Certainty in Heisenberg's uncertainty principle: Revisiting definitions for estimation errors and disturbance

Abstract

We revisit the definitions of error and disturbance recently used in error-disturbance inequalities derived by Ozawa and others by expressing them in the reduced system space. The interpretation of the definitions as meansquared deviations relies on an implicit assumption that is generally incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-semidefinite joint quasiprobability distribution. For unbiased measurements, the error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to directly measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect distributional estimations can have nonzero error according to this measure. We conclude that the inequalities using these definitions do not capture the spirit of Heisenberg's eponymous inequality, but do indicate a qualitatively different relationship between dispersion and disturbance that is appropriate for ensembles being probed by all outcomes of an apparatus. To reconnect with the discussion of Heisenberg, we suggest alternative definitions of error and disturbance that are intrinsic to a single apparatus outcome. These definitions naturally involve the retrodictive and interdictive states for that outcome, and produce complementarity and error-disturbance inequalities that have the same form as the traditional Heisenberg relation.

On the relation between measurement outcomes and physical properties

“…It has also been shown that the measurement error can be observed experimentally by performing a weak measurement ofÊ a =| a a | on the input state [3,5], and the present discussion was originally motivated by an attempt to understand the physics of these experimental results better [15]. Here, we would like to address the problem that such measurement results cannot be interpreted as a relative frequency of joint outcomes, since the individual weak measurements represent only a statistical preference for the outcome a and not an exclusive selection of a [12]. It is therefore necessary to consider the role of the outcomes a in the experiments more carefully.…”

“…In principle, Eq. (12) indicates that any projective measurement {| m } can have a measurement error of ε 2 (A) = 0 if the corresponding set of weak values of m are chosen for the estimatesà m . However, weak values are generally complex, and the imaginary part is usually obtained in a dynamical response of the system that is not directly connected to the quantity represented by [34,35].…”

“…It is this non-classical quantitative relation between physical properties that is expressed by the error free weak value estimates of Eq. (12). In particular, the initial state | ψ always corresponds to a known physical propertyB that evaluates the quantitative difference ( −M ) between the target observable and the measured quantityM .…”

“…Recently, new insights into measurement uncertainties have emerged, initiated by Ozawa's analysis of the relation between errors and disturbance in quantum measurements [1][2][3][4][5][6][7][8][9]. In the context of these new results it is interesting to note that there is no clear consensus about the physics involved in quantum measurements [10][11][12][13][14]. It seems that too much of the focus has been on quantitative evaluations of the uncertainties, and too little attention has been paid to the astonishing fact that quantum mechanics does not provide a clear explanation of the measurement process itself.…”

Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat

“…The weak value's property of allowing information to be extracted from a quantum system with minimal disturbance is applied as a new method for estimation of quantum states [47][48][49][50] (see [51] for a recent review). This particular property of the weak value is also utilized in experimental tests of a reformulation of Heisenberg's uncertainty principle [52][53][54][55] (an overview of different viewpoints of that topic can be found in [56]). In addition, the weak value and weak measurements have been successfully applied to quantum paradoxes such as the three-box problem [57] and Hardy's paradox [58][59][60].…”