# 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 $$x_{1,n},x_{2,n-1},...,x_{n,1}$$.)

There is nothing special about the anti-diagonal you've specified. As far as quantum mechanics goes, all off-diagonal elements are equally important.

(For an easy way to see this, consider a basis transformation which simply permutes all the elements of your basis in some arbitrary way. The result will be that the main diagonal will be mapped to itself, with the permutation acting within it, while the anti-diagonal you've specified will be scattered into a random selection of off-diagonal matrix elements.)

Generally speaking:

• 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.