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

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 mean-squared 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.

Figure 1

Detector picture of an indirect measurement, highlighted in gray, with time increasing along the vertical axis. (See [53] for a more detailed description of the graphical notation used throughout this paper.) An initially uncorrelated system state ${\widehat{\rho }}_S$ and detector state ${\widehat{\rho }}_D$ are coupled with a unitary interaction $\mathcal{U}$. After the reference time indicated by the black dots, a joint observable $\widehat{1}\otimes \widehat{M}\left[m\right]$ with spectral function $m_k$ over the detector outcomes indexed by $k$ is projectively measured. No further measurement is considered on the system space, which is indicated by the identity $\widehat{1}$ in the joint observable $\widehat{1}\otimes \widehat{M}\left[m\right]$. This procedure is formally equivalent to the measurement of $\widehat{M}\left[m\right]$ directly after the preparation of a reduced detector state ${\widehat{\rho }}_D^{\prime }$ that contains correlated information about the initial system state ${\widehat{\rho }}_S$.

Figure 2

System picture of the same indirect measurement as in Fig. 1, highlighted in gray. The joint measurement procedure is equivalent to a quantum instrument $\mathcal{A}\left[m\right]$ (see Sec. 2b) that acts directly on the system after a reference time indicated by the black dots. Since no subsequent measurements are considered (as indicated by the identity operator $\widehat{1}$), then this instrument induces a weighted POM ${\widehat{A}}_e\left[m\right]$ as the system observable that is effectively measured after the preparation of the initial state ${\widehat{\rho }}_S$, as shown in Eq. (9).

Figure 3

Weak measurement used to approximate the quasiprobabilities $\tilde{p}(a,k)$ according to Eqs. (23) and (24), highlighted in gray. A weak interaction ${\mathcal{U}}_2$ couples a second detector state ${\widehat{\rho }}_{D2}$ to the system, which is followed by a measurement of the detector observable ${\widehat{M}}_2\left[n^{\left(a\right)}\right]$ with assigned values $n_{\ell }^{\left(a\right)}$. This joint procedure is equivalent to acting with a second quantum instrument $\mathcal{D}\left[n^{\left(a\right)}\right]$ on the system prior to measuring ${\widehat{P}}_k$, with operations ${\mathcal{D}}_{\ell }$ corresponding to each outcome $\ell$. This detector is tuned to measure a spectral projection ${\mathcal{D}}^{*}\left[n^{\left(a\right)}\right]\left(\widehat{1}\right)={\widehat{\Pi }}_a$ of $\widehat{A}$ when the ${\widehat{P}}_k$ measurement is ignored. However, subsequently measuring each $k$ results in the averages ${\sum }_{\ell }{n}_{\ell }^{\left(a\right)}p(\ell ,k)\approx \tilde{p}(a,k)$ that approximate a quasiprobability distribution when ${\mathcal{D}}_{\ell }$ is nearly the identity for all $\ell$. Changing the assigned values $n_{\ell }^{\left(a\right)}$ retargets the detector for different spectral projectors ${\widehat{\Pi }}_a$.

Figure 4

Sequential measurements. After the indirect measurement illustrated in Fig. 2 and highlighted in gray, another system observable $\widehat{B}$ is measured. The procedure produces correlated pairs of measurement outcomes $(m_k,B_b)$. The quantum instrument $\mathcal{A}\left[m\right]$ is needed to fully describe these correlations in the reduced system space. The effective system observable ${\widehat{A}}_e\left[m\right]$ from Fig. 2 reappears from the sequential observable ${\mathcal{A}}^{*}\left[m\right]\left(\widehat{B}\right)$ only when the correlations from the measurement of $\widehat{B}$ are ignored.

Figure 5

Disturbance induced by a nonselective measurement. After the nonselective measurement highlighted in gray, another system observable $\widehat{B}$ is measured. Unlike in Fig. 4, this procedure ignores the correlations between specific pairs of measurement outcomes, so will measure an average perturbed observable ${\widehat{B}}^{\prime }={\mathcal{A}}^{*}\left[1\right]\left(\widehat{B}\right)$.

