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Method 1:

(i) Expand the density operator in the Fock basis as $\rho=\sum_{m,n}{\rho_{mn}|m\rangle\langle n|}$.

(ii) Purity = $tr{\rho^2}=\sum_{m,n,f,g, \lambda}{\rho_{mn}\rho_{fg}\langle \lambda|m\rangle\langle n|f\rangle\langle g|\lambda\rangle}$

(iii) Three out of five sums vanish due to the inner products

(iv) Therefore, Purity = $\sum_{m,n}{\rho_{mn}\rho_{nm}}$

Method 2: This method uses the Wigner distributions of the Fock states. (Ref: What is the Wigner function of $|n\rangle\langle m|$?)

(i) Wigner distribution of the state can be written as $W = \sum_{m,n}{\rho_{mn} W_{nm}} $

(ii) Purity = $2\pi \int{W^2dq dp}= \sum_{m,n,f,g}\rho_{mn}\rho_{fg}2\pi\int{ W_{mn}W_{fg} dq dp}$

(iii) $2\pi\int{ W_{mn}W_{fg} dq dp} = \delta_{m,n,f,g} $ which collapses three out of four sums and causes a mismatch with respect to the result from method 1. This step must be incorrect, but I cannot determine the problem using the Wigner distributions.

Saurabh Shringarpure
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  • $2\pi \int{W_{mn}W_{fg} dqdp = \delta_{m,g}\delta_ {n,f}}$ which solves the issue. This was using $W_{mn}(q,p)=\dfrac{(-1)^{\min(m,n)}}{\pi}\sqrt{\dfrac{\min(n,m)!}{\max(n,m)!}}e^{-(q^2+p^2)}(\sqrt{2}(q+(-1)^{n-m}ip))^{|m-n|}L^{|m-n|}_{\min(n,m)}(2(q^2+p^2))$ – Saurabh Shringarpure May 03 '23 at 10:11
  • Sorry, the $q+(-1)^{n-m}ip$ term (in the brackets in the middle) should be replaced with $q+sgn(n-m) ip$ – Saurabh Shringarpure May 03 '23 at 23:12

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