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.
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.
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.
We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always attain the optimal bounds in the many-copy scenario. Adaptive homodyning schemes do attain the bounds for displaced thermal states, but for squeezed states they yield fidelities that are insensitive to temperature variations and are, therefore, suboptimal. Finally, we find that heterodyne measurements perform very poorly for pure states but can attain the optimal bounds for sufficiently mixed states. We apply our results to investigate the influence of losses in an optical metrology experiment. In the presence of losses squeezed states cease to provide the Heisenberg limited precision, and their performance is close to that of coherent states with the same mean photon number.
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.
Quantum states and measurements are two central physical objects in the theory of quantum mechanics. The preparation, control and measurement of quantum states lies at the heart of modern quantum information processing protocols ranging from secure communication to quantum computation and precision sensing. And yet, the reliable operation of these advanced protocols ultimately rests on a far more essential prerequisite - our ability to calibrate, certify and benchmark the quantum states we prepare and the quantum measurements we perform. Quantum information theory provides a natural framework for these tasks, meaningfully connecting the resources we expend for characterisation with the accuracy and precision we can expect in return. While decades of advances in quantum information have refined our understanding of quantum characterisation and sharpened our practical ability to calibrate with enhanced precision, important questions about the fundamental limits of quantum characterisation remain unanswered. Although incorporating entanglement at the measurement stage is known to enhance the estimation of unknown quantum states, the extent of this improvement remains poorly understood. This leaves open a fundamental question on the utility of complex entangling measurement strategies, impeding progress towards bridging the significant gap between practically achievable precision and the ultimate theoretical quantum limit. Complementary to quantum state estimation, which is relatively well-established in theory and practice, the characterisation of quantum measurements lacks a rigorous information-theoretic foundation, leading to a dearth of principled benchmarks and provably optimal strategies despite growing experimental interest. This thesis develops and applies information-theoretic tools to answer questions in quantum characterisation, with an eye to enabling practical improvements beyond current techniques in the field. First, we introduce a computable measure of entanglement enhancement for quantum state estimation and study its variation across general multi-parameter estimation problems for higher-dimensional quantum states. We show that the entanglement enhancement is maximised for full qudit state tomography but grows only linearly with system dimension even when allowing an asymptotically large number of entangling operations. Our proposed tight upper bound to entanglement enhancement therefore constitutes a strong limitation on the utility of such entangling measurements. Next, we formulate the dual problem of quantum detector estimation, developing a comprehensive framework for maximally precise calibration centred around a newly introduced quantity - the detector quantum Fisher information. This information-theoretic measure stipulates the ultimate precision attainable for measurement calibration and identifies the optimal input states that achieve this precision. Further, our framework enables the first provably optimal detector estimation experiment and reveals the scope for entanglement-based enhancements, thereby placing measurement characterisation on equal footing with state characterisation. Considered together, these results suggest two contrasting conclusions about quantum characterisation. On the one hand, they show that for quantum state estimation, increasingly powerful entangling measurements offer only diminishing returns at the ultimate precision limit. On the other hand, they highlight practical quantum advantages available for detector estimation by providing a framework that can improve calibration performance and guide future experimental design. More broadly, these results complete and advance the triad of state, process, and detector characterisation, positioning quantum information measures as not merely abstract bounds, but practical guides for deciding when complex quantum resources are genuinely worth deploying and where near-term quantum advantages are most likely to be realised.
Precise measurement of physical quantities plays a crucial role in the development of science and technology.The main purpose of the dissertation is two-fold: to investigate the ultimate precision for estimation of physical quantities using Gaussian states and to propose an efficient method for certification of Bayesian error region in general quantum parameter estimation.In the first part, we begin with analyzing sensitivity for estimating a phase difference in an optical interferometer.Optical interferometry is widely used in science and industry for measuring small displacements.Recently, a large-scale optical interferometer socalled the Laser interferometer Gravitational-Wave Observatory (LIGO) has succeeded in detecting a gravitational wave, the signal of which is extremely small.On the other hand, it has been shown that a non-classical feature of quantum states can improve the sensitivity of estimation, such as in optical interferometer, including the LIGO.From a practical point of view, we inspect the practically achievable precision using non-classical Gaussian states in Mach-Zehnder interferometer with feasible measurements and realistic photon loss.We then investigate the precision of single-mode phase estimation using Gaussian measurement, which can be realized by using homodyne detection, and i show that non-Gaussian measurement is necessary to utilize the power of Gaussian input probes optimally.Finally, we find the optimal measurement for general Gaussian quantum metrology and identify three distinct optimal measurements corresponding to different circumstances.In the second part, we study the Bayesian error region, which is a crucial concept for a general estimation process.When estimating a physical quantity, one has to supply the error interval (single-parameter) or error region (multi-parameter) as well as the estimate.However, it has been shown that as the dimension of quantum systems of interest grows, it becomes intractable to calculate the size and credibility of Bayesian error regions.As an alternative, we derive an analytical expression for the properties, the size and credibility, of Bayesian error regions, in an asymptotic regime.We then propose an efficient numerical method to calculate them for high-dimensional quantum systems even in a non-asymptotic regime.
Teleportation and storage of continuous variable states of light and atoms are essential building blocks for the realization of large scale quantum networks. Rigorous validation of these implementations require identifying, and surpassing, benchmarks set by the most effective strategies attainable without the use of quantum resources. Such benchmarks have been established for special families of input states, like coherent states and particular subclasses of squeezed states. Here we solve the longstanding problem of defining quantum benchmarks for general pure Gaussian single-mode states with arbitrary phase, displacement, and squeezing, randomly sampled according to a realistic prior distribution. As a special case, we show that the fidelity benchmark for teleporting squeezed states with totally random phase and squeezing degree is 1/2, equal to the corresponding one for coherent states. We discuss the use of entangled resources to beat the benchmarks in experiments.