Neutron optical test of completeness of quantum root-mean-square errors

One of the major problems in quantum physics has been to generalize the classical root-mean-square error to quantum measurements to obtain an error measure satisfying both soundness (to vanish for any accurate measurements) and completeness (to vanish only for accurate measurements). A noise-operator based error measure has been commonly used for this purpose, but it has turned out incomplete. Recently, Ozawa proposed a new definition for a noise-operator based error measure to be both sound and complete. Here, we present a neutron optical demonstration for the completeness of the new error measure for both projective (or sharp) as well as generalized (or unsharp) measurements.


INTRODUCTION
The notion of the mean error of any measurement is a well-defined quantity in classical physics. However, extending the classical notion of root-mean-square (rms) error, which has been broadly accepted as the standard definition for the mean error of a measurement, to quantum measurements is a highly challenging and a non-trivial task [1][2][3][4][5][6][7][8]. A straightforward generalization is represented by the noise-operator based quantum root-meansquare (q-rms) error, defined as the root-mean-square of the noise operator, where the noise-operator indicates how closely a meter observable "tracks" the observable to be measured. The noise-operator was first used in attempts to prove Heisenberg's uncertainty relation [9] for approximate simultaneous measurements of pairs of non-commuting observables [10][11][12].
However, Busch, Heinonen, and Lahti (BHL) [25] raised a completeness problem for the noise-operator based quantum root-mean-square error, which eluded a solution for a decade and brought about a debate on the use of the noise-operator [7], until Ozawa [26] recently brought a satisfactory solution. It is the purpose of this work to briefly recapitulate the argument of BHL [25], Ozawa's new definition of a sound and complete noise-operator based q-rms error [26], and to present a neutron optical experiment which demonstrates the completenesss of the new noise-operator based q-rms error. At this point we want to emphasize that the new error notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations [26].

THEORY
The noise-operator based quantum root-meansquare (q-rms) error [27] of a measuring process M, on quantum instrument I, is denoted as ε NO (A) = ψ, ξ|N (A, M) 2 |ξ, ψ 1/2 , where the noiseoperator N (A, M) describes how accurately the value of an observable A is transferred to the meter observable M A , during the evolution U (t) of the composite system: Here A is an observable of a system S in state |ψ of Hilbert space H, and M A is the observable representing the meter of the observer in the probe system (measurement device) P in initial state |ξ of Hilbert space K. Moreover, U (t) is the unitary evolution of the composite quantum system S + P. This concept, introduced in [28], is usually referred to as indirect measurement model of measuring process M and schematically illustrated in Fig. 1. In the Heisenberg picture, we shall write A(0) = A ⊗ 1 1 and M A (t) = U (t) † (1 1 ⊗ M A )U (t). The POVM Π of the measuring process M is defined by Π(x) = ξ|P M A (t) (x)|ξ , where P M A (t) (x) is the spectral projection of M A (t) for eigenvalues x. The moment operator M of the POVM Π is defined by M = x x Π(x), and the second moment operator M (2) of the POVM Π is defined by M (2) = x x 2 Π(x). The measurement is called a sharp measurement of M if Π is projection-valued. In this case, we have Π(x) = P M (x) and M (2) = M 2 . Otherwise, the measurement is called an unsharp (or generalized) measurement of M ; in this case we have M (2) > M 2 . An important property of ε NO (A) is that it is determined by (moment operators of) the POVM Π of M in such a way that (1) This property and its consequence are to be studied in the present experiment.

