We propose an error-disturbance relation for general observables on finite dimensional Hilbert spaces based on operational notions of error and disturbance. For two-dimensional systems we derive tight inequalities expressing the trade-off between accuracy and disturbance.
The formulation of our error-disturbance relation is closest in spirit to the one by Bush, Lahti, Pearson and Werner @cite_16 @cite_10 @cite_7 for canonical position and momentum operators, but a direct comparison is not possible, since we work with finite dimensional systems. Here we will briefly compare our proposed inequality with two other proposals.
@cite_16 @cite_10 @cite_7
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While the slogan “no measurement without disturbance” has established itself under the name of the Heisenberg effect in the consciousness of the scientifically interested public, a precise statement of this fundamental feature of the quantum world has remained elusive, and serious attempts at rigorous formulations of it as a consequence of quantum theory have led to seemingly conflicting preliminary results. Here we show that despite recent claims to the contrary [L. , Phys. Rev. Lett. 109 100404 (2012)], Heisenberg-type inequalities can be proven that describe a tradeoff between the precision of a position measurement and the necessary resulting disturbance of momentum (and vice versa). More generally, these inequalities are instances of an uncertainty relation for the imprecisions of any joint measurement of position and momentum. Measures of error and disturbance are here defined as figures of merit characteristic of measuring devices. As such they are state independent, each giving worst-case estimates across all states, in contrast to previous work that is concerned with the relationship between error and disturbance in an individual state. Abstract We investigate the recovery of almost s -sparse vectors x ∈ C N from undersampled and inaccurate data y = A x + e ∈ C m by means of minimizing ‖ z ‖ 1 subject to the equality constraints A z = y . If m ≍ s ln ( N s ) and if Gaussian random matrices A ∈ R m × N are used, this equality-constrained l 1 -minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m ≍ s ln ( N s ) and if Weibull random matrices are used, we prove here that the equality-constrained l 1 -minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the l 1 -norm and the outer norm is replaced by a norm comparable to the l 2 -norm. For the l 1 -minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables. We critically revisit the definitions of mean-squared estimation error and disturbance recently used in error-disturbance inequalities derived by Ozawa, Hall, Branciard, and by expressing them in the reduced system space. The interpretation of the definitions as mean-squared deviations relies on a hidden assumption that is incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-definite joint quasiprobability distribution. For unbiased measurements, the estimation error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect estimations can have nonzero estimation 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 applies to ensembles of measurements. To reconnect with the discussion of Heisenberg, we suggest alternate 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. Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report improved new versions of various error-disturbance tradeoff relations by decomposing the measurement errors into two different components, namely, operator bias and fuzziness. Our improved uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance.