In this PhD thesis we address the problem of characterizing quantum states and parameters of systems that are of particular interest for quantum technologies. In the first part we consider continuous variable systems and in particular Gaussian states; we address the estimation of quantities characterizing single-mode Gaussian states as the displacement and squeezing parameter and we study the improvement in the parameter estimation by introducing a Kerr nonlinearity. Moreover, we address the discrimination of noisy channels by means of Gaussian states as probe states considering two problems: the detection of a lossy channel against the alternative hypothesis of an ideal lossless channel and the discrimination of two Gaussian noisy channels. In the last part of the thesis, we consider the one dimensional quantum Ising model in a transverse magnetic field. We exploit the recent results about the geometric approach to quantum phase transitions to derive the optimal estimator of the coupling constant of the model at zero and finite temperature in both cases of few spins and in the thermodynamic limit. We also analyze the effects of temperature and the scaling properties of the estimator of the coupling constant. Finally, we consider the discrimination problem for two ground states or two thermal states of the model.
In this PhD thesis we address the problem of characterizing quantum states and parameters of systems that are of particular interest for quantum technologies. In the first part we consider continuous variable systems and in particular Gaussian states; we address the estimation of quantities characterizing single-mode Gaussian states as the displacement and squeezing parameter and we study the improvement in the parameter estimation by introducing a Kerr nonlinearity. Moreover, we address the discrimination of noisy channels by means of Gaussian states as probe states considering two problems: the detection of a lossy channel against the alternative hypothesis of an ideal lossless channel and the discrimination of two Gaussian noisy channels. In the last part of the thesis, we consider the one dimensional quantum Ising model in a transverse magnetic field. We exploit the recent results about the geometric approach to quantum phase transitions to derive the optimal estimator of the coupling constant of the model at zero and finite temperature in both cases of few spins and in the thermodynamic limit. We also analyze the effects of temperature and the scaling properties of the estimator of the coupling constant. Finally, we consider the discrimination problem for two ground states or two thermal states of the model.
In this thesis I study optimal strategies to estimate or characterize quantum states and processes (quantum channels). I consider the problem of estimating a phase encoded on a gaussian state. Phase estimation is at the heart of many quantum metrology applications, such as gravitational wave detectors or clock synchronization. I address the problem to the phase estimation fidelity for pure and mixed gaussian states and study how temperature affects the precision of estimation. Although the case of pure states has been studied before, very little is known about estimation in mixed states. The interest in mixed gaussian states relies on the fact that these are the states one has in real experiments with light due to the unavoidable presence of noise. Displaced and squeezed thermal states are studied. For the case of displaced states, I find that fidelity estimation gets worse when the temperature increases. On the contrary, for squeezed thermal states I find the surprising result that a larger temperature yields better estimation. This apparently paradoxical feature, also appears when I derive the estimation accuracy that can be obtained when an asymptotically large number of copies of the state is available. I also study quantum benchmarks that can be applied to a wide variety of protocols, such us quantum storage or quantum teleportation. A very important question in QIT is to know whether a particular quantum protocol can be realized by classical means with the same efficiency. After all, QIT lives from the promise of this quantum advantage. This question becomes specially relevant in experimental (real) implementations of quantum schemes. In real experiments there are always losses and the ideal quantum protocols become imperfect. It is then necessary to assess whether the same experimental result could be obtained by classical (less costly) means. So, for example, in a particular teleportation experiment -which involves generation of entanglement, complicated Bell measurements, fighting decoherence, etc.- one might ask whether one could achieve the same (i.e. mapping the state of a system to a second system located in a different place or time) by measuring the system, transmitting the collected information to the other location, and finally preparing the system accordingly. It is hence of paramount importance to give bounds on the efficiency of these measure and prepare (or entanglement breaking) channels. For this purpose we propose a threshold based on a phase-covariant family of states. Besides of being a family with a straightforward group-theoretical structure, which gives obvious computational advantages, this family is interesting because it can be tested experimentally also in a straightforward way. A crucial feature for the practical use of our benchmark is that it does not require pure (ideal) input states. I finally consider the problem of estimating the parameter of the amplification channel. The amplification channel is interesting by itself but also by the fact that the Hawking-Unruh effect can be formally modeled by it. The Unruh-Hawking effect is a relativistic effect and as such it is hard to measure and detect in current experiments. I investigate whether quantum strategies can provide an improvement over classical protocols in the estimation of the amplification parameter. If the field is inertially prepared in the vacuum one could measure its temperature as detected by a noninertial observer and extract information about the parameter. But I would like to know if the vacuum state is the optimal probe state in an inertial frame for such a task or a better sensitivity could be achieved if the field were in a state of nonzero energy (mean photon number) from an inertial perspective. The aim is to determine the optimal input states, so that, by optimally measuring the output, one can construct an estimator for amplification channel parameter which is unbiased and has the minimum possible variance.