Counter-example
It is shown by BHL [25] that there exists a measuring process M with ε NO (A, Π, |ψ ) = 0, whereas M does not accurately measure A. However, a vanishing error is only expected for an accurate measurement for the completeness of the error measure. Here, we do not give the original counter-example but the slightly simplified version as stated in [26]. Consider a measurement of the observable A in a two-level system in the initial state |ψ with measuring process described by a POVM Π with the moment operator M given as follows.
First, we consider the sharp measurement of M with the POVM Π 1 . In this case, one obtains Π 1 (x) = P M (x) and M (2) = M 2 , so that However, this particular measurement is not accurate, since A and M have disjoint spectra. The operator A has spectral decomposition A = i a i |a i a i |, with eigenvalues a i = {2, 0} and normalized eigenvectors With P A (2), being the projector associated with eigenvalue 2, that is |+x +x| ≡ P σx (1), which finally gives +z| P σx (1) |+z = 1 2 . We can then write ψ|P A (2)|ψ = 1 2 = ψ|Π 1 (2)|ψ = 0 to express the inaccuracy of the measurement. Thus, the measurement with the POVM Π 1 = P M does not accurately measure A but ε NO (A, Π 1 , |ψ ) = 0.

Requirements
To resolve this inconsistency, Ozawa introduced four requirements for a valid definition of error measure ε generalizing the classical rms error [26]: (i) Operational definability: The error measure is definable by he POVM Π of measuring process M with A and |ψ , i.e., ε = ε(A, Π, |ψ ).
(ii) Correspondence principle: If A(0) and M A (t) commute, ε(A, Π, |ψ ) equals the classical rms error determined by the joint probability distribution of A(0) and M (t).
(iii) Soundness: The error measure ε should vanish for any accurate measurements.
(iv) Completeness: The converse of soundness -a measurement should be accurate if the error measure ε vanishes.
Ozawa [26] showed that the noise-operator-based quantum rms error satisfies requirements (i) -(iii) so that it is a sound generalization of the classical rms error, and proposed a modification of it to satisfy all the requirements (i) -(iv) including completeness. In addition to (i) -(iv), the new error measure is shown to have the following two properties: (v) Dominating property: The error measure ε dominates the noise-operator based q-rms error, that is (vi) Conservation property for dichotomic measurements: The error measure ε coincides with the noise-operator based q-rms error ε NO for dichotomic measurements, i.e., ε NO (A, Π, |ψ ) = ε(A, Π, |ψ ) if Thus the new notion maintains all previously obtained universally valid uncertainty relations and their experimental confirmations [13][14][15][16][17][18] without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable [7].
Definition & predictions of locally uniform quantum root-mean-square error For any t ∈ R the quantum root-mean-square (q-rms) error profile ε t for A and Π in |ψ is defined as In order to obtain a numerical error measure the locally uniform q-rms errorε is given bȳ Thenε is a sound and complete q-rms error, satisfying both the dominating property, and the conservation property for dichotomic measurements. For the given example from Eq.(2), with A, Π 1 , and |ψ , we get (7) for the sharp M measurement described by the POVM Π 1 . The relationε(A, Π, |ψ ) = 2 correctly indicates that the measurement of A described in the example above is not an accurate measurement. For the unsharp M measurement described by the POVM Π 2 , one gets for the completeness of the error measure¯ .

EXPERIMENTAL
Here, we present a neutron polarimetric measurement of the quantum root-mean-square (q-rms) error profile ε α , resulting in determination of the locally uniform qrms error¯ , for the POVM Π 1 (the sharp measurement of M ) and the POVM Π 2 (an unsharp measurement of M ), as given in Eqs. (7), (8) to demonstrate the completeness property and thereby confirm the resolution of the inconsistency in question.

Experimental setup
The experiment was performed at the polarimeter instrument NepTUn (NEutron Polarimeter TU wieN), located at the tangential beam port of the 250 kW TRIGA , |ψ(α + π) and |+x , respectively. Projective (sharp) measurement are realized by applying projectors P M (± √ 2) and generalized (unsharp) in terms of POVM by randomized sequences of P M and P σx . Bloch spheres above setup indicate evolution of initial state |ψ(α) and measured projectors P M and P σx .
Mark II research reactor at the Atominstitut -TU Wien, in Vienna, Austria. A schematic illustration of the setup is given in Fig. 2. An incoming monochromatic neutron beam, reflected from a pyrolytic graphite crystal, with mean wavelength λ 2.02Å (∆λ/λ 0.02) is polarized along the vertical (+z) direction by refraction from a CoTi multilayer array, hence on referred to as supermirror. The neutron polarimetric setup consists of three stages, as indicated in Fig. 2. The blue stage indicates the preparation of the incident state |ψ = |+z , which is reflected from the polarizer (first super mirror). In the red stage the state evolution of initial state |ψ = |+z as |ψ(t) = e −itA |ψ → e (iασx)/2 |ψ ≡ |ψ(α) is induced, due to rotation by angle α about the x-axis (note that the error profile α is a function of the rotation angle α). The Larmor precession inside direct current (DC) coil 1 is induced by the static magnetic field B (α) x . In the green stage, a projective (or sharp) measurement of M is performed first, in order to demonstrate the counter example ε t (A, Π 1 , |ψ(t) ) = 2| sin t| from [26]. The Π 1 measurement has two possible outcomes, namely m = + √ 2 and m = − √ 2, corresponding to measure- The combined action of DCcoil 2 and the analyzer (second super mirror) realizes the respective projector.

