# Physical interpretation of the density matrix in specific representation

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Any textbook gives the interpretation of the density matrix in a SINGLE continuous basis $|\alpha\rangle$:

• The diagonal elements $\rho(\alpha,\alpha) = \langle\alpha| \hat{\rho} |\alpha\rangle$ give the populations.
• The off-diagonal elements $\rho(\alpha,\alpha') = \langle\alpha| \hat{\rho} |\alpha'\rangle$ give the coherences.

But what is the physical interpretation (if any) of the density matrix $\rho(\alpha,\beta) = \langle\alpha| \hat{\rho} |\beta\rangle$ for a DOUBLE continuous basis $|\alpha\rangle$, $|\beta\rangle$?

I already know that when the double basis are momentum $|p\rangle$ and position $|x\rangle$, then $\rho(p,x)$ (the well-known Wigner function) is interpreted as a pseudo-probability. I may confess that I have never completely understood the concept of pseudo-probability [*], but I would like to know if this physical interpretation as pseudo-probability can be extended to arbitrary continuous basis $|\alpha\rangle$, $|\beta\rangle$ for non-commuting operators $\hat{\alpha}$, $\hat{\beta}$ and if a probability interpretation holds for commuting operators.

I.e. can $\rho(\alpha,\beta)$ be interpreted as a pseudo-probability for arbitrary non-commuting operators beyond x and p? Can $\rho(\alpha,\beta)$ be interpreted as a probability for arbitrary commuting operators?

[*] Specially because $\rho(p,x)$ is bounded and cannot be 'spike'.

Edited by juanrga

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