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

Journal name:
Nature Physics
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Published online
Corrected online
Corrected online

The uncertainty principle generally prohibits simultaneous measurements of certain pairs of observables and forms the basis of indeterminacy in quantum mechanics1. Heisenberg’s original formulation, illustrated by the famous γ-ray microscope, sets a lower bound for the product of the measurement error and the disturbance2. Later, the uncertainty relation was reformulated in terms of standard deviations3, 4, 5, where the focus was exclusively on the indeterminacy of predictions, whereas the unavoidable recoil in measuring devices has been ignored6. A correct formulation of the error–disturbance uncertainty relation, taking recoil into account, is essential for a deeper understanding of the uncertainty principle, as Heisenberg’s original relation is valid only under specific circumstances7, 8, 9, 10. A new error–disturbance relation, derived using the theory of general quantum measurements, has been claimed to be universally valid11, 12, 13, 14. Here, we report a neutron-optical experiment that records the error of a spin-component measurement as well as the disturbance caused on another spin-component. The results confirm that both error and disturbance obey the new relation but violate the old one in a wide range of an experimental parameter.

At a glance


  1. Concept of the experiment.
    Figure 1: Concept of the experiment.

    A successive measurement scheme of observables A and B is exploited for the demonstration of the error–disturbance uncertainty relation. After preparing an initial state , apparatus M1 is assumed to measure an observable A (red region). The error ε(A) and the disturbance η(B) are experimentally controlled by detuning apparatus M1 to make a projective measurement of OA instead of A(light red). The disturbance η(B) on the observable B is caused by the measurement made by apparatus M1, which randomly projects the state onto one of the eigenstates of OA, and is quantified using the B measurement carried out by apparatus M2 (yellow region). The successive measurements of OA and B result in four possible outcomes, denoted as (++), (+−), (−+) and (−−), from which error ε(A) and disturbance η(B) are quantitatively determined.

  2. Illustration of the experimental apparatus.
    Figure 2: Illustration of the experimental apparatus.

    The set-up is designed for demonstration of the universally valid uncertainty relation for error and disturbance in neutron spin measurements. The neutron optical set-up consists of three stages: preparation (blue region), apparatus M1 making the measurement of observable OA=σϕ (red region) and apparatus M2 carrying out the measurement of observable B=σy (yellow region). A monochromatic neutron beam is polarized in the +z direction by passing through a supermirror spin polarizer. By combining the action of four d.c. coils, the magnetic guide field B0 and the analysing supermirrors, the successive measurements of σϕ and σy are made for the required states (see Methods for details). The error ε(A) and the disturbance η(B), as well as the standard deviations of σ(A) and σ(B), are determined from the expectation values of the successive measurements.

  3. Experimental results.
    Figure 3: Experimental results.

    Normalized intensity of the successive measurements carried out by apparatus M1 and M2. The successive measurements of M1 and M2 have four outcomes, denoted as (++), (+−), (−+) and (−−). Intensities, according to the corresponding outcomes, are depicted for each initial state, that is , , and . Three sets for detuning parameter ϕ=0°,40° and 90° are plotted. The error ε(A) and the disturbance η(B) are determined from these 16 intensities, for each setting of the detuning parameter ϕ. Error bars represent ±1s.d. of the normalized intensities. Some error bars are smaller than the size of the markers.

  4. Trade-off relation between error and disturbance.
    Figure 4: Trade-off relation between error and disturbance.

    a,b, Error ε(A) (a) and disturbance η(B) (b) both as a function of the detuning angle ϕ with the plots of the values determined by the experiment. The theory predicts ε(A)=2sin(ϕ/2) and . The experimental data are in good agreement with theory, showing the trade-off: the less the error, the more the disturbance. Error bars represent ±1s.d.

  5. Experimentally determined values of the universally valid uncertainty relation.
    Figure 5: Experimentally determined values of the universally valid uncertainty relation.

    a, The universally valid expression ε(A)η(B)+ε(A)σ(B)+σ(A)η(B) (orange) and the actual Heisenberg product ε(A)η(B) (red) as a function of the detuning angle ϕ with theoretical predictions. The Heisenberg product is always smaller than the calculated limit (1/2)|left fenceψ|[A,B]|ψright fence|=1 (depicted as the dashed line). In contrast, the universally valid expression is always larger than the limit. b, The two additional product terms σ(A)η(B) (green) and ε(A)σ(B) (blue) in the universally valid expression together with the theoretical predictions. ε(A), η(B) and σ(A) (σ(B)) represent error, disturbance and standard deviations, respectively. Error bars represent ±1s.d.

Change history

Corrected online 12 March 2012
In the version of this Letter originally published, in Fig. 5a a factor of /2 appeared incorrectly against the label 'Heisenberg lower limit' and in the figure caption. This error has been corrected in the HTML and PDF versions of the Letter.
Corrected online 01 August 2012
In the version of this Letter originally published, in the Methods section under the heading 'Error and disturbance in spin measurements: theoretical determination' the equation defining η(B) was incorrect — the term σϕ should have been divided by 2. This error has been corrected in the HTML and PDF versions of the Letter.


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  1. Atominstitut, Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria

    • Jacqueline Erhart,
    • Stephan Sponar,
    • Georg Sulyok,
    • Gerald Badurek &
    • Yuji Hasegawa
  2. Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan

    • Masanao Ozawa


J.E., G.S. and S.S. carried out the experiment and analysed the data; G.B. contributed to the development at the early stage of the experiments; M.O. supplied the theoretical part and conceived the experiment; Y.H. conceived and carried out the experiment; all authors co-wrote the paper.

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