In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.
In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.
In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be non-negative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.
Quantum measurement is ultimately a physical process, resulting from an interaction between the measured system and a measurement apparatus. Considering the physical process of measurement within a thermodynamic context naturally raises the following question: how can the work and heat resulting from the measurement process be interpreted? In the present manuscript, we model the measurement process for an arbitrary discrete observable as a measurement scheme. Here, the system to be measured is first unitarily coupled with a measurement apparatus, and subsequently the apparatus is measured by a pointer observable, thus producing a definite measurement outcome. The work can therefore be interpreted as the change in internal energy of the compound of system-plus-apparatus due to the unitary coupling. By the first law of thermodynamics, the heat is the subsequent change in internal energy of this compound due to the measurement of the pointer observable. However, in order for the apparatus to serve as a stable record for the measurement outcomes, the pointer observable must commute with the Hamiltonian, and its implementation must be repeatable. Given these minimal requirements, we show that the heat will necessarily be a classically fluctuating quantity.
It has been established that the inclusive work for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the conditional stochastic work for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the energy conditioned on the initial energy surface. As main results we obtain a generalized Jarzynski equality and a sharper maximum work theorem, which account for how non-adiabatic the process is. Our findings are illustrated with the parametric harmonic oscillator.
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.
Considering a general microscopic model for a quantum measuring apparatus comprising a quantum probe coupled to a thermal bath, we analyze the energetic resources necessary for the realization of a quantum measurement, which includes the creation of system-apparatus correlations, the irreversible transition to a statistical mixture of definite outcomes, and the apparatus resetting. Crucially, we do not resort to another quantum measurement to capture the emergence of objective measurement results, but rather exploit the properties of the thermal bath which redundantly records the measurement result in its degrees of freedom, naturally implementing the paradigm of quantum Darwinism. In practice, this model allows us to perform a quantitative thermodynamic analysis of the measurement process. From the expression of the second law, we show how the minimal required work depends on the energy variation of the system being measured plus information-theoretic quantities characterizing the performance of the measurement – efficiency and completeness. Additionally, we show that it is possible to perform a thermodynamically reversible measurement, thus reaching the minimal work expenditure, and provide the corresponding protocol. Finally, for finite-time measurement protocols, we illustrate the increasing work cost induced by rising entropy production inherent in finite-time thermodynamic processes. This highlights an emerging trade-off between velocity of the measurement and work cost, on top of a trade-off between efficiency of the measurement and work cost. We apply those findings to bring new insights in the thermodynamic balance of the measurement-powered quantum engines.
Various approaches of defining and determining work performed on a quantum system are compared. Any operational definition of work, however, must allow for two facts, first, that work characterizes a process rather than an instantaneous state of a system, and, second, that quantum systems are sensitive to the interactions with a measurement apparatus. We compare different measurement scenarios on the basis of the resulting post-measurement states and the according probabilities for finding a particular work value. In particular, we analyze a recently proposed work-meter for the case of a Gaussian pointer state and compare it with the results obtained by two projective and, alternatively, two Gaussian measurements. In the limit of a strong effective measurement strength the work distribution of projective two energy measurements can be recovered. In the opposite limit the average of work becomes independent of any measurement. Yet the fluctuations about this value diverge. The performance of the work-meter is illustrated by the example of a spin in a suddenly changing magnetic field.