Graph States as a Resource for Quantum Metrology.

In quantum metrology, achieving optimal simultaneous multiparameter estimation is of great significance but remains highly challenging. The research approach involving evolution on dynamics provides a framework to investigate simultaneous multiparameter estimation within graph states. For single‐parameter estimation, it is observed that the precision limit exceeds the Heisenberg limit in higher‐dimensional spin systems. For multiparameter estimation, two scenarios are considered: one with commutative Hamiltonian operators and another with non‐commutative Hamiltonian operators. The results demonstrate that the global estimation precision exceeds the local estimation precision. Under the conditions of parameter limit, the precision of parameter estimation for simultaneously estimating each parameter is equal to that of single‐parameter estimation. Furthermore, a precision‐enhancement scheme has been identified that depends on the dynamics of . The smaller the value of in the dynamic evolution, the higher the precision of the parameter estimation. Finally, it is demonstrated that graph states serve as optimal states in quantum metrology. A set of optimal measurement bases is also identified, and it is illustrated that the precision limit of multiparameter estimation can attain the quantum Cramér‐Rao bound.

Quantum entanglement lies at the heart of quantum information and quantum metrology. In quantum metrology, with a colossal amount of quantum Fisher information (QFI), entangled systems can be ameliorated to be a better resource scheme. However, noisy channels affect the QFI substantially. This research work seeks to investigate how QFI of N-qubit Greenberger-Horne-Zeilinger (GHZ) state is affected when subjected to decoherence channels: bit-phase flip (BPF) and generalize amplitude damping (GAD) channels, which can be induced experimentally. We determine the evolution under these channels, deduce the eigenvalues, and then derive the QFI. We found that when there is no interaction with the environment, the Heisenberg limit can be achieved via rotations along the z direction. It has been shown that in BPF channel, the maximal mean QFI of the N-qubit GHZ state ({bar{F}}_{max}) dwindles as decoherence rate (pB) increases due to flow of information from the system to the environment, until pB = 0.5, then revives to form a symmetric around pB = 0.5. Thus, pB > 0.5 leads to a situation where more noise yields more efficiency. We found that in GAD channel, at finite temperature, QFIs decay more rapidly than at zero temperature. Our results also reveal that QFI can be enhanced by adjusting the temperature of the environment.

We propose a scheme to obtain the Heisenberg-limited parameter estimation precision by immersing atoms in a thermally equilibrated quasi-one-dimensional dipolar Bose-Einstein condensate reservoir. We show that the collisions between the dipolar atoms and the immersed atoms can result in a controllable nonlinear interaction through tuning the relative strength and the sign of the dipolar and contact interaction. We find that the repulsive dipolar interaction reservoir is preferential for the spin squeezing and the appearance of an entangled non-Gaussian state. As a useful resource for quantum metrology, we also show that the non-Gaussian state results in the phase estimation precision in the Heisenberg scaling, outperforming that of the spin-squeezed state.

Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.

The quantum Fisher information, the quantum analog of the classical Fisher information, is a central quantity in quantum metrology and quantum sensing due to its connection to parameter estimation and fidelity susceptibility. Using numerically exact methods applied to a paradigmatic open quantum system, the spin-boson model, we calculate both static and dynamical quantum Fisher information matrix elements with respect to spin-bath couplings and magnetic field strengths. As the spin-bath interaction increases, we first show that the coupling-coupling matrix elements relative to the ground state of the Hamiltonian are linked to the entanglement growth and signal the Berezinskii-Kosterlitz-Thouless quantum phase transition through their nonmonotonic behavior. We also point out that the static quantum Fisher information exhibits a nonperturbative behavior in the zero-coupling limit, which we justify with an analytic argument. Furthermore, we demonstrate that the time-dependent matrix elements can reveal non-Markovian effects as well as the transition from the coherent to incoherent regime at the Toulouse point, remaining robust under pure dephasing noise. Nonmonotonic signatures of the quantum Fisher information matrix reflect changes in quantum resources such as entanglement and coherence, quantify non-Markovian behavior, and enable criticality-enhanced quantum sensing, thereby shedding light on key features of open quantum systems.

In this paper we reconsider the single-parameter quantum Fisher information (QFI) and compare it with the two-parameter one. We find simple relations connecting the single-parameter QFI (both in the asymmetric and symmetric phase shift cases) to the two-parameter Fisher matrix coefficients. Following some clarifications about the role of an external phase [Phys. Rev. A 85, 011801(R) (2012)], the single-parameter QFI and its overoptimistic predictions have been disregarded in the literature. We show in this paper that both the single- and two-parameter QFI have physical meaning, and their predicted quantum Cram\'er-Rao bounds are often attainable with the appropriate experimental setup. Moreover, we give practical situations of interest in quantum metrology, where the phase sensitivities of a number of input states approach the quantum Cram\'er-Rao bound induced by the single-parameter QFI, outperforming the two-parameter QFI.