Exemplified by the Heisenberg uncertainty principle, quantum mechanics imposes a fundamental limit to the precision we can measure complementary properties of an object, for example position and momentum. Similarly, quantum mechanics tells us there is a limit to which we can discriminate between two quantum states that are nonorthogonal. Therefore, it is an interesting question to ask, given the laws of quantum mechanics, what exactly is the best measurement outcome we can achieve? This thesis examines various problems related to quantum state discrimination and quantum parameter estimation of continuous variable quantum states. We first examine quantum illumination, which can be regarded as a quantum state discrimination problem, where a quantum probe state is used to determine the presence or absence of an object in a noisy environment. If we use a probe that is quantum entangled with another probe, we can obtain a better performance outcome than is possible by using a probe without entanglement. We show that the source of enhancement is due to quantum discord. We also investigate what is the best probe state when entanglement is not used (conventional illumination) and when entanglement is used (quantum illumination). We analytically prove the optimal probe for various cases, and also provide some numerical results. We also investigate the optimal estimation of parameters that parametrise a quantum state. We provide methods of calculating the Holevo Cramer Rao bound, which is a useful quantity for quantum parameter estimation. The Holevo Cramer Rao bound not only is a good lower bound to the sum of the mean squared error, in some cases it can provide a measurement that obtains it. We consider the example of the estimation of conjugate quadrature parameters of a displacement experienced by one mode of a two-mode EPR state. We not only prove that the near-optimal measurement scheme proposed by previous authors is optimal for high squeezing, we find a different measurement that is optimal for low squeezing. Finally, we investigate using a probabilistic amplifier for quantum state discrimination and parameter estimation for coherent states. We find that the probabilistic amplifier results in improvement in discrimination and parameter estimation of coherent states under certain conditions such as when there is a high cost associated with measuring a sample.
Gaussian quantum channels are relevant operations in continuous variable systems. In general, given an arbitrary state, the action on it is well known, provided that the quantum channels are completely characterized. In this work, we have studied the role played by quantum coherence in the estimation of Gaussian channel parameters. For this purpose, we start by considering an initial single-mode vacuum state where coherence is generated through displaced or/and squeezing operations. Then, we consider a probe state with two resources, the average thermal number and the quantum coherence. The results are computed using the quantum Fisher information (QFI) as well as some gain functions defined in the work. We applied the estimation protocol for two paradigmatic bosonic Gaussian channels: the thermal attenuator and the thermal amplifier. Our results show that for the vacuum initial state, coherence can be exploited to obtain a high QFI. Moreover, we can state two important facts: the first being that for a small parameter region of a channel, the QFI depends on the mechanism used to generate the coherence, while the second is that the QFI depends not only on the amount of coherence but also on how it changes as a function of the parameter to be estimated. For the probe state, including also an average thermal number, the two examples show that thermal resources are not always beneficial to the QFI and that this depends on the channel structure. For the attenuator channel, the optimal probe state is only limited by the ability to implement an average thermal number and generate an amount of coherence, while for the amplifier channel, the vacuum probe state with coherence is the optimal state. Finally, we obtain a direct relation between the quantum Fisher information and the relative entropy of coherence, allowing, in principle, an experimental verification based on the measurement of the covariance matrix of the probe system.
This PhD thesis contains results about two different main topics. The first part deals with the application of continuously monitored quantum systems to high precision quantum metrology. A continuous in time measurement on a quantum system is a kind indirect measurement, which only weakly perturbs the system and leaves room for it to evolve under its dynamics. This time-continuous measurement allows one to collect information about some interesting parameter characterizing the dynamics. In this thesis we show how to apply the theory of quantum parameter estimation to continuously monitored quantum systems. In particular, we study the estimation of a magnetic field applied to an ensemble of two level atoms; we show that by continuously monitoring the system we can obtain a quadratic scaling of the precision with the number of atoms, in two different physical settings (dynamically generated entanglement or initial entanglement). In the second part we study different aspects of nonclassicality of continuous variable quantum systems (bosonic degree of freedoms). They can be described by distributions (in particular, the Wigner function) on a classical phase space, which however can take negative values, the hallmark of nonclassicality. In this context, states with a Gaussian distribution are very useful and very well studied; however, on a fundamental level they must be considered classical. We present several studies connected to the vast topic of non-Gaussian states, starting from an application to parameter estimation, as a link to the first part. We study the relationships between anharmonic Hamiltonians and the nonclassicality of their ground states; we also explore the connection between a quantum effect called `backflow of probability' and the negativity of the Wigner function. We end by showing that quantum operations made out of Gaussian building blocks give rise to a well-defined resource theory of Wigner negativity and non-Gaussianity.
