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Non-Markovian dynamics in open quantum systems arise when the system's evolution is influenced by its past interactions with the environment. Here, we present a novel metric for quantifying non-Markovianity based on local quantum Fisher information (LQFI). The proposed metric offers a distinct perspective compared to existing measures, providing a deeper understanding of information flow between the system and its environment. By comparing the LQFI-based measure to the LQU-based measure, we demonstrate its effectiveness in detecting non-Markovianity and its ability to capture the degree of non-Markovian behavior in various quantum channels. Furthermore, we show that a positive time derivative of LQFI signals the flow of information from the environment to the system, providing a clear interpretation of non-Markovian dynamics. Finally, the computational efficiency of the LQFI-based measure makes it a practical tool for characterizing non-Markovianity in diverse physical systems.

The purpose of the present work is to construct estimators for the random effects in a fractional diffusion model using a hybrid estimation method where we combine parametric and nonparametric thechniques. We precisely consider $n$ stochastic processes $\left\{X_t^j,\ 0\leq t\leq T\right\}$, $j=1,\ldots, n$ continuously observed over the time interval $[0,T]$, where the dynamics of each process are described by fractional stochastic differential equations with drifts depending on random effects. We first construct a parametric estimator for the random effects using the techniques of maximum likelihood estimation and we study its asymptotic properties when the time horizon $T$ is sufficiently large. Then by taking into account the obtained estimator for the random effects, we build a nonparametric estimator for their common unknown density function using Bernstein polynomials approximation. Some asymptotic properties of the density estimator, such as its asymptotic bias, variance and mean integrated squared error, are studied for an infinite time horizon $T$ and a fixed sample size $n$. The asymptotic normality and the uniform convergence of the estimator are investigated for an infinite time horizon $T$, a high frequency and as the order of Bernstein polynomials is sufficiently large. Some numerical simulations are also presented to illustrate the performance of the Bernstein polynomials based estimator compared to standard Kernel estimator for the random effects density function.

Improving interferometric phase sensitivity is crucial for high-precision measurements in rapidly developing quantum technologies. The Mach-Zehnder interferometer (MZI) is a versatile tool for analyzing this phenomenon. By splitting and recombining a light beam using beam splitters, MZIs allow for precise phase sensitivity analysis using tools like the quantum Cram\'er-Rao bound (QCRB) and the quantum Fisher information (QFI). This paper analyzes the phase sensitivity of a MZI in various scenarios using different detection schemes and input states. We compare the single- and two-parameter quantum estimation and their associated QCRB for three phase-shift situations: in both arms, only in the upper arm (asymmetric), and in both arms symmetrically. We then investigate the phase sensitivity under three detection schemes: difference intensity, single-mode intensity, and balanced homodyne. Additionally, we explore the use of Perelomov and Barut-Girardello coherent states, two types of SU(1,1) coherent states, in all scenarios. Notably, we demonstrate that under optimal conditions, all detection schemes can achieve the QCRB by utilizing entangled SU(1,1) coherent states as input states.

Cyclic quantum teleportation schemes requires at least the existence of three collaborators acting all as senders and receivers of quantum information, each one of them has an information to be transmitted to the next neighbour in a circular manner. Here, new cyclic quantum teleportation scheme is proposed for perfectly transmitting cyclically three arbitrary unknown two-qubit states ($\alpha$, $\beta$ and $\gamma$) among the three collaborators. In this scheme, Alice can send to Bob the quantum information contained in her two-qubit state $\alpha$ and receive from Charlie the quantum information contained in the two-qubit state in his possession $\gamma$ and similarly, Bob can transmit to Charlie the quantum information contained in his two-qubit state $\beta$ through a quantum channel of twelve-qubit state consisting of a six-qubit cluster state and a six-qubit entangled state by sequentially and cyclically performing Bell state measurements. Subsequently, each one of the three participants can afterwards retrieve his own desired two-qubit state using classical channel and by performing appropriate unitary Pauli operators and we have shown that our proposed scheme performs efficiently.

