# On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings

@inproceedings{Burgdorf2015OnTC,
title={On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings},
author={Sabine Burgdorf and Monique Laurent and Teresa Piovesan},
booktitle={TQC},
year={2015}
}
• Published in TQC 10 February 2015
• Mathematics
We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+^n$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set $\mathcal Q$ of bipartite quantum correlations, as projection of an affine section of it. We have two main…

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## References

SHOWING 1-10 OF 38 REFERENCES

### Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone

• Mathematics
SIAM J. Optim.
• 2015
This new cone is investigated, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size) and is used to model quantum analogues of the classical independence and chromatic graph parameters.

### Approximation of the Stability Number of a Graph via Copositive Programming

• Mathematics, Computer Science
SIAM J. Optim.
• 2002
This paper shows how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method).

### Linear conic formulations for two-party correlations and values of nonlocal games

• Mathematics
Math. Program.
• 2017
This work shows that the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models can be expressed as projections of affine sections of appropriate convex cones, and shows that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lovász is in fact an upper Bound to the quantum value of the game.

### A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations

• Computer Science
• 2008
It is shown that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests, and used in particular to bound the quantum violation of various Bell inequalities.

### Quantum Bilinear Optimization

• Computer Science
SIAM J. Optim.
• 2016
An asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization of entangled value of two-prover games, entanglement-assisted coding for classical channels, and quantum-proof randomness extractors is introduced.

### On the accuracy of uniform polyhedral approximations of the copositive cone

• E. Yildirim
• Computer Science, Mathematics
Optim. Methods Softw.
• 2012
A hierarchy of increasingly better outer polyhedral approximations to the copositive cone is proposed, and it is established that the sequence of approxIMations is exact in the limit.

### Constrained trace-optimization of polynomials in freely noncommuting variables

• Mathematics
J. Glob. Optim.
• 2016
This paper presents Lasserre’s relaxation scheme for trace optimization based on semidefinite programming (SDP) and demonstrates its convergence properties, and employs a noncommutative version of the randomization technique championed by Nie to enforce flatness.

### On the Quantum Chromatic Number of a Graph

• Mathematics
Electron. J. Comb.
• 2007
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a