## Abstract

In the light of the Busch, Lathi and Werner proposal, we explore, for the first time, the joint measurements and confirmation of uncertainty relations for three incompatible observables that reflect the original spirit proposed by Heisenberg in 1927. We first develop the error trade-off relations theoretically and then demonstrate the first experimental witness of joint measurements using a single ultracold ^{40}Ca^{+} ion trapped in a harmonic potential. In addition, we report, that in contrast to the case of two observables, scarifying accuracy of any one of the three observables the rest of two can be measured with ultimate accuracy.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The uncertainty principle provides a fundamental limitation in quantum measurements, which refers to the error bounds associated with the joint measurements of non-commuting observables in preparing/measuring the quantum states [1]. This principle was initially expostulated by Heisenberg in 1927. However, the mathematical forms of uncertainty relations derived by Kennard [2], Weyl [3] and Robertson [4] are basically different, which are based on the deviation in measurement statistics of two complementary observables. Physically, these forms of uncertainty relations concern separate measurements of the two complementary observables performed on two ensembles of identically prepared quantum systems, which is conceptually different from the theme of Heisenberg’s idea of a trade-off for the errors of approximate simultaneous or successive measurements performed on the same system [5].

From the experimental point of view, measuring two complementary observables $A$ and $B$ inevitably involves disturbance, e.g., a single measurement of $A$ can only be accomplished with an imminent disturbance in any subsequent or simultaneous measurement of $B$. As a result, treatment of the inaccuracy in an approximate measurement of $A$ and the disturbance incorporated in a subsequent or simultaneous measurement of $B$ originates a trade-off, which is known as an error-disturbance relation. Based on this idea, new inequalities for uncertainty relations have been independently proposed over past years [6–14]. Although some of these inequalities were verified experimentally [15–22], the debates still have been lasting due to existing disagreements on appropriate quantification of error and disturbance. In particular, the approach proposed by Busch, Lahti and Werner (BLW) [14] has artfully employed compatible observables, which are noncommuting but own common eigenvectors. With direct comparison between the probability distributions of incompatible observables and their relevant compatible counterparts across all states, the combined approximation errors, detected as the worst-case estimate of the inaccuracy, are nothing to do with any concrete system or measurement, but only relevant to figures of merit characterizing the performance of the measuring device. This idea reflects the foundation of Heisenberg’s uncertainty relation, and has recently been confirmed experimentally [23,24].

The present work concerns a more complicated situation, that is, triplewise joint measurements of three incompatible observables in a qubit. For a single qubit, e.g., a spin-1/2 particle, the spin operators given by three Pauli operators $\sigma _{k}~(k=x,y,z)$, constitute the fundamental representation of SU(2). Although it was mathematically proven that a pair of complementary observables, e.g., two of the three Pauli operators, could completely parameterize a quantum degree of freedom [25], there is no reason to restrict the uncertainty relations stated only with two observables. Besides, it seems natural to extend the uncertainty relations for all components of Pauli matrix simultaneously, for the sense of generality and/or for the rotational symmetry of the problem of interest. In this context, we have noticed the previous attempts to explore the uncertainty principle encompassing three or more observables [26–46]. Except [46] considering the measurement uncertainty, others have merely concerned preparation or entropy uncertainty relations. In particular, the necessary and sufficient condition of the lower bound is considered analytically in [46] for triplewise joint measurability of incompatible observables.

By extending the BLW proposal in [14] to three compatible observables approximating, respectively, three incompatible observables (See Fig. 1), we derive theoretically and then confirm experimentally, both for the first time, the underlying lower bounds of the error-disturbance relations at the fundamental level of a single spin. The essential point, following the idea in [14], is the execution of a joint measurement operator $\mathcal {M}$, which, in our case, outcomes eight measurement operators. Thus approximate compatible observables could be simultaneously acquired from the eight outcome operators by marginal relations. However, since the prevailing situation is much more complicated than considered in [14], we will only focus on some typical cases as detailed later, which provide more alluring physical picture than the general cases for the concerned problem without any resultant damage in the context of our insight into the problem’s essence. Besides, we resort to numerical solutions quite often since such a complicated problem can hardly be solved analytically. Moreover, like in Refs. [23,24], our experiment verifying the theoretical results also utilizes a single qubit encoded in a trapped ultracold $^{40}$Ca$^{+}$ ion, which could be manipulated by high-level control of lasers. As specified in [23], the joint measurement operator $\mathcal {M}$, as a positive operator-valued measure (POVM) operator, could be carried out in a single qubit, although with loss of generality of the POVM, to assess the combined approximation errors indicated by BLW scheme. So by unitary operations under carrier transitions, we witness in a single spin the lower bounds of Heisenberg uncertainty relations from the triplewise joint measurements and discover some novel characteristics completely beyond the usual considerations with two incompatible observables. Our study makes the test of the measurement uncertainty relation more general and complete.

The paper is organized as follows. We first describe the theoretical scheme regarding the joint measurements of three incompatible observables, and then present experimental measurements in verifying the predicted lower bounds. A concise discussion and a short summary are given before the end of the main text. Some details of the theoretical derivations and experimental implementation can be found in Appendix.

