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 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.
Heisenberg originally envisioned the error-disturbance relation under the premise that the state after measurement must remain consistent with the Kennard-Robertson uncertainty relation. This implies that the measurement error must be formulated via a posterior observable
x^t, rather than a prior observable
x^0, because the posterior measurement error determines the post-measurement uncertainty of the electron. Building upon the preliminary conceptual foundation reported in arXiv:1504.03779, this paper presents a rigorous derivation of the error-disturbance principle formulated from the unitary transformation equations of observables. Each readout
X of a posterior probe observable
X^t completes a single measurement event, producing a specific conditional object state. The Kennard-Robertson uncertainty relation must hold for these states. Specifically, we show that the uncertainty relation between the error
εX(x^t) and the disturbance
ηX(p^0) holds strictly at the level of individual measurements, characterized by the specific readout
X. We also verify that conventional error-disturbance relations, where the errors are evaluated by averaging over the unconditioned state, hold true. Our results provide a refined theoretical basis for understanding the fundamental trade-off in individual measurement outcomes.
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 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.
A universal formulation of the quantum uncertainty regarding quantum indeterminacy, quantum measurement, and its inevitable observer effect is presented with additional focus on the representability of quantum observables over a given state. The operational tangibility of the framework assures that the resultant general relations admit natural operational interpretations and characterisations, and thereby perhaps most importantly, their experimental verifiability. In view of the universal formulation, Heisenberg's original philosophy of the uncertainty principle, most typically exemplified in his famous gamma-ray microscope Gedankenexperiment, is revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. The relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the standard Kennard-Robertson relation, the Arthurs-Kelly-Goodman relations for joint measurement as well as the Ozawa and the Watanabe-Sagawa-Ueda relations for error and error-disturbance.