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We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group [Formula: see text], which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of [Formula: see text] that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.This article is part of the themed issue 'Second quantum revolution: foundational questions'.
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We present the first complete optimization of quantum tomography, for states, positive operator value measures, and various classes of transformations, for arbitrary prior ensemble and arbitrary representation, giving corresponding feasible experimental schemes in terms of random Bell measurements.
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We present a method for optimizing quantum circuits architecture, based on the notion of a quantum comb, which describes a circuit board where one can insert variable subcircuits. Unexplored quantum processing tasks, such as cloning and storing or retrieving of gates, can be optimized, along with setups for tomography and discrimination or estimation of quantum circuits.
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We address the problem of removing correlation from sets of states while preserving as much local quantum information as possible. We prove that states obtained from universal cloning can only be decorrelated at the expense of complete erasure of local information (i.e., information about the copied state). We solve analytically the problem of decorrelation for two qubits and two qumodes (harmonic oscillators in Gaussian states), and provide sets of decorrelable states and the minimum amount of noise to be added for decorrelation.
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We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data processing of the outcomes of a quantum measurement described by a positive operator-valued measure. We determine the optimal processing that minimizes the statistical error of the estimation.
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We propose a covariant protocol for transmitting reference frames encoded on N spins, achieving sensitivity N-2 without the need of a preestablished reference frame and without using entanglement between sender and receiver. The protocol exploits the use of equivalent representations that were overlooked in the previous literature.
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We show that, contrary to the widespread belief, in quantum mechanics repeatable measurements are not necessarily described by orthogonal projectors--the customary paradigm of observable. Nonorthogonal repeatability, however, occurs only for infinite dimensions. We also show that, when a nonorthogonal repeatable measurement is performed, the measured system retains some "memory" of the number of times that the measurement has been performed.