In this work, we consider \textit{quantum unital Otto heat engines}. The latter refers to the fact that both the unitaries of the adiabatic strokes and the source of the heat provided to the engine preserve the maximally mixed state. We show how to compute the cumulants of either the dephased or undephased engine. For a qubit, we give the analytical expressions of the averages and variances for arbitrary unitaries and unital channels. We do a detailed comparative study between the dephased and undephased heat engines. More precisely, we focus on the effect of the parameters on the average work and its reliability and efficiency. As a case study of unital channels, we consider a quantum projective measurement. We show on which basis we should projectively measure the qubit, either the dephased or undephased heat engine, to extract higher amounts of work, increase the latter's reliability, and increase efficiency. Further, we show that non-adiabatic transitions \textit{are not always detrimental} to thermodynamic quantities. Our results, we believe, are important for heat engines fueled by \textit{quantum measurement}.

We investigate the dynamics of non-classical correlations and quantum coherence in open quantum systems by employing metrics like local quantum Fisher information, local quantum uncertainty, and quantum Jensen-Shannon divergence. Our focus here is on a system of two qubits in two distinct physical situations: the first one when the two qubits are coupled to a single-mode cavity, while the second consists of two qubits immersed in dephasing reservoirs. Our study places significant emphasis on how the evolution of these quantum criterion is influenced by the initial state's purity (whether pure or mixed) and the nature of the environment (whether Markovian or non-Markovian). We observe that a decrease in the initial state's purity corresponds to a reduction in both quantum correlations and quantum coherence, whereas higher purity enhances these quantumness. Furthermore, we establish a quantum teleportation strategy based on the two different physical scenarios. In this approach, the resulting state of the two qubits functions as a quantum channel integrated into a quantum teleportation protocol. We also analyze how the purity of the initial state and the Markovian or non-Markovian regimes impact the quantum teleportation process.

In quantum phase estimation, the Heisenberg limit provides the ultimate accuracy over quasi-classical estimation procedures. However, realizing this limit hinges upon both the detection strategy employed for output measurements and the characteristics of the input states. This study delves into quantum phase estimation using $s$-spin coherent states superposition. Initially, we delve into the explicit formulation of spin coherent states for a spin $s=3/2$. Both the quantum Fisher information and the quantum Cramer-Rao bound are meticulously examined. We analytically show that the ultimate measurement precision of spin cat states approaches the Heisenberg limit, where uncertainty decreases inversely with the total particle number. Moreover, we investigate the phase sensitivity introduced through operators $e^{i\zeta{S}_{z}}$, $e^{i\zeta{S}_{x}}$ and $e^{i\zeta{S}_{y}}$, subsequently comparing the resultants findings. In closing, we provide a general analytical expression for the quantum Cramer-Rao boundary applied to these three parameter-generating operators, utilizing general $s$-spin coherent states. We remarked that attaining Heisenberg-limit precision requires the careful adjustment of insightful information about the geometry of $s$-spin cat states on the Bloch sphere. Additionally, as the number of $s$-spin increases, the Heisenberg limit decreases, and this reduction is inversely proportional to the $s$-spin number., Comment: 15 pages, 63 figures

Beam splitters are optical elements widely used in modern technological applications to split the initial light beam into a required number of beams and they play a very promising role for generating entangled optical states. Here, a potential scheme is proposed to generate Bell coherent-states superpositions through the action of a beam splitter when a Glauber coherent state is injected on one input mode and vacuum state is incident on the other one. Different quantifiers are used to measure the quantumness in the output state such as concurrence entanglement, entropic quantum discord, quantum coherence, geometric measure of quantum discord, local quantum uncertainty (LQU) and local quantum Fisher information. Thereby, we derive their analytical formulas and focus more on the behavior and bounds of each measure. Besides, we have introduced the notion of "weak measurement-induced LQU" captured by weak measurements as the generalization of normal LQU defined for standard projective measurement, and we investigate the effect of the measurement strength on the estimated phase enhancement if the generated Bell cat states are the probe states in quantum metrology. Our results suggest that the sensitivity of the interferometric phase estimation depends on how strongly one perturbs the probe state and that a weak measurement does not necessarily capture more quantumness in composite system., Comment: 15 pages and 10 figures, published version in the Journal of the Optical Society of America B