## 2. Theoretical scheme

In this work we consider three incompatible observables $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$, and explore their uncertainty relations under triplewise joint measurements. This basically refers to extending the BLW proposal of pairwise joint measurements, as detailed in [14], to the triplewise counterparts, as sketched in Fig. 1, where three compatible observables $\mathcal {D}$, $\mathcal {E}$, $\mathcal {F}$ represent the corresponding approximations of the target observables $\mathcal {A}$, $\mathcal {B}$, $\mathcal {C}$. For clarity, we first introduce the three compatible observables as well as their measurements. Then three incompatible observables, as the targets of the three compatible observables to approach are elucidated.

#### 2.1 Joint measurements of three compatible observables

An arbitrary observable $\mathcal {O}$ for a qubit can be written as $\mathcal {O}=\boldsymbol{o}\cdot \boldsymbol{\sigma }$, where $\boldsymbol{o}=(o_x,o_y,o_z)$ is a general real vector with $|\boldsymbol{o}|\leq 1$ and $\boldsymbol{\sigma }=(\sigma _x,\sigma _y,\sigma _z)$ is the vector regarding Pauli operators. The observable $\mathcal {O}$ has two positive operators $O_{\pm }=(1\pm \boldsymbol{o}\cdot \boldsymbol{\sigma })/2$. Three observables $\mathcal {D}$, $\mathcal {E}$ and $\mathcal {F}$ are called *triplewise jointly measurable* if there is a joint measurement observable $\mathcal {M}$ with eight measurement operators $M_{\boldsymbol{\mu }}$ ($=M_{\mu _1 \mu _2 \mu _3}$ with $\mu _1,\mu _2,\mu _3=\pm$) making the three given observables as the marginals, e.g., $D_{\mu _1}=\sum _{\mu _2 \mu _3}M_{\mu _1 \mu _2 \mu _3}$. This implies that the observables $\mathcal {D}$, $\mathcal {E}$ and $\mathcal {F}$ can be indirectly measured by means of measuring $\mathcal {M}$, as detailed later.

Three observables to be triplewise jointly measurable should satisfy [47],

where $\boldsymbol{\Lambda }_{\textrm {FT}}$ denotes the Fermat-Toricelli (FT) vector of four vectors $\boldsymbol{\Lambda }_0=\sum _{k=1}^3\boldsymbol{\lambda }_k$ and $\boldsymbol{\Lambda }_k=2\boldsymbol{\lambda }_k-\boldsymbol{\Lambda }_0 \ (k=1,2,3)$ with $\boldsymbol{\lambda }_{1,2,3}$ the vectors regarding the observables $\mathcal {D}$, $\mathcal {E}$ and $\mathcal {F}$, respectively. The FT vector for a set of three or more vectors $\boldsymbol{v}_a$ in Euclidean space, which is unique and always exists, represents the vector $\boldsymbol{v}$ with the total distance $\sum _a|\boldsymbol{v}_a-\boldsymbol{v}|$ minimized [48]. As such, the measurement operators of $\mathcal {M}$ for three triplewise joint measurement observables take the following form [47],In order to understand Eq. (2) in a more simplified way, we take into account two special cases as explained below. The first case encapsulates the situation of three observables existing in mutual orthogonality, for which the satisfying condition in Eq. (1) reduces to [47,49,50],

We have to mention that the joint measurement operators given by Eq. (2) are incompact although they satisfy Eq. (3). A proper compact form of $\mathcal {M}$ could be written as [49,50],The second special case deals with a reduction from three observables to two. Normally, a pair of observables $\mathcal {D}$ and $\mathcal {E}$ defined by two vectors $\boldsymbol{\lambda }_{1}$ and $\boldsymbol{\lambda }_{2}$ are *jointly measurable* if and only if [14,23,24,49]

#### 2.2 Uncertainty relations for three incompatible observables

In general, the three incompatible observables $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$ do not satisfy the triplewise jointly measurable condition under the limitation imposed by Eq. (1) unless they are colinear. In order to undertake a quantitative treatment, we consider $\mathcal {A}=\boldsymbol{a}\cdot \boldsymbol{\sigma }$, $\mathcal {B}=\boldsymbol{b}\cdot \boldsymbol{\sigma }$ and $\mathcal {C}=\boldsymbol{c}\cdot \boldsymbol{\sigma }$ with $|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=1$. Then we assume the three compatible observables to be $\mathcal {D}=\boldsymbol{d}\cdot \boldsymbol{\sigma }$, $\mathcal {E}=\boldsymbol{e}\cdot \boldsymbol{\sigma }$ and $\mathcal {F}=\boldsymbol{f}\cdot \boldsymbol{\sigma }$ with $|\boldsymbol{d}|,|\boldsymbol{e}|, |\boldsymbol{f}|\leq 1$ as the approximations of $\mathcal {A}$, $\mathcal {B}$ and $\mathcal {C}$, respectively. By extending the trade-off idea in [14], we define the state-independent uncertainty relation under the triplewise joint measurements as

For the set $\Xi$ consisting of all the groups of three observables satisfying the triplewise joint measurement condition, the lower bound of the uncertainty relation, based on the minimization of the vector distances regarding the incompatible and compatible observables, is defined as

