Scattering and Perturbation Theory for Discrete-Time Dynamics.

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasienergy which is defined only modulo 2π. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative quantum field theory. We use this formalism to compare the scattering amplitudes of a continuous-time model and of the corresponding discretized one. We give a rigorous assessment of the comparison for the case of bounded free Hamiltonian, as in a lattice theory with a bounded number of particles. Our framework can be applied to a wide class of quantum simulators, like quantum walks and quantum cellular automata. As a case study, we analyze the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.

Fermionic State Discrimination by Local Operations and Classical Communication.

We consider the problem of local operations and classical communication (LOCC) discrimination between two bipartite pure states of fermionic systems. We show that, contrary to the case of quantum systems, for fermionic systems it is generally not possible to achieve the ideal state discrimination performances through LOCC measurements. On the other hand, we show that an ancillary system made of two fermionic modes in a maximally entangled state is a sufficient additional resource to attain the ideal performances via LOCC measurements. The stability of the ideal results is studied when the probability of preparation of the two states is perturbed, and a tight bound on the discrimination error is derived.

Theoretical framework for higher-order quantum theory.

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.

Quantum Information and Foundations.

The new era of quantum foundations, fed by the quantum information theory experience and opened in the early 2000s by a series of memorable papers [...].

Solutions of a Two-Particle Interacting Quantum Walk.

We study the solutions of an interacting Fermionic cellular automaton which is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particle solutions of the automaton recently in the literature, which exploits the symmetries of the evolution operator. In the two-particle sector, the evolution operator is given by the sequence of two steps, the first one corresponding to a unitary interaction activated by two-particle excitation at the same site, and the second one to two independent one-dimensional Dirac quantum walks. The interaction step can be regarded as the discrete-time version of the interacting term of some Hamiltonian integrable system, such as the Hubbard or the Thirring model. The present automaton exhibits scattering solutions with nontrivial momentum transfer, jumping between different regions of the Brillouin zone that can be interpreted as Fermion-doubled particles, in stark contrast with the customary momentum-exchange of the one-dimensional Hamiltonian systems. A further difference compared to the Hamiltonian model is that there exist bound states for every value of the total momentum and of the coupling constant. Even in the special case of vanishing coupling, the walk manifests bound states, for finitely many isolated values of the total momentum. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.

Quantum walks, deformed relativity and Hopf algebra symmetries.

We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras.

Discord and nonclassicality in probabilistic theories.

Quantum discord quantifies nonclassical correlations in quantum states. We introduce discord for states in causal probabilistic theories, inspired by the original definition proposed by H. Ollivier and W. H. Zurek [Phys. Rev. Lett. 88, 017901 (2001)]. We show that the only probabilistic theory in which all states have null discord is classical probability theory. Non-null discord is then not just a quantum feature, but a generic signature of nonclassicality.

No signaling, entanglement breaking, and localizability in bipartite channels.

A bipartite quantum channel represents the interaction between systems, generally allowing for the exchange of information. A special class of bipartite channels is the no-signaling ones, which do not allow for communication. Piani et al. [Phys. Rev. A 74, 012305 (2006)] conjectured that all no-signaling channels are mixtures of entanglement breaking and localizable channels, which require only local operations and entanglement. Here we provide the general realization scheme, and give a counterexample to the conjecture, achieving no-signaling superquantum correlations while preserving entanglement.

Memory effects in quantum channel discrimination.

We consider quantum-memory assisted protocols for discriminating quantum channels. We show that for optimal discrimination of memory channels, memory assisted protocols are needed. This leads to a new notion of distance for channels with memory, based on the general theory of quantum testers. For discrimination and estimation of sets of independent unitary channels, we prove optimality of parallel protocols among all possible architectures.

Optimal cloning of unitary transformation.

After proving a general no-cloning theorem for black boxes, we derive the optimal universal cloning of unitary transformations, from one to two copies. The optimal cloner is realized by quantum channels with memory, and greatly outperforms the optimal measure-and-reprepare cloning strategy. Applications are outlined, including two-way quantum cryptographic protocols.

Secret quantum communication of a reference frame.

We propose quantum-cryptographic protocols to secretly communicate a reference frame--unspeakable information in the sense it cannot be encoded into a string of bits. Two distant parties can secretly align their Cartesian axes by exchanging N spin-1/2 particles, achieving the optimal accuracy 1/N. A possible eavesdropper cannot gain any information without being detected.

Superbroadcasting of mixed states.

We derive the optimal universal broadcasting for mixed states of qubits. We show that the no-broadcasting theorem cannot be generalized to more than a single input copy. Moreover, for four or more input copies it is even possible to purify the input states while broadcasting. We name such purifying broadcasting superbroadcasting.

Efficient universal programmable quantum measurements.

A universal programmable detector is a device that can be tuned to perform any desired measurement on a given quantum system, by changing the state of an ancilla. With a finite dimension d for the ancilla only approximate universal programmability is possible, with size d=f(epsilon(-1)) increasing the function of the "accuracy" epsilon(-1). In this Letter we show that, much better than the exponential size known in the literature, one can achieve polynomial size. An explicit example with linear size is also presented. Finally, we show that for covariant measurements exact programmability is feasible.