What is a basis (not necessarily orthogonal) of Herm(A) consisting of pure density matrices in D(A)?($A cong mathbb{C}^{n}$)

Question

Let $A cong mathbb{C}^{n}$ be a Hilbert space $A,$ and let $operatorname{Herm}(A)$ be the Hilbert space consisting of all Hermitian matrices on $A$. Give an example of a basis (not necessarily orthogonal) of Herm (A) consisting of pure density matrices in $mathfrak{D}(A)$.
$mathbf Amathbf tmathbf tmathbf emathbf mmathbf pmathbf t$:
I Started with the case $n=2$ and considered the vectors $|0rangle,|1rangle,|+rangle$ and $|+irangle$ , Because I think $left|0rightrangleleftlangle 0right|$, $left|1rightrangleleftlangle 1right|$, $left|+rightrangleleftlangle +right|$, and $left|+irightrangleleftlangle +iright|$ are a non-orthogonal basis and all of them are pure density matrices. And as you know,
$|+rangle=frac{1}{sqrt{2}}(|0rangle+|1rangle)$
$|+irangle=frac{1}{sqrt{2}}(|0rangle+i|1rangle)$.
But my question is how can I prove or show that they are form a basis for $operatorname{Herm}(A)$, in this case $A cong mathbb{C}^{2}$ and how can I expand it to $A cong mathbb{C}^{n}$? I mean how can I determine a basis for $A cong mathbb{C}^{n}$ which are pure density matrices and how can I show that it is a basis?

tsgeorgios · Accepted Answer

For $ n = 2 $, it is known that the Pauli matrices together with the identity matrix $ I $ form a basis. Now observe that we can write:

$ I = |0 rangle langle 0| + |1 rangle langle 1| $
$ sigma_z = 2 cdot |0 rangle langle 0| - I $
$ sigma_x = 2 cdot |+ rangle langle +| - I $
$ sigma_y = 2 cdot |+i rangle langle +i| - I $

This means that also the pure density matrices $|0 rangle langle 0|, hspace{0.3em} |1 rangle langle 1|, hspace{0.3em} |+ rangle langle +|, hspace{0.3em} |+i rangle langle +i| $ are a basis (not orthogonal).

For the general case, the matrices $ H_{a,b} $, with $ 1 leq a,b leq n $, form an orthogonal basis for Herm$(A)$ (see section 1.4.2) $$ H_{a,b} = begin{cases} E_{a,a} & text{if $a = b $} E_{a,b} + E_{b,a} & text{if $a < b$} i (E_{a,b} - E_{b,a}) & text{if $a > b$} end{cases} $$ where $ E_{a,b} = |e_a rangle langle e_b| $ and $ |e_a rangle $ a state with 1 in the $a$-th entry and all other entries zeros.

Now define the states: $$ |psi_{a,b} rangle = begin{cases} |e_a rangle & text{if $a = b $} frac{1}{sqrt{2}} (|e_a rangle + |e_b rangle) & text{if $a < b$} frac{1}{sqrt{2}} (i|e_a rangle + |e_b rangle) & text{if $a > b$} end{cases} $$ and the pure density matrices $ rho_{a,b} = |psi_{a,b} rangle langle psi_{a,b}| $. After some calculations we get

$ H_{a, a} = rho_{a,a} $
$ H_{a, b} = 2 rho_{a,b} - rho_{a,a} - rho_{b,b} $

so $ rho_{a,b} $ form a basis.

KAJ226 · Answer

I will try to take a stab at it from my understanding of your question:
The basis for the space of $2 times 2$ Hermitian matrices over $mathbb{R}$ is:
begin{equation}
begin{pmatrix} 1 & 0 0 & 0 end{pmatrix}   begin{pmatrix} 0 & 0 0 & 1 end{pmatrix}   begin{pmatrix} 0 & 1 1 & 0 end{pmatrix}   begin{pmatrix} 0 & i -i & 0 end{pmatrix}
end{equation}
But from my understanding, you want to restrict the basis set  to consist of only rank 1 matrices. Is that right? You are considering the basis set
begin{equation}
|0ranglelangle 0| = begin{pmatrix} 1 & 0 0 & 0 end{pmatrix}     |1ranglelangle 1| = begin{pmatrix} 0 & 0 0 & 1 end{pmatrix}     |+ranglelangle +| =dfrac{1}{2}begin{pmatrix} 1 & 1 1 & 1 end{pmatrix}     |iranglelangle i| =dfrac{1}{2}begin{pmatrix} 1 & i i & 1 end{pmatrix}
end{equation}

Well, if we take $H$ to be the Hermitian matrix
$$ H = dfrac{1}{2}begin{pmatrix} 1 & i -i & 1 end{pmatrix}$$
Can you form this Hermitian matrix $H$ from your supposedly basis set?

update: As commented, I made a wrong calculation, as $|irangle langle i|$ should be
$$
|iranglelangle i| =dfrac{1}{2}begin{pmatrix} 1 & -i i & 1 end{pmatrix}
$$
and therefore it can be written as $H = dfrac{1}{2}|0ranglelangle 0| + dfrac{1}{2}|1ranglelangle 1| - |iranglelangle i | $
And it turns out that the basis set in consideration is actually correct as now pointed out by the other answer! Thanks for bringing up this problem though.

What is a basis (not necessarily orthogonal) of Herm(A) consisting of pure density matrices in D(A)?($A cong mathbb{C}^{n}$)

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