## 3. Experimental system and basic operations

Our experiment is carried out on a single ultracold $^{40}$Ca$^{+}$ ion confined in a linear Paul trap as employed previously [23,24], which is constituted by four parallel blade-like electrodes and two end-cap electrodes. Constant voltage applied to the end-caps ensures axial confinement and the rf potential applied to the blade electrodes via helical resonator corresponds to $\Omega _{rf}/2\pi = 20.6$ MHz with the power of 5.5 W. Under the pseudo-potential approximation, we have the axial and radial frequencies of the trap potential to be, respectively, $\omega _z/2\pi =1.0$ MHz and $\omega _r/2\pi =1.2$ MHz. To contrive an intrinsic Zeeman substructure and hence to ascertain a peculiar quantization axis, we employ a magnetic field of 0.6 mT directed in axial orientation, yielding the ground state $4^2S_{1/2}$ and the metastable state $3^2D_{5/2}$ split into two and six hyperfine energy levels, respectively. We encode qubit $\mid \downarrow \rangle$ in $|4^{2}S_{1/2}, m_{J}=+1/2\rangle$ and $\mid \uparrow \rangle$ in $|3^{2}D_{5/2}, m_{J}=+3/2\rangle$ with $m_{J}$ being the magnetic quantum number.

After Doppler cooling and resolved sideband cooling, the $z$-axis motional mode of the ion is cooled down to its vibrational ground state with the final average phonon number of $0.030(7)$. A narrow-linewidth 729-nm laser radiates the ultracold ion with an incident angle of $22.5^{\circ }$ between the laser and the $z$-axis, yielding the carrier-transition Hamiltonian $H_c=\Omega (\sigma _+e^{i\phi _{L}}+\sigma _-e^{-i\phi _{L}})/2$, with the Rabi frequency $\Omega$ as the laser-ion coupling strength in units of $\hbar =1$ and the laser phase $\phi _{L}$. The system evolution under the execution of the carrier-transition operator is given by [23],

Prior to proceeding to the substantial experimental operations to witness the uncertainty relation lower bounds, we are required to prepare an optimal state $\rho _{\textrm {op}}$ which is to maximize the state-dependent uncertainty $\Delta _{\rho }(\mathcal {A},\mathcal {D})$ with $\boldsymbol{r}=(\boldsymbol{a}-\boldsymbol{d})/|\boldsymbol{a}-\boldsymbol{d}|$. Starting from this state, we check Eq. (7) experimentally. The required operations include execution of coupling of $|\uparrow \rangle$ and $|\downarrow \rangle$ by a 729-nm laser under a unitary evolution $U_{C}(\theta _{L1},\phi _{L1})$ (defined below) and measurement of the necessary observables $A$, $B$, $C$ and $M$ under another evolution $U_{C}(\theta _{L2},\phi _{L2})$ (defined below). Finally, detection is made on the state $|\uparrow \rangle$.

The preparation of an optimal state is achieved by the unitary operator in Eq. (11) with $\rho _{\textrm {op}}=U_{C}(\theta _{L1},\phi _{L1})\mid \downarrow \rangle \langle \downarrow \mid U^{\dagger }_{C}(\theta _{L1},\phi _{L1})$, that is,

where the angles will be further clarified later.The measurement operator $M$ is a positive operator-valued measure. As proven in [23], such a positive operator-valued measure with rank of $1$ can be measured by a single qubit through unitary transformations, at the expense of losing generality, i.e., no possibility to observe the region above the lower bound. Nevertheless, in this way, the lower bound, corresponding to optimal approximations, can be observed as desired in a single qubit. As such, the measurement operator $M=\textrm {Tr[M]}(1+\boldsymbol{m}\cdot \boldsymbol{\sigma })/2$ with $\boldsymbol{m}$ having rank $1$ can be rewritten as $M=\textrm {Tr}[M] U_{C}^{\dagger }(\theta _{L2},\phi _{L2})\mid \uparrow \rangle \langle \uparrow \mid U_{C}(\theta _{L2},\phi _{L2})$ [23]. Thus we have

For convenience of the experimental implementation in our system, we choose $\theta _{L2}=\arccos m_z$, which means $\sin (\theta _{L2})\geq 0$. Thus, $\phi _{L2}$, which only depends on $m_x$ and $m_y$, is given by, where $\textrm {sign}(m_y)=1$ if $m_y>0$ and $\textrm {sign}(m_y)=-1$ if $m_y<0$. In the case of $m_y=0$, the situation is beyond Eq. (14), that is, $\phi _{L2}=\pi /2$ if $m_x>0$ and $\phi _{L2}=-\pi /2$ if $m_x<0$. In the case of both $m_x=0$ and $m_y=0$, we have $\phi _{L2}=0$. Correspondingly, for those angles in initial state preparation, we may choose $\theta _{L1}=\arccos (-m_z)$. Due to $\sin (\theta _{L1})\geq 0$, $\phi _{L1}$ has the same form as $\phi _{L2}$ in Eq. (14).In our experiment, there are six operators required to be measured, i.e., $A_+$, $B_+$, $C_+$, $D_+$, $E_+$ and $F_+$. This is due to the fact that our computation of Wasserstein distances encapsulates an added privilege of diminished experimental operations. For example, due to the fact that $|p_{\rho }^{X_{+}}-p_{\rho }^{Y_{+}}| = |p_{\rho }^{X_{-}}-p_{\rho }^{Y_{-}}|$, we are merely required to carry out experimental manipulations for the determination of former terms $|p_{\rho }^{X_{+}}-p_{\rho }^{Y_{+}}|$. Moreover, $D_+$, $E_+$ and $F_+$ cannot be measured directly by our experimental exertions, but alternatively measured by joint measurement operator $M_{\boldsymbol{\mu }}$. For instance, to determine $D_+$, we employ the marginal relation $D_+=\sum _{\mu _2 \mu _3}M_{+\mu _2 \mu _3}$ by explicitly measuring the four operators $M_{+++}$, $M_{++-}$, $M_{+-+}$ and $M_{+--}$.

