Verifying Heisenberg’s error-disturbance relation using a single trapped ion

We report experimental verification of Heisenberg’s EDR by a single trapped ⁴⁰Ca⁺ ion, following the BLW proposal. The atomic ion is confined in a harmonic trap, that is, within the Lamb-Dicke regime of a linear Paul trap, with an axial frequency of ω_z/2π = 1.01 MHz and a radial frequency of ω_r/2π = 1.2 MHz. We encode a qubit into two electronic levels $$|\downarrow \rangle \equiv |{\mathit{S}}_{1/2},{\mathit{m}}_{\mathrm{J}}=+1/2\rangle$$ and $$|\uparrow \rangle \equiv |{\mathit{D}}_{5/2},{\mathit{m}}_{\mathrm{J}}=+3/2\rangle$$, where m_J is the magnetic quantum number (see Fig. 2A). Doppler cooling and resolved-sideband cooling are performed mainly along the axial direction, yielding the final average phonon number $$\overline{\mathit{n}}<$$0.1 along the axial direction with the Lamb-Dicke parameter η ~0.09. Together with the optical pumping, the system is initially prepared in $$|\downarrow \rangle$$. We carry out the unitary rotations between the two encoded levels and implement projective measurement on $$|\uparrow \rangle$$ by the electron shelving technique (see the Supplementary Materials).

Before presenting our experimental results, we first specify some important points in our experimental scheme. We consider the positive operators $${\mathit{A}}_{\pm }=(\mathit{I}\pm \vec{\mathit{a}}\cdot \vec{\mathrm{\sigma }})/2$$ and $${\mathit{B}}_{\pm }=(\mathit{I}\pm \vec{\mathit{b}}\cdot \vec{\mathrm{\sigma }})/2$$ regarding $$\mathcal{A}$$ and $$\mathcal{B}$$, respectively, where $$\vec{\mathit{a}}$$ and $$\vec{\mathit{b}}$$ are unit vectors and $$\vec{\mathrm{\sigma }}=({\mathrm{\sigma }}_{\mathit{x}},{\mathrm{\sigma }}_{\mathit{y}},{\mathrm{\sigma }}_{\mathit{z}})$$ represents a vector associated with the usual Paul matrices. To be jointly measurable, the compatible observables $$\mathcal{C}$$ and $$\mathcal{D}$$, as the approximation of $$\mathcal{A}$$ and $$\mathcal{B}$$, own the positive operators $${\mathit{C}}_{\pm }=(\mathit{I}\pm \vec{\mathit{c}}\cdot \vec{\mathrm{\sigma }})/2$$ and $${\mathit{D}}_{\pm }=(\mathit{I}\pm \vec{\mathit{d}}\cdot \vec{\mathrm{\sigma }})/2$$, which satisfy $$\parallel \vec{\mathit{c}}\parallel \le 1$$, $$\parallel \vec{\mathit{d}}\parallel \le 1,$$ and $$\parallel \vec{\mathit{c}}+\vec{\mathit{d}}\parallel +\parallel \vec{\mathit{c}}-\vec{\mathit{d}}\parallel \le 2$$.

The essential step of our execution is the joint measurement G on the compatible observables $$\mathcal{C}$$ and $$\mathcal{D}$$. In the study of Busch et al. (23), G is associated with the POVM operators G_±,± commonly owned by C_± and D_± with the marginality relation C_± = G_±,+ + G_±,− and D_± = G_+,± + G_−,±. Generally speaking, the POVM can be achieved in a qubit with the assistance of another auxiliary qubit, implying the requirement of two qubits for implementing the operations. However, in our experiment, we construct the POVMs by single-qubit operations, and thus, the BLW scheme can be verified on a single qubit. To this end, the POVMs constructed are not general but satisfy the restricted conditionwhich means that only some special POVMs are achievable in the single-qubit system. The condition also implies that we have to involve a prefactor Tr[G] in the measurement of the POVM operator. Besides, in our ion trap system, the projective measurement is performed on $$|\uparrow \rangle$$. Thus, we have to first rotate the POVMs unitarily to be in line with $$|\uparrow \rangle$$ before making the measurements.

