#### Abstract

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 another measurement. 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.

#### 1. 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 employed—that 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 and the disturbance 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 , 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
Equation (2) was proved by Kennard in 1927 [3] for the standard deviations and of the position observable and the momentum observable , given by . But this is a different physical situation: here statistical distributions of not a joint but a single measurement of either or are considered. Kennard’s relation addresses an intrinsic uncertainty which every quantum system must possess, independent of whether it is measured or not. The unavoidable recoil caused by the measuring device is ignored here. Later Robertson generalized Kennards relation between standard deviations to arbitrary pairs of observables and :
Robertson’s relation (3) has been confirmed by many different experiments [4–6] and is uncontroversial.

A corresponding generalized form of Heisenberg’s original error-disturbance uncertainty relation would read However, certain measurements do not obey (4) [7–9], 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 and 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 [12–14] 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 trade-off relation between error and disturbance : with . Experimental demonstrations of (6) using photons are reported in [16, 17].

#### 2. 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, is omitted for each spin component.) The error and the disturbance are defined for a joint measurement apparatus, so that apparatus measures the observable with error and disturbs the observable thereby with disturbance during the measurement (here and denote the Pauli matrices). Finally apparatus measures . To control the error and the disturbance , apparatus 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 measuring an observable of an object system aswhere is the state before the measurement of system , which is described by a Hilbert space , and and are the initial state of the probe system (in Hilbert space ) and an observable , referred to as* meter observable*, of which accounts for the meter of the apparatus. A unitary operator on describes the time evolution of the composite system 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: . A schematic illustration of a measurement apparatus is given in Figure 3.

A nondegenerate meter observable has a spectral decomposition , where varies over eigenvalues of , and then the apparatus has a family of operators, called the* measurement operators*, acting on and defined as . Hence, the error is given by . If are mutually orthogonal, projection operators sum and norm can be exchanged and the error can be written in compact form as , where is the* output operator* given as . The disturbance can be written as . All these calculations are elaborated in detail in [18].

In our experiment, the measuring apparatus is considered to carry out a projective spin measurement along a distinct axis denoted by (where ) instead of precisely . In order to detect the disturbance on the observable , induced by measuring , apparatus 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 defined as and , measurement error and disturbance are given by

The expectation values of in a state (see (8a) and (8b)), necessary for the determination of error , are derived from the intensities at the four possible output ports, depicted in Figure 4, denoted by , , , and . The expectation value is obtained from the following combination of count rates: , using intensities at the four possible output ports, indicating which projections have been carried out. As already discussed due to the prior measurement of , the operator of apparatus is modified from to , with the corresponding expectation value expressed as , required to determine the disturbance . Consequently all expectation values necessary to determine error and disturbance can be derived from the intensities in the three input states , , and , , , respectively.

These states are generated by spinor rotations within DC-1 and induced by the guide field, due to an appropriate coil position within the preparation section (blue) of the neutron-optical setup depicted in Figure 2. The projective measurement of (apparatus , light red in Figure 2) consists of two sequential steps: first the initially prepared state is projected onto the eigenstates of by DC-2, which rotates the respective spin component of belonging to in direction. Then, in order to complete the projective measurement the spin, which is pointing in after the analyzer, has to be prepared in an eigenstate of . This is achieved by proper positioning and magnetic field of DC-3 (thereby applying the same procedure as that for DC-1 in the initial state preparation). Finally the measurement is performed (apparatus , green in Figure 2) utilizing DC-4 and the second analyzer. Unlike the -measurement, subsequent preparation of the eigenstates of is not necessary since the detector is insensitive to the spin state. For the measurement of the standard deviations of the observables and , which are also required to test Ozawa’s relation (see (5)), the two measurement apparatuses are used individually.

#### 3. Results and Discussion

The experiment was carried out at the polarimeter beam line of the tangential beam port of the research reactor facility TRIGA Mark II at the Atominstitut, Vienna University of Technology, where mainly fundamental aspects of quantum mechanics are investigated [22–25]. The experimental settings for initial state and observables and require the auxiliary input states , , and finally to be prepared. In Figure 4, explicit examples of related intensity sets for different values of are depicted. Standard deviations yield and the right-hand side of the uncertainty relations gives a lower bound of .

In a first experimental run, is varied along the equator () parameterized by its azimuthal angle . The theory curves for and are then given by
For , the error vanishes and the disturbance is maximal. The disturbance vanishes for ( 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 , 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 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 (see (5)) is always above the lower bound defined by the expectation value of the commutator demonstrating the validity of Ozawa’s new relation.

**(a)**

**(b)**

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 and .

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 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).

#### 4. 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.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors thank M. Ozawa for valuable discussion. This work has been supported by the Austrian Science Fund (FWF), Projects nos. P25795-N02 and P24973-N20.