## 4. Experimental implementation

In the following section we are proceeding to rigorously elucidate the experimental verification of the uncertainty relations perceived in the proceeding section by considering three typical cases concerning the three incompatible observables.

#### 4.1 Three orthogonal incompatible observables

In this section, we proceed to demonstrate the case for three incompatible observables when they are oriented orthogonally. As presented in Fig. 2(a) the mutually perpendicular incompatible observables can be given as $\mathcal {A}=\sigma _z$, $\mathcal {B}=\sigma _y$ and $\mathcal {C}=\sigma _x$ such that the corresponding $\boldsymbol{a},~\boldsymbol{b}$ and $\boldsymbol{c}$ are $(0,0,1), (0,1,0)$ and $(1,0,0)$ respectively. Following the steps as presented in Appendix B, we can write the approximations $\boldsymbol{d},\boldsymbol{e}$ and $\boldsymbol{f}$ as

Keeping in view the constraints in a single qubit-system, we need to scrutinize the possibility for the predominating boundary conditions. The optimal approximations are always obtained at the boundary of Eq. (3) with $k=1$. Utilizing Eq. (4), we can obtain the joint measurement operators for $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$, from which one can readily obtain $G=1+\sum _{j>j}\mu _i\mu _j\boldsymbol{\lambda }_i\cdot \boldsymbol{\lambda }_j=1$. Considering $\textrm {Rank}[M_{\mu }]=1$ as the paramount condition referring to the measurement prospects of $M_{\boldsymbol{\mu }}$ in one-qubit system as detailed in [23], we have,

where $\lambda _i^j$ denotes the $j$th element of the vector $\boldsymbol{\lambda }_i$. Additionally from the specific forms of $\bf {d}$, $\bf {e}$ and $\bf {f}$, we know $\sum _{j=1,2,3}(\sum _i\mu _i\lambda _i^j)^2=k^2=1$, which implies $k=1$, consistent with the lower bound condition in Eq. (17). Consequently, the lower bound in this case could be reached by the one-qubit measurements.In our experiment we illustrate the case related to $\varphi =\pi /4$, where the lower bound of the uncertainty relation exists along the black line as plotted in Fig. 2(b). From Eq. (8) and Eq. (16), we can easily find that the optimal states for $\Delta _{\rho }(\mathcal {A},\mathcal {D})$, $\Delta _{\rho }(\mathcal {B},\mathcal {E})$ and $\Delta _{\rho }(\mathcal {C},\mathcal {F})$ are $\rho _1=(1+\sigma _z)/2$, $\rho _2=(1+\sigma _y)/2$ and $\rho _3=(1+\sigma _x)/2$, respectively. In this case, we have analytical results for the experimentally measurable quantities, i.e., $p_{\rho _1}^{K_{+}}=1$ with $K=A,B,C$, $p_{\rho _1}^{M_{+\mu _2 \mu _3}}=(1+k\sin \varphi \sin \phi )/8$, $p_{\rho _2}^{M_{\mu _1 + \mu _3}}=(1+k\cos \varphi \sin \phi )/8$, and $p_{\rho _3}^{M_{\mu _1 \mu _2 +}}=(1+k\cos \phi )/8$ for all the possibilities of $\mu _1,\mu _2,\mu _3=\pm$. In particular, for the given angle $\varphi =\pi /4$ (i.e., the black line in Fig. 2(b)), we have $p_{\rho _1}^{M_{+\mu _2 \mu _3}}=p_{\rho _2}^{M_{\mu _1 +\mu _3}}$.

With the laser irradiation as set by the parameter values listed in Table 1, we have experimentally measured the uncertainty relations. Figure 2(c) demonstrates that the separate terms of Eq. (7), i.e., $\Delta (\mathcal {A},\mathcal {D})$ and $\Delta (\mathcal {B},\mathcal {E})$, behave nearly the same whereas $\Delta (\mathcal {C},\mathcal {F})$ varies very differently. This observation is resulted from the spatial asymmetry of $\boldsymbol{f}$ with respect to $\boldsymbol{d}$ and $\boldsymbol{e}$. Nevertheless, the three curves have a point of intersection at $\phi =\arccos \sqrt {1/3}$, at which the three approximate observables have the corresponding vectors of the same length $1/\sqrt {3}$ and deviate from their target incompatible observables with the same uncertainty. Moreover, as plotted in Fig. 2(d), the lower bound also occurs at $\phi =\arccos \sqrt {1/3}$, which agrees with the prediction of Eq. (16).