Quantum teleportation has become a fundamental building block of quantum technologies, playing a vital role in the development of quantum communication networks. Here, we present a bidirectional quantum teleportation (BQT) protocol that enables even and odd coherent states to be transmitted and reconstructed over arbitrary distances in two directions. To this end, we employ the multipartite Glauber coherent state, comprising the Greenberger-Horne-Zeilinger, ground and Werner states, as a quantum resource linking distant partners Alice and Bob. The pairwise entanglement existing in symmetric and antisymmetric multipartite coherent states is explored, and by controlling the overlap and number of probes constructing various types of quantum channels, the teleportation efficiency of teleported states in both directions may be maximized. Besides, Alice's and Bob's trigger phases are estimated to explore their roles in our protocol using two kinds of quantum statistical speed referred to as quantum Fisher information (QFI) and Hilbert-Schmidt speed (HSS). Specifically, we show that the lower bound of the statistical estimation error, quantified by QFI and HSS, corresponds to the highest fidelity from Alice to Bob and conversely from Bob to Alice, and that the choice of the pre-shared quantum channel has a critical role in achieving high BQT efficiency. Finally, we show how to implement the suggested scheme on current experimental tools, where Alice can transfer her even coherent state to Bob, and at the same time, Bob can transfer his odd coherent state to Alice.

Recently, measurement-based quantum thermal machines have drawn more attention in the field of quantum thermodynamics. However, the previous results on quantum Otto heat engines were either limited to special unital and non-unital channels in the bath stages, or a specific driving protocol at the work strokes and assuming the cycle being time-reversal symmetric i.e. $V^{\dagger}=U$ (or $V=U$). In this paper, we consider a single spin-1/2 quantum Otto heat engine, by first replacing one of the heat baths by an arbitrary unital channel and then we give the exact analytical expression of the characteristic function from which all the cumulants of heat and work emerge. We prove that under the effect of monitoring, $\nu_{2}>\nu_{1}$ is a necessary condition for positive work, either for a symmetric or asymmetric-driven Otto cycle. Furthermore, going beyond the average we show that the ratio of the fluctuations of work and heat is lower and upper-bounded when the system is working as a heat engine. However, differently from the previous results in the literature, we consider the third and fourth cumulants as well. It is shown that the ratio of the third (fourth) cumulants of work and heat is not upper-bounded by unity nor lower-bounded by the third (fourth) power of the efficiency, as is the case for the ratio of fluctuations. Finally, we consider applying a specific unital map that plays the role of a heat bath in a coherently superposed, manner and we show the role of the initial coherence of the control qubit on efficiency, on the average work and its relative fluctuations.

Recently, the Hilbert-Schmidt speed, as a special class of quantum statistical speed, has been reported to improve the interferometric phase in single-parameter quantum estimation. Here, we test this concept in the multiparameter scenario where two laser phases are estimated in a theoretical model consisting of a three-level atom interacting with two classical monochromatic fields. When the atom is initially prepared in the lower bare state taking into account the detuning parameters, we extract an exact analytical solution of the atomic density matrix in the case of two-photon resonant transition. Further, we compare the performance of laser phase parameters estimation in individual and simultaneous metrological strategies, and we explore the role of quantum coherence in improving the efficiency of unknown multi-phase shift estimation protocols. The obtained results show that the Hilbert-Schmidt speed detects the lower bound on the statistical estimation error as well as the optimal estimation regions, where its maximal corresponds to the maximal quantum Fisher information, the performance of simultaneous multiparameter estimation with individual estimation inevitably depends on the detuning parameters of the three-level atom, and not only the quantum entanglement, but also the quantum coherence is a crucial resource to improve the accuracy of a metrological protocol.

With the growth of geometric science, including the methods of exploring the world of information by means of modern geometry, there has always been a mysterious and fascinating ambiguous link between geometric, topological and dynamical characteristics with quantum entanglement. Since geometry studies the interrelations between elements such as distance and curvature, it provides the information sciences with powerful structures that yield practically useful and understandable descriptions of integrable quantum systems. We explore here these structures in a physical system of $N$ interaction spin-$1/2$ under all-range Ising model. By performing the system dynamics, we determine the Fubini-Study metric defining the relevant quantum state space. Applying Gaussian curvature within the scope of the Gauss-Bonnet theorem, we proved that the dynamics happens on a closed two-dimensional manifold having both a dumbbell-shape structure and a spherical topology. The geometric and topological phases appearing during the system evolution processes are sufficiently discussed. Subsequently, we resolve the quantum brachistochrone problem by achieving the time-optimal evolution. By restricting the whole system to a two spin-$1/2$ system, we investigate the relevant entanglement from two viewpoints; The first is of geometric nature and explores how the entanglement level affects derived geometric structures such as the Fubini-Study metric, the Gaussian curvature, and the geometric phase. The second is of dynamic nature and addresses the entanglement effect on the evolution speed and the related Fubini-Study distance. Further, depending on the degree of entanglement, we resolve the quantum brachistochrone problem., Comment: Published version: Phys. J. Plus, 139 (2024) 771

