According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
A projective measurement of energy (PME) on a quantum system is a quantum measurement determined by the Hamiltonian of the system. PME protocols exist when the Hamiltonian is given in advance. Unknown Hamiltonians can be identified by quantum tomography, but the time cost to achieve a given accuracy increases exponentially with the size of the quantum system. In this Letter, we improve the time cost by adapting quantum phase estimation, an algorithm designed for computational problems, to measurements on physical systems. We present a PME protocol without quantum tomography for Hamiltonians whose dimension and energy scale are given but which are otherwise unknown. Our protocol implements a PME to arbitrary accuracy without any dimension dependence on its time cost. We also show that another computational quantum algorithm may be used for efficient estimation of the energy scale. These algorithms show that computational quantum algorithms, with suitable modifications, have applications beyond their original context.
Quantum measurements are known to affect the state of the measured system. Such a measurement backaction is due to the interaction of the quantum system with the measuring device. When the measured observable and the system's Hamiltonian do not commute, the backaction can lead to a mean energy change of the system. This energy change has been dubbed "quantum heat" and, in association with feedback processes and/or the interaction with thermal bath(s), proves to be a genuinely quantum resource to fuel new kinds quantum engines. We propose such a measurement powered engine exploiting the non-commutativity of local and non-local operations on a bipartite quantum system.We then investigate the origin of the quantum heat. Even if one cannot model the full dynamics of a measurement due to the collapse of the wavefunction, the energetic aspects can still be investigated by modeling the so-called pre-measurement. During this step, the system interacts and gets correlated with a quantum system which can be viewed as a small part of the measuring apparatus and called the quantum meter. Ideally, at the end of this process, the system's reduced state will be fully decohered in the measurement basis, thus corresponding to the averaged collapsed output states given by the measurement postulate. We find that the quantum heat received by the measured system corresponds to the energy necessary to turn off the coupling between the meter and the system. Using generalized definitions of heat and work, we moreover characterize the work and heat nature of these energy exchanges.Tracing the source of this energy one step further, we propose an autonomous description of a measurement. This is done by considering a flying qubit measured by the quantum field of the cavity it crosses. The interaction being position dependent, the kinetic degree of freedom is providing the required energy to switch on and off the interaction between the qubit and the meter field. A full quantum treatment allows us to evaluate the impact of the finite spatial and momentum extension of the qubit wavepacket. Since the kinetic degree of freedom can be affected by the interaction, it does not simply generate a time dependent Hamiltonian for the other degrees of freedom. We find the correction to this ideal dynamic and analyse its consequences on the nature of the energy fluxes. %spread/extension/broadnessTaking the opposite point of view, we compare the measurement cost of a qubit by a field propagating in a waveguide, depending on the initial state of the field. In the context of a circuit quantum electrodynamics setup, we find that thermal and coherent fields induce the same backaction on the qubit given the same energy constraints.The results presented and discussed in this thesis contribute to unravel the mysterious effects and mechanisms due to quantum measurements. Notably, they give an outlook to further analyse quantum measurement-based engines and to understand the energetic resource that measurements cost and constitute at the same time.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.
Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\ensuremath{\Delta}t$ that obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\ensuremath{\Delta}t$ which obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.
This thesis investigates three distinct facets of the theory of quantum information. The first two, quantum state estimation and quantum process estimation, are closely related and deal with the question of how to estimate the classical parameters in a quantum mechanical model. The third attempts to bring quantum theory as close as possible to classical theory through the formalism of quasi-probability. \n \nBuilding a large scale quantum information processor is a significant challenge. First, we require an accurate characterization of the dynamics experienced by the device to allow for the application of error correcting codes and other tools for implementing useful quantum algorithms. The necessary scaling of computational resources needed to characterize a quantum system as a function of the number of subsystems is by now a well studied problem (the scaling is generally exponential). However, irrespective of the computational resources necessary to just write-down a classical description of a quantum state, we can ask about the experimental resources necessary to obtain data (measurement complexity) and the computational resources necessary to generate such a characterization (estimation complexity). These problems are studied here and approached from two directions. \n \nThe first problem we address is that of quantum state estimation. We apply high-level decision theoretic principles (applied in classical problems such as, for example, universal data compression) to the estimation of a qubit state. We prove that quantum states are more difficult to estimate than their classical counterparts by finding optimal estimation strategies. These strategies, requiring the solution to a difficult optimization problem, are difficult to implement in practise. Fortunately, we find estimation algorithms which come close to optimal but require far fewer resources to compute. Finally, we provide a classical analog of this quantum mechanical problem which reproduces, and gives intuitive explanations for, many of its features, such as why adaptive tomography can quadratically reduce its difficulty. \n \nThe second method for practical characterization of quantum devices takes is applied to the problem of quantum process estimation. This differs from the above analysis in two ways: (1) we apply strong restrictions on knowledge of various estimation and control parameters (the former making the problem easier, the latter making it harder); and (2) we consider the problem of designing future experiments based on the outcomes of past experiments. We show in test cases that adaptive protocols can exponentially outperform their off-line counterparts. Moreover, we adapt machine learning algorithms to the problem which bring these experimental design methodologies to realm of experimental feasibility. \n \nIn the final chapter we move away from estimation problems to show formally that a classical representation of quantum theory is not tenable. This intuitive conclusion is formally borne out through the connection to quasi-probability -- where it is equivalent to the necessity of negative probability in all such representations of quantum theory. In particular, we generalize previous no-go theorems to arbitrary classical representations of quantum systems of arbitrary dimension. We also discuss recent progress in the program to identify quantum resources for subtheories of quantum theory and operational restrictions motivated by quantum computation.
We present a method to estimate the probabilities of outcomes of a quantum observable, its mean value, and higher moments by measuring any other observable. This method is general and can be applied to any quantum system. In the case of estimating the mean energy of an isolated system, the estimate can be further improved by measuring the other observable at different times. Intuitively, this method uses interplay and correlations between the measured observable, the estimated observable, and the state of the system. We provide two bounds: one that is looser but analytically computable and one that is tighter but requires solving a nonconvex optimization problem. The method can be used to estimate expectation values and related quantities such as temperature and work in setups where performing measurements in a highly entangled basis is difficult, finding use in state-of-the-art quantum simulators. As a demonstration, we show that in Heisenberg and Ising models of ten sites in the localized phase, performing two-qubit measurements excludes 97.5% and 96.7% of the possible range of energies, respectively, when estimating the ground-state energy.