#### 4.2 Three coplanar incompatible observables

In this section, we consider three coplanar incompatible observables generally described by,

with $x=a,b, c$ satisfying $|\boldsymbol{x}|=1$. By virtue of unitary transformation, any set of three coplanar observables in the Bloch sphere can be transformed to attain the above representation. In contrast to the above subsection, no analytical solution can be found in this case to compute the lower bound. As a result, here we implement a different way of finding the solution for the optimal function $\bar {\Delta }_{lb2}$ associated with the lower bound, based on Eq. (33) in Appendix C. So the uncertainty relation is given by In our experiment, we exemplify the case for which $\mathcal {A}=\sigma _z$, $\mathcal {B}=\sin \varphi \sigma _y+\cos \varphi \sigma _z$ and $\mathcal {C}=\sin \phi \sigma _y+\cos \phi \sigma _z$, implyingFigure 3(b) indicates that the uncertainty increases with $\varphi$ and $\phi$, where the maximum uncertainty appears along the diagonal line $\phi =\varphi$. With parameters set as in Table 2, the numerical result shows that the maximum uncertainty appears in the case of $\bar {\Delta }_{lb2}=2$ at the position $\phi =\varphi =\pi /3$, implying that $\Delta (\mathcal {A},\mathcal {B},\mathcal {C})\geq 2$ with $\mathcal {A}=\sigma _z$, $\mathcal {B}=\sigma _z/2+\sqrt {2}\sigma _y/\sqrt {3}$ and $\mathcal {C}=\sigma _z/2-\sqrt {2}\sigma _y/\sqrt {3}$. This indicates that $\boldsymbol{a}=\boldsymbol{b}+\boldsymbol{c}$, forming a regular triangle. In fact, at this point we also have the relation $\Delta (\mathcal {A},\mathcal {D})=\Delta (\mathcal {B},\mathcal {E})=\Delta (\mathcal {C},\mathcal {F})=2/3$. In addition, we mention that validity of the coplanar condition for $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ has been checked in the numerical calculation for Fig. 3(b), which shows the module $|\boldsymbol{d}\times \boldsymbol{e}\cdot \boldsymbol{f}|<10^{-5}$.

Our experimental measurement takes the case regarding the diagonal line $\phi =\varphi$ as an example to verify the uncertainty relation. The data sets of the probabilities regarding the observables have been measured, as presented in the panels of Fig. 3(c). We find that, due to the identical change of $\phi$ and $\varphi$, $\Delta (\mathcal {B},\mathcal {E})$ and $\Delta (\mathcal {C},\mathcal {F})$ behave nearly the same as parabolas whereas $\Delta (\mathcal {A},\mathcal {D})$ varies differently as a monotonous increase with $\phi$. The three curves have the common point at $\phi =0$ which means a trivial solution since the three curves are collinear and zero uncertainty exists in this case. The three curves intersect at $\phi =\pi /3$, which, from the state-independent Heisenberg uncertainty relation plotted in Fig. 3(d), is the point corresponding to the maximum of the lower bound.

The experimental results plotted in Fig. 3(c) confirm the numerical prediction for the maximum uncertainty. Since we suppose that $\boldsymbol{b}$ and $\boldsymbol{c}$ are positioned in opposite sides of $\boldsymbol{a}$, we find the optimal approximation of the vector $\boldsymbol{d}$ to the target vector $\boldsymbol{a}$ always existing along the direction of $\boldsymbol{a}$. As such, we may only consider the case of $\alpha =0$ in Fig. 3(a). In this case, when $\phi =0$ turns to be $\phi \neq$0, the measurement values of $A_+$ keep constant and $D_+$ decreases slightly, whereas for other observables, including $B_+$, $C_+$, $E_+$ and $F_+$, the measured possibilities drop largely, see shown in Figs. 3(c4)-(c6). The latter is due to the fact that the optimal vector representing $E_+$ ($F_+$) should have a large deviation from the vector $\boldsymbol{b}$ ($\boldsymbol{c}$), which actually reflects the restriction imposed by the uncertainty relation (See more analysis in Appendix C). Nevertheless, as the dropping occurs in $B_+$ ($C_+$) and $E_+$ ($F_+$) nearly identically, no discontinuity can be found in the uncertainty relations $\Delta (\mathcal {B},\mathcal {E})$ and $\Delta (\mathcal {C},\mathcal {F})$ in Figs. 3(c2) and (c3). As such, we have a complete and continuous observation of the Heisenberg uncertainty relation in Fig. 3(d).

#### 4.3 Three incompatible observables with one observable orthogonal to the other two

This subsection deals with the three incompatible observables with one of them orthogonal to the other two, whose substantial representation is as follows,

with $x=a,b$ satisfying $|\boldsymbol{x}|=1$. The uncertainty relation in this case is assumed to be where $\bar {\Delta }_{lb3}$ is the optimal function associated with the lower bound and can be numerically solved by Eq. (35) in Appendix D. For the sake of convenience of the experimental demonstration, we consider, as sketched in Fig. 4(a), the three incompatible observables with the following formAs shown in Fig. 4(b), the uncertainty varies with $\varphi$ and $\phi$, reaching both the maximum and minimum along the diagonal line $\phi =\varphi$. This is due to the symmetry of the vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ in $\sigma _y-\sigma _z$ plane. As such, we have two maxima of the uncertainty, i.e., $\bar {\Delta }_{lb3}^{\max }=2\sqrt {3}(\sqrt {3}-1)$ at $\phi =\varphi =\pi /4$ and $3\pi /4$, which actually corresponds to the situation where all of the three observables are in mutual orthogonal orientation. The minima of the uncertainty appear at $\phi =\varphi =0$ and $\pi /2$, where the vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are collinear and $\bar {\Delta }_{lb3}^{\min }=1.55$. We also have to mention that we have found in the calculation for Fig. 4(b) the module $|\boldsymbol{d}\cdot \boldsymbol{f}|+|\boldsymbol{e}\cdot \boldsymbol{f}|<10^{-4}$, implying $\boldsymbol{f}\perp \boldsymbol{d},\boldsymbol{e}$ is satisfied.