Relative phase distribution and the precision of optical phase sensing in quantum metrology

Geometric Properties of Evolutionary Graph States and Their Detection on a Quantum Computer

Entanglement is widely regarded as an essential resource for a number of tasks and can in some cases be quantified by figures of merit related to those tasks. In quantum metrology, this is showcased by the connections between the quantum Fisher information (QFI), providing a bound to the precision, and multipartite entanglement quantifiers such as the entanglement depth. However, a connection between the QFI and entanglement monotones, i.e., functions that do not increase under Local Operations and Classical Communications, has so far remained elusive. In this Letter, we fill this gap by introducing a family of uncertainty relations that bound bipartite entanglement monotones from below via elements of a quantum Fisher information matrix. To further emphasize the significance of our results, we connect these relations to the achievable precision in multiparameter estimation. Considering a system split into two parts with arbitrary dimension, we also show that, while two-dimensional entanglement is sufficient to estimate a single parameter with maximal precision, genuine high-dimensional entanglement is required for multiparameter estimation. We conclude by illustrating how our method extends naturally to a multipartite splitting.

The enhancement of the precision of phase estimation in quantum metrology is investigated by employing weak measurement (WM) and quantum measurement reversal (QMR). We derive the exact expressions of the optimal quantum Fisher information (QFI) and success probability of phase estimation for an exactly solving model consisting of a qubit interacting with a structured reservoir. We show that the QFI can be obviously enhanced by means of the WM and QMR in different regimes. In addition, we also show that the magnitude of the decoherence involved in the WM and QMR can be a general complex number, which extends the applicable scope of the WM and QMR approach.

In this paper, our prime objective is to apply the techniques of parameter estimation theory and the concept of Quantum Metrology in the form of Fisher Information to investigate the role of certain physical quantities in the open quantum dynamics of a two entangled qubit system under the Markovian approximation. There exist various physical parameters which characterize such system, but can not be treated as any quantum mechanical observable. It becomes imperative to do a detailed parameter estimation analysis to determine the physically consistent parameter space of such quantities. We apply both Classical Fisher Information (CFI) and Quantum Fisher Information (QFI) to correctly estimate these parameters, which play significant role to describe the out-of-equilibrium and the long range quantum entanglement phenomena of open quantum system. Quantum Metrology, compared to classical parameter estimation theory, plays a two-fold superior role, improving the precision and accuracy of parameter estimation. Additionally, in this paper we present a new avenue in terms of Quantum Metrology, which beats the classical parameter estimation. We also present an interesting result of revival of out-of-equilibrium feature at the late time scales, arising due to the long range quantum entanglement at early time scale and provide a physical interpretation for the same in terms of Bell's Inequality Violation in early time scale giving rise to non-locality.

We consider the probability distribution when the monotonic function $F(X)$ of the independent variable $X$ takes the maximum or minimum expected value under the two constraints of a certain probability and a certain expected value of the independent variable $X$. We proposed an equal probability and equal expected value splitting method. With this method, we proved four inequalities, and two of them can be reduced to Jensen's inequalities. Subsequently, we find that after dividing the non-monotone function $H(X)$ into multiple monotone intervals, the problem of solving the maximum and minimum expected values of $H(X)$ can be transformed into the problem of solving the extreme value of a multiple-variable function. Finally, we apply the proved theory to solve three problems in quantum information processing. When studying the quantum parameter estimation in Mach-Zehnder interferometer, for an equal total input photon number, we find an optimal path-symmetric input state that makes the quantum Fisher information take the maximum value, and we prove that the NOON state is the path-symmetric state that makes the quantum Fisher information takes the minimum value. When studying the quantum parameter estimation in Landau-Zener-Jaynes-Cummings model, we find the optimal initial state of the cavity field that makes the system obtain the maximum quantum Fisher information. Finally, for an equal initial average photon number, we find the optimal initial state of the cavity field that makes the Tavis-Cummings quantum battery have the maximum stored energy and the maximum average charging power.

Quantum sensing is one of the arenas that exemplifies the superiority of quantum technologies over their classical counterparts. Such superiority, however, can be diminished due to unavoidable noise and decoherence of the probe. Thus, metrological strategies to fight against or profit from decoherence are highly desirable. This is the case of certain types of decoherence-driven many-body systems supporting dissipative phase transitions, which might be helpful for sensing. Boundary time crystals are exotic dissipative phases of matter in which the time-translational symmetry is broken, and long-lasting oscillations emerge in open quantum systems at the thermodynamic limit. We show that the transition from a symmetry unbroken into a boundary time crystal phase, described by a second-order transition, reveals quantum-enhanced sensitivity quantified through quantum Fisher information. We also determine the critical exponents of the system and establish their relationship. Our scheme is indeed a demonstration of harnessing decoherence for achieving quantum-enhanced sensitivity. From a practical perspective, it has the advantage of being independent of initialization and can be captured by a simple measurement.

We define two ways of quantifying the quantum correlations based on quantum Fisher information (QFI) in order to study the quantum correlations as a resource in quantum metrology. By investigating the hierarchy of measurement-induced Fisher information introduced in Lu et al. [X. M. Lu, S. Luo, and C. H. Oh, Phys Rev. A 86, 022342 (2012)], we show that the presence of quantum correlation can be confirmed by the difference of the Fisher information induced by the measurements of two hierarchies. In particular, the quantitative quantum correlations based on QFI coincide with the geometric discord for pure quantum states.

We present an entanglement generation scheme which allows arbitrary graph states to be efficiently created in a linear quantum register via an auxiliary entangling bus (EB). The dynamical evolution of the EB is described by an effective non-interacting fermionic system undergoing mirror-inversion in which qubits, encoded as local fermionic modes, become entangled purely by Fermi statistics. We discuss a possible implementation using two species of neutral atoms stored in an optical lattice and find that the scheme is realistic in its requirements even in the presence of noise.