Abstract
Nonlocality is one of the most important resources for quantum information protocols. The observation of nonlocal correlations in a Bell experiment is the result of appropriately chosen measurements and quantum states. We quantify the minimal purity to achieve a certain Bell value for any Bell operator. Since purity is the most fundamental resource of a quantum state, this enables us also to quantify the necessary coherence, discord, and entanglement for a given violation of two-qubit correlation inequalities. Our results shine new light on the CHSH inequality by showing that for a fixed Bell violation an increase in the measurement resources does not always lead to a decrease of the minimal state resources.
Highlights
It is arguably one of the most astonishing features of quantum theory that local measurements performed on certain quantum states can lead to the phenomenon of quantum nonlocality [1]
This approach based on the Bell operator makes use of the full information available and allows us to study in a simple way how the required state resources depend on the chosen measurements
We have shown that the minimal purity necessary to achieve a certain Bell value for the most general Bell operator can be found analytically via an accessible criterion
Summary
We derive from the spectrum of any given Bell operator an analytical expression for the minimal purity of a quantum state that is needed to achieve some fixed amount of nonlocality in terms of a Bell inequality violation. This result is general, i.e., it holds for any dimension, any number of parties, measurement settings, and outcomes. As an application of our results, we present a closed expression for the maximal possible violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality [23] given some fixed amount of entanglement or purity and a given level of measurement incompatibility This enables us to establish a surprising link between the incompatibility of quantum measurements and the minimal entanglement needed. We are considering Hermitian Bell operators of the form
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