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arXiv:1406.0270

On repeated (continuous) weak measurements of a single copy of an unknown quantum state

N. D. Hari Dass

Published Jun 2, 2014• 0 citations
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Abstract

In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.

DOI
10.48550/arxiv.1406.0270

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In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.
In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.
Three recent results on weak measurements are presented. They are: i) repeated measurements on a single copy can not provide any information on it and further, that in the limit of very large such measurements, weak measurements have exactly the same characterstics as strong measurements, ii) the apparent non-invasiveness of weak measurements is \emph{illusory} and they are no more advantageous than strong measurements even in the specific context of establishing Leggett-Garg inequalities, when errors are properly taken into account, and, finally, iii) weak value measurements are optimal, in the precise sense of Wootters and Fields, when the post-selected states are mutually unbiased with respect to the eigenstates of the observable whose weak values are being measured. Notion of weak value coordinates for state spaces are introduced and elaborated.
The reconstruction of quantum states from a sufficient set of experimental data can be achieved with arbitrarily weak measurement interactions. Since such weak measurements have negligible back action, the quantum state reconstruction is also valid for the postselected subensembles usually considered in weak measurement paradoxes. It is shown that postselection can then be identified with a statistical decomposition of the initial density matrix into transient density matrices conditioned by the anticipated measurement outcomes. This result indicates that it is possible to ascribe the properties determined by the final measurement outcome to each individual quantum system before the measurement has taken place. The ``collapse'' of the pure state wave function in a measurement can then be understood in terms of the classical ``collapse'' of a probability distribution as new information becomes available.
Weak measurements can provide a complete characterization of post-selected ensembles, including the uncertainties of observables. Interestingly, the average uncertainties for pure initial and final states are always zero, suggesting the kind of complete knowledge that would allow a knowledge of past, presence and future in the sense of Laplace's demon. However, the quantum version actually describes cancellations of positive and negative uncertainties made possible by the strangeness of weak values. In this paper, I take a closer look at the relation between statistics and causality in quantum mechanics, in an attempt to recover the traces of classical determinism in the statistical relations of quantum measurement outcomes.
Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigen state will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. In this work, I revisit the quantum trajectory framework that models quantum measurement as a continuous nonlinear stochastic process. I describe the ensemble of quantum trajectories as noise fluctuations on top of geodesics that attract the quantum state towards the measured operator eigen states. In this effective theory framework for the ensemble of quantum trajectories, the measurement interaction is specific to each system-apparatus pair—a context necessary for understanding weak measurements. Also in this framework, the constraint to reproduce projective measurement as per the Born rule in the appropriate limit, requires that the magnitudes of the noise and the attraction are precisely related, in a manner reminiscent of the fluctuation dissipation relation. This relation implies that both the noise and the attraction have a common origin in the underlying measurement interaction between the system and the apparatus. I analyse the quantum trajectory ensemble for the scenarios of quantum diffusion and binary quantum jump, and show that the ensemble distribution is completely determined in terms of a single evolution parameter. I test the trajectory ensemble distribution predicted by the quantum diffusion model against the experimental data for weak measurement of superconducting transmon qubits. There is a good fit between theory and experiment for different initial states and several weak measurement couplings. This test vindicates the continuous stochastic measurement framework for quantum state collapse, where the rate of collapse is a characteristic parameter for each system-apparatus pair and is not a universal constant. Furthermore, it implies that the environment can influence the measurement outcomes only via the apparatus and not directly. These are important clues in construction of a complete theory of quantum measurement. The framework of weak measurements can also be used to construct quantum error correction protocols that protect a quantum state from external disturbances. Unlike projective measurements, one can extract only partial information about the error syndrome from the encoded state using weak measurements. I construct a feedback protocol that probabilistically corrects the error based on the extracted information. Using numerical simulations of one-qubit error correction codes, I show that the error correction succeeds for a range of the weak measurement strength, where (a) the error rate is below the threshold beyond which multiple errors dominate, and (b) the…
It is widely assumed that the quantum fluctuations of the meter readout in a weak measurement make it impossible to identify the contributions originating from the individual values of the physical property observed in the measurement. Here, we show that a careful analysis of the quantum dynamics of the meter system allows us to identify a universal relation between quantum interference in the post-selection probability, and quantum interference in the statistics of the meter readout. The analysis reveals that quantum interference modifies the readout distribution of the meter in two ways, a diffusion term that identifies the appropriate operator ordering in the post-selected variance of the observed system property, and a wavefunction-dependent update of the initial meter statistics based on the effect of back action on the post-selection probability. Our results show that the statistical patterns described by meter interference provide important details about the physics of post-selection in weak measurements, allowing us to exclude the possibility of statistical artefacts in the experimental observation of weak values.