# How to know if a given 2-qubit gate is separable into two single-qubit gates?

This question seems fairly trivial but I'm struggling to find an elegant way to determine such. Suppose that you have a unitary matrix $$U$$ (not necessarily a density matrix) and you want to know if $$U=U_1\otimes U_2$$, for a given $$U_1$$ and $$U_2$$. This question is not trivial in the sense that if it was easy to determine such for any dimension of $$U,U_1$$ and $$U_2$$, then it would be easy to know if a density matrix is separable, nevertheless it remains as an NP-hard problem.

However, for the case where $$U$$ is a two qubit gate it should be relatively simple. Is there a criteria I could use to show if $$U$$ (4x4) is is separable into two single-qubit gates?

The only thing I could get is to diagonalize $$U$$ and check if there are some common factors.

• Jul 18, 2022 at 12:19

This is relatively straightforward to do in practice. Remember that if $$U$$ has the structure you want, it has a block-matrix form (change to whatever size you want) $$U=\left(\begin{array}{cccc} U^1_{00}U^2 & U^1_{01}U^2 & U^1_{02}U^2 & U^1_{03}U^2 \\ U^1_{10}U^2 & U^1_{11}U^2 & U^1_{12}U^2 & U^1_{13}U^2 \\ U^1_{20}U^2 & U^1_{21}U^2 & U^1_{22}U^2 & U^1_{23}U^2 \\ U^1_{30}U^2 & U^1_{31}U^2 & U^1_{32}U^2 & U^1_{33}U^2 \\ \end{array}\right).$$ So, all you have to do is look at $$U$$ and extract each block, and ask if they are all identical up to a scale factor for each block. If yes, then the matrix has the desired decomposition, otherwise not.

Notation: target decomposition is $$U=U^1\otimes U^2$$. Matrix $$U^1$$ has matrix elements $$U^1_{ij}$$.

A more procedural version. Let $$U$$ be $$d^2\times d^2$$ where we're expecting each matrix to be $$d\times d$$.

• Look at the top-left $$d\times d$$ block. Do all of the rows and columns have the same sum mod-square? No: $$U$$ not separable. Yes: Let $$U^2$$ be that block, renormalised so that sum-mod square of all rows and columns is 1.
• Calculate $$U^1=\text{Tr}_2\left((I\otimes U^2)\cdot U\right)/d$$
• Compare $$U^1\otimes U^2$$ to $$U$$. If equal, $$U$$ was separable. If not, $$U$$ was not separable.
• Straightforward but there should be a more systematic procedure than to look carefully. Maybe that can get systematized by a trace operation? Jul 18, 2022 at 11:11
• Also could you describe the notation? it is unclear as it is. Jul 18, 2022 at 11:12
• Of course you can turn this into a systematic procedure. Take your matrix $U$. divide it up into blocks by slicing (or whatever the terminology in your preferred language). Do element-wise division between one block and all the others. Are the resulting matrices constant? Jul 18, 2022 at 11:18
• I will leave this question open, I am looking for a more elegant solution (maybe there is none as this problem gets harder in higher dimensions). Jul 18, 2022 at 11:48