# Is this a known operator decomposition?

Consider a Hermitian operator $H_\mathcal{AB}$ acting on the Hilbert space $\mathcal{A}\otimes\mathcal{B}$. What is the smallest commuting set of separable operators in the form $A_k\otimes B_k$, such that $$H_\mathcal{AB}=\sum_k A_k\otimes B_k?$$ As pointed out in the comments, a set like this might not even exist.

It would also be interesting to constrain $A_k$ and $B_k$ to be Hermitian themselves. Note that if we didn't require commutativity, we could use the operator singular value decomposition.

• In fact, I'm not convinced that it's necessarily possible to do this with any number of commuting operators of that form. – Omnomnomnom May 1 '17 at 3:46
• What is a separable operator ? – user42761 May 1 '17 at 9:53
• @AndréS. In this context $O$ is separable if it can be written $O=A\otimes B$, with $A$ and $B$ acting on $\mathcal A$ and $\mathcal B$. – Ziofil May 1 '17 at 15:32
• @Omnomnomnom, are there explicit examples of operators that cannot be decomposed using a commuting set? – Ziofil May 8 '17 at 18:25
• @Ziofil I don't know whether the statement is true, let alone whether there are explicit examples. – Omnomnomnom May 8 '17 at 18:28