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arXiv:1310.4762 · Physical Review A

Robertson-Schrödinger-type formulation of Ozawa's noise-disturbance uncertainty principle

Catarina Bastos, Alex E. Bernardini, Orfeu Bertolami A, Nuno Costa Dias, João Nuno Prata

Published Apr 22, 2014• 29 citations
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Abstract

In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schr\"odinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that for certain types of measurement interactions it is covariant with respect to linear symplectic transformations of the noise and disturbance operators. Finally, we also study the tightness of our matrix inequality.

DOI
10.1103/physreva.89.042112

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In this work we derive a matrix formulation of a noise-disturbance uncertainty relation, which is akin to the Robertson-Schr"odinger uncertainty principle. Our inequality is stronger than Ozawa's uncertainty principle and takes noise-disturbance correlations into account. Moreover, we show that for certain types of measurement interactions it is covariant with respect to linear symplectic transformations of the noise and disturbance operators. Finally, we also study the tightness of our matrix inequality.
We establish the tightest possible Robertson-type preparation uncertainty relation, which explicitly depends on the eigenvalues of the quantum state. The conventional constant $ \tfrac{1}{4} $ is replaced by the state-dependent coefficient (λmax⁡+λmin⁡)24(λmax⁡−λmin⁡)2, \frac{(λ_{\max} + λ_{\min})^2}{4(λ_{\max} - λ_{\min})^2}, 4(λmax​−λmin​)2(λmax​+λmin​)2​, where $ λ_{\max} $ and λmin⁡λ_{\min}λmin​ denote the largest and smallest eigenvalues of the density operator $ ρ$, respectively. We prove that this coefficient is optimal among all Robertson-type generalizations and does not admit further improvement. We also obtain a complete characterization of all nontrivial equality cases. Our relation becomes more pronounced as the quantum state becomes more mixed, capturing a trade-off in quantum uncertainty that the conventional Robertson's relation fails to detect. In addition, our result also provides a strict generalization of the Schrödinger's uncertainty relation, showing that the uncertainty trade-off is governed by the sum of the covariance term and a state-dependent improvement over the Robertson bound. As applications, we also refine error-disturbance trade-offs by incorporating spectral information of both the system and the measuring apparatus, thereby generalizing the Arthurs-Goodman and Ozawa inequalities.
It has been pointed out that for some types of measurement the Heisenberg uncertainty relation seems to be violated. In order to save the situation a new uncertainty relation was proposed by Ozawa. Here we introduce revised definitions of error and disturbance taking into account the gain associated with generalized measurement interactions. With these new definitions, the validity of the Heisenberg inequality is recovered for continuous linear measurement interactions. We also examine the changes in distribution functions caused by the general measurement interaction and clarify the physical meanings of infinitely large errors and disturbances.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement and the resultant observation (observer) effect of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional na{ï}ve bound ℏ/2\hbar/2ℏ/2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our error-disturbance relation admits a parallel formulation to our relation for errors, which also embraces the standard Kennard--Robertson (Schr{ö}dinger) relation as a special case; this attains a unified picture of the three orthodox realms of uncertainty regarding quantum indeterminacy, measurement, and observation effect within a single framework.
We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.