Grothendieck's constant and local models for noisy entangled quantum states

@article{Acn2006GrothendiecksCA,
  title={Grothendieck's constant and local models for noisy entangled quantum states},
  author={Antonio Ac{\'i}n and Nicolas Gisin and Ben Toner},
  journal={Physical Review A},
  year={2006},
  volume={73},
  pages={062105}
}
We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states rho p W =p|psi–><psi–|+(1–p)[openface 1]/4, we show that there is a local model for projective measurements if and only if p<=1/KG(3), where KG(3) is Grothendieck's constant of order 3. Known bounds on KG(3) prove the existence of this model at least for p<~0.66, quite close to the current region of Bell violation, p~0.71… 

Figures from this paper

A generalized Grothendieck inequality which lower bounds the entanglement required to play nonlocal games
Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if
General Method for Constructing Local Hidden Variable Models for Entangled Quantum States.
TLDR
The first general test to decide whether a quantum state is local, and it is shown that the test can be implemented by semidefinite programing and can be applied to any given state.
Noise robustness of the nonlocality of entangled quantum states.
TLDR
This work constructs a local model for the case in which rho is maximally entangled and p is at or below a certain bound, and extends the model to arbitrary rho, providing bounds on the resistance to noise of the nonlocal correlations of entangled states.
The Werner gap in the presence of simple coloured noise
The ‘Werner gap’ is the range of relevant parameters characterizing a quantum state for which it is both entangled and admits a local hidden variable model. Werner showed that the gap becomes maximal
Quantifying quantum nonlocality
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question
A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement
Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if
A generalized Grothendieck inequality and entanglement in XOR games
Suppose Alice and Bob make local two-outcome measurements on a shared entangled state. For any d, we show that there are correlations that can only be reproduced if the local dimension is at least d.
Algorithmic Construction of Local Hidden Variable Models for Entangled Quantum States.
TLDR
This work presents a simple method for building LHV models, applicable to any entangled state and considering continuous sets of measurements, which leads to a sequence of tests which fully captures the set of quantum states admitting a LHV model.
Unbounded Violation of Tripartite Bell Inequalities
We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 44 REFERENCES
Symmetric extensions of quantum States and local hidden variable theories.
TLDR
A simple and efficient algorithmic approach for the problem of constructing local hidden variable theories for quantum states based on constructing a so-called symmetric quasiextension of the quantum state that gives rise to a localhidden variable model with a certain number of settings for the observers Alice and Bob.
Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality
We present a local-hidden-variable model for positive-operator-valued measurements (an LHVPOV model) ) on a class of entangled generalized Werner states, thus demonstrating that such measurements do
Largest separable balls around the maximally mixed bipartite quantum state
For finite-dimensional bipartite quantum systems, we find the exact size of the largest balls, in spectral ${l}_{p}$ norms for $1l~pl~\ensuremath{\infty},$ of separable (unentangled) matrices around
Distinguishing separable and entangled states.
TLDR
This work shows how to design families of operational criteria that distinguish entangled from separable quantum states, and provides an explicit construction of the corresponding entanglement witnesses.
Purification of noisy entanglement and faithful teleportation via noisy channels.
TLDR
Upper and lower bounds on the yield of pure singlets ($\ket{\Psi^-}$) distillable from mixed states $M$ are given, showing $D(M)>0$ if $\bra{Psi-}M\ket-}>\half$.
A relevant two qubit Bell inequality inequivalent to the CHSH inequality
We computationally investigate the complete polytope of Bell inequalities for two particles with small numbers of possible measurements and outcomes. Our approach is limited by Pitowsky's connection
Bell's Inequalities and Density Matrices: Revealing "Hidden" Nonlocality.
  • Popescu
  • Physics
    Physical review letters
  • 1995
TLDR
For mixed states, correlations arising from a single ideal measurement on each system may obey standard Bell inequalities, yet when each system is subjected to a sequence of ideal measurements the correlations are nonlocal.
Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model.
  • Werner
  • Physics
    Physical review. A, General physics
  • 1989
TLDR
Any classically correlated state can be modeled by a hidden-variable theory and hence satisfies all generalized Bell's inequalities and the converse of this statement is false.
Bell Inequalities, Grothendieck's Constant, and Root Two
TLDR
A series of elementary examples are provided, which yield lower bounds on $K_{k(k - 1)}$ that approach 3/2 as $k$ gets large.
...
1
2
3
4
5
...