Error-Disturbance Uncertainty Relations in Neutron-Spin Measurements

In his seminal paper, which was published in 1927, Heisenberg originally introduced a relation between the precision of a measurement and the disturbance it induces onto anothermeasurement. Here, we report a neutron-optical experiment that records the error of a spin-component measurement as well as the disturbance caused on a measurement of another spin-component to test error-disturbance uncertainty relations (EDRs). We demonstrate that Heisenberg’s original EDR is violated and the Ozawa and Branciard EDRs are valid in a wide range of experimental parameters.


Introduction
The uncertainty principle represents, without any doubt, one of the most important cornerstones of the Copenhagen interpretation of quantum theory.In his celebrated paper from 1927 [1], Heisenberg gives at least two distinct statements about the limitations on preparation and measurement of physical systems: (i) incompatible observables cannot be measured with arbitrary accuracy: a measurement of one of these observables disturbs the other one accordingly, and vice versa; (ii) it is impossible to prepare a system such that a pair of noncommuting (incompatible) observables are arbitrarily well defined.In [1], the observables are represented by position and momentum.
In his original paper [1], Heisenberg proposed a reciprocal relation for measurement error and disturbance by the famous -ray microscope thought experiment: "At the instant when the position is determined-therefore, at the moment when the photon is scattered by the electron-the electron undergoes a discontinuous change in momentum.This change is the greater the smaller the wavelength of the light employedthat is, the more exact the determination of the position. .." [1].Heisenberg follows Einstein's realistic view, that is, to base a new physical theory only on observable quantities (elements of reality), arguing that terms like velocity or position make no sense without defining an appropriate apparatus for a measurement.By solely considering the Compton effect, Heisenberg gives a rather heuristic estimate for the product of the inaccuracy (error) of a position measurement  1 and the disturbance  1 induced on the particles momentum, denoted by According to (1), it can be referred to as a measurement uncertainty (i) or as an error-disturbance uncertainty relation (EDR).Heisenberg's original formulation [1,2] can be read in modern treatment as ()() ≥ ℏ/2, for error () of a measurement of the position observable  and disturbance () of the momentum observable  induced by the position measurement.However, most modern textbooks introduce the uncertainty relation in terms of a preparation uncertainty (ii) relation denoted by (2) Equation ( 2) was proved by Kennard in 1927 [3] for the standard deviations Δ() and Δ() of the position observable  and the momentum observable , given by However, certain measurements do not obey (4) [7][8][9], proving (4) to be formally incorrect.
In 2003, Ozawa introduced the correct form of a generalized error-disturbance uncertainty relation based on rigorous theoretical treatments of quantum measurements: where () denotes the root-mean-square (r.m.s.) error of an arbitrary measurement for an observable , () is the r.m.s.disturbance on another observable  induced by the measurement, and Δ() and Δ() are the standard deviations of  and  in the state |⟩ before the measurement.Ozawa's inequality (5) was tested experimentally with neutronic [10,11] and photonic [12][13][14] systems.Though universally valid, Ozawa's relations (5) are not optimal.Recently, Branciard [15] has revised Ozawa's EDR, resulting in a tight EDR, describing the now optimal tradeoff relation between error () and disturbance (): with Experimental demonstrations of ( 6) using photons are reported in [16,17].

