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arXiv:1910.12189

Revisiting weak measurement in light of thermodynamics

Daiki Ueda, Hikaru Ohta, Sumito Yokoo

Published Oct 27, 2019• 0 citations
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Abstract

We investigate a weak measurement described by a von Neumann type interaction $\hat{A}\otimes \hat{p}^2$, where $\hat{A}$ is a system observable and $\hat{p}^2$ is a measurement pointer observable. We consider the weak measurement in terms of thermodynamics by adopting a mixed Gaussian state as a quantum state of the measurement pointer. We show that Maxwell's demon appears as a measure who carries out post-selections. It is found that, even if the demon only knows the weak value, a difference in the von Neumann entropy between the initial and final system can be the QC mutual information contents, which is the maximum amount of obtainable information by a measurement. Besides, our study indicates that a temperature of the system described by this interaction is amplified by weak value amplification. In addition, we show that this demon can be realized in an atomic system.

DOI
10.48550/arxiv.1910.12189

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We investigate a weak measurement described by a von Neumann type interaction A^⊗p^2\hat{A}\otimes \hat{p}^2A^⊗p^​2, where A^\hat{A}A^ is a system observable and p^2\hat{p}^2p^​2 is a measurement pointer observable. We consider the weak measurement in terms of thermodynamics by adopting a mixed Gaussian state as a quantum state of the measurement pointer. We show that Maxwell's demon appears as a measure who carries out post-selections. It is found that, even if the demon only knows the weak value, a difference in the von Neumann entropy between the initial and final system can be the QC mutual information contents, which is the maximum amount of obtainable information by a measurement. Besides, our study indicates that a temperature of the system described by this interaction is amplified by weak value amplification. In addition, we show that this demon can be realized in an atomic system.
We investigate a weak measurement described by a von Neumann type interaction A^⊗p^2\hat{A}\otimes \hat{p}^2A^⊗p^​2, where A^\hat{A}A^ is a system observable and p^2\hat{p}^2p^​2 is a measurement pointer observable. We consider the weak measurement in terms of thermodynamics by adopting a mixed Gaussian state as a quantum state of the measurement pointer. We show that Maxwell's demon appears as a measure who carries out post-selections. It is found that, even if the demon only knows the weak value, a difference in the von Neumann entropy between the initial and final system can be the QC mutual information contents, which is the maximum amount of obtainable information by a measurement. Besides, our study indicates that a temperature of the system described by this interaction is amplified by weak value amplification. In addition, we show that this demon can be realized in an atomic system.
Weak measurements allow a probe system to encode the transitionary weak value of a desired observable on a signal system that is both pre- and post-selected. Such interactions alter the state and measurement statistics of the probe system. We establish that by encoding the imaginary part of a weak value onto an initially entangled probe, we can modify its entanglement content, provided the probe observable can distinguish between system states of different von Neumann entropies. Apart from fundamental interest, this result illustrates the utility of the imaginary weak value as a calculational tool in certain entanglement concentration protocols.
We discuss the thermodynamic aspects of a single qubit based device, powered by weak quantum measurements, and feedback controlled by a quantum Maxwell's demon. We discuss both discrete and time-continuous operation of the measurement based device at finite temperature of the reservoir. In the discrete example where a demon acquires information via discrete weak measurements, we find that the thermodynamic variables including the heat exchanged, extractable work, and the entropy produced are completely determined by an information theoretic measure of the demon's perceived arrow of time. We also discuss a realistic time-continuous operation of the device where the feedback is applied after a sequence of weak measurements. In the time-continuous limit, we derive the exact finite-time statistics of work, heat and entropy changes along individual quantum trajectories of the quantum measurement process, and relate them to the demon's arrow of time.
This work will incorporate a few related tools for addressing the conceptual difficulties arising from sewing together classical and quantum mechanics: deterministic operators, weak measurements and post-selection. Weak Measurement, based on a very weak von Neumann coupling, is a unique kind of quantum measurement with numerous theoretical and practical applications. In contrast to other measurement techniques, it allows to gather a small amount of information regarding the quantum system, with only a negligible probability of collapsing it. A single weak measurement yields an almost random outcome, but when performed repeatedly over a large ensemble, the averaged outcome becomes increasingly robust and accurate. Importantly, a long sequence of weak measurements can be thought of as a single projective measurement. I claim in this work that classical variables appearing in the macro-world, such as centre of mass, moment of inertia, pressure and average forces, result from a multitude of quantum weak measurements performed in the micro-world. Here again, the quantum outcomes are highly uncertain, but the law of large numbers obliges their convergence to the definite quantities we know from our everyday lives. By augmenting this description with a final boundary condition and employing the notion of "classical robustness under time-reversal" I will draw a quantitative borderline between the classical and quantum regimes. I will conclude by analyzing the role of macroscopic systems in amplifying and recording quantum outcomes.
Weak measurement is a potentially powerful tool for studying post-selected quantum systems. The original approach to weak measurement requires a different von Neumann interaction between the quantum system and measurement pointer for weak values of every different observable. When studying weak values of properties of more than one particle, which is necessary for studying entanglement, these interactions can range from difficult to impossible to implement experimentally. For example, the measurement of properties on a pair of optical photons requires a strong Kerr-like nonlinearity at the two-photon level. In a recent work, we described a new method for extracting weak values of operators that can be expressed as products of two local, single-particle operators by studying the correlations in experimentally feasible local, single-particle measurements. In the present work, that theoretical approach is generalized so that weak values of products of N local, single-particle operators can be extracted from the correlations in N single-particle measurements.
Abstract Quantum metrology is being gradually studied for weak measurement systems. For weak measurement systems with a thermal state pointer, we find that in the displacement space corresponding to imaginary weak values, the maximal quantum Fisher information (QFI) after the postselection for an accepted event (conversely, the postselection for a rejected event) can attain a level of thermal fluctuation, without surpassing the total QFI, and further, that QFI which increases with increasing temperature can constantly improve measurement precision. These results are much better than those of weak measurement with a pure state (i.e. Gaussian state) pointer. On the other hand, in Kerr nonlinear interaction systems with weak measurement, and by using a thermal state pointer, we obtain in the phase space the accepted postselection and the postselected measurement both achieving the Heisenberg limit of quantum metrology, and showing that weak measurement with thermal states only obtain classical Fisher information (CFI) which increases with increasing temperature and achieves classical-enhanced scaling of N 2 . Moreover, weak measurement with thermal states has an advantage over that with coherent states or mixed states of light because generating these states with larger uncertainty is limited under current technology, but thermal states with larger uncertainly are very easy to achieve with increasing temperature in nature, such as for optical systems and optomechanical systems.