Extremal generalized quantum measurements

A measurement on a section K of the set of states of a finite dimensional C*-algebra is defined as an affine map from K to a probability simplex. Special cases of such sections are used in description of quantum networks, in particular quantum channels. Measurements on a section correspond to equivalence classes of so-called generalized POVMs, which are called quantum testers in the case of networks. We find extremality conditions for measurements on K and characterize generalized POVMs such that the corresponding measurement is extremal. These results are applied to the set of channels. We find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1-tester such that the corresponding measurement is not extremal.Comment: 13 pages. The paper was rewritten, reorganized and shortened, the title changed, references were added. Main results are the sam

Does cointegration matter? An analysis in a RBC perspective

The aim of this paper is to verify if a proper SVEC representation of a standard Real Business Cycle model exists even when the capital stock series is omitted. The argument is relevant as the common unavailability of su¢ ciently long medium-frequency capital series prevent researchers from including capital in the widespread structural VAR (SVAR) representations of DSGE models - which is supposed to be the cause of the SVAR biased estimates. Indeed, a large debate about the truncation and small sample bias a¤ecting the SVAR performance in approximating DSGE models has been recently rising. In our view, it might be the case of a smaller degree of estimates distorsions when the RBC dynamics is approximated through a SVEC model as the information provided by the cointegrating relations among some variables might compensate the exclusion of the capital stock series from the empirical representation of the model.RBC, SVAR, SVEC model, cointegration

Quantum conditional operations

An essential element of classical computation is the "if-then" construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.Comment: 7 page

The Dirac Quantum Cellular Automaton in one dimension: Zitterbewegung and scattering from potential

We study the dynamical behaviour of the quantum cellular automaton of Refs. [1, 2], which reproduces the Dirac dynamics in the limit of small wavevectors and masses. We present analytical evaluations along with computer simulations, showing how the automaton exhibits typical Dirac dynamical features, as the Zitterbewegung and the scattering behaviour from potential that gives rise to the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical and atomic-physics quantum simulator.Comment: 8 pages, 7 figure

Quantum Field as a quantum cellular automaton: the Dirac free evolution in one dimension

We present a quantum cellular automaton model in one space-dimension which has the Dirac equation as emergent. This model, a discrete-time and causal unitary evolution of a lattice of quantum systems, is derived from the assumptions of homogeneity, parity and time-reversal invariance. The comparison between the automaton and the Dirac evolutions is rigorously set as a discrimination problem between unitary channels. We derive an exact lower bound for the probability of error in the discrimination as an explicit function of the mass, the number and the momentum of the particles, and the duration of the evolution. Computing this bound with experimentally achievable values, we see that in that regime the QCA model cannot be discriminated from the usual Dirac evolution. Finally, we show that the evolution of one-particle states with narrow-band in momentum can be effi- ciently simulated by a dispersive differential equation for any regime. This analysis allows for a comparison with the dynamics of wave-packets as it is described by the usual Dirac equation. This paper is a first step in exploring the idea that quantum field theory could be grounded on a more fundamental quantum cellular automaton model and that physical dynamics could emerge from quantum information processing. In this framework, the discretization is a central ingredient and not only a tool for performing non-perturbative calculation as in lattice gauge theory. The automaton model, endowed with a precise notion of local observables and a full probabilistic interpretation, could lead to a coherent unification of an hypothetical discrete Planck scale with the usual Fermi scale of high-energy physics.Comment: 21 pages, 4 figure