Natural Framework for Device-Independent Quantification of Quantum Steerability, Measurement Incompatibility, and Self-Testing

Abstract

We introduce the concept of assemblage moment matrices, i.e., a collection of matrices of expectation values, each associated with a conditional quantum state obtained in a steering experiment. We demonstrate how it can be used for quantum states and measurements characterization in a device-independent manner, i.e., without invoking any assumption about the measurement or the preparation device. Specifically, we show how the method can be used to lower bound the steerability of an underlying quantum state directly from the observed correlation between measurement outcomes. Combining such device-independent quantifications with earlier results established by Piani and Watrous [Phys. Rev. Lett. 114, 060404 (2015)], our approach immediately provides a device-independent lower bound on the generalized robustness of entanglement, as well as the usefulness of the underlying quantum state for a type of subchannel discrimination problem. In addition, by proving a quantitative relationship between steering robustness and the recently introduced incompatibility robustness, our approach also allows for a device-independent quantification of the incompatibility between various measurements performed in a Bell-type experiment. Explicit examples where such bounds provide a kind of self-testing of the performed measurements are provided.

N-term pairwise-correlation inequalities, steering, and joint measurability

“…In recent years there is a surge of research activity dedicated to explore the notion of measurement incompatibility and its connection with the counter-intuitive quantum notions like non-locality, contextuality and nonmacrorealism [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In particular, it is known that measurement incompatibility plays a key role in bringing to surface the violations of the so-called no-go theorems in the quantum world.…”

Exploring Bell inequalities for the device-independent certification of multipartite entanglement depth

“…Here, we provide the analytic expression of the optimal parameter φ * n alluded to right after Eq. (14), and the corresponding maximal quantum value S Q, * n,γ for n = 2, 3, 4, and 5.…”

Exploring the framework of assemblage moment matrices and its applications in device-independent characterizations

“…In particular, among existing formulations of such measurements [26][27][28], any measurements that are incompatible-in the sense of being nonjointly-measurable [29]-can always be used [30,31] to demonstrate the steerability of some quantum states. In fact, the incompatibility robustness [32]-a quantifier for measurement incompatibility-has even been shown to be lower bounded [33,34] by the steering robustness [21] -a quantifier for quantum steerability.…”

“…In the context of DI quantum information, a moment matrix, i.e., a matrix composed of a set of expectation values of observables, is known to play a very important role. In particular, the hierarchy of moment matrices due to Navascués, Pironio, and Acín (NPA) [35] not only has provided the only known effective characterization (more precisely, approximation) of the quantum set, but also has found applications in DI entanglement detection [36,37], quantification [33,38,39], dimensionwitnessing [13,40,41], self-testing [42,43] etc. Similarly, some other variants [17,44] of the NPA hierarchy have also found applications in the context of one-sided DI quantum information.…”