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

@article{Laurent2015ConicAT, title={Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone}, author={Monique Laurent and Teresa Piovesan}, journal={SIAM J. Optim.}, year={2015}, volume={25}, pages={2461-2493} }

We investigate the completely positive semidefinite cone ${\mathcal{CS}_{+}^n}$, a new matrix cone consisting of all $n\times n$ matrices that admit a Gram representation by positive semidefinite matrices (of any size). In particular, we study relationships between this cone and the completely positive and the doubly nonnegative cone, and between its dual cone and trace positive noncommutative polynomials. We use this new cone to model quantum analogues of the classical independence and…

## 46 Citations

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

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- 2017

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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

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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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- 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.

### Positive semidefinite rank

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

### Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

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- 2018

A hierarchy of semidefinite programming lower bounds is constructed and convergence to a new parameter is shown, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free.

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ABSTRACT A symmetric matrix X is completely positive semidefinite (cpsd) if there exist positive semidefinite matrices (for some ) such that for all . The of a cpsd matrix is the smallest for which…

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- Mathematics, Computer ScienceThe Electronic Journal of Linear Algebra
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It is demonstrated that there are graphs for which the above bound for the quantum independence number $\alpha_q$ is not exact with any Hermitian weight matrix, for $\alpha(G)$ and $\alpha_(G)$.

### Graph isomorphism: Physical resources, optimization models, and algebraic characterizations

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- 2020

In the $(G,H)$-isomorphism game, a verifier interacts with two non-communicating players (called provers) by privately sending each of them a random vertex from either $G$ or $H$, whose aim is to…

## References

SHOWING 1-10 OF 99 REFERENCES

### Sums of Hermitian Squares and the BMV Conjecture

- Mathematics
- 2008

We show that all the coefficients of the polynomial $$\mathop{\mathrm{tr}}((A+tB)^{m})\in\mathbb{R}[t]$$ are nonnegative whenever m≤13 is a nonnegative integer and A and B are positive semidefinite…

### 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.

### 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.

### Computing Semidefinite Programming Lower Bounds for the (Fractional) Chromatic Number Via Block-Diagonalization

- Computer Science, MathematicsSIAM J. Optim.
- 2008

The numerical results indicate that the new bounds can be much stronger than the Lovasz theta number, and for some of the DIMACS instances the best known lower bounds significantly are improved.

### Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2013

A quantum version of Lovász' famous ϑ function on general operator systems is defined, as the norm-completion of a “naive” generalization of ϑ, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero- error capacities can be formulated.

### Constrained Polynomial Optimization Problems with Noncommuting Variables

- MathematicsSIAM J. Optim.
- 2012

C constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball is studied, and several examples pertaining to matrix inequalities are given to illustrate the results.

### 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.

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

- Mathematics, Computer ScienceSIAM 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).

### Trace-positive polynomials, sums of hermitian squares and the tracial moment problem

- Mathematics
- 2011

A polynomial in non-commuting variables is trace-positive if all its evaluations by symmetric matrices have positive trace. The investigation of trace-positive polynomials is related to two famous…