Abstract We suggest generalized robustness for quantifying nonlocality and derive its equivalence to the maximum violation ratio of Bell inequalities defined as vectors with non-negative elements. We investigate its properties by comparing it with white-noise and standard robustness measures. As a result, we show that white-noise robustness does not fulfill monotonicity under local operations and shared randomness, whereas the other measures do. To compare the standard and generalized robustness measures, we introduce the concept of inequivalence, which indicates a reversal in the order relationship depending on the choice of monotones. From an operational perspective, the inequivalence of monotones for resourceful objects implies the absence of free operations that connect them. Applying this concept, we find that standard and generalized robustness measures are inequivalent between even- and odd-dimensional cases up to eight dimensions. This is obtained using randomly performed Collins–Gisin–Linden–Massar–Popescu measurement settings in a maximally entangled state. This study contributes to the resource theory of nonlocality and sheds light on comparing monotones by using the concept of inequivalence valid for all resource theories.
Abstract We suggest generalized robustness for quantifying nonlocality and derive its equivalence to the maximum violation ratio of Bell inequalities defined as vectors with non-negative elements. We investigate its properties by comparing it with white-noise and standard robustness measures. As a result, we show that white-noise robustness does not fulfill monotonicity under local operations and shared randomness, whereas the other measures do. To compare the standard and generalized robustness measures, we introduce the concept of inequivalence, which indicates a reversal in the order relationship depending on the choice of monotones. From an operational perspective, the inequivalence of monotones for resourceful objects implies the absence of free operations that connect them. Applying this concept, we find that standard and generalized robustness measures are inequivalent between even- and odd-dimensional cases up to eight dimensions. This is obtained using randomly performed Collins–Gisin–Linden–Massar–Popescu measurement settings in a maximally entangled state. This study contributes to the resource theory of nonlocality and sheds light on comparing monotones by using the concept of inequivalence valid for all resource theories.
The robustness of the violation of the Collins-Linden-Gisin-Masser-Popescu (CGLMP) inequality is investigated from the dual perspectives of noise in measurements and in states. To quantify it, we introduce a quantity called the area of the nonlocal region which reveals a dimensional advantage. Specifically, we report that with an increase in dimension, the maximally violating states show greater enhancement in the area of the nonlocal region in comparison to the maximally entangled states and the scaling of the increment in this case grows faster than visibility. Moreover, we examine the robustness of the sequential violation of the CGLMP inequality using weak measurements and find that even for higher dimensions, the simultaneous violation of the CGLMP inequalities of two observers as obtained for two-qubit states persists. We notice that the complementarity between information gain and disturbance from measurements is manifested by the decrease in the visibility in the first round and the increase in the same in the second round with dimensions. Furthermore, the amount of white noise that can be added to a maximally entangled state so that it has two rounds of the violation decreases with the dimension, while it does not appreciably change for the maximally violating states.
We compare entanglement with quantum nonlocality employing a geometric structure of the state space of bipartite qudits. The central object is a regular simplex spanned by generalized Bell states. The Collins–Gisin–Linden–Massar–Popescu–Bell inequality is used to reveal states of this set that cannot be described by local-realistic theories. Optimal measurement settings necessary to ascertain nonlocality are determined by means of a recently proposed parameterization of the unitary group combined with robust numerical methods. The main results of this paper are descriptive geometric illustrations of the state space that emphasize the difference between entanglement and quantum nonlocality. Namely, it is found that the shape of the boundaries of separability and Bell inequality violation are essentially different. Moreover, it is also shown that for mixtures of states sharing the same amount of entanglement, Bell inequality violations and entanglement measures are non-monotonically related.
The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarse-graining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as Clauser-Horne-Shimony-Holt (CHSH) Bell inequality violations or Einstein-Podolsky-Rosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noise-based quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson's bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.
Abstract Treating robustness as a key measure for various quantum resources, this work proposes a general quantum resource theory (GQRT) for a quantum system (GQRT4QS), which was formally a pair consisting of a free-state set and a free-operation set. Here, the free-state set whose elements are called free states is defined as any compact convex set of quantum states of a quantum system that includes the maximally random state, while the free-operation set whose elements are called free operations is a semigroup consisting of quantum operations of the system that map free states to free states. Within this GQRT4QS framework, we first introduce the unified robustness of quantum resources and rigorously prove its validity as a quantum resource measure. Subsequently, we define the unified geometric robustness of quantum resources and further demonstrate that it also satisfies the criteria for a valid quantum resource measure. For practical applications, we derive a series of resource-specific robustness measures, encompassing the robustness and geometric robustness of entanglement, Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering, and quantum coherence.
While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality. In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.
We introduce the witness robustness of quantum measurements, a resource quantifier whose admissible noise consists of tuples of free-state witnesses rather than physical measurements. We establish its operational interpretation: it quantifies the maximal advantage that a measurement can provide over free measurements in discriminating an ensemble composed entirely of free states. Unlike the standard and generalized robustnesses, the witness robustness is not faithful in general, reflecting the fact that a resourceful measurement need not be useful when only free states can be prepared. We identify conditions under which faithfulness is recovered and show that, in resource theories admitting a resource-destroying map, the witness robustness vanishes for every measurement. We also establish fundamental properties, including convexity and monotonicity. Finally, we derive analytical results for projective measurements in single-qubit magic and for binary pure-state projective measurements in the two-qubit PPT entanglement theory.