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arXiv:1211.4169 · Physical Review A

Uncertainties in successive measurements

Jacques Distler B, Sònia Paban

Published Jun 21, 2013• 29 citations
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Abstract

When you measure an observable, $A$, in quantum mechanics, the state of the system changes. This, in turn, affects the quantum-mechanical uncertainty in some noncommuting observable, $B$. The standard uncertainty relation puts a lower bound on the uncertainty of $B$ in the initial state. What is relevant for a subsequent measurement of $B$, however, is the uncertainty of $B$ in the postmeasurement state. We re-examine this problem, both in the case where $A$ has a pure point spectrum and in the case where $A$ has a continuous spectrum. In the latter case, the need to include a finite detector resolution, as part of what it means to measure such an observable, has dramatic implications for the result of successive measurements. Ozawa, [Phys. Rev. A 67, 042105 (2003)] proposed an inequality satisfied in the case of successive measurements. Among our results, we show that his inequality is ineffective (can never come close to being saturated). For the cases of interest, we compute a sharper lower bound.

DOI
10.1103/physreva.87.062112

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When you measure an observable, AAA, in quantum mechanics, the state of the system changes. This, in turn, affects the quantum-mechanical uncertainty in some noncommuting observable, BBB. The standard uncertainty relation puts a lower bound on the uncertainty of BBB in the initial state. What is relevant for a subsequent measurement of BBB, however, is the uncertainty of BBB in the postmeasurement state. We re-examine this problem, both in the case where AAA has a pure point spectrum and in the case where AAA has a continuous spectrum. In the latter case, the need to include a finite detector resolution, as part of what it means to measure such an observable, has dramatic implications for the result of successive measurements. Ozawa, [Phys. Rev. A 67, 042105 (2003)] proposed an inequality satisfied in the case of successive measurements. Among our results, we show that his inequality is ineffective (can never come close to being saturated). For the cases of interest, we compute a sharper lower bound.
We previously remarked that when an observable A has a continuous spectrum, then von Neumann's formula for the post-measurement state needs to be extended and the correct formula ineluctably involves the resolution of the detector used in the measurement. We generalize previous results to compute the uncertainties in successive measurements of more general pairs of observables. We also show that this extended von Neumann's formula for the post-measurement state is a completely positive map and, moreover, that there is a completely-positive interpolation between the pre- and post-measurement states. Weinberg has advocated that the time-evolution during the measurement process should be modeled as an open quantum system and governed by a Lindblad equation. We verify that this is indeed the case for an arbitrary observable, A, and a fairly general class of interpolations.
We study theoretically a weak measurement of an observable AAA and its impact on a subsequent strong measurement of another observable BBB. We assume that both measurements are described by the Gaussian point detectors. It is shown that a decrease of the system-detector interaction during the measurement of AAA results in an increase of the fluctuations of AAA, while the mean value of AAA is kept unaltered. The ratio ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}/{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ turns out to be universal, i.e., independent of the physical system, its quantum state, and the quantity measured. Most of our attention is focused on analyzing outcomes of the subsequent strong measurement of the quantity BBB, namely, on the mean value B‾\overline{B}B and the variance ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$. We derive explicit analytic corrections for B‾\overline{B}B and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ arising due to the prior weak measurement of AAA. Importantly, the leading-order impact of the weak measurement of AAA on B‾\overline{B}B and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ is encoded in the terms ${\phantom{\rule{0.16em}{0ex}}}_{s}\ensuremath{\langle}\ensuremath{\psi}|[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}^{k}]]{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$, where k=1,2k=1,2k=1,2 and ${|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$ is the quantum state vector of our system before the measurements. Our additional contribution consists of deriving Heisenberg-like uncertainty relations for the product ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}\phantom{\rule{0.28em}{0ex}}{(\mathrm{\ensuremath{\Delta}}B)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ and analyzing the case of minimum uncertainty. Finally, our theoretical developments are illustrated numerically via studying the sss states of the hydrogen atom, assuming $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}$ (atomic radius) and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (energy).
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables A^\hat{A}A^ and B^\hat{B}B^, with eigenvalues ana_nan​ and bmb_mbm​, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable Q2Q_2Q2​ conditioned on the first {\em probe} measurement yielding Q1Q_1Q1​ we obtain information on the statistical distribution of the {\em system} variable bmb_mbm​ conditioned on having found the system variable ana_nan​ in the interval δaδaδa around a(n)a^{(n)}a(n). The width of this statistical distribution as function of δaδaδa constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.
Uncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg–Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. The Rényi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive measurements of canonically conjugate operators. Entropic uncertainty relations for the cases of position and momentum are separately discussed. image
Measurement uncertainty relations are quantitative bounds on the errors in an approximate joint measurement of two observables. They can be seen as a generalization of the error/disturbance tradeoff first discussed heuristically by Heisenberg. Here we prove such relations for the case of two canonically conjugate observables like position and momentum, and establish a close connection with the more familiar preparation uncertainty relations constraining the sharpness of the distributions of the two observables in the same state. Both sets of relations are generalized to means of order α rather than the usual quadratic means, and we show that the optimal constants are the same for preparation and for measurement uncertainty. The constants are determined numerically and compared with some bounds in the literature. In both cases, the near-saturation of the inequalities entails that the state (resp. observable) is uniformly close to a minimizing one.
This chapter reviews the main points in the theory of instantaneous measurements. A description of continuous measurements as limits of series of instantaneous measurements turns out to be very cumbersome and in fact inadequate. A characteristic feature of quantum systems is that measurements in them unavoidably affect their dynamics. The well known manifestation of this back reaction is the uncertainty principle, discussion of which can be found in any textbook on quantum mechanics. The operator plays the role of a propagator for a quantum system under continuous measurement. The quantum Zeno effect is an example of a continuous measurement giving a trivial result. However, this is a consequence of the very special character of the measurement procedure described by a discrete set of orthogonal projectors each of which is a bounded operator. The description of approximate measurements was derived for the case when precise measurement is a discrete-spectrum measurement.