Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$

Abstract

We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state $\rho =v|{\psi }_{-}><{\psi }_{-}|+(1-v)\frac{1}{4}$ via a local hidden variable (LHV) model, where $|{\psi }_{-}>$ denotes the singlet state. We show analytically that these correlations are local for $v=999\times 689\times 10^{-6}$ ${\cos }^4(\pi /50)\simeq 0.6829$. In turn, as this problem is closely related to a purely mathematical one formulated by Grothendieck, our result implies a new bound on the Grothendieck constant $K_G(3)\le 1/v\simeq 1.4644$. We also present a LHV model for reproducing the statistics of arbitrary POVMs on the Werner state for $v\simeq 0.4553$. The techniques we develop can be adapted to construct LHV models for other entangled states, as well as bounding other Grothendieck constants.

12 pages, typos corrected

References in corpus (10)

• Steering, Entanglement, Nonlocality, and the EPR Paradox
• Grothendieck's constant and local models for noisy entangled quantum states
• Simulating positive-operator-valued measures with projective measurements
• More efficient Bell inequalities for Werner states
• Noise robustness of the nonlocality of entangled quantum states
• Einstein-Podolsky-Rosen steering and the steering ellipsoid
• Local hidden--variable models for entangled quantum states
• Necessary and sufficient condition for steerability of two-qubit states by the geometry of steering outcomes
• Towards Grothendieck Constants and LHV Models in Quantum Mechanics
• Convex separation from convex optimization for large-scale problems

Cited by in corpus (65)

• Quantum Steering
• Entanglement Certification $-$ From Theory to Experiment
• Simulating positive-operator-valued measures with projective measurements
• Incompatible measurements in quantum information science
• Device-independent entanglement certification of all entangled states
• All entangled pure quantum states violate the bilocality inequality
• Self-testing non-projective quantum measurements in prepare-and-measure experiments
• Demonstration of Einstein-Podolsky-Rosen Steering with Enhanced Subchannel Discrimination
• Quantum measurement incompatibility does not imply Bell nonlocality
• Self-testing of Pauli observables for device-independent entanglement certification
• Quantum states with a positive partial transpose are useful for metrology
• Measurement incompatibility does not give rise to Bell violation in general
• Operational framework for quantum measurement simulability
• Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering
• Set coherence: basis-independent quantification of quantum coherence
• Semidefinite programming relaxations for quantum correlations
• Geometry of two-qubit states with negative conditional entropy
• Tripartite entanglement: Foundations and applications
• Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution protocols
• Improved DIQKD protocols with finite-size analysis
• Simulating all quantum measurements using only projective measurements and postselection
• Simulability of observables in general probabilistic theories
• Qutrit witness from the Grothendieck constant of order four
• Multipartite nonlocality and random measurements
• Semi-device-independent characterisation of multipartite entangled states and measurements
• Fidelity deviation in quantum teleportation
• Unified picture for spatial, temporal and channel steering
• Improved local models and new Bell inequalities via Frank-Wolfe algorithms
• Bell nonlocality with a single shot
• Simple and maximally robust processes with no classical common-cause or direct-cause explanation
• Implementation of quantum measurements using classical resources and only a single ancillary qubit
• Exact Steering Bound for Two-Qubit Werner States
• Measurement compatibility in Bell nonlocality tests
• Simulability of high-dimensional quantum measurements
• Algorithmic construction of local models for entangled quantum states: optimization for two-qubit states
• Compatibility of Generalized Noisy Qubit Measurements
• Deep learning the hierarchy of steering measurement settings of qubit-pair states
• General Method for Classicality Certification in the Prepare and Measure Scenario
• Single-copy activation of Bell nonlocality via broadcasting of quantum states
• How much randomness can be generated from a quantum black-box device?
• Strength and typicality of nonlocality in multisetting and multipartite Bell scenarios
• Family of Bell inequalities violated by higher-dimensional bound entangled states
• Experimental detection of steerability in Bell local states with two measurement settings
• Spreading Nonlocality in Quantum Network
• Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant
• Optimal and tight Bell inequalities for state-independent contextuality sets
• Quantum steering of Bell-diagonal states with generalized measurements
• Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles
• Device-independent certification of non-classical joint measurements via causal models
• The grasshopper problem
• Keyring models: an approach to steerability
• Device-independent and semi-device-independent entanglement certification in broadcast Bell scenarios
• Certifying measurement incompatibility in prepare-and-measure and Bell scenarios
• Semi-device-independent information processing with spatiotemporal degrees of freedom
• Discovering Local Hidden-Variable Models for Arbitrary Multipartite Entangled States and Arbitrary Measurements
• Improved algorithms and novel applications of the FrankWolfe.jl library
• Revealing the Boundary between Quantum Mechanics and Classical Model by EPR-Steering Inequality
• Grothendieck constant is norm of Strassen matrix multiplication tensor
• Symmetric Grothendieck inequality
• Cyclic Einstein-Podolsky-Rosen Steering
• Certifying the activation of Bell nonlocality with finite data
• Semi-device-independent certification of number of measurements
• Better bounds on finite-order Grothendieck constants
• Robust certification of non-projective measurements: theory and experiment
• Trade-off between Information Gain and Disturbance in Local Discrimination of Entangled Quantum States