Experimental test of an entropic measurement uncertainty relation for arbitrary qubit observables

A tight information-theoretic measurement uncertainty relation is experimentally tested with neutron spin-1/2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e., positive-operator valued measures), as predicted theoretically.

The uncertainty principle was one of the first quantum phenomena discovered without any classical analogue. In 1927 Heisenberg presented his γ-ray microscope Gedankenexperiment [1] demonstrating that the position and momentum of an electron cannot be determined simultaneously with arbitrary precision due to the Compton recoil of the electron interacting with the photon. The uncertainty relation for the standard deviations ∆(·) of the canonically conjugate observables of position Q and momentum P ∆(Q)∆(P ) ≥ 2 (1) was proven soon thereafter by Kennard for arbitrary states [2]. This relation, however, quantifies the accuracy with which a state can be prepared with respect to the observables of interest, rather than the ability to jointly measure them. This shift of perspective persisted and, for decades, research on the uncertainty principle focused on these new types of relations, nowadays referred to as preparation uncertainty relations.
With the advent of information theory came novel approaches to quantifying uncertainty. The Shannon entropy [3] emerged as a powerful measure to quantify information content as a degree of uncertainty, with many applications in telecommunication, data compression, cryptography and coding theory [4]. It is not surprising, then, that entropic uncertainty relations were formulated soon after [5][6][7]. In particular, for finite dimensional systems, uncertainty relations such as Deutsch's inequality [8] and the Maassen and Uffink relation [9] were discovered, and presented advantages over Robertson's relation [10] ∆(A)∆(B) ≥ 1 2i ψ|[A, B]|ψ , such as their inherent state-independence. Entropic uncertainty relations have moreover proven useful in quantum cryptography [11,12], entanglement witnessing [13], complementarity [14] and other topics in quantum information theory [15], where entropy is a natural quantity of interest.
In recent years measurement uncertainty relations, in the spirit of Heisenberg's original proposal, have received renewed attention. Such uncertainty relations can be further divided into two classes: noise-disturbance relations, which quantify the fact that the more accurately a measurement determines the value of one observable, the more it disturbs the state of the measured system with respect to another incompatible observable; and noise-noise relations, which quantify the tradeoff between how accurately a measurement can jointly determine the values of two non-commuting observables. Noise-disturbance relations have received much attention, and many different measures of noise and disturbance have been proposed [16,17], subjected to experimental tests [18][19][20][21][22][23], and refined over time [24][25][26][27]. Information-theoretic measures of noise and disturbance have received significant recent interest, first in the framework introduced by Buscemi et al. [28] based on conditional Shannon entropies, and subsequently in several alternative approaches [29][30][31][32]. A major challenge in the derivation of entropic measurement inequalities based on such measures is to determine to what extent they are tight. Relations for even the simplest quantum systems can be delicate, as demonstrated in [33], where an allegedly tight noise-disturbance relation for orthogonal qubit observables was given and tested experimentally. In a recent publication [34], however, a counterexample was found, showing that for non-projective measurements the given relation can be violated. In this letter we focus on related noise-noise uncertainty relations, and we experimentally test the joint-measurement noise tradeoff for a range of (not necessarily orthogonal) Pauli observables. By implementing 4-outcome general quantum measurements, we confirm that they allow the tight qubit noise-noise relations to be saturated, while projective measurements generally do not.
Theoretical framework. Noise-noise uncertainty relations place fundamental restrictions on how accurately a quantum measurement device can jointly measure arXiv:1711.05023v1 [quant-ph] 14 Nov 2017 two (incompatible) quantum observables. As long as we are not concerned with how such a device disturbs the measured system-as one is for noise-disturbance relations-it can most generally be represented as a positive-operator valued measure (POVM). Recall that a POVM M is a set {M m } m of Hermitian positive semidefinite operators satisfying m M m = 1 (where 1 is the identity); the probability of obtaining outcome m when measuring M on a state ρ is given by Tr(M m ρ). Intuitively, if a device accurately measures an observable A it should be able to reliably distinguish between the eigenstates {|a } a of A [35], whereas a noisy measurement will fail to do so. Buscemi et al. proposed to quantify this noise by considering a scenario in which the eigenstates of A are prepared uniformly at random with probability p(a) = 1 d (where d is the Hilbert space dimension) and looking at the information that the measurement result m provides as to which eigenstate was prepared, in terms of conditional Shannon entropy [28]. Denoting the random variables associated with a and m as A and M, respectively, the noise of M on A is thus defined as where the joint probability p(a, m) may be calculated as p(a, m) = p(a)p(m|a) = 1 d Tr(M m |a a|). If A and B are two incompatible observables, the noises N (M, A) and N (M, B) (defined similarly) cannot both be zero, and, in general, there is a tradeoff between how accurately M can jointly measure these two observables. This tradeoff is completely characterized by the set of obtainable noise values. Noise-noise uncertainty relations are inequalities satisfied by all points in the noise-noise region R(A, B), such as the relation [9,28] which is valid for any observables. In general such relations may not be tight unless they coincide with the lower boundary of R(A, B), which is often difficult to characterize.
