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. A typical bipartite quantum protocol, such as EPRsteering, relies on two quantum features, entanglement of states and incompatibility of measurements. Noise can delete both of these quantum features. In this work we study the behavior of incompatibility under noisy quantum channels. The starting point for our investigation is the observation that compatible measurements cannot become incompatible by the action of any channel. We focus our attention to channels which completely destroy the incompatibility of various relevant sets of measurements. We call such channels incompatibility breaking, in analogy to the concept of entanglement breaking channels. This notion is relevant especially for the understanding of noise-robustness of the local measurement resources for steering.
Abstract. An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Schrödinger and diffusion equations, also higher order hyperbolic and parabolic equations.Key words: integration theory via linear functionals, measure theory on infinite dimensional spaces, probabilistic representation of solutions of PDEs, stochastic processes, Feynman path integrals.
We consider questions related to quantizing complex valued functions defined on a locally compact topological group. In the case of bounded functions, we generalize R. Werner's approach to prove the characterization of the associated normal covariant quantization maps.
The term Einstein-Podolsky-Rosen steering refers to a quantum correlation intermediate between entanglement and Bell nonlocality, which has been connected to another fundamental quantum property: measurement incompatibility. In the finite-dimensional case, efficient computational methods to quantify steerability have been developed. In the infinite-dimensional case, however, less theoretical tools are available. Here, we approach the problem of steerability in the continuous variable case via a notion of state-channel correspondence, which generalizes the well-known Choi-Jamio?kowski correspondence. Via our approach we are able to generalize the connection between steering and incompatibility to the continuous variable case and to connect the steerability of a state with the incompatibility breaking property of a quantum channel, with applications to noisy NOON states and amplitude damping channels. Moreover, we apply our methods to the Gaussian steering setting, proving, among other things, that canonical quadratures are sufficient for steering Gaussian statesauthorsversionPeer reviewe
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