Quantum resource theories allow us to quantify a useful quantum phenomenon, to develop new protocols for its detection and determine the exact processes that maximize its use for practical tasks. These theories aim at transforming physical phenomena, such as entanglement and quantum coherence, into useful properties for the execution of concrete tasks related to quantum information. In this thesis, we focus on the resource theories of entanglement, discord-like quantum correlations, and quantum coherence, the most intriguing quantum phenomena exploited so far in quantum information theory. We begin by presenting in detail the theoretical tools of these quantum resources, focusing on the most remarkable techniques and computational problems. In this sense, we discuss several mathematical methods that solve some problems related to their quantifications, and some analytical results for bipartite quantum systems are given. We also examine the intrinsic connections between these quantum resources by extracting the links that unite the corresponding measures. In contrast, the revolution of quantum technology has led to a growing interest in quantum metrology, and quantum entanglement has been employed to overcome the classical limit in several quantum estimation protocols. In this work, we analyze the role of quantum correlations beyond entanglement in improving the accuracy of an unknown parameter. According to our results, correlations can be captured using quantum Fisher information, and quantum discord correlations can be exploited to ensure the accuracy of phase estimation protocols. This thesis includes also the contributions on the dynamics of these quantum resources in various models of open quantum systems. Among our objectives, is to study the effects of the environment on these quantum resources and to obtain techniques to protect them from the effects of intrinsic decoherence., Comment: PhD Dissertation, in French language. Successfully defended on September 11, 2021

Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we have proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-$\mathtt{s}$ system, we explore the relevant entanglement from two different perspectives; The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree., Comment: Published version in Phys. Rev. A

In statistical estimation theory, it has been shown previously that the Wigner-Yanase skew information is bounded by the quantum Fisher information associated with the phase parameter. Besides, the quantum Cram\'er-Rao inequality is expressed in terms of skew information. Since these two fundamental quantities are based on the concept of quantum uncertainty, we derive here their analytical formulas for arbitrary two qubit $X$-states using the same analytical procedures. A comparison of these two informational quantifiers for two quasi-Werner states composed of two bipartite superposed coherent states is examined. Moreover, we investigated the decoherence effects on such quantities generated by the phase damping, depolarization and amplitude damping channels. We showed that decoherence strongly influences the quantum criteria during the evolution and these quantities exhibit similar dynamic behaviors. This current work is characterized by the fact that these two concepts play the same role and capture similar properties in quantum estimation protocols., Comment: Published version in Physica A

Quantum teleportation of an unknown state basing on the interaction between discrete-valued states (DV) and continuous-valued states (CV) presented a particular challenge in quantum technologies. Here we consider the problem of controlled quantum teleportation of an amplitude-matched CV qubit, encoded by a coherent state of a varied phase as a superposition of the vacuum- and single-photon optical states among two distant partners Alice and Bob, with the consent of controller, Charlie. To achieve this task, we use an hybrid tripartite entangled state (interaction between the discrete and continuous variables states) as the quantum resource where the coherent part belongs to Alice, while the single-photon belongs to Bob and Charlie and the CV qubit is at the disposal of Alice. The discrete-continuous interaction is realized on highly transmissive beam-splitter. We have shown that the perfectly of teleportation fidelity depends on the phase difference between the phase of the state to teleport and the phase of the sender's mode, we found that for a difference which approaches 0 or $\pi$, near perfect controlled quantum teleportation can be obtained in terms of the fidelity and independently of the amplitude $\alpha$ and the squeezing parameter $\zeta$. Experimentally, this proposed scheme has been implemented using linear optical components such as beam splitter, phase shifters and photon counters., Comment: Published version in Physica Scripta