Density Matrix
Wiki: Density matrix is used to describe the quantum state (pure or miexed state) of a physical system. It allows for calculation of the probabilities of the outcomes of any measurement performed upon a system.
The density matrix is a representation of the linear density operator.
For details of the following contents, please refer to the Reference.
A mixed state can be viewed as a mixture of pure collections which is characterized by one normalized state \(|\psi_i\rangle\)，and the fractional populations of mixed system satisfy the condition:
\[\sum_ip_i=1
\]
And therefore measurement on the observable quantity \(\Omega\) of a mixed state yeilds the average value: (insert the completeness relation)
\[|\Omega|=\sum_ip_i\langle\psi_i|\Omega|\psi_i\rangle\\
=\sum_{i,n,m}p_i\langle\psi_i|\phi_n\rangle\langle\phi_n|\Omega|\phi_m\rangle\langle\phi_m|\psi_i\rangle\\
=\sum_{i,n,m}p_i[\langle\phi_m|\psi_i\rangle\langle\psi_i|\phi_n\rangle]\langle\phi_n|\Omega|\phi_m\rangle
\]
We consider the following quantity:
\[\rho\equiv\sum_ip_i|\psi_i\rangle\langle\psi_i|
\]
to be the density operator for the mixed system.
The corresponding \((m,n)\)th element of matrix representation in orthonormal basis \(\{|\phi_n\rangle\}\) is:
\[\langle\phi_m|\rho|\phi_n\rangle=\sum_ip_i\langle\phi_m|\psi_i\rangle\langle\psi_i|\phi_n\rangle
\]
Therefore the average of physical quantity \(\Omega\) can be rewrited as:
\[|\Omega|=\sum_{m,n}\langle\phi_m|\rho|\phi_n\rangle\langle\phi_n|\Omega|\phi_m\rangle\\
=\sum_m\langle\phi_m|\rho\Omega|\phi_m\rangle\\
=\text{Tr}(\rho\Omega)
\]
The Hermitian conjugate of density operator is:
\[\rho^{\dagger}=(\sum_ip_i|\psi_i\rangle\langle\psi_i|)^{\dagger}\\
= \sum_i (p_i)^{\dagger}(|\psi_i\rangle)^{\dagger}(\langle\psi_i|)^{\dagger}\\
=\sum_ip_i|\psi_i\rangle\langle\psi_i|=\rho\\
\Rightarrow \rho=\rho^{\dagger}
\]
Therefore the density operator/matrix is Hermitian.
Consider the trace of the density operator in orthonormal basis \(\{|\phi_n\rangle\}\) : ( remember that the pure state \(|\psi_i\rangle\) is normalized and fractional populations of mix state equal to 1.)
\[\text{Tr}(\rho)=\sum_n\langle\phi_n|\rho|\phi_n\rangle\\
=\sum_n\langle\phi_n|\bigg(\sum_ip_i|\psi_i\rangle\langle\psi_i|\bigg)|\phi_n\rangle\\
=\sum_{i,n}p_i\langle\phi_n|\psi_i\rangle\langle\psi_i|\phi_n\rangle\\
=\sum_i p_i\bigg(\sum_n\langle\psi_i|\phi_n\rangle\langle\phi_n|\psi_i\rangle\bigg)\\
=\sum_ip_i\langle\psi_i|\psi_i\rangle\\
=\sum_ip_i=1\\
\Rightarrow \text{Tr}(\rho)=1
\]
In situations wherein the normalization does not hold, the average of phyical quantity is given as:
\[|\Omega|=\frac{\sum_ip_i\langle\psi_i|\Omega|\psi_i\rangle}{\sum_ip_i}=\frac{\text{Tr}(\rho\Omega)}{\text{Tr}(\rho)}
\]
Then we calculate the trace of the square of a density operator:
\[\text{Tr}(\rho^2)=\sum_m\langle\phi_m|\rho^2|\phi_m\rangle\\
=\sum_{m,n}\langle\phi_m|\rho|\phi_n\rangle\langle\phi_n|\rho|\phi_m\rangle\\
=\sum_{m,n}\bigg[\langle\phi_m|\bigg(\sum_ip_i|\psi_i\rangle\langle\psi_i|\bigg)|\phi_n\rangle\bigg]\bigg[\langle\phi_n|\bigg(\sum_jp_j|\psi_j\rangle\langle\psi_j|\bigg)|\phi_m\rangle\bigg]\\
=\sum_{i,j,m,n}p_ip_j\langle\phi_m|\psi_i\rangle\langle\psi_i|\phi_n\rangle\langle\phi_n|\psi_j\rangle\langle\psi_j|\phi_m\rangle\\
=\sum_{i,j,m}p_ip_j\langle\phi_m|\psi_i\rangle\bigg(\sum_n\langle\psi_i|\phi_n\rangle\langle\phi_n|\psi_j\rangle\bigg)\langle\psi_j|\phi_m\rangle\\
\text{suppose that pure states }|\psi_i\rangle \text{ are orthonormal,then:}\\
=\sum_{i,j,m}p_ip_j\langle\phi_m|\psi_i\rangle\delta_{ij}\langle\psi_j|\phi_m\rangle\\
=\sum_ip_i^2\bigg(\sum_m\langle\psi_i|\phi_m\rangle\langle\phi_m|\psi_i\rangle\bigg)\\
=\sum_ip_i^2\langle\psi_i|\psi_i\rangle\\
=\sum_ip_i^2\leq\bigg(\sum_ip_i\bigg)^2\\
\Rightarrow \text{Tr}(\rho^2)\leq[\text{Tr}(\rho)]^2
\]
For a pure state, the density operator is: （\(p_i=1\))
\[\rho=|\psi_{i0}\rangle\langle\psi_{i0}|
\]
Therefore:
\[\rho^2=|\psi_{i0}\rangle\langle\psi_{i0}|\psi_{i0}\rangle\langle\psi_{i0}|=|\psi_{i0}\rangle\langle\psi_{i0}|=\rho\\
\text{Tr}(\rho^2)=\text{Tr}(\rho)=1
\]
The above suggests that \(\text{Tr}(\rho^2)\) has its maximum of 1 when the ensemble is pure and normalized. For a mixed ensemble, \(\text{Tr}(\rho^2)\) is a positive number whose value is less than 1.
