Optimal estimation of ensemble averages from a quantum measurement

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 POVM. After reviewing the optimization of data processing that minimizes the statistical error of the estimation, we provide a compact formula for the evaluation of the estimation error.Comment: 4 pages, contribution for the proceedings of the QCMC06 at Tsukuba, Japan. qcmc06.st

Quantum cellular automata and free quantum field theory

In a series of recent papers it has been shown how free quantum field theory can be derived without using mechanical primitives (including space-time, special relativity, quantization rules, etc.), but only considering the easiest quantum algorithm encompassing a countable set of quantum systems whose network of interactions satisfies the simple principles of unitarity, homogeneity, locality, and isotropy. This has opened the route to extending the axiomatic information-theoretic derivation of the quantum theory of abstract systems to include quantum field theory. The inherent discrete nature of the informational axiomatization leads to an extension of quantum field theory to a quantum cellular automata theory, where the usual field theory is recovered in a regime where the discrete structure of the automata cannot be probed. A simple heuristic argument sets the scale of discreteness to the Planck scale, and the customary physical regime where discreteness is not visible is the relativistic one of small wavevectors. In this paper we provide a thorough derivation from principles that in the most general case the graph of the quantum cellular automaton is the Cayley graph of a finitely presented group, and showing how for the case corresponding to Euclidean emergent space (where the group resorts to an Abelian one) the automata leads to Weyl, Dirac and Maxwell field dynamics in the relativistic limit. We conclude with some perspectives towards the more general scenario of non-linear automata for interacting quantum field theory.Comment: 10 pages, 2 figures, revtex style. arXiv admin note: substantial text overlap with arXiv:1601.0483

Cellular automata in operational probabilistic theories

The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion of locality for update rules. Cellular automata are then defined as homogeneous and local update rules. Finally, we prove a general version of the wrapping lemma, that connects CA on different Cayley graphs sharing some small-scale structure of neighbourhoods.Comment: Updated version: the only change consists in the extension of the proof of lemma

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 completeness of quantum theory for predicting measurement outcomes

The predictions that quantum theory makes about the outcomes of measurements are generally probabilistic. This has raised the question whether quantum theory can be considered complete, or whether there could exist alternative theories that provide improved predictions. Here we review recent work that considers arbitrary alternative theories, constrained only by the requirement that they are compatible with a notion of "free choice" (defined with respect to a natural causal order). It is shown that quantum theory is "maximally informative", i.e., there is no other compatible theory that gives improved predictions. Furthermore, any alternative maximally informative theory is necessarily equivalent to quantum theory. This means that the state a system has in such a theory is in one-to-one correspondence with its quantum-mechanical state (the wave function). In this sense, quantum theory is complete.Comment: 15 pages, 4 figures. This is an expanded and more pedagogical version of arXiv:1005.5173 and arXiv:1111.6597 that discusses in detail the relation to other result

Quantum Walks, Weyl equation and the Lorentz group

Quantum cellular automata and quantum walks provide a framework for the foundations of quantum field theory, since the equations of motion of free relativistic quantum fields can be derived as the small wave-vector limit of quantum automata and walks starting from very general principles. The intrinsic discreteness of this framework is reconciled with the continuous Lorentz symmetry by reformulating the notion of inertial reference frame in terms of the constants of motion of the quantum walk dynamics. In particular, among the symmetries of the quantum walk which recovers the Weyl equation--the so called Weyl walk--one finds a non linear realisation of the Poincar\'e group, which recovers the usual linear representation in the small wave-vector limit. In this paper we characterise the full symmetry group of the Weyl walk which is shown to be a non linear realization of a group which is the semidirect product of the Poincar\'e group and the group of dilations.Comment: 9 pages, 2 figure

Causal influence in operational probabilistic theories

We study the relation of causal influence between input systems of a reversible evolution and its output systems, in the context of operational probabilistic theories. We analyse two different definitions that are borrowed from the literature on quantum theory -- where they are equivalent. One is the notion based on signalling, and the other one is the notion used to define the neighbourhood of a cell in a quantum cellular automaton. The latter definition, that we adopt in the general scenario, turns out to be strictly weaker than the former: it is possible for a system to have causal influence on another one without signalling to it. We stress that, according to our definition, it is impossible anyway to have causal influence in the absence of an interaction, e.g.~in a Bell-like scenario. We study various conditions for causal influence, and introduce the feature that we call {\em no interaction without disturbance}, under which we prove that signalling and causal influence coincide.Comment: 11 pages, no figures, many diagram

Superbroadcasting of conjugate quantum variables

We consider the problem of broadcasting arbitrary states of radiation modes from N to M>N copies by a map that preserves the average value of the field and optimally reduces the total noise in conjugate variables. For N>=2 the broadcasting can be achieved perfectly, and for sufficiently noisy input states one can even purify the state while broadcasting--the so-called superbroadcasting. For purification (i.e. M<=N), the reduction of noise is independent of M. Similar results are proved for broadcasting with phase-conjugation. All the optimal maps can be implemented by linear optics and linear amplification.Comment: 7 pages, 1 eps figures. Accepted for publication on Europhysics Letter