Recently, it was shown [34] that for qubit measurements R(A, B) can be related to the set where conv denotes the convex hull and H(A|ρ) = − a Tr(|a a| ρ) log Tr(|a a| ρ) is the Shannon entropy of the measurement statistics for A on the state ρ. In fact, the noise-noise region coincides with E(A, B) when one restricts oneself to projective qubit measurements M in Eq. (3). Thus, when E(A, B) is non-convex, noise values contained in R(A, B) \ E(A, B) can only be obtained by non-projective measurements M [34]. For qubit measurements, E(A, B) has been characterized fully [37], allowing R(A, B) to similarly be characterized, and tight noise-noise uncertainty relations were given in Ref. [34]. There, it was shown that if A = a · σ and B = b · σ are Pauli observables [with a, b two unit vectors on the Bloch sphere and σ = (σ x , σ y , σ z ) the vector of Pauli matrices] then R(A, B) has the form where g is the inverse of the binary entropy function h(x) defined for x ∈ [0, 1] as When | a · b| 0.391, E(A, B) is convex and one obtains explicitly the tight uncertainty relation whereas for | a · b| 0.391 it is non-convex [37][38][39]. In this case, the lower boundary of R(A, B) can be obtained by measuring 4-outcome POVMs corresponding to randomly choosing between two different projective measurements [34], thus allowing the noise-noise tradeoff to be saturated. Note that, for orthogonal Pauli observables (i.e. a · b = 0), Eq. (7) gives the linear inequality N (M, A) + N (M, B) ≥ 1, which corresponds to Eq. (4). In the present experiment we probe the noise-noise tradeoff for a range of Pauli observables A and B by implementing both projective measurements and POVMs, thereby highlighting the benefits obtainable by POVMs when E(A, B) is non-convex. Experiment. In our experiment we utilize neutron spins as qubits. The neutrons are produced by nuclear fission at the TRIGA Mark II reactor of the Vienna University of Technology, where they are first monochromatized to an average wavelength of λ = 2.02Å and then polarized by reflection on a Co-Ti supermirror. The particles entering the beam line are guided by a vertical magnetic field which determines the quantization axis and specifies the incident spin as |+z , the eigenvector of the Pauli matrix σ z . The experiment in its essence has to infer the unknown quantum state from the measurement procedure. For that we implemented the POVM where P ± ( r i ) = 1 2 (1 ± r i · σ) and r 1 , r 2 are two unit vectors on the Bloch sphere. For q = 1 this reduces to a single projective spin measurement along the direction r 1 , see Fig. 1(a), while for a general value of q between 0 and 1 this corresponds to a mixture of projective spin measurements in the directions r 1 and r 2 taken with probabilities q and 1 − q, see Fig. 1(b). In the experimental setup the mixing parameter q is given by the probabilistic transmission through a filter, realized by the combination of DC-Coil 1 and Analyzer 1 seen in Fig. 1(c). The purpose of the magnetic coil is to induce a unitary Larmor precession of the initial up-spin U (α) |+z = exp(iασ x ) |+z , where the rotation angle α is proportional to the applied magnetic field B QCT x (α) in the x -direction. Hereby, it is possible to generate two mutually orthogonal states |+ψ = U (α) |+z and |−ψ = U (α + π) |+z such that the probabilities of passing the analyzer, are given by q = | +z| + ψ | 2 and 1 − q = | +z| − ψ | 2 . The states emerging after the analyzer, which are projected onto |+z in both cases, are thus divided into two sub-ensembles with weights q, 1−q. The observables that the experiment aims at jointly (but noisily) measuring are the Pauli observables A = a · σ and B = b · σ. In each run of the experiment, one of their eigenstates |a or |b (with eigenvalues a, b = ±1) is generated uniformly at random by inducing an appropriately chosen rotation at DC-Coil 2. For simplicity we choose a = e z (the unit vector in the z direction) and consider various choices of b in the yz-plane. is applied: first the Bloch vector b is fixed, which determines the value | a · b| characterizing the noise-noise tradeoff. Initially we choose q = 1 so that only the operators P ± ( r 1 ) in Eq. (10) appear with a nonzero weight, corresponding