Figure 6

Weak measurement used to approximate the quasiprobabilities $\tilde{p}(b,b^{\prime })$ in Eq. (37), highlighted in gray. A second quantum instrument $\mathcal{D}\left[n^{\left(b\right)}\right]$ is coupled to the system prior to the nonselective measurement $\mathcal{A}\left[1\right]$. This detector is tuned to measure a spectral projection ${\mathcal{D}}^{*}\left[n^{\left(b\right)}\right]\left(\widehat{1}\right)={\widehat{\Pi }}_b$ of $\widehat{B}$ when all subsequent measurements are ignored. However, measuring each $b^{\prime }$ after $\mathcal{A}\left[1\right]$ results in the averages ${\sum }_{\ell ,k}{n}_{\ell }^{\left(b\right)}p(\ell ,k,b^{\prime })\approx \tilde{p}(b,b^{\prime })$ that approximate a quasiprobability distribution when ${\mathcal{D}}_{\ell }$ is nearly the identity for all $\ell$. Changing the assigned values $n_{\ell }^{\left(b\right)}$ retargets the detector for different spectral projectors ${\widehat{\Pi }}_b$.

Figure 7

Operational procedure for determining the estimation error ${\epsilon }_{A,k}$. For every preparation ${\widehat{\Pi }}_a$ in the spectral basis of $\widehat{A}$ made with probability $p\left(a\right)$, the system is prepared in the initial state ${\widehat{\Pi }}_a$ and subsequently conditioned on the successful measurement of a particular outcome $k$, which produces the measurable probabilities $p\left(k\right|a)$. Applying Bayes' rule yields the retrodictive probabilities $p\left(a\right|k)=p\left(k\right|a)p\left(a\right)/{\sum }_{a}p\left(k\right|a)p\left(a\right)$. For a uniform preparation with constant $p\left(a\right)$, this procedure produces $p\left(a\right|k)={\text{Tr}}_S\left[{\widehat{\Pi }}_a{\check{\rho }}_k\right]$, which effectively normalizes the POM element ${\widehat{P}}_k$ to produce the retrodictive state ${\check{\rho }}_k$ in Eq. (43) that averages each possible preparation ${\widehat{\Pi }}_a$. These probabilities define ${\epsilon }_{A,k}$ according to Eq. (44). The assumption of uniform $p\left(a\right)$ indicates that a single independent event $k$ cannot be assumed to correspond to any particular source preparation a priori. The resulting probabilities $p\left(a\right|k)$ are thus the best retrodictive inference about the preparation that one can make using only information about a single isolated detector outcome $k$.

Figure 8

Operational procedure for determining the measurement disturbance ${\eta }_{B,k}$. For each projection ${\widehat{\Pi }}_b$ in the spectral basis of $\widehat{B}$, the system in prepared in the initial state ${\widehat{\Pi }}_b$ with probability $p\left(b\right)$, the detector outcome $k$ is measured, then the outcome ${\widehat{\Pi }}_{b^{\prime }}$ is measured, producing the measurable probabilities $p(k,b^{\prime }|b)$. Conditioning on the successful measurement of a particular intermediate outcome $k$ produces the interdictive joint probabilities according to Bayes' rule $p(b,b^{\prime }|k)=p(k,b^{\prime }|b)p\left(b\right)/{\sum }_{b,b^{\prime }}p(k,b^{\prime }|b)p\left(b\right)$. When the preparation $p\left(b\right)$ is uniform, this yields $p(b,b^{\prime }|k)={\text{Tr}}_S\left[{\widehat{\Pi }}_{b^{\prime }}{\tilde{\rho }}_k\left({\widehat{\Pi }}_b\right)\right]$, which effectively normalizes the operation ${\mathcal{A}}_k$ to produce the interdictive state operation ${\tilde{\rho }}_k$ in Eq. (46) that correlates each preparation ${\widehat{\Pi }}_b$ and postselection ${\widehat{\Pi }}_{b^{\prime }}$. These probabilities define ${\eta }_{B,k}$ according to Eq. (47). As with the procedure in Fig. 7, the assumption of uniform $p\left(b\right)$ indicates the lack of a priori bias that can be inferred by a single isolated outcome $k$.