Our experimental measurements in this case exemplify the case of the diagonal line $\phi =\varphi$. The measured data sets for the observables’ probabilities in Fig. 4(c) demonstrate that all the three terms of the uncertainty relation vary in symmetry with respect to $\phi =\pi /2$, where $\Delta (\mathcal {A},\mathcal {D})$ and $\Delta (\mathcal {B},\mathcal {E})$ behave nearly the same whereas $\Delta (\mathcal {C},\mathcal {F})$ varies very differently. This is due to the symmetry of $\boldsymbol{a}$ and $\boldsymbol{b}$ in $\sigma _y-\sigma _z$ plane and the orthogonality of $\boldsymbol{c}$ with $\boldsymbol{a},\boldsymbol{b}$. We find some peculiar characteristics in Figs. 4(c1)-(c6). In Figs. 4(c4) and (c5), $P_{K_+}$ with $K=D,E$ keep almost constant for most values of $\phi$, while jumping to the maximum around $\phi /\pi =1/2$ and 1. Correspondingly, both $\Delta (\mathcal {A},\mathcal {D})$ and $\Delta (\mathcal {B},\mathcal {E})$ in Figs. 4(c1) and (c2) behave like the absolute value of sine function with respect to $\phi =\pi /2$. In contrast, $\Delta (\mathcal {C},\mathcal {F})$ exhibits a different behavior, but with similarity to $P_{F_+}$. This observation reflects the fact, as displayed in Fig. 4(a), that $\boldsymbol{f}$ is always along the direction of $\boldsymbol{c}$, whereas $\boldsymbol{d}$ and $\boldsymbol{e}$ have variational intersection angles with $\boldsymbol{a}$ and $\boldsymbol{b}$, respectively. With combination of the results in Figs. 4(c1)-(c3), we obtain the state-independent Heisenberg uncertainty relation in Fig. 4(d), which is in good agreement with the prediction in Fig. 4(b).

## 5. Discussion

For general cases, whose arbitrary group of three incompatible observables can be described as

Our demonstration as detailed above reveals that the triplewise joint measurements concern much more complex situation than the pairwise counterparts investigated previously [14,23]. For the case involving only two incompatible observables, there is no possibility to observe a simultaneous change for two different uncertainties denoted by Wesserstein distances, which is a meticulous obligation to the realization of the fundamental limitation imposed by Heisenberg uncertainty, i.e., when one observable is more precisely measured, the other is more perturbed and fuzzy. In contrast, Heisenberg uncertainty by triplewise joint measurements works under the condition of no simultaneous change for all the uncertainties regarding the three incompatible observables. As a result, two uncertainties, e.g., $\Delta (\mathcal {B},\mathcal {E})$ and $\Delta (\mathcal {C},\mathcal {F})$, could perhaps behave identically. The idea provided in the current work is more fascinating to furnish a new possibility in the precision measurement, such that we may have two incompatible observables precisely measured by sacrificing the precision of the third incompatible observable, which is legitimately a quintessential realization of essence of Heisenberg uncertainty. In this context, we have to mention that Eq. (7) is formulated by summing the independently maximized Wasserstein distances following the original idea of the BLW proposal [14]. In fact, this error trade-off relation could also be modified as a maximization of the summed Wasserstein distances [23], which gives the lower bound of the uncertainty to be lower by $\sqrt {2}$ with respect to the idea in [14]. Nevertheless, there is no essential difference between the two treatments in understanding Heisenberg uncertainty relations by the idea of error trade-off.

Moreover, a clear understanding of the operational imperfections is essential to the precise experimental observation of the uncertainty relations. In our experimental implementation, some imperfections from, e.g., initial-state preparation and final-state detection as well as dephasing due to thermal phonons from the radial direction, could be partially calibrated by practical methods [51]. The error bars reflect the statistical errors due to quantum projection noise and some other experimental errors resulted from the fluctuation as a consequence of instability of laser power and magnetic field. In addition, some observed data points, e.g., in Fig. 4, deviated somewhat largely from the numerical curves are due to accumulation of the errors regarding measurements of Wesserstein distances. Nevertheless, all the deviations in our observation are non-sequential and almost completely within the allowed regions of experimental errors, as denoted by the error bars.

