Writing a random 2N by 2N matrix in terms of Pauli Matrices

[sokrates]

Hi,

Wasn't sure if I should post this to Linear Algebra or here.

My question is really simple:

Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as:

$H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z$

where A,B,C,D are all N by N matrices, while the sigma's are the Pauli spin matrices.

My question is, as long as A,B,C,D are random and complex Hermitian matrices of size N by N, do I cover the
whole 2N by 2N complex Hermitian space with this representation?

If yes, do you know a reference, a theorem, or a simple proof of this?A very simple case is when N = 1 , and I know that any 2 x 2 complex , Hermitian matrix can be written as a linear combination of Pauli Matrices.

Many thanks,
sokrates.

 

[dauto]

Yes, it is. It's just a matter of counting degrees of freedom

 

[wle]

My question is really simple:

Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as:

$H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z$

where A,B,C,D are all N by N matrices, while the sigma's are the Pauli spin matrices.

Yes, because the Pauli operators form a basis of the set of operators acting on qubit (two-dimensional Hilbert) spaces. If you're not convinced then note that you can always write your operator in the form

$$H = A_{00} \otimes \lvert 0 \rangle \langle 0 \rvert + A_{01} \otimes \lvert 0 \rangle \langle 1 \rvert + A_{10} \otimes \lvert 1 \rangle \langle 0 \rvert + A_{11} \otimes \lvert 1 \rangle \langle 1 \rvert$$​

just by writing it out explicitly in some basis and collecting the terms in $\lvert 0 \rangle \langle 0 \rvert$, $\lvert 0 \rangle \langle 1 \rvert$, etc., and then substituting

$$\begin{eqnarray} \lvert 0 \rangle \langle 0 \rvert &=& \tfrac{1}{2} ( \mathbb{I} + \sigma_{z} ) \,, \\ \lvert 0 \rangle \langle 1 \rvert &=& \tfrac{1}{2} ( \sigma_{x} + i \sigma_{y} ) \,, \\ \lvert 1 \rangle \langle 0 \rvert &=& \tfrac{1}{2} ( \sigma_{x} - i \sigma_{y} ) \,, \\ \lvert 1 \rangle \langle 1 \rvert &=& \tfrac{1}{2} ( \mathbb{I} - \sigma_{z} ) \,. \end{eqnarray}$$​

This works for any operator. If $H$ happens to be Hermitian then this imposes additional constraints. For instance, as you pointed out, the $A$, $B$, $C$, and $D$ from your post must also be Hermitian in that case.

 

[sokrates]

Hi , Thank you for the responses ... However, I still don't understand it from a matrix point of view.

Let's take N = 2 , and have a 4x4 H matrix ... can one prove that my representation will always cover the full space ?

I didn't follow it from the Dirac notation,

Many thanks for responses.