Talk:Schmidt decomposition

From Wikipedia, the free encyclopedia

I haven't changed it because I'm not sure, but I guess that in the following paragraph ej should be fj. So:

The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases \{ e_1, \cdots, e_n \} \subset H_1 and \{ f_1, \cdots, f_m \} \subset H_2. We can identify an elementary tensor e_i \otimes e_j with the matrix e_i e_j ^T, where e_j ^T is the transpose of ej. A general element of the tensor product

v = \sum _{1 \leq i \leq n, 1 \leq j \leq m} \beta _{ij} e_i \otimes e_j


Should be:


The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases \{ e_1, \cdots, e_n \} \subset H_1 and \{ f_1, \cdots, f_m \} \subset H_2. We can identify an elementary tensor e_i \otimes f_j with the matrix e_i f_j ^T, where f_j ^T is the transpose of fj. A general element of the tensor product

v = \sum _{1 \leq i \leq n, 1 \leq j \leq m} \beta _{ij} e_i \otimes f_j

Please have a look and change it if I'm rigth or correct me here if I'm wrong. 130.102.2.60 03:34, 12 April 2007 (UTC)

yes, you're right. thanks for pointing it out. Mct mht 07:14, 12 April 2007 (UTC)

[edit] quibbles

Why call it a statement and not a theorem, lemma, or proposition?

In the statement you use αi, while at the end of the proof you use σi. It might not be clear to all readers that the αi in the statement are equal to the σi in the proof. It may be worthwhile to keep the separate variables names to maintain the distinction between singular values and Schmidt coefficients, however, at some point it should be made explicit that αi = σi.

It seems to me that the Schmidt rank should be defined as the number of nonzero Schmidt coefficients, otherwise the Schmidt rank is always m, in which case the definition is unnecessary. Furthermore, it would imply that every v is an entangled state, which is clearly not true.

In the last paragraph in the section on Schmidt rank and entanglement, the symbol v is used in two different ways, once as an element of H_1\otimes H_2 and once in an expression involving the tensor product of two vectors.

Mathmoose 14:23, 21 July 2007 (UTC)

well, all valid points (modulo "come on, man..." :-) ). wanna go ahead and make the changes you propose? Mct mht 03:59, 25 July 2007 (UTC)

Sure. How do I do that, or where do I found out how to do that? (It's my first contribution to Wikipedia.)

Mathmoose 23:11, 31 July 2007 (UTC)

on the article, do you see the "edit this page" button on top? double click that and edit away. :-) Mct mht 03:07, 1 August 2007 (UTC)

I can see the button, but when I click it, it takes me to the discussion window, that very same window than I am entering this message in.

Mathmoose 22:36, 5 August 2007 (UTC)

hm, that can't be right. you sure you're on this page when you click it? Mct mht 22:39, 5 August 2007 (UTC)

That does it! I should have read your instructions as carefully as I read the article.

Mathmoose 19:03, 6 August 2007 (UTC)