**Product rules in Semidefinite Programming**

*Mario Szegedy*

In recent years
we witness the proliferation of semidefinite
programming

bounds in combinatorial optimization,
quantum computing, and even in

complexity theory. In these bounds
the key observation is often the fact

that the new parameter in question is
multiplicative with respect to the

product of the problem instances. Our goal
is to classify those semidefinite

programming instances for which the optimum is
multiplicative under a

naturally defined product operation. The
product operation we define

generalizes the ones used in *L. Lovasz*
and

XOR game paper of *R. Cleve, W. Slofstra,
F. Unger* and *S. Upadhyay*.

We find conditions under which the
product rule always holds and give

examples for cases when the product rule
does not hold.

(Joint with *Rajat** Mittal* and *Troy Lee*)