## 6. Conclusion

In summary, at the fundamental level of a single spin, we have demonstrated the first exploration, both theoretically and experimentally, of error trade-off relations from the triplewise joint measurement of three incompatible observables in a pure quantum regime. In contrast to most of previous theoretical proposals and experimental tests based on state preparation and entropy, our investigation for Heisenberg uncertainty relies on a direct comparison of the approximating and target observables, which could furnish more insight into the realistic operations and imprecision in joint measurements. In addition to experimental confirmation of the uncertainty relations for three observables in three different orientations we have witnessed an interesting phenomenon, that is, in contrast to the case of conventional two observables, scarifying the accuracy of one of the three observables, the rest of two can be measured with definite accuracy which provided a clue of speedup along with enhanced accuracy in context of multi observables joint measurement. Subsequently, which could be of relative importance for the quantum community dealing with fast and joint measurement of enhanced number of observables. Recent advances in precision measurement, particularly in the area of quantum information science, have led to heightened interest in the fundamental limitations on the achievable quantum measurement accuracy. Thus Heisenberg uncertainty relations have again become an attractive topic, which could help for further understanding the foundation of quantum mechanics, such as the deeper reason for nonlocality [52,53] and entanglement [54]. Besides, the uncertainty relations constitute an important ingredient of many device-independent security proofs [55,56] and quantum memory [57], which belong to technological foundation in quantum information processing. Accurate measurement of the quantum state is crucial in regard to get the maximum available information to an exceptional accuracy, particularly in the situation of three or more observables. In this context, our methods and results for the triplewise joint measurements, involving additional dimensions of incompatible variables in precision measurements, could be helpful to enrich our knowledge and give a more complete perception in understanding the uncertainty relations and joint quantum measurement.

## Appendix A: Solution to Eq. (9)

In order to numerically solve Eq. (9), we transform the constrained extreme-value problem into the unconstrained one, that is, the optimal problem in Eq. (9) is changed to the following,

The penalty factor used in Eqs. (25) and (26) is relevant to the accuracy of the numerical calculation for $\Delta _{lb}$, that is, a larger value of $N_p$ makes $\tilde {\Delta }_{lb}$ closer to $\Delta _{lb}$, i.e., $\lim _{N_p\rightarrow \infty }\tilde {\Delta }_{lb}=\Delta _{lb}$. The main idea of this solution is from the penalty function approach to effectively solving constrained minimax problems by an unconstrained minimax problem [58]. However, the larger value of $N_p$ makes the solution more time-consuming. As a result, we have to find a modest value of $N_p$ satisfying $|\Delta _{lb}-\tilde {\Delta }_{lb}|\leq \varepsilon$ with $\varepsilon$ being a small positive number, i.e., $< 10^{-4}$.

Moreover, an inlaid optimization process is involved in Eq. (26) for solving the FT vector $\boldsymbol{\Lambda }_{\textrm {FT}}$ of the four different vectors. This inlaid optimization process brings in a numerical obstacle for fast solving Eq. (26), which makes the solution time-consuming and inaccurate. Therefore, we have to only consider some special cases in our experimental demonstration of Heisenberg uncertainty relations for the three incompatible observables.

## Appendix B: Details for three orthogonal incompatible observables

The analytical result can be found in the case of three orthogonal incompatible observables $\mathcal {A},\mathcal {B},\mathcal {C}$. In comparison with the case of two incompatible observables with the optimal approximations along their corresponding directions [23,24], we consider that the three incompatible observables under our consideration should behave similarly. Therefore, we employ the orthogonal triplewise-jointly-measurable observables as the approximations of the orthogonal incompatible observables $\mathcal {A},\mathcal {B},\mathcal {C}$, which means $\boldsymbol{d},\boldsymbol{e},\boldsymbol{f}$ obeying the constraint condition in Eq. (3). Thus we assume,

As an analytical solution, Eq. (28) could be used to check the efficiency and correctness of the optimization approach based on the penalty factor $N_p$, which is helpful for justifying the purely numerical treatments for the other two cases discussed below. As presented in Fig. 5, the numerical calculation following the steps of the optimization approach shows a rapid decrease of $|\tilde {\Delta }_{lb}-\Delta _{lb}^{op}|$ with the increase of $N_p$, indicating $\Delta _{lb}^{op}=\Delta _{lb}$. Therefore, from both the analytical and numerical results, we consider that the uncertainty relation for three orthogonal incompatible observables can be stated as

For experimental demonstration of Eq. (29), we choose $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ to satisfy Eq. (27). Under the condition that the FT vector always satisfies $\boldsymbol{\Lambda }_{\textrm {FT}}=0$, the objective function in Eq. (26) is reduced to## Appendix C: Details for three coplanar incompatible observables

Following the idea in above section, we assume that, for three coplanar incompatible observables, the corresponding optimal approximations $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ should also lie in a plane. In this case, the triplewise jointly measurable condition in Eq. (1) can be simplified. If $\boldsymbol{f}$ lies in the triangle formed by $\boldsymbol{d}$ and $\boldsymbol{e}$, the triplewise jointly measurable condition is reduced to

Otherwise, the triplewise jointly measurable condition is written asSimilar to the results in above section, the lower bound $\bar {\Delta }_{lb2}$ can be obtained by an appropriate value of $\bar {N}_p$. To justify Eq. (33), we have carried out numerical simulation for $|\bar {\Delta }_{lb2}-\tilde {\Delta }_{lb2}|$ to make sure the difference between $\bar {\Delta }_{lb2}$ and $\tilde {\Delta }_{lb2}$ to be smaller than $10^{-3}$, implying $\bar {\Delta }_{lb2}=\tilde {\Delta }_{lb2}$. Moreover, we have also checked the vector product $|\boldsymbol{d}\times \boldsymbol{e}\cdot \boldsymbol{f}|< 10^{-5}$, implying $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ being coplanar. Therefore, Eq. (33) could be reasonably employed to calculate the lower bound of Heisenberg uncertainty relation in comparison with the experimental measurements.