to projective measurements. The vector r 1 = cos(ϑ 1 ) e z +sin(ϑ 1 ) e y is rotated in the interval ϑ 1 ∈ [0 • , 180 • ] with increments of ∆ϑ 1 ∼ = 10 • (see Fig. 1(a)). The variation of the polar angle changes the conditional probabilities p(a|m) and p(b|m) and thus the noises. The tradeoff attained this way saturates the lower-left boundary of E(A, B) (when the vector r 1 is in between a and b), i.e., the subset of the region R(A, B) obtainable with projective measurements. The upper-right boundary of this region can be obtained by rotating r 1 out of the plane spanned by a and b by a range of azimuthal angles ϕ, increasing the noise with respect to both A and B. Experimentally, this variation of ϕ is accomplished by a displacement of DC-Coil 3, denoted by y 0 (ϕ) in Fig. 1(c).
For | a · b| 0.391 the above approach already saturates the tight relation (9). To saturate the lower-left boundary of R(A, B) for | a · b| 0.391, the mixing parameter q is varied to implement the full 4-outcome POVM M. This is done by mixing the projectors P ± ( r 1 ) and P ± ( r 2 ) where the angles ϑ 1 and ϑ 2 defining these projectors are chosen as those for which the corresponding projective measurements give noise values that lie on the lower-left boundary of the shaded purple region. The optimal choice of projectors to mix corresponds to ϑ 1 = 0 and ϑ 2 = π 2 ; as q is varied, new points are obtained which are color coded from red to dark yellow, marking the change from q = 1 to q = 0. We see that in this case the improvement on the uncertainty relation by the POVMs is substantial, which implies that the ability to correctly infer the unknown eigenstates has been substantially increased by performing general measurements compared to projective measurements, which previous experiments had focused on [33].
When the eigenstates of B begin to approach those of A, as is the case in Fig. 2(b), the lower-left boundary of R(A, B) and, more noticeably, the purple region E(A, B), start to move towards the origin. This becomes more apparent in Fig. 2(c), where the optimal . By realizing the POVM accordingly for a range of q values we again succeeded in surpassing the limit of projective measurements and saturating the noise-noise tradeoff. In Fig. 3 we present two cases with inner products (a) a · b ∼ = 0.35 and (b) a · b ∼ = 0.5, on either side of the critical value of a · b ∼ = 0.391 at which the region E(A, B) (in purple) becomes convex. In Fig. 3(a) the dashed orange line from (N (M, A), N (M, B)) ∼ = (0.17, 0.70) to (0.70, 0.17) implies that projective measurements are theoretically not sufficient to saturate the noise-noise tradeoff, but the size of the error bars means that there is no chance of resolving this difference in our experiment. In Fig. 3(b) the region R(A, B) is already convex and can be fully attained with projective measurements, hence improvements by general POVMs are no longer possible.

Conclusion.
A measurement device cannot jointly measure two non-commuting observables with arbitrary precision, and thus there is a tradeoff between the accuracy with which they can both be measured. By quantifying this (im)precision via entropic definitions of noise, this tradeoff can be captured formally by state-independent measurement uncertainty relations. Using a definition of noise that quantifies how well a measurement device can distinguish eigenstates of incompatible observables [28], we experimentally tested tight entropic noise-noise uncertainty relations for qubits [34] for various Pauli spin observables. If these observables are close enough (with an inner product between the corresponding Bloch vectors larger than 0.391), we saw that the uncertainty relation could be saturated with simple projective measurements. However, we verified experimentally that this is not generally the case, and that four outcome POVMs allow better joint measurements that saturate the uncertainty relation when projective measurements cannot. Our study paves the way for further experiments that may also consider how a measurement disturbs a quantum state, so as to test noise-disturbance relations-which, in the framework of Ref. [28] considered here, are more difficult to obtain and to prove to be tight [34].