Matrix form of Density Operator

[Peter Yu]

Hi All,
I have spent hours trying to understand the matrix form of Density Operator. But, I fail. Please see page 2 of the attached file. (from the book "Quantum Mechanics - The Theoretical Minimum" page 199).
Most appreciated if someone could enlighten me this.
Many thanks in advance.
Peter Yu

 

[ShayanJ]

I find the notation in that book really awful so let me change the notation. Imagine I have a set of basis vectors $\{|\varphi_i\rangle\}$ and I prepared my system in the mixed state $\rho=\sum_i p_i |\psi_i\rangle\langle \psi_i |$ where $|\psi_i\rangle=\sum_k c_{ik}|\varphi_k\rangle$.

The matrix element of an operator L in the above basis are defined to be $L_{mn}=\langle \varphi_m|L|\varphi_n\rangle$.

About the second equation, we have $Tr(\rho L)=\sum_i \langle \varphi_i | \rho L |\varphi_i\rangle$. Now if I put the above expression of $\rho$ in the formula for trace, I'll have:

$Tr(\rho L)=\sum_i \langle \varphi_i | (\sum_i p_i |\psi_i\rangle\langle \psi_i |) L |\varphi_i\rangle=\sum_i \sum_j p_j\langle \varphi_i |\psi_j\rangle\langle \psi_j | L |\varphi_i\rangle= \\ \\ \sum_i \sum_j p_jc_{ji}(\sum_k c_{jk}^* \langle \varphi_k |) | L |\varphi_i\rangle=\sum_i \sum_j \sum_k p_jc_{ji}c_{jk}^* \langle \varphi_k | L |\varphi_i\rangle= \\ \\ \sum_i \sum_j \sum_k p_jc_{ji}c_{jk}^* L_{ki}=\sum_i\sum_k (\sum_j p_j c_{ji}c_{jk}^*) L_{ki}=\sum_i\sum_k \rho_{ik} L_{ki}$

 

[Dyatlov]

Hello!
Density matrix is a matrix that describes the expectation value(L) of a of a state vector from a quantum subsystem which is a part of composed entangled system.
If you have two systems, each with it's own space of states H_A and H_B, the density matrix simply gives you the information of one of the subsystems from it's own point of view, the second system being just a simple observer and not interfering with any measurement done on the first.

For determining <L>_A from H_A X H_B (note the product state of the entangled system): <L>_A = <ψ(x’y’)|L|ψ(xy)>.
<L>_A = Σ_xy ψ*(x’y’)ψ(xy)L_x’x. Here <x’|L|x> = L_x’x = are just the matrix elements of the observable. If we apply δy’y for <L>_A (this means we just replace y' with y), we get Σ_xy ψ*(x’y)ψ(xy)L_x’x. Now we sum over the eigenvalues of the eigenvectors of the second system y (the observer) just so the first system won't have a dependency on those values, so we can truly measure L_A. Therefore<L>_A = Σ_y ψ*(x’y)ψ(xy) L_x’x. ρ_x’x = Σ_y ψ*(x’y)ψ(xy) is the density matrix and the more non-zero eigenvectors you find inside the matrix of ρ the more entangled the system is. Maximum entanglement is when all eigenvalues are non-zero, equal and of the 1/n form (n being the dimentionality of the space).
NOTE: Σ_y = sum over y, since I don't know how to put the value under the sum.
Hope this helped!

 

[Peter Yu]

Hi Shyan and Dyatloc,
Heartfelt thank for your assistance. I need some time to absorb your explanation.
Many Thanks
Peter