arXiv subjects

Paolo Perinotti

Publications and source records attributed to Paolo Perinotti.

At least 37 records · Page 2Linked to original sources

arXiv2023

Unifying different notions of quantum incompatibility into a strict hierarchy of resource theories of communication

While there is general consensus on the definition of incompatible POVMs, moving up to the level of instruments one finds a much less clear situation, with mathematically different and logically independent definitions of incompatibility. Here we close this gap by introducing the notion of q-compatibility, which unifies different notions of POVMs, channels, and instruments incompatibility into one hierarchy of resource theories of communication between separated parties. The resource theories that we obtain are complete, in the sense that they contain complete families of free operations and monotones providing necessary and sufficient conditions for the existence of a transformation. Furthermore, our framework is fully operational, in the sense that free transformations are characterized explicitly, in terms of local operations aided by causally-constrained directed classical communication, and all monotones possess a game-theoretic interpretation making them experimentally measurable in principle. We are thus able to pinpoint exactly what each notion of incompatibility consists of, in terms of information-theoretic resources.

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arXiv2022

Incompatibility of observables, channels and instruments in information theories

Every theory of information, including classical and quantum, can be studied in the framework of operational probabilistic theories--where the notion of test generalizes that of quantum instrument, namely a collection of quantum operations summing to a channel, and simple rules are given for the composition of tests in parallel and in sequence. Here we study the notion of compatibility for tests of an operational probabilistic theory. Following the quantum literature, we first introduce the notion of strong compatibility, and then we illustrate its ultimate relaxation, that we deem weak compatibility. It is shown that the two notions coincide in the case of observation tests--which are the counterpart of quantum POVMs--while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a theory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.

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arXiv2022

Shannon theory beyond quantum: information content of a source

The information content of a source is defined in terms of the minimum number of bits needed to store the output of the source in a perfectly recoverable way. A similar definition can be given in the case of quantum sources, with qubits replacing bits. In the mentioned cases the information content can be quantified through Shannon's and von Neumann's entropy, respectively. Here we extend the definition of information content to operational probabilistic theories, and prove relevant properties as the subadditivity, and the relation between purity and information content of a state. We prove the consistency of the present notion of information content when applied to the classical and the quantum case. Finally, the relation with one of the notions of entropy that can be introduced in general probabilistic theories, the maximum accessible information, is given in terms of a lower bound.

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arXiv2022

Classicality without local discriminability: decoupling entanglement and complementarity

An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed. The theory is endowed with a rule for composing an arbitrary number of systems, and with a nontrivial set of transformations. Hence, we demonstrate that the presence of entanglement is independent of the existence of incompatible measurements. We then study a variety of phenomena occurring in the theory -- some of them contradicting both Classical and Quantum Theories -- including: cloning, entanglement swapping, dense coding, additivity of classical capacities, non-monogamous entanglement, hypersignaling. We also prove the existence, in the theory, of a universal processor. The theory is causal and satisfies the no-restriction hypothesis. At the same time, it violates a number of information-theoretic principles enjoyed by Quantum Theory, most notably: local discriminability, purity of parallel composition of states, and purification. Moreover, we introduce an exhaustive procedure to construct generic operational probabilistic theories, and a sufficient set of conditions to verify their consistency. In addition, we prove a characterisation theorem for the parallel composition rules of arbitrary theories, and specialise it to the case of bilocal-tomographic theories. We conclude pointing out some open problems. In particular, on the basis of the fact that every separable state of the theory is a statistical mixture of entangled states, we formulate a no-go conjecture for the existence of a local-realistic ontological model.

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arXiv2021

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. Remarkably, the counterexample comes from classical theory, where the proposed notion of causal influence determines a redefinition of the neighbourhood of a cell in cellular automata. 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 no interaction without disturbance, under which we prove that signalling and causal influence coincide. The proposed definition has interesting consequences on the analysis of causal networks, and leads to a revision of the notion of neighbourhood for classical cellular automata, clarifying a puzzle regarding their quantisation that apparently makes the neighbourhood larger than the original one.

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arXiv2021

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.

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arXiv2021

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 quasi-energy which is defined only modulo $2 π$. 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 analyse the scattering properties of a one-dimensional cellular automaton with locally interacting fermions.

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arXiv2021

Shannon theory for quantum systems and beyond: information compression for fermions

We address the task of compression of fermionic quantum information. Due to the parity superselection rule, differently from the case of encoding of quantum information in qubit states, part of the information carried by fermionic systems is encoded in their delocalised correlations. As a consequence, reliability of a compression protocol must be assessed in a way that necessarily accounts also for the preservation of correlations. This implies that input/output fidelity is not a satisfactory figure of merit for fermionic compression schemes. We then discuss various aspects regarding the assessment of reliability of an encoding scheme, and show that entanglement fidelity in the fermionic case is capable of evaluating the preservation of correlations, thus revealing itself strictly stronger than input/output fidelity, unlike the qubit case. We then introduce a fermionic version of the source coding theorem showing that, as in the quantum case, the von Neumann entropy is the minimal rate for which a fermionic compression scheme exists, that is reliable according to the entanglement fidelity criterion.

