Geometric Event-Based Relativistic Quantum Mechanics

Universita’ di Pavia
We propose a special relativistic framework for quantum mechanics. It is based on introducing a Hilbert space for events. Events are taken as primitive notions (as customary in relativity), whereas quantum systems (e.g. fields and particles) are emergent in the form of joint probability amplitudes for position and time of events. Textbook relativistic quantum mechanics and quantum field theory can be recovered by dividing the event Hilbert spaces into space and time (a foliation) and then conditioning the event states onto the time part. Our theory satisfies the full Poincare’ symmetry as a ‘geometric’ unitary transformation, and possesses observables for space (location of an event) and time (position in time of an event).

The p-adic qubits

Within the framework of quantum mechanics over (a quadratic extension of) the non-
Archimedean field of p-adic numbers Qp, we provide a general definition of a quantum state
relying on an algebraic approach and on a suitable p-adic model of probability theory. As in
the standard complex case, a distinguished class of physical states are related to a notion of a trace for a bounded operator, and one can define a suitable class of trace class operators in the non-Archimedean setting. Eventually, we will particularize our discussion to two-dimensional systems, thus obtaining a p-adic model of the qubit.

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 oper- ations 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 compatibil- ity for tests an operational probabilistic theory. Following the quantum litera- ture, 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 counter- part of quantum POVMs—while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a the- ory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.

Multi-parameter digital quantum estimation

Global quantum estimation strategies allow to extract information on a
phase or a set of phases encoded on a quantum state without any prior
knowledge about them. As far as we know, unlike the local estimation case,
a general model encompassing all of the global strategies does not exist
yet. We show that Holevo’s estimation theory provides a good candidate,
i.e. a model that allows to retrieve both parallel and sequential
strategies. Moreover, it yields a framework for multiparameter global
estimation, since the model can be straightforwardly generalized for the
estimation of more than one phase. In particular, we focus on the case of
double-parameter estimation and explore the advantages of multiparameter
estimation with respect to multiple single-parameter estimations in terms
of mutual information.

Here we summarize recent theoretical studies on the dynamical Casimir effects (DCEs) in optomechanical systems.

We studied the DCE using a fully quantum-mechanical description and without linearizing the dynamics. We have shown that the resonant generation of pho- tons from the vacuum is determined by a ladder of mirror-field vacuum Rabi split- ting. We find that vacuum emission can originate from the free evolution of an initial pure mechanical excited state, in analogy with the spontaneous emission from excited atoms. We also show that the DCE can also be driven by incoherent mechanical pumping. We then applied this framework to study the interaction of two mechanical oscillators mediated by the exchange of virtual photon pairs. Specifically, we demonstrated that mechanical quantum excitations can be co- herently transferred among spatially separated mechanical oscillators, through a dissipationless quantum bus, due to the exchange of virtual photon pairs. This system can also operate as a mechanical parametric downconverter.

Cavity optomechanics with levitated nanosphere: Quantum signatures in hybrid light-mechanical states and advances in two-dimensional cooling

Optically levitated nanospheres in high-finesse cavities offer a unique platform for the study of macroscopic quantum mechanics in all three spatial dimensions and the achievement of quantum coherent control of their motion, with appli- cations ranging from quantum foundations and information processing to direc- tional quantum sensing. We report on cavity-optomechanical experiments in which the motion of a nanosphere levitated in high vacuum is strongly coupled to a single cavity mode by coherent scattering of the tweezer photons. The two- dimensional motion on the plane orthogonal to the tweezer axis and the optical cavity mode define an optomechanical system with three degrees of freedom. In the quantum-coherent strong-coupling regime, i.e. when the optomechani- cal coupling exceeds the total decoherence rate, we observe the formation of hybrid light-mechanical states with a peculiar vectorial nature. Such states give rise to polaritonic dispersion relations characterized by two avoided crossings at different frequencies, unambiguous signature of the strong three-body interac- tions. For appropriate frequencies and polarization of the tweezer beam, the motion of the particle is strongly cooled in the plane orthogonal to the tweezer axis. We demonstrate a regime in which the fully 2D dynamics of the nanopar- ticle exhibits strong non-classical properties, and we introduce an indicator that quantifies how close the system is to a minimum uncertainty state. These find- ings pave the way to novel protocols for the transfer of quantum information between photonic and phononic components and represent an important step towards the demonstration of optomechanical entangled states at room tem- perature.

