suppose the density matrix $\rho_{ABC}$ with the subsystems {A,B,C}

can we write $\rho_{ABC}$ as below?

$\rho_{ABC}=\rho_A \otimes \rho_B \otimes \rho_C $

if the answer is yes please share a reference.


1 Answer 1


No. The case you describe does not even include all possible separable states, and certainly cannot describe an entangled state. For example, $$ |\psi\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle) $$ cannot be written in this way.

A simple proof is using the partial transpose criterion. If you transpose the density matrix of just one system, then if it can be written as $\rho_A\otimes\rho_B\otimes\rho_C$, then you'll get $\rho_A\otimes\rho_B\otimes\rho_C^T$, and this is still a valid quantum state. Whereas, if I take $\rho=|\psi\rangle\langle\psi|$ and take the partial transpose, I'll find it has a negative eigenvalue, so it certainly isn't a valid density matrix.


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