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Register a new userNon-ideal classical measurements and quantum measurements: a comparative study
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Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using a measurement probe with finite resolution. We use the von Neumann interaction model to represent the interaction between system and probe. We find that in reality classical systems are affected by measurement in a similar manner as quantum systems. In particular, we derive classical equivalents of Luders rule, the collapse postulate, and the Lindblad equation.
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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.
With the advent of gravitational wave detectors employing squeezed light, quantum waveform estimation---estimating a time-dependent signal by means of a quantum-mechanical probe---is of increasing importance. As is well known, backaction of quantum measurement limits the precision with which the waveform can be estimated, though these limits can in principle be overcome by quantum nondemolition (QND) measurement setups found in the literature. Strictly speaking, however, their implementation would require infinite energy, as their mathematical description involves Hamiltonians unbounded from below. This raises the question of how well one may approximate nondemolition setups with finite energy or finite-dimensional realizations. Here we consider a finite-dimensional waveform estimation setup based on the quasi-ideal clock and show that the estimation errors due to approximating the QND condition decrease slowly, as a power law, with increasing dimension. As a result, we find that good QND approximations require large energy or dimensionality. We argue that this result can be expected to also hold for setups based on truncated oscillators or spin systems.
From the perspective of quantum thermodynamics, realisable measurements cost work and result in measurement devices that are not perfectly correlated with the measured systems. We investigate the consequences for the estimation of work in non-equilibrium processes and for the fundamental structure of the work fluctuations when one assumes that the measurements are non-ideal. We show that obtaining work estimates and their statistical moments at finite work cost implies an imperfection of the estimates themselves: more accurate estimates incur higher costs. Our results provide a qualitative relation between the cost of obtaining information about work and the trustworthiness of this information. Moreover, we show that Jarzynskis equality can be maintained exactly at the expense of a correction that depends only on the systems energy scale, while the more general fluctuation relation due to Crooks no longer holds when the cost of the work estimation procedure is finite. We show that precise links between dissipation and irreversibility can be extended to the non-ideal situation.
Non-classical state generation is an important component throughout experimental quantum science for quantum information applications and probing the fundamentals of physics. Here, we investigate permutations of quantum non-demolition quadrature measurements and single quanta addition/subtraction to prepare quantum superposition states in bosonic systems. The performance of each permutation is quantified and compared using several different non-classicality criteria including Wigner negativity, non-classical depth, and optimal fidelity with a coherent state superposition. We also compare the performance of our protocol using squeezing instead of a quadrature measurement and find that the purification provided by the quadrature measurement can significantly increase the non-classicality generated. Our approach is ideally suited for implementation in light-matter systems such as quantum optomechanics and atomic spin ensembles, and offers considerable robustness to initial thermal occupation.
We show that it is impossible to perform ideal projective measurements on quantum systems using finite resources. We identify three fundamental features of ideal projective measurements and show that when limited by finite resources only one of these features can be salvaged. Our framework is general enough to accommodate any system and measuring device (pointer) models, but for illustration we use an explicit model of an $N$-particle pointer. For a pointer that perfectly reproduces the statistics of the system, we provide tight analytic expressions for the energy cost of performing the measurement. This cost may be broken down into two parts. First, the cost of preparing the pointer in a suitable state, and second, the cost of a global interaction between the system and pointer in order to correlate them. Our results show that, even under the assumption that the interaction can be controlled perfectly, achieving perfect correlation is infinitely expensive. We provide protocols for achieving optimal correlation given finite resources for the most general system and pointer Hamiltonians, phrasing our results as fundamental bounds in terms of the dimensions of these systems.
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