Effective Number Theory: Counting the Identities of a Quantum State

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes, and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a “total” that takes into account their varied relevance. For example, such an effective count of position states available to a lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of a quantum computation relates to the efficiency of the quantum algorithm. Despite a broad need for effective counting, a well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of the participation number or exponentiated entropies, are used in some areas. Here, we develop the additive theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of the measure, our results may find applications not only in quantum physics, but also in other quantitative sciences

Counting the Identities of a Quantum State

Quantum physics frequently involves a need to count the states, subspaces, measurement outcomes and other elements of quantum dynamics. However, with quantum mechanics assigning probabilities to such objects, it is often desirable to work with the notion of a "total" that takes into account their varied relevance so acquired. For example, such effective count of position states available to lattice electron could characterize its localization properties. Similarly, the effective total of outcomes in the measurement step of quantum computation relates to the efficiency of quantum algorithm. Despite a broad need for effective counting, well-founded prescription has not been formulated. Instead, the assignments that do not respect the measure-like nature of the concept, such as versions of participation number or exponentiated entropies, are used in some areas. Here we develop and solve the theory of effective number functions (ENFs), namely functions assigning consistent totals to collections of objects endowed with probability weights. Our analysis reveals the existence of a minimal total, realized by the unique ENF, which leads to effective counting with absolute meaning. Touching upon the nature of measure, our results may find applications not only in quantum physics but also in other quantitative sciences.Comment: 19 pages, 1 figure; v2: some restructuring and presentation improvements, 15 pages, 1 figur

Strong Non-Ultralocality of Ginsparg-Wilson Fermionic Actions

It is shown that it is impossible to construct a free theory of fermions on infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally nonsymmetric part of its propagator is local, and (d) describes single species of massless Dirac fermions in the continuum limit. This establishes non-ultralocality for arbitrary doubler-free Ginsparg-Wilson fermionic action with hypercubic symmetries ("strong non-ultralocality"), and complements the earlier general result on non-ultralocality of infinitesimal Ginsparg-Wilson-Luscher symmetry transformations ("weak non-ultralocality").Comment: 21 pages, 1 figure, LATEX. Few typos corrected; few sentences reformulated; figure centere

A Different Angle on Quantum Uncertainty (Measure Angle)

The uncertainty associated with probing the quantum state is expressed as the effective abundance (measure) of possibilities for its collapse. New kinds of uncertainty limits entailed by quantum description of the physical system arise in this manner.Comment: 10 pages, 3 figures. Talk at the 7th International Conference on New Frontiers in Physics (ICNFP2018), 4-12 July 2018, Kolymbari, Cret

Counting-Based Effective Dimension and Discrete Regularizations

Effective number theory determines all additive ways to assign counts to collections of objects with probabilities or other additive weights. Here we construct all counting-based schemes to select effective supports on such collections, and show that it leads to a unique notion of effective dimension. This effective counting dimension (ECD) specifies how the number of objects in a support scales with their total number, and its uniqueness means that all schemes yield the same value. Hence, ECD is well defined and can be used to characterize targets of discrete regularizations in physics and other quantitative sciences. Given its generality, ECD may help to connect and interpret results from widely distinct areas. Our analysis makes recent studies of effective spatial dimensions in lattice quantum chromodynamics and Anderson localization models well founded. We address the reliability of regularization removals in practice and perform the respective numerical analysis in the context of 3d Anderson criticality. Our arguments suggest that measure-based dimensions (Minkowski, Hausdorff) of fixed sets have good probabilistic extensions.Comment: 5 pages, 1 figur