# How to tell if a density matrix is separable?

Consider the following 4-qubit entangled state $$\left|\psi\right>=\left|0000\right>+\left|1110\right>+\left|1101\right>+\left|1011\right>$$

By tracing out qubits A and B (where I am using $\left|ABCD\right>$), we are left with the reduced density matrix $$\rho^{CD} = \left|00\right>\left<00\right|+\left|11\right>\left<11\right|+\left|01\right>\left<01\right|+\left|01\right>\left<10\right|+\left|10\right>\left<01\right|+\left|10\right>\left<10\right|,$$ or, in a later more useful way, $$\rho^{CD} =\big(\left|0\right>\left<0\right|\otimes\left|0\right>\left<0\right|\big)+\big(\left|1\right>\left<1\right|\otimes\left|1\right>\left<1\right|\big)+\big(\left|0\right>\left<0\right|\otimes\left|1\right>\left<1\right|\big)+\big(\left|0\right>\left<1\right|\otimes\left|1\right>\left<0\right|\big)+\big(\left|1\right>\left<0\right|\otimes\left|0\right>\left<1\right|\big)+\big(\left|1\right>\left<1\right|\otimes\left|0\right>\left<0\right|\big)$$

I was hoping that this would lead to an evidently separable solution, but that does not seem to be the case.

What I would like to know is if there is any generic way of checking if such a density matrix is separable or not, rather than simply looking at the expression. This because the process becomes near impossible for a density matrix of more than two qubits.

• But is this not a sum of 2-qubit density matrices? If yes (as I think it is) then it's a simple application of the Peres-Horodecki criterion en.wikipedia.org/wiki/Peres%E2%80%93Horodecki_criterion Commented Mar 17, 2017 at 19:58
• For two qubits (or one qubit and one qutrit), you can use the PPT criterion. But generally, this problem is computationally hard. There should be an answer somewhere here; if you can't find it, I can try to write on later. Of course, there are always criteria which are necessary or sufficient. Commented Mar 17, 2017 at 20:01
• Thank you! This is exactly what I was looking for. Since I am new to the topic, I had never heard of the Peres-Horodecki (PPT) criterion. It turns out that the state is indeed separable.
– Rmaa
Commented Mar 17, 2017 at 21:41
• If I may add a follow up question... Were this a three qubit density matrix, the criterion would still be a necessary condition, but not sufficient, correct?
– Rmaa
Commented Mar 17, 2017 at 21:42