I use this decomposition all the time, but I have never read a paper solely devoted to the topic. From my experience a complete characterization of the constraints on ti1,t2,..tn is tricky, and so if you want to be sure ρ is physical you should calculate the density matrix and its eigenvalues.
However, there are a lot of necessary conditions that have a useful form in this decomposition. For example, for a positive unit-trace Hermitian operator ρ is follows that
|ti1,i2,..in|≤1
tr(ρ2)=12n∑i1,i2,..int2i1,i2,..in≤1
The above condition tells us that if we think of t as a vector in a real vector space, then the physical states live within the unit sphere. This is a bit like the Bloch sphere for 1 qubit but for many qubits we have some other constraints that take the form of hyperplanes. For every |ψ⟩ expressed in the same form
|ψ⟩⟨ψ|=12n∑i1,i2,...inQi1,i2,...inσi1⊗σi2...σin we require that
⟨ψ|ρ|ψ⟩≥0 and so
∑Qi1,i2,...inti1,i2,...in≥0
which defines a hyperplane.
The problem is you have a hyperplane for every ψ so that requiring t to satisfy every inequality one of the infinite hyperplanes is impossible to check by brute force. If you want sufficient conditions for positivity of ρ I suspect you have to calculate eigenvalues.
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