7 research outputs found
Finite convergence of sum-of-squares hierarchies for the stability number of a graph
We investigate a hierarchy of semidefinite bounds for
the stability number of a graph , based on its copositive
programming formulation and introduced by de Klerk and Pasechnik [SIAM J.
Optim. 12 (2002), pp.875--892], who conjectured convergence to in
steps. Even the weaker conjecture claiming finite convergence
is still open. We establish links between this hierarchy and sum-of-squares
hierarchies based on the Motzkin-Straus formulation of , which we
use to show finite convergence when is acritical, i.e., when
for all edges of . This relies, in
particular, on understanding the structure of the minimizers of Motzkin-Straus
formulation and showing that their number is finite precisely when is
acritical. As a byproduct we show that deciding whether a standard quadratic
program has finitely many minimizers does not admit a polynomial-time algorithm
unless P=NP.Comment: We removed the material from section 7 about rank 0 graphs which will
be included in separate forthcoming wor
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Bounding the separable rank via polynomial optimization
We investigate questions related to the set SEPd consisting of the linear maps Ï acting on CdâCd that can be written as a convex combination of rank one matrices of the form xxââyyâ. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank ranksep(Ï), defined as the smallest number of rank one states xxââyyâ entering the decomposition of a separable state Ï. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter Ïsep(Ï), a natural convexification of the combinatorial parameter ranksep(Ï). A distinguishing feature is exploiting the positivity constraint Ï âxxââyyââ0 to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set SEPd and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from Doherty et al. (2002) [16]
An Improved Semidefinite Programming Hierarchy for Testing Entanglement
© 2017, Springer-Verlag Berlin Heidelberg. We present a stronger version of the DohertyâParriloâSpedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing that is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints