# 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} }

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…

## 21 Citations

Tilburg University On the closure of the completely positive semidefinite cone and linear approximations to quantum

- Mathematics
- 2017

The structural properties of the completely positive semidefinite cone CS+, consisting of all the n×n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any…

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

- MathematicsSIAM 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.

Positive semidefinite rank

- MathematicsMath. Program.
- 2015

The main mathematical properties of psd rank are surveyed, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

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

- MathematicsMath. 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.

Quantum entanglement: insights via graph parameters and conic optimization

- Computer Science
- 2016

This thesis proposes a novel approach to the study of these quantum graph parameters using the paradigm of conic optimization, and introduces and study the completely positive semidefinite cone, a new matrix cone consisting of all symmetric matrices that admit a Gram representation by positive semidfinite matrices.

Quantum Bilinear Optimization

- Computer ScienceSIAM 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.

THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED

- MathematicsForum of Mathematics, Pi
- 2019

We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert…

Completely positive semidefinite rank

- Mathematics, Computer ScienceMath. Program.
- 2018

The cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix and every doubly nonnegative matrix whose support is given by G is cPSd, and it is shown that a graph is cpsD if and only if it has no odd cycle of length at least 5 as a subgraph.

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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.

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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.

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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.

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