DISCUSSION & CONCLUSION
As seen already from Eq.(1) the error ε(A, Π, |ψ ) depends on the choice of the respective POVM Π that realizes a particular measurement. From a physical point of view one might ask which measurement is optimal ? Although individual expectation values (mean values) are the same for sharp (projective) and unsharp (POVM) realizations of the same measurement M , regarding measurement error of single measurements, sharp measurements are always superior compared to unsharp measurements; the reason a clearly seen from Eq.(1). However in the case of joint, simultaneous (or successive) measurements, where an optimal error-error (or errordisturbance) trade-off is required, unsharp measurements are able to outperform sharp measurements [17,29,30].
To conclude, despite numerous successful experimental demonstrations of error-disturbance uncertainty relations based on the noise-operator based q-rms error commonly used as a sound generalization of the classical rms error [1,2,13,19,20,27], Busch, Heinonen, and Lahti [25] pointed out the incompleteness of this error measure. A new definition for a noise-operator based errormeasure remedying the incompleteness was recently proposed by Ozawa [26]. It is important to note that the new error-notion affects only non-dichotomic measurements and consequently all experimentally obtained results up to date remain valid. The completeness behavior of the new root-mean-square definition of the error has been observed in detail along with the counter-example given in Ref. [26].

METHODS
In order to experimentally demonstrate the completeness of locally uniform q-rms errorε, Eqs. (7), (8) need to be expressed in terms of experimentally accessible quantities, i.e., expectation values. This can be achieved by applying the well known three-state-method [27] for generalized measurements [2] (p.383) to obtain the statedependent q-rms error profile ε t (A, Π, |ψ(t) ) from where M (2) denotes the second moment of Π, given by . The first term of Eq.(9) can be symmetrized, applying the operator identity which gives The q-rms error-profile for all evolved states |ψ(α) ) is calculated using the three-state method (see Supplementary Information [31] for the individual measurement results of all terms from Eq. (11)). Next, we analyze the time evolution of the initial state |ψ = (1, 0) T ≡ |+z , dependent on A, as expressed in Eq. (5). The observable A can be decomposed as A = 1 1 + σ x . Hence, the time evolution of the initial state yields which is simply a rotation about the x-axis by an angle α (see Bloch sphere in Fig. 2). Thus the parametrization has changed from time t to an (experimentally adjustable) spinor rotation angle α.

Sharp M measurement
In order to demonstrate the counter example a sharp measurement of M is required. The decomposition of M into projectors is denoted as where with Therefore, the error-profile ε 2 α (A, Π 1 , |ψ(α) ) yields which finally gives with locally uniform q-rms errorε(A, Π 1 , |ψ ) = 2, as predicted in [26]. Note that only for dichotomic measurements the first two terms of Eq.(16) are unity and the error profiles becomes α-independent (see Supplementary Information [31] for experimental details and results of all individual expectation values of the sharp M -measurement).
Experimental results of individual expectation values for projective measurements and POVMs.

(S. 2)
For the projective (sharp) measurement we simply get with projectors P ± m from Eq. (14). The experimental results for the third term from Eq.(11) are depicted in Fig. S. 3 (left) for generalized (unsharp) measurement via POVM decomposition, and (right) for a projective (sharp) measurement.