Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Conditionally valid uncertainty relations†
K. Fujikawa∗1
It is shown that the well-deﬁned unbiased measurement or disturbance of a dynamical variable is not
maintained for the precise measurement of the conjugate variable, independently of uncertainty relations.
The conditionally valid uncertainty relations on the basis of those additional assumptions, which include mostof the familiar Heisenberg-type relations, thus become
singular for the precise measurement. We clarify some
contradicting conclusions in the literature concerning
those conditionally valid uncertainty relations: The
failure of a naive Heisenberg-type error-disturbance
relation and the modiﬁed Arthurs-Kelly relation in
the recent spin measurement is attributed to thissingular behavior. The naive Heisenberg-type errordisturbance relation is formally preserved in quantum
estimation theory, which is shown to be based on thestrict unbiased measurement and disturbance, but it
leads to unbounded disturbance for bounded operatorssuch as spin variables. In contrast, the Heisenbergtype error-error uncertainty relation and the ArthursKelly relation, as conditionally valid uncertainty relations, are expected to be consistently maintained.
A recent experiment1) , which invalidated a naive
Heisenberg-type error-disturbance relation2) , revived
our interest in the subject of uncertainty relations. In contrast to the naive Heisenberg-type errordisturbance relation, the relations which are based on
only the positive deﬁnite Hilbert space and naturalcommutator algebra are expected to be valid as longas quantum mechanics is valid, namely, ”universallyvalid”2)3) . It was recently shown4) that all the known
universally valid uncertainty relations are derived from
Robertson’s relation written for suitable combinationsof operators. It is important to distinguish the uncertainty relations which are universally valid from thoserelations based on additional assumptions and thus
only conditionally valid.
In this paper, we analyze the implications of the assumptions of unbiased joint measurements or unbiased
measurement and disturbance which are widely usedin the formulation of uncertainty relations5) . We clarify the origin of quite diﬀerent conclusions concerningthe conditionally valid Heisenberg-type relations in themeasurement operator formalism2) and in the quantum estimation theory6) which is a new approach touncertainty relations.
We ﬁrst note that the well-deﬁned unbiased measurement or disturbance of a quantum mechanical op†∗1

erator is not maintained for the precise measurementof the conjugate operator in the framework of the ordinary measurement theory. For example, those assumptions lead to⟨[M out , N out ]⟩ = ⟨[A, B]⟩,⟨[M out , B out ]⟩ = ⟨[A, B]⟩.

(1)

We work in the Heisenberg picture and the variables without any suﬃx stand for the initial variables; A, B stand for dynamical variables and M, N
stand for the corresponding measurement operators,respectively. The variables M out = U † (1 ⊗ M )U andN out = U † (1 ⊗ N )U stand for the variables after themeasurement, and B out = U † (B ⊗ 1)U stands for thevariable B after the measurement of A. By assumption, ⟨[M out , N out ]⟩ = ⟨[M out , B out ]⟩ = 0, and thus
relations in (1) are contradictions.
The conditionally valid uncertainty relation such asnaive Heisenberg-type error-disturbance relation1)2) ,1|⟨[A, B]⟩|,(2)2which is based on the assumptions of unbiased measurement and disturbance, thus fails if one formulates
the relation in terms of well-deﬁned bounded operators. The naive Heisenberg-type error-disturbance
relation is formally preserved in quantum estimation
theory, but the disturbance of the bounded operator
is forced to be singular and divergent for the precise
measurement of the conjugate variable6) .
In contrast, the Heisenberg-type error-error uncertainty relationσ(M out − A)σ(B out − B) ≥σ(M out − A)σ(N out − B) ≥1|⟨[A, B]⟩|,2(3)

and the Arthurs-Kelly relation,σ(M out )σ(N out ) ≥ |⟨[A, B]⟩|,(4)

as conditionally valid uncertainty relations, are expected to be consistently maintained.
References

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