13 research outputs found
Two-Site Quantum Random Walk
We study the measure theory of a two-site quantum random walk. The truncated
decoherence functional defines a quantum measure on the space of
-paths, and the in turn induce a quantum measure on the
cylinder sets within the space of untruncated paths. Although
cannot be extended to a continuous quantum measure on the full -algebra
generated by the cylinder sets, an important question is whether it can be
extended to sufficiently many physically relevant subsets of in a
systematic way. We begin an investigation of this problem by showing that
can be extended to a quantum measure on a "quadratic algebra" of subsets of
that properly contains the cylinder sets. We also present a new
characterization of the quantum integral on the -path space.Comment: 28 page
A characterisation of ordered abstract probabilities
In computer science, especially when dealing with quantum computing or other
non-standard models of computation, basic notions in probability theory like "a
predicate" vary wildly. There seems to be one constant: the only useful example
of an algebra of probabilities is the real unit interval. In this paper we try
to explain this phenomenon. We will show that the structure of the real unit
interval naturally arises from a few reasonable assumptions. We do this by
studying effect monoids, an abstraction of the algebraic structure of the real
unit interval: it has an addition which is only defined when
and an involution which make it an effect algebra, in
combination with an associative (possibly non-commutative) multiplication.
Examples include the unit intervals of ordered rings and Boolean algebras.
We present a structure theory for effect monoids that are -complete,
i.e. where every increasing sequence has a supremum. We show that any
-complete effect monoid embeds into the direct sum of a Boolean algebra
and the unit interval of a commutative unital C-algebra. This gives us from
first principles a dichotomy between sharp logic, represented by the Boolean
algebra part of the effect monoid, and probabilistic logic, represented by the
commutative C-algebra. Some consequences of this characterisation are that
the multiplication must always be commutative, and that the unique
-complete effect monoid without zero divisors and more than 2 elements
must be the real unit interval. Our results give an algebraic characterisation
and motivation for why any physical or logical theory would represent
probabilities by real numbers.Comment: 12 pages. V2: Minor change
Sensitivity of entanglement measures in bipartite pure quantum states
Entanglement measures quantify the amount of quantum entanglement that is contained in quantum states. Typically, different entanglement measures do not have to be partially ordered. The presence of a definite partial order between two entanglement measures for all quantum states, however, allows for meaningful conceptualization of sensitivity to entanglement, which will be greater for the entanglement measure that produces the larger numerical values. Here, we have investigated the partial order between the normalized versions of four entanglement measures based on Schmidt decomposition of bipartite pure quantum states, namely, concurrence, tangle, entanglement robustness and Schmidt number. We have shown that among those four measures, the concurrence and the Schmidt number have the highest and the lowest sensitivity to quantum entanglement, respectively. Further, we have demonstrated how these measures could be used to track the dynamics of quantum entanglement in a simple quantum toy model composed of two qutrits. Lastly, we have employed state-dependent entanglement statistics to compute measurable correlations between the outcomes of quantum observables in agreement with the uncertainty principle. The presented results could be helpful in quantum applications that require monitoring of the available quantum resources for sharp identification of temporal points of maximal entanglement or system separability
Quantum probability
Quantum probability is a subtle blend of quantum mechanics and classical probability theory. Its important ideas can be traced to the pioneering work of Richard Feynman in his path integral formalism.Only recently have the concept and ideas of quantum probability been presented in a rigorous axiomatic framework, and this book provides a coherent and comprehensive exposition of this approach. It gives a unified treatment of operational statistics, generalized measure theory and the path integral formalism that can only be found in scattered research articles.The first two chapters survey the n