Uncertainty relations: An operational approach to the error-disturbance tradeoff

Abstract

The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englert's wave-particle duality relation [Phys. Rev. Lett. 77, 2154(1996] can be viewed as an error-disturbance uncertainty relation.

Uncertainty and trade-offs in quantum multiparameter estimation

“…In Section II C we demonstrated this by relating the algebraic form of the Holevo Cramér-Rao bound to the strength of the resulting trade-off curve. Equations (16) and (19) show how as the expectation value Tr[L i , L j ]ρ 0 approaches zero, the corresponding trade-off curve becomes closer and closer to the trivial one. We have also provided two examples of systems-the qubit (Section III) and qutrit (Section IV) models-where the commutation condition is satisfied only between some pairs of parameters, and demonstrated how this reflects in their trade-off surfaces.…”

“…6. I addition we mention that it has been recently shown that the bound in Eq (19). is always greater or equal than the Holevo Cramér-Rao bound, and that the Holevo Cramér-Rao bound is less or equal than two times the SLD bound Eq.…”

“…Combining these observations we can use Eq. (19) to understand the trade-offs which the Holevo Cramér-Rao bound exhibit in this model. We can treat the matrices appearing in Eq.…”

“…We can treat the matrices appearing in Eq. (19) as block diagonal. In H −1 the only block which contains off diagonal terms corresponds to the parameters θ 3 , θ 8 .…”

State-independent uncertainty relations

“…The former refers to the variance of the outcomes of measurements of different observables performed on different systems prepared in identical states. The latter, which has been the subject of a lively debate recently [20,[34][35][36][37][38], refers to the relation between errors and post-measurement disturbance in an apparatus. We underline these are very different notions [20,34,[36][37][38][39], even if this difference is often obscured in the literature.As our relations are based on the sum of variances, they easily relate more than two observables and possess a simple physical interpretation as the diagonal of the uncertainty volume, depicted in Fig.…”