# Does the anti-diagonal of a density matrix have any special interpretation?

Suppose a density matrix $$\rho= \begin{bmatrix} x_{11} & x_{12} & x_{13} & \cdots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \cdots & x_{2n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_{n1} & x_{n2} & x_{n3} & \cdots & x_{nn} \end{bmatrix}$$

It's known that $$\mathrm{Tr}(\rho)=\sum_ix_{i,i}$$ carries important meaning in probability distribution.

However, does the other diagonal, $$\sum_{i}x_{i,n+1-i}$$, carry any meaning as well? Do they represent asymmetry or the maximum interference?

(The other diagonal means $$x_{1,n},x_{2,n-1},...,x_{n,1}$$.)

• the diagonal elements of the density matrix $$\rho_{ii}$$ are known as the populations, as they specify the probability that the matrix will be found in state $$|i⟩$$ if we measure the system in the basis $$\{|i⟩\}$$
• the off-diagonal elements $$\rho_{ij}$$ (with $$i\neq j$$) are known as the coherences, and they specify how much interference you'll be able to observe in a measurement that mixes the states $$|i⟩$$ and $$|j⟩$$.
Generally speaking, the requirement that the density matrix be positive-semidefinite implies that each coherence is bounded above by some function of the relevant populations. (For a two-level system, $$\rho_{12}\leq \sqrt{\rho_{11}\rho_{22}}$$, but it gets more complicated for larger dimensions.) The off-diagonal elements saturate those bounds at pure superposition states, at which the interference is maximal. If the off-diagonal elements are zero, you won't be able to observe any interference at all.