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arXiv2021

Unambiguous discrimination of Fermionic states through local operations and classical communication

The paper studies unambiguous discrimination of Fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two Fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two Fermionic modes, independently of the dimension of the system, prepared in a maximally entangled state: in this case, LOCC protocols achieve the optimal success probability.

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arXiv2020

Information and disturbance in operational probabilistic theories

Any measurement is intended to provide information on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. The traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. Moreover, since in a theory without causality information can be gathered also on the effect, we generalise the notion of no-information test. We then prove an equivalent condition for no-information without disturbance-atomicity of the identity-namely the impossibility of achieving the trivial evolution-the identity-as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. Based on this, we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. As a consequence, a theory is such that any information can be extracted without disturbance only if all its systems are classical. Finally, we show via concrete examples that no-information without disturbance is independent of both local discriminability and purification.

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arXiv2020

Classical theories with entanglement

We investigate operational probabilistic theories where the pure states of every system are the vertices of a simplex. A special case of such theories is that of classical theories, i.e. simplicial theories whose pure states are jointly perfectly discriminable. The usual Classical Theory satisfies also local discriminability. However, simplicial theories---including the classical ones---can violate local discriminability, thus admitting of entangled states. First, we prove sufficient conditions for the presence of entangled states in arbitrary probabilistic theories. Then, we prove that simplicial theories are necessarily causal, and this represents a no-go theorem for conceiving non-causal classical theories. We then provide necessary and sufficient conditions for simplicial theories to exhibit entanglement, and classify their system-composition rules. We conclude proving that, in simplicial theories, an operational formulation of the superposition principle cannot be satisfied, and that---under the hypothesis of $n$-local discriminability---no mixed state admits of a purification. Our results hold also in the general case where the sets of states fail to be convex.

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arXiv2020

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.

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arXiv2019

Chirality from quantum walks without quantum coin

Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom---called the coin and represented by a finite-dimensional Hilbert space---is associated to each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.

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arXiv2019

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.

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arXiv2019

Symmetries of the Dirac quantum walk and emergence of the de Sitter group

A quantum walk describes the discrete unitary evolution of a quantum particle on a discrete graph. Some quantum walks, referred to as the Weyl and Dirac quantum walks, provide a description of the free evolution of relativistic quantum fields in a regime where the wave-vectors involved in the particle state are small. The clash between the intrinsic discreteness of quantum walks and the symmetries of special relativity can be resolved by rethinking the notion of a change of inertial reference frame. We give here a definition of the latter that avoids a pre-defined space-time geometry, in terms of a change of values of the constants of motion that leaves the walk operator unchanged. Starting from the family of 1+1 dimensional Dirac quantum walks with all possible values of the mass parameter, we introduce a unique walk encompassing the latter as an extra degree of freedom, and we derive its group of changes of inertial frames. This symmetry group contains a non linear realization of $SO^+(2,1) \ltimes \mathbb{R}^3$; since one of the two space-like dimensions does not correspond to an actual spatial degree of freedom but rather the mass, we interpret it as a 2+1 dimensional de-Sitter group. This group group contains also a non-linear realisation of the proper orthochronous Poincaré group $SO^+(1,1) \ltimes \mathbb{R}^2$ in 1+1 dimension, as the ones considered within the framework of doubly special relativity, which recovers the usual relativistic symmetry of the Dirac Equation in the limit of small wave-vectors and masses. Surprisingly, if one considers the Dirac walk with a fixed value of the mass parameter, the group of allowed changes of reference frame does not have a consistent interpretation in the relativistic limit of small wave-vectors.

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arXiv2018

Scalar fermionic cellular automata on finite Cayley graphs

A map on finitely many fermionic modes represents a unitary evolution if and only if it preserves canonical anti-commutation relations. We use this condition for the classification of fermionic cellu- lar automata (FCA) on Cayley graphs of finite groups in two simple but paradigmatic case studies. The physical properties of the solutions are discussed. Finally, features of the solutions that can be extended to the case of cellular automata on infinite graphs are analyzed.

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arXiv2018

Solutions of a two-particle interacting quantum walk

We study the solutions of the interacting Fermionic cellular automaton introduced in Ref. [Phys Rev A 97, 032132 (2018)]. The automaton is the analogue of the Thirring model with both space and time discrete. We present a derivation of the two-particles solutions of the automaton, which exploits the symmetries of the evolution operator. In the two-particles 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 even for vanishing coupling constant. As a complement to the analytical derivations we show numerical simulations of the interacting evolution.

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arXiv2017

Isotropic quantum walks on lattices and the Weyl equation

We present a thorough classification of the isotropic quantum walks on lattices of dimension $d=1,2,3$ for cell dimension $s=2$. For $d=3$ there exist two isotropic walks, namely the Weyl quantum walks presented in Ref. [G. M. D'Ariano and P. Perinotti, Phys. Rev. A 90, 062106 (2014)], resulting in the derivation of the Weyl equation from informational principles. The present analysis, via a crucial use of isotropy, is significantly shorter and avoids a superfluous technical assumption, making the result completely general.

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