Here we analyze the discontinuities displayed in the panels (c5,c6,c8,c9) of Fig. 3, which occur due to the uncertainty relations and our special choice of the vectors. For $\varphi =0$, the vectors $\boldsymbol{a}$, $\boldsymbol{b}$ and $\boldsymbol{c}$ are co-linear and compatible, implying no action of the uncertainty principle. But once $\varphi$ turns to be nonzero, the error trade-off relation starts working. For the vectors exemplified in our case here, the optimal state vector $\boldsymbol{r_1}$ regarding the difference between $\boldsymbol{a}$ and $\boldsymbol{d}$ is still along $\boldsymbol{a}$, but the vectors for other compatible observables different from the corresponding incompatible counterparts would change significantly under the government of the uncertainty principle. As a specific description, we plot Fig. 7 for the case of $\varphi /\pi \ll 1$, where $\boldsymbol{e}$ and $\boldsymbol{b}$ are of almost equal length, and the optimal state vector $\boldsymbol{r_2}$ regarding the difference between $\boldsymbol{b}$ and $\boldsymbol{e}$ is $\boldsymbol{r_2}=(\boldsymbol{b}-\boldsymbol{e})/\vert \boldsymbol{b}-\boldsymbol{e}\vert$. We schematically demonstrate in Fig. 7(b) the rapid change of the angle $\theta$ between $\boldsymbol{e}$ and $\boldsymbol{r_2}$ in the variation from $\varphi /\pi =$0 to $0< \varphi /\pi \ll 1$. The discontinuities observed in $P_{B_+}$ and $P_{E_+}$ are resulted from this rapid change of $\theta$. Due to the symmetrical choice of $\boldsymbol{c}$ in our case, $P_{C_+}$ and $P_{F_+}$ are also observed with discontinuity.

## Appendix D: Details for three incompatible observables with one orthogonal to the other two

In this case, the corresponding optimal approximations $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ should satisfy both the orthogonal condition that one of the three observables is orthogonal to the others and the triplewise jointly measurable condition. In the case of $\boldsymbol{f}$ orthogonal to both $\boldsymbol{d}$ and $\boldsymbol{e}$, the jointly measurable condition is reduced to

Similar to the above sections, the lower bound $\bar {\Delta }_{lb3}$ could be obtained by choosing a reasonable large value of $\bar {N}_p$. We have numerically checked $|\bar {\Delta }_{lb3}-\tilde {\Delta }_{lb3}|$ to make sure the difference to be smaller than $10^{-3}$. The optimal approximate observables $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ also satisfy the orthogonal relation $\boldsymbol{f}\perp \boldsymbol{d},\boldsymbol{e}$, whereas they might have intersection angles with $\boldsymbol{a}$, $\boldsymbol{b}$ and $\boldsymbol{c}$, respectively. Moreover, the FT vector of $\boldsymbol{d}$, $\boldsymbol{e}$ and $\boldsymbol{f}$ in this case can be analytically solved as

Here we briefly analyze the physics relevant to the discontinuous characteristics in Fig. 4. This reflects the fact that each of the cases $\phi =0$, $\pi /2$ and $\pi$ corresponds to the situation similar to two complementary observables, since $\mathcal {A}$ and $\mathcal {B}$ are pointing in parallel or anti-parallel. Consequently, the discontinuities mean the error trade-off relation transited between the two complementary observables and the three counterparts.

## Appendix E: Some details of experimental manipulation

In our experiment with the single ultracold ion, the state manipulation and experimental evolution are performed by an ultra-stable narrow linewidth Ti:Sapphire 729-nm laser corresponding to linewidth (FWHM) of 7 Hz, as measured via the heterodyne beat note method with respect to another laser system. The 729-nm laser is locked to a high-finesse ultra-low expansion cavity with the long-term drift to be 0.06 Hz/s.

For producing well-ordered laser pulses in the implementation, all the lasers are controlled by the acousto-optic modulators (AOM) by passing through the AOMs before irradiating the ion. The operational systematic rf signals applied to all the AOMs are being supplied by the direct digital synthesizer (DDS) which is controlled by a field programmable gate array. The DDS functions as the phase and frequency control of all the lasers during the consecutive experimental progressions. A typical experimental sequence involves more than 300 optical pulses within a time slot of about 40 ms.

Implementing the required evolutions comprises two kinds of physical operations. For preparing a required state, we execute carrier transitions for different time spans by the 729-nm laser pulses with the mutual phase difference of 0, $\pi$ or any other desired value. For measuring the observables, we first generate a superposition of the states $\mid \downarrow \rangle$ and $\mid \uparrow \rangle$, and then the detection is made by applying the cooling lasers again and counting the emitted photons for 6 ms by the photon multiplier tube. Each data point of the observables in the scheme is typically measured for 20,000 times.

## Funding

National Natural Science Foundation of China (11674360, 11734018, 11804375, 61862014, Nos. 11835011); Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB21010100); National Key Research and Development Program of China (No. 2017YFA0304503); Guangxi Key Laboratory of Trusted Software (No. kx201505).

## Acknowledgments

LLY and MF appreciate insightful discussion with Paul Busch before his sad demise. K.R. thankfully acknowledges support from CAS-TWAS president’s fellowship. KR and TPX contributed equally to this work.

## Disclosures

The authors declare no conflicts of interest.

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