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.

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.

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.

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

In the nonrelativistic setting with finitely many canonical degrees of freedom, a shiftcovariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty.We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for p = 1, 2, ∞ of a more general notion of p-regularity defined as the norm density of the span of translates of the operator in the Schatten-p class. We show that the relation between zero sets and p-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis. C 2012 American Institute of Physics. [http://dx.

A system prepared in an unstable quantum state generally decays following an exponential law, as environmental decoherence is expected to prevent the decay products from recombining to reconstruct the initial state. Here we show the existence of deviations from exponential decay in open quantum systems under very general conditions. Our results are illustrated with the exact dynamics under quantum Brownian motion and suggest an explanation of recent experimental observations.The exponential decay law of unstable systems is ubiquitous in Nature and has widespread applications [1][2][3]. Yet, in isolated quantum systems deviations occur at both short and long times of evolution [4][5][6]. Short time deviations underlie the quantum Zeno effect [7, 8], ubiquitously used to engineer decoherence free-subspaces and preserve quantum information. Long-time deviations are expected in any nonrelativistic systems with a ground state; they slow down the decay and generally manifest as a power-law in time [9]. Both short and long-time deviations are present as well in manyparticle systems [11][12][13][14][15][16]. Indeed, the latter signal the advent of thermalization in isolated many-body systems [17,18]. In quantum cosmology, power-law deviations constrain the likelihood of scenarios with eternal inflation [10]. They also rule the scrambling of information as measured by the decay of the form factor [19][20][21][22] in blackhole physics and strongly coupled quantum systems described by AdS/CFT, that are believed to be maximally chaotic [23].Given a unstable quantum state |Ψ 0 prepared at time t = 0, it is customary to describe the closed-system decay dynamics via the survival probability, which is the fidelity between the initial state and its time evolution S(t) := |A(t)| 2 = | Ψ 0 |Ψ(t) | 2 .Explicitly, the survival amplitude reads A(t)is the time evolution operator generated by the Hamiltonian of the systemĤ. Short time deviations are associated with the quadratic decayand are generally suppressed by the coupling to an environment that induces the appearance of a term linear in t, see, e.g. [2, 3,24,25]. The origin of the long-time deviations can be appreciated using the Ersak equation for the survival amplitude [4,26,27]that follows from the unitarity of time evolution in isolated quantum systems. The memory term readsHere, we denote the projector onto the space spanned by the initial state byP ≡ |Ψ 0 Ψ 0 | and its orthogonal complement byQ ≡ 1 −P. As a result, the memory term m(t, t ) represents the formation of decay products at an intermediate time t and their subsequent recombination to reconstruct the initial state |Ψ 0 . The suppression of this term leads to the exponential decay law for A(t) and S(t), as an ansatz of the form A(t) = e −γt is a solution of Eq. (3) with m(t, t ) = 0, i.e, A(t) = A(t − t )A(t ). [4,27]. In addition, using the definition of the survival probability and Eq. (3), it has been demonstrated that the long-time non-exponential behavior of S(t) is dominated by |m(t, t )| 2 . The onset of lon...

We consider the problem of identifying and estimating dynamical parameters of an ergodic quantum Markov chain, when only the stationary output is accessible for measurements. The starting point of the analysis is the fact that the knowledge of the output state completely fixes the dynamics up to an equivalence class of ?coordinate transformation? consisting of a multiplication by a phase and a unitary conjugation of the Kraus operators. Assuming that the dynamics depends on an unknown parameter, we show that the latter can be estimated at the ?standard? rate n ?1/2, and give an explicit expression of the (asymptotic) quantum Fisher information of the output, which is proportional to the Markov variance of a certain ?generator?. More generally, we show that the output is locally asymptotically normal, i.e., it can be approximated by a simple quantum Gaussian model consisting of a coherent state whose mean is related to the unknown parameter. As a consistency check, we prove that a parameter related to the ?coordinate transformation? unitaries has zero quantum Fisher informationPeer reviewe