This thesis studies quantum-enhanced phase change detection using continuous-variable Gaussian states. The main goal is to understand how different probe states and receiver designs affect the detection of a phase change in a noisy optical channel. The phase change problem is written as a quantum state discrimination problem. The pre-change and post-change states are Gaussian states described by displacement vectors and covariance matrices. A thermal-loss channel is used to model optical loss and background noise, and a phase shift is applied to the signal mode. The main quantum benchmark is the quantum relative entropy (QRE) between the post-change and pre-change states. A pure two-mode Gaussian probe is studied in detail. The probe energy is split between squeezing and coherent displacement. This split is controlled by a power allocation ratio ?. By optimizing over ?, the probe can balance quantum correlations and coherent amplitude under a fixed photon-number constraint. Amplified spontaneous emission (ASE), coherent state, and two-mode squeezed vacuum (TMSV) probes are also considered as comparison cases. Practical receiver designs are then studied using classical relative entropy (CRE). The receivers include ideal photon-number-resolving (PNR) detection, finite-resolution PNR detection, a binary single-photon detector (SPD), an ASE receiver, and a homodyne jointdetection receiver. The finite-resolution study distinguishes binary SPD, with outcomes 0 and ? 1, from 1-PNR, with outcomes 0, 1, and ? 2. The receiver-level classical relative entropy is compared with the quantum relative entropy bound to show how much information each receiver keeps after measurement. Finally, the thesis connects these relative entropy results to the cumulative sum (CUSUM) and quantum CUSUM (QUSUM) frameworks for quickest change detection. In this framework, a larger relative entropy means a shorter detection delay for the same false alarm requirement. The results show that both probe design and receiver design are important for quantum-enhanced phase change detection.
This thesis explores the topics of parameter estimation and model reduction in the context of quantum filtering. The last is a mathematically rigorous formulation of continuous quantum measurement, in which a stream of auxiliary quantum systems is used to infer the state of a target quantum system. Fundamental quantum uncertainties appear as noise which corrupts the probe observations and therefore must be filtered in order to extract information about the target system. This is analogous to the classical filtering problem in which techniques of inference are used to process noisy observations of a system in order to estimate its state. Given the clear similarities between the two filtering problems, I devote the beginning of this thesis to a review of classical and quantum probability theory, stochastic calculus and filtering. This allows for a mathematically rigorous and technically adroit presentation of the quantum filtering problem and solution. Given this foundation, I next consider the related problem of quantum parameter estimation, in which one seeks to infer the strength of a parameter that drives the evolution of a probe quantum system. By embedding this problem in the state estimation problem solved by the quantum filter, I present the optimal Bayesian estimator for a parameter when given continuous measurements of the probe system to which it couples. For cases when the probe takes on a finite number of values, I review a set of sufficient conditions for asymptotic convergence of the estimator. For a continuous-valued parameter, I present a computational method called quantum particle filtering for practical estimation of the parameter. Using these methods, I then study the particular problem of atomic magnetometry and review an experimental method for potentially reducing the uncertainty in the estimate of the magnetic field beyond the standard quantum limit. The technique involves double-passing a probe laser field through the atomic system, giving rise to effective non-linearities which enhance the effect of Larmor precession allowing for improved magnetic field estimation. I then turn to the topic of model reduction, which is the search for a reduced computational model of a dynamical system. This is a particularly important task for quantum mechanical systems, whose state grows exponentially in the number of subsystems. In the quantum filtering setting, I study the use of model reduction in developing a feedback controller for continuous-time quantum error correction. By studying the propagation of errors in a noisy quantum memory, I present a computation model which scales polynomially, rather than exponentially, in the number of physical qubits of the system. Although inexact, a feedback controller using this model performs almost indistinguishably from one using the full model. I finally review an exact but polynomial model of collective qubit systems undergoing arbitrary symmetric dynamics which allows for the efficient simulation of spontaneous-emission and related open quantum system phenomenon.