A density matrix can be expanded in a sum of Pauli strings: $$\rho = \sum c_i P_i.$$ where $P_i$ are Pauli strings, i.e. a tensor product of Pauli matrices (e.g. $X\otimes Y \otimes 1 \otimes Z$).
How do we need to choose the coefficients $c_i$ so that $\rho$ is positive semi-definite ?
For a $N$ qubit system, we have that $$c_i=\frac{1}{2^N}Tr(\rho P_i)$$ and $Tr(\rho P_i)$ is the expectation value of an operator that has eigenvalues $\{-1,1\}$ so $$-\frac{1}{2^N}\leq c_i \leq \frac{1}{2^N}.$$ Moreover, from $Tr(\rho)=1$ we get $c_1=\frac{1}{2^N}$ where $c_1$ is the coefficient of the identity string.
These conditions are not sufficient to ensure that $\rho$ is positive (I found some numerical counter examples).
What is a sufficient set of conditions on the $c_i$ so that $\rho$ is positive ?
