Experimental demonstration of a universally valid error–disturbance uncertainty relation in spin measurements

Abstract

The author of the comment [1] claimed that the experimental results for a universally valid uncertainty relation cannot be recognized to establish a violation of the Heisenberg-type uncertainty relation. However, after careful examination of the author's argument, we regard the author's argument to be on an improper basis, thus disagree with his opinion. Hereby, we provide arguments to disprove his objection.According to his statement "To insist the violation of the Heisenbergtype uncertainty relation, ǫ(Q) = 0 or η(P ) = 0, one has to prove | [N (Q), D(P )] | = 0," he appears to formulate the violation to be the case where ǫ(Q) = 0 or η(P ) = 0. However, in our paper the violation is taken to be the cases where the relation ǫ(A)η(B) ≥ 1 2 | [A, B] | does not hold, and we observed the violation for all the parameter values of φ actually tested.The author conclude the comments by writing "In conclusion, the experimental set-up of ref.[2] is not satisfied the necessary condition to be true for the UVUR proved by Ozawa in [3,4], because the neutron spin measurement ("projective measurement" as well) does not have the unitary operator to defined the equal-time commutation relation".In contrast to what the author claimed, the projective measurement is defined by a family of projection operators M m , instead of a unitary operator U , and the root-mean-square error is defined by equation (5) in Ref.[2] using the projection operators M m , instead of using the unitary operator U . It is well-known that if the measurement is defined by a unitary operator U , then the definition of the root-mean-square error using the unitary operator U and the definition (5) in Ref.[2] using only projection operators M m are equivalent. A detailed account on those equivalences between definitions are given in [M. Ozawa, Uncertainty relations for noise and disturbance in generalized quantum measurements, Ann. Phys. (N.Y.) 311, 350-416 (2004)]. Thus, the author's criticism based on the absence of the unitary operator is also irrelevant.

Optimized entropic uncertainty for successive projective measurements

“…Two input states |x+ and |y+ are considered. By inspectio n, we find that the LHS of EUR for successive measurement (12) always has larger value than the LHS of (16) due to subadditivity (14). The conditional entropy has a smaller value.…”

“…According to the sub-additivity inequality, the joint entropy has a relation with the entropy of marginal distributions of the joint probability, i.e. the LHS of the entropic uncertainty relation (12), as…”

Quantum back-action in measurements of zero-point mechanical oscillations

“…For this reason, we can expand the measurement errorˆ into two parts, i.e.,ẑ is the sensing 1 We note that Ozawa has developed a different formalism to quantify the issues that arise when attempts are made to measure noncommuting observables such asx q (t) [9,10]. However, we have chosen to adopt the Braginsky-Khalili approach [11] because it is immediately applicable when the noncommuting observable is acting as a probe for an external classical force.…”

Heisenberg's uncertain legacy

“…Only in recent years has it been challenged by the suggestion that the precision of measurement can do better 2 -an idea prompted originally by attempts to estimate the feasibility of gravitational-wave detection. Opposed points of view, and experimental measurements of the revised error-disturbance relation, have been put forward [3][4][5][6] . The upshot is that whether or not you think Heisenberg's relation was correct depends on what you think he meant by it: whether he was talking about one-shot measurements or averages.…”