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Register a new userSemispectral measures as convolutions and their moment operators
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The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are then applied to conveniently determine the moment operators of the Cartesian margins of the phase space observables.
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We develop a framework to extend resource measures from one domain to a larger one. We find that all extensions of resource measures are bounded between two quantities that we call the minimal and maximal extensions. We discuss various applications of our framework. We show that any relative entropy (i.e. an additive function on pairs of quantum states that satisfies the data processing inequality) must be bounded by the min and max relative entropies. We prove that the generalized trace distance, the generalized fidelity, and the purified distance are optimal extensions. And in entanglement theory we introduce a new technique to extend pure state entanglement measures to mixed bipartite states.
An entanglement measure for a bipartite quantum system is a state functional that vanishes on separable states and that does not increase under separable (local) operations. It is well-known that for pure states, essentially all entanglement measures are equal to the v. Neumann entropy of the reduced state, but for mixed states, this uniqueness is lost. In quantum field theory, bipartite systems are associated with causally disjoint regions. There are no separable (normal) states to begin with when the regions touch each other, so one must leave a finite safety-corridor. Due to this corridor, the normal states of bipartite systems are necessarily mixed, and the v. Neumann entropy is not a good entanglement measure in the above sense. In this paper, we study various entanglement measures which vanish on separable states, do not increase under separable (local) operations, and have other desirable properties. In particular, we study the relative entanglement entropy, defined as the minimum relative entropy between the given state and an arbitrary separable state. We establish rigorous upper and lower bounds in various quantum field theoretic (QFT) models, as well as also model-independent ones. The former include free fields on static spacetime manifolds in general dimensions, or integrable models with factorizing $S$-matrix in 1+1 dimensions. The latter include bounds on ground states in general conformal QFTs, charged states (including charges with braid-group statistics) or thermal states in theories satisfying a nuclearity condition. Typically, the bounds show a divergent behavior when the systems get close to each other--sometimes of the form of a generalized area law--and decay when the systems are far apart. Our main technical tools are of operator algebraic nature.
In a quantum system with d-dimensional Hilbert space, the Q-function of a Hermitian positive semidefinite operator ?, is defined in terms of the d2 coherent states in this system. The Choquet integral CQ of the Q-function, is introduced using a ranking of the values of the Q-function, and Mobius transforms which remove the overlaps between coherent states. It is a figure of merit of the quantum properties of Hermitian operators, and it provides upper and lower bounds to various physical quantities in terms of the Q-function. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar operators. Comonotonic operators are shown to be bounded, with respect to an order based on Choquet integrals. Applications of the formalism to the study of the ground state of a physical system, are discussed. Bounds for partition functions, are also derived.
In this paper a connection between Hamburger moment sequences and their moment subsequences is given and the determinacy of these problems are related.
We study the randomness properties of reals with respect to arbitrary probability measures on Cantor space. We show that every non-computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees.
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