Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Heisenberg uncertainty relation revisited†
K. Fujikawa∗1
Kennard and Robertson formulated the uncertainty
relation which appears in any textbook on quantum
mechanics
σ(A)σ(B) ≥1|⟨[A, B]⟩|.
2⟨M out ⟩ = ⟨A⟩(2)

for any state of the system ψ in the total Hilbert spaceof the system and apparatus |ψ⟩⊗|ξ⟩ in the Heisenberg
picture. Here variables M and N (and also A and B)
stand for the variables before the measurement, andthe variable M out = U † M U stands for the apparatus
M after measurement.
Traditionally, it has been common to take the relation2)
1|⟨[A, B]⟩|2(3)as the naive Heisenberg error-disturbance relation; we
use the adjective ”naive” since no reliable derivationof this relation is known. An elegant experiment ofspin measurement by J. Erhart et al.3) , invalidated
the naive Heisenberg-type error-disturbance relation,
which initiated the recent activities on uncertainty relations.
It is shown that all the uncertainty relations are derived from suitably defined Robertson’s relation4) . Westart with Robertson’s relationσ(M out − A)σ(B out − B)1≥ |⟨[M out − A, B out − B]⟩|
2{σ(M out − A) + σ(A)}{σ(B out − B) + σ(B)}≥ |⟨[A, B]⟩|.

∗1(7)We thus conclude that all the known universally valid
relations are the secondary consequences of Robertson’s relation. Also, the saturation of Robertson’s relation is a necessary condition of the saturation of universally valid uncertainty relations. If one assumes theunbiased measurement and disturbance, one obtains(3).
By assuming unbiased joint measurements, we conclude6)⟨[A, B]⟩ = ⟨[M out , N out ]⟩ = 0(8)which is a contradiction since ⟨[A, B]⟩ ≠ 0 in general.
Similarly, one concludes6)⟨[A, B]⟩ = ⟨[M out , B out ]⟩ = 0(9)if one assumes the precise measurement of A and theunbiased disturbance of B which implies ⟨B out − B⟩ =0 for all ψ. Here B out = U † (B ⊗ 1)U stands for thevariable B after the measurement of A. Note that
[M out , B out ] = [M, B] = 0.
We interpret the algebraic inconsistency (9) as anindication of the failure of the assumption of unbiased
disturbance of B for the precise projective measurement of A, if all the operators involved are well-defined.
Thus the naive relation (3) fails. On the other hand,the Heisenberg error-error relationϵ(M out − A)ϵ(N out − B) ≥1|⟨[A, B]⟩|2σ(M out )σ(N out ) ≥ |⟨[A, B]⟩|

(4)

(10)

(5)

(11)

are expected to be valid as conditionally valid uncertainty relations. In this case the apparatus variable
N out becomes singular for the precise measurement ofA, namely, M out −A → 0 if the unbiasedness condition⟨N out − B⟩ = 0 is imposed.
Referenceswhere we used [M out , B out ] = [M, B] = 0. Using thevariations of Robertson’s relation, we obtain2)†(6)

and the Arthurs-Kelly relation

and use the triangle inequalityσ(M out − A)σ(B out − B)1≥ {|⟨[A, B]⟩| − |⟨[A, B out − B]⟩|
2−|⟨[M out − A, B]⟩|},

1|⟨[A, B]⟩|,
2and5)(1)

Another important development in the history of uncertainty relations is the analysis of Arthurs andKelly1) . They introduce the measuring apparatus Mfor A, and N for B, respectively, with [M, N ] = 0. The
notion of unbiased measurement is important in theiranalysis, which is defined byσ(M out − A)σ(B out − B) ≥+σ(A)σ(B out − B) ≥σ(M out − A)σ(B out − B) + σ(M out − A)σ(B)Condensed from the article in Int. J. Mod. Phys. A29
1450016 (2014). Invited talk given at Dyson Festivity, Singapore, August 26-29 (2013).
RIKEN Nishina Center- 122 -1) E. Arthurs and J.L. Kelly Jr, Bell. Syst. Tech. J. 44,725 (1965).
2) M. Ozawa, Phys. Rev. A67, 042105 (2003).
3) J. Erhart, et al., Nature Phys. 8, 185 (2012).
4) K. Fujikawa and K. Umetsu, Prog. Theor. Exp. Phys.,
013A03 (2013).
5) K. Fujikawa, Phys. Rev. A85, 062117 (2012).
6) K. Fujikawa, Phys. Rev. A88, 012126 (2013).