Materials and Methods
In our experiment the error-disturbance uncertainty relations, as defined in ( 5) and ( 6), are tested via a successive measurement for spin observables  and .The experimental scheme is depicted in Figure 1.The observables  and  are set as the  and  components of the neutron 1/2 spin.(For simplicity, ℏ/2 is omitted for each spin component.)The error () and the disturbance () are defined for a joint measurement apparatus, so that apparatus A1 measures the observable  =   with error () and disturbs the observable  =   thereby with disturbance () during the measurement (here   and   denote the Pauli matrices).Finally apparatus A2 measures  =   .To control the error () and the disturbance (), apparatus A1 is designed to actually carry out not the maximally disturbing projective measurement  =   , but the projective measurement along a distinct axis ⃗   (, ) denoted by   = ⃗   (, )⋅ ⃗ , where ⃗  = (  ,   ,   )  .Here ,  denote polar and azimuthal angle of the measurement direction ⃗   and are experimentally controlled detuning parameters, so that () and () are determined as a function of  and .A schematic illustration of the experimental apparatus for successive neutron-spin measurements is given in Figure 2.
For ( 5) and ( 6), error () and disturbance () are defined via an indirect measurement model for an apparatus A measuring an observable  of an object system S as where |⟩ is the state before the measurement of system S, which is described by a Hilbert space H obj , and |⟩ and  are the initial state of the probe system P (in Hilbert space H pro ) and an observable , referred to as meter observable, of P which accounts for the meter of the apparatus.A unitary operator  on H obj ⊗H pro describes the time evolution of the composite system S + P during the measurement interaction.
Here the Euclidean norm is used where the norm of a state vector in Hilbert space |⟩ is given by the square root of its inner product: ‖|⟩‖ = ⟨| † |⟩ 1/2 .A schematic illustration of a measurement apparatus A is given in Figure 3.
A nondegenerate meter observable  has a spectral decomposition  = ∑  |⟩⟨|, where  varies over eigenvalues of , and then the apparatus A has a family {  } of operators, called the measurement operators, acting on H obj and defined as   = ⟨||⟩.Hence, the error is given by () 2 = ∑  ‖  ( − )|⟩‖ 2 .If   are mutually orthogonal, projection operators sum and norm can be exchanged and the error can be written in compact form as The setup is divided into three stages: state preparation (blue region), apparatus A1 carrying out the measurement of observable   = ⃗   (, ) ⋅ ⃗  (red region), and apparatus A2 performing the measurement of observable  =   (green region).All required terms of (5), that is, error () and disturbance () as well as the standard deviations Δ() and Δ(), are determined from the expectation values of the successive measurement. .All these calculations are elaborated in detail in [18].
In our experiment, the measuring apparatus A1 is considered to carry out a projective spin measurement along a distinct axis ⃗   (, ) denoted by   = ⃗   (, )⋅ ⃗  =  +1 − −1 (where  ±1 = 1/2(1± ⃗   (, )⋅ ⃗ )) instead of precisely  =   .In order to detect the disturbance () on the observable , induced by measuring   , apparatus A2 carries out the projective measurement of  =   in the state just after the first measurement.Though claimed to be experimentally inaccessible [19,20], in the case of projection operators error () and disturbance () can be expressed as a sum of expectation values in three different states, applying the method proposed in [21].Using the modified output operators of the apparatus A2 defined as  For  = 0, the error () vanishes and the disturbance () is maximal.The disturbance () vanishes for   =  ( = /2) and reaches a second maximum for   = −.Note that at this point also the error () has its (only) maximum.The famous trade-off relation, that is, the reciprocal relation for error and disturbance, only holds for −/2 ≤  ≤ /2, which can be seen in Figure 5(a).The product of error and disturbance ()()-left-hand side of (4) or Heisenberg term-is below the limit given by (1/2)⟨|[, ]|⟩ in a wide range of -values, thereby revealing a violation of the generalized Heisenberg relation (see (4)).On the contrary, the left-hand side of Ozawa's relation ()() + ()Δ() + Δ()() (see ( 5)) is always above the lower bound defined by the expectation value of the commutator demonstrating the validity of Ozawa's new relation.
In the following experimental setting,   is rotated out of the equatorial plane, when the evolution is on circles of latitude on the Bloch sphere (fixed polar angle ), which yields The observed values are depicted in Figure 5(b).Now neither the error () nor the disturbance () vanishes, since they never coincide with , , or −.This behaviour affects the curves in such a way that they are now shrunken from below.The smaller  is, the less regions the polar angle of   gets where the Heisenberg term ()() remains below the limit.Ozawa's inequality is again fulfilled over the entire range of .The relations shown in Figure 5 are verified for all directions of .
A modification of the measurement apparatus allows for reducing the disturbance and saturating Branciard's EDR given in (6).If we apply an arbitrary unitary rotation after the first measurement, the error remains unchained but the disturbance is altered.By investigating all possible rotation axes and angles, one finds out that   =  −((  −)/2)  minimises the disturbance yielding where   is the relative angle between the  and  measurement direction in the equatorial plane.Note that this particular rotation just generates the eigenstates of observable , that is, | ± ⟩, making the result of the optimisation procedure more intuitive.For a detailed calculation see [15].This is experimentally achieved by an appropriate displacement of DC-3, such that the required rotation is induced, and by additional Larmor precision in the guide field.The results, both for modified and for original apparatuses, are plotted in Figure 6, demonstrating the tightness of Branciard's inequality, defined in (6).

Conclusions
To summarize, we have experimentally tested the Ozawa and Branciard error-disturbance uncertainty relations in successive neutron-spin measurements.Our experimental results clearly demonstrated the validity of Ozawa and Branciard EDRs and that the original Heisenberg EDR is violated throughout a wide range of experimental parameters.Blue curve: the Branciard bound as defined in (6).Blue marker: experimental results using the modified apparatus for () and (), as defined in (10).Orange curve error and disturbance as in (9).Green curve: bound imposed by Heisenberg's original error-disturbance relation ()() ≥ |  |, which is violated by our experimental results.Red curve: Ozawa's relation (5), which is indeed satisfied but is not saturated.

Figure 1 :
Figure 1: A successive measurement scheme of observables  and  exploited for the demonstration of the error-disturbance uncertainty relation.After state preparation (blue) apparatus A1 carries out a projective measurement of   instead of  (light red), thereby disturbing observable  which is detected by apparatus A2 (green), error () and disturbance () are quantitatively determined by the four possible outcomes denoted by (++), (+−), (−+), and (− −).

4 (Figure 2 :
Figure 2: Neutron polarimetric setup for demonstration of the universally valid uncertainty relation for error and disturbance in neutronspin measurements.The setup is divided into three stages: state preparation (blue region), apparatus A1 carrying out the measurement of observable   = ⃗  (, ) ⋅ ⃗  (red region), and apparatus A2 performing the measurement of observable  =   (green region).All required terms of(5), that is, error () and disturbance () as well as the standard deviations Δ() and Δ(), are determined from the expectation values of the successive measurement.

Figure 5 :
Figure 5: Experimentally determined values of ()Δ(), Δ()(), and ()().This last term corresponds to the left-hand side of the Heisenberg relation (4), and the sum of the three terms corresponds to the left-hand side of Ozawa's relation(5), including Bloch sphere representation of observables and initial state.

Figure 6 :
Figure6: Results of error and disturbance plotted as   versus ().Blue curve: the Branciard bound as defined in(6).Blue marker: experimental results using the modified apparatus for () and (), as defined in(10).Orange curve error and disturbance as in(9).Green curve: bound imposed by Heisenberg's original error-disturbance relation ()() ≥ |  |, which is violated by our experimental results.Red curve: Ozawa's relation(5), which is indeed satisfied but is not saturated.