Two-membrane cavity optomechanics and Quantum Technologies

The linear and non-linear dynamics of an optomechanical system made of a two- membrane ethalon in a high-finesse Fabry-Pérot cavity is presented. This two- membrane setup has the capacity to enhance the single-photon optomechanical coupling, and in the linearized interaction regime to cool simultaneously two me- chanical oscillators. The experimental characterization of the optical, mechan- ical, and especially optomechanical properties of a two-membrane sandwich within an optical cavity will be presented. In the non-linear regime, a truthful de- tection of membrane displacements much above the usual linear sensing limited by the cavity linewidth is presented. The non-linear dynamics of the mechanical oscillator provides a novel procedure for the determination of the single-photon optomechanical coupling rate, that is the optomechanical interaction strength of the system. In the second part of the talk, we will illustrate the QUANTum Exper- imental Platform (QUANTEP) INFN project, which aims at the development and implementation of a complete Silicon Photonics Integrated platform for Quan- tum Computation with linear optics circuits, focusing on its recent achievements.

Quantum-jump trajectories from non-linear rate operators: continuous measurements and non-Markovianity

Stochastic methods with quantum jumps are routinely used to describe open quantum system dynamics, both for the advantage they provide from a computa- tional point of view, and for their insight into fundamental topics, such as the role of measurements in quantum mechanics and the description of non-Markovian memory effects. In this talk, I will present a recently introduced quantum-jump approach [1], named rate operator quantum jumps (ROQJ), whose quantum- jump trajectories are defined in terms of state-dependent, non-linear rate op- erators. I will first show that ROQJ is associated with a systematic continuous- measurement scheme for a wide and physically relevant class of dynamics, in- cluding a set of master equations with negative decay rates, where the standard Monte Carlo wave function (MCWF) approach [2] to quantum jumps does not apply. I will then discuss how ROQJ can be extended to deal with general non- Markovian evolutions, going beyond the current non-Markovian generalizations of MCWF [3] and helping clarify the different types of memory effects within the context of quantum jumps. Finally, I will show that different quantum-jump pic- tures can be formulated by exploiting the freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the trajectories, which can result in a significant simplification of the corresponding stochastic description [4]. [1] A. Smirne, M. Caiaffa, and J. Piilo, Phys. Rev. Lett. 124, 190402 (2020) [2] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992) [3] J. Piilo, S. Maniscalco, K. Härkönen, and K.A. Suominen, Phys. Rev. Lett. 100,180402 (2008) [4] D. Chruscinski, K. Luoma, J. Piilo, A. Smirne, arXiv:2009.11312v2 (2021)

Quantum transfer of interacting qubits

The transfer of quantum information between different locations is key to many quantum information processing tasks. Whereas, the transfer of a single qubit state has been extensively investigated, the transfer of a many-body system con- figuration has insofar remained elusive. We address the problem of transferring the state of n interacting qubits. Both the exponentially increasing Hilbert space dimension, and the presence of interactions significantly scale-up the complexity of achieving high-fidelity transfer. By employing tools from random matrix the- ory and using the formalism of quantum dynamical maps, we derive a general expression for the average and the variance of the fidelity of an arbitrary quan- tum state transfer protocol for n interacting qubits. Finally, by adopting a weak- coupling scheme in a spin chain, we obtain the explicit conditions for high-fidelity transfer of 3 and 4 interacting qubits. https://doi.org/10.48550/arXiv.2205.01579

Exact solution for the quantum and private capacities of bosonic dephasing channels

The capacities of noisy quantum channels capture the ultimate rates of infor- mation transmission across quantum communication lines, and the quantum ca- pacity plays a key role in determining the overhead of fault-tolerant quantum computation platforms. In the case of bosonic systems, central to many applica- tions, no closed formulas for these capacities were known for bosonic dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise affecting superconducting circuits or fiber-optic communication channels. Here we pro- vide the first exact calculation of the quantum, private, two-way assisted quan- tum, and secret-key agreement capacities of all bosonic dephasing channels. We prove that that they are equal to the relative entropy of the distribution underly- ing the channel to the uniform distribution. Our result solves a problem that has been open for over a decade, having been posed originally by [Jiang & Chen, Quantum and Nonlinear Optics 244, 2010].