^\dagger P_j[/tex]

but all I could think of. :(
 
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  • #2
It's easy if you think in terms of eigenvalues and eigenvectors.
 
  • #3
At this point in the book (An Introduction to Quantum Computing by P. Kaye, R. Laflamme, M. Mosca) the connection between trace and the sum of the eigenvalues has not been made.

I was hoping some could point out a more direct proof using only the 3 facts I listed.

It looks like there won't be an answer in time for my exam, but I'm still interested in a solution at any time if someone knows one !
 
  • #4
The eigenvalues of a projection map are either 1 or 0. What are the eigenvectors?
 
  • #5
The essential fact you need is that any Pi has eigenvalues 0 and 1 and you can make a basis out of those eigenvectors. Given this can you show that <v,Pi(v)> >= 0, with equality holding only if Pi(v)=0? Once you've done that, to show PjPi=0 test it on an eigenbasis of Pi. If Pi(v)=0, you're done, if Pi(v)=v then PjPi(v)=Pj(v). If that's not equal to zero then you have Pi(v)=v and Pj(v) nonzero. Now write <Iv,v>=<v,v> and substitute the sum of the P's for I.
 
  • #6
Woops, not sure why I mentioned trace. Must have been all the studying. :)

Dick, that's really helpful. I appreciate it! Is it customary/ok to post the solution? (I couldn't find any rules on the forum)
 
  • #7
Martin Muñoz said:
Woops, not sure why I mentioned trace. Must have been all the studying. :)

Dick, that's really helpful. I appreciate it! Is it customary/ok to post the solution? (I couldn't find any rules on the forum)

Posting a complete solution is a no-no. But I put more detail than usual into this one since your exam is over - and I actually found it harder to get the details right than I expected. But it's not complete. You still have to show the <Pi(v),v> >= 0 part and use the operator sum to get the final result.
 
  • #8
Dick said:
Posting a complete solution is a no-no.

Just to clarify, I meant to ask if it is ok for ME to post the full solution (assuming I solve it - I have to study for a complex analysis exam first :smile:)
 
  • #9
If you want to post a solution for discussion, sure. The rule applies to solving somebody else's problem for them. Of course you can solve your own problem. :)
 
  • #10
Martin Muñoz said:
At this point in the book (An Introduction to Quantum Computing by P. Kaye, R. Laflamme, M. Mosca) the connection between trace and the sum of the eigenvalues has not been made.

How are you liking that book so far? I'm planning on studying QC on my own in the fall. Would you recommend it for self study?
 
  • #11
Technically, this book is suitable for anyone who has taken a couple Linear Algebras. Realistically, some experience with Hilbert Spaces, group theory, dirac notation, tensor products, minimal quantum mechanics would help - the authors do spend the first two chapters bringing you up to speed though.

I find it difficult to say that a book is "bad" when I have not written a book on the subject.

With that said, objectively, few theorems in this book are book are proven and there are few exercises (and no solutions available). This can be a good or bad thing, I guess. I personally like to see more things proved, because the exercises generally require steps/logic/ideas that are seen in the proofs of theorems. I think they were trying to keep the book short and concise though.

Feel free to check out my course's assignments while they're still up:

www dot math dot mcmaster dot ca/courses/term2/math3qc3/Assignment%20Questions%20and%20Solutions.htm

(Sorry, I'm not allowed to post a URL until I have 15 posts)
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves operations on vectors and matrices, and is widely used in various fields such as physics, engineering, and computer science.

What is orthogonality in linear algebra?

In linear algebra, orthogonality refers to the concept of perpendicularity between two vectors. Two vectors are considered orthogonal if their dot product is zero, meaning they are at right angles to each other.

What are Hermitean projectors?

Hermitean projectors are matrices that are both Hermitian and idempotent. This means that they are equal to their own conjugate transpose and when multiplied by themselves, they give the same matrix. They are used in linear algebra to project a vector onto a subspace.

How is orthogonality related to Hermitean projectors?

In linear algebra, Hermitean projectors are used to represent orthogonal projections onto subspaces. This means that they can be used to decompose a vector into its component parts along orthogonal directions.

What are some applications of orthogonality and Hermitean projectors?

Orthogonality and Hermitean projectors have various applications in fields such as signal processing, data compression, and quantum mechanics. They are also used in machine learning algorithms and image processing techniques.

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