Karush-Kuhn-Tucker conditions for non-commutative optimization problems
arxiv(2023)
摘要
We consider the problem of optimizing the state average of a polynomial of
non-commuting variables, over all states and operators satisfying a number of
polynomial constraints, and over all Hilbert spaces where such states and
operators are defined. Such non-commutative polynomial optimization (NPO)
problems are routinely solved through hierarchies of semidefinite programming
(SDP) relaxations. By phrasing the general NPO problem in Lagrangian form, we
heuristically derive, via small variations on the problem variables, state and
operator optimality conditions, both of which can be enforced by adding new
positive semidefinite constraints to the SDP hierarchies. State optimality
conditions are satisfied by all Archimedean (that is, bounded) NPO problems,
and allow enforcing a new type of constraints: namely, restricting the
optimization over states to the set of common ground states of an arbitrary
number of operators. Operator optimality conditions are the non-commutative
analogs of the Karush-Kuhn-Tucker (KKT) conditions, which are known to hold in
many classical optimization problems. In this regard, we prove that a weak form
of non-commutative operator optimality holds for all Archimedean NPO problems;
stronger versions require the problem constraints to satisfy some qualification
criterion, just like in the classical case. We test the power of the new
optimality conditions by computing local properties of ground states of
many-body spin systems and the maximum quantum violation of Bell inequalities.
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