In a single-qubit system, for each POVM element operator G_μ,ν (μ, ν = ±) applied on a density operator ρ, the measurement result p_ρ(G_μ,ν) and the normalized density operator $${\mathcal{K}}_{\mathrm{\rho }}({\mathit{G}}_{\mathrm{\mu },\mathrm{\nu }})$$ correspond, respectively, to p_ρ(G_μ,ν) = Tr[G_μ,νρ] and $${\mathcal{K}}_{\mathrm{\rho }}({\mathit{G}}_{\mathrm{\mu },\mathrm{\nu }})=\sqrt{{\mathit{G}}_{\mathrm{\mu },\mathrm{\nu }}}\mathrm{\rho }\sqrt{{\mathit{G}}_{\mathrm{\mu },\mathrm{\nu }}}/{\mathit{P}}_{\mathrm{\rho }}({\mathit{G}}_{\mathrm{\mu },\mathrm{\nu }}),$$ with ∑_μ,νG_μ,ν = 1. To simplify the description below, we rewrite G_μ,ν as G by neglecting the subscripts μ and ν. Defining a pure-state measurement basis $$|\mathrm{\varphi }\rangle$$, we obtain a measurement M along the same direction as $$|\mathrm{\varphi }\rangle$$ satisfying $$\mathit{M}\equiv \text{Tr}[\mathit{M}]|\mathrm{\varphi }\rangle \langle \mathrm{\varphi }|$$ and Tr[M] = Tr[G]. If there is a unitary operation U mapping G to M with M = UGU^†, the density operator changes accordingly as $$\mathrm{\rho }\prime =\mathit{U}\mathrm{\rho }{\mathit{U}}^{†}$$. Therefore, we reach important relations as belowwhich are one-by-one mappings between the POVM and the projective measurement on $$|\mathrm{\varphi }\rangle$$. Because no unitary transformation changes the rank of an operator, the POVM operator G can be achieved by a pure-state–relevant positive operator, strictly obeying the condition in Eq. 1.

In our case with a single qubit consisting of the upper level $$|\uparrow \rangle$$ and the lower level $$|\downarrow \rangle$$, we assume that $$\mathit{G}={\mathit{g}}_0\mathit{I}+\vec{\mathit{g}}\cdot \vec{\mathrm{\sigma }}$$. Provided that $${\parallel \vec{\mathit{g}}\parallel }^2={\mathit{g}}_0^2$$ is satisfied and the ranks of all the POVM operators are units, the operators can be directly measured by combining a unitary operation and a projective measurement on $$|\uparrow \rangle$$. In addition, the marginality relations between G_±,± and C_±, D_± imply that the condition of $${\parallel \vec{\mathit{g}}\parallel }^2={\mathit{g}}_0^2$$ is equivalent to $$\parallel \vec{\mathit{c}}+\vec{\mathit{d}}\parallel +\parallel \vec{\mathit{c}}-\vec{\mathit{d}}\parallel =2$$, that is, $$1+\vec{\mathit{c}}\cdot \vec{\mathit{d}}=\parallel \vec{\mathit{c}}+\vec{\mathit{d}}\parallel$$ and $$1-\vec{\mathit{c}}\cdot \vec{\mathit{d}}=\parallel \vec{\mathit{c}}-\vec{\mathit{d}}\parallel$$, under which finding optimal approximations to $$\mathcal{A}$$ and $$\mathcal{B}$$ are always available (see the Supplementary Materials). In the trapped ion system, the unitary operators under the government of carrier transitions are accomplished by tuning the evolution time and the laser phase as explained in Fig. 2B. Thus, we obtain the Wasserstein distances (23) between $$\mathcal{A}$$, $$\mathcal{C}$$ and $$\mathcal{B}$$, $$\mathcal{D}$$, in association with Heisenberg’s EDR. Then, we examine the maximal uncertainty for various states of the system and different choices of $$\mathcal{C}$$ and $$\mathcal{D}$$.

In our implementation, we consider $$\mathcal{A}={\mathrm{\sigma }}_{\mathit{y}}$$ with A_± = (I ± σ_y)/2 and $$\mathcal{B}={\mathrm{\sigma }}_{\mathit{z}}$$ with B_± = (I ± σ_z)/2. As the approximation of $$\mathcal{A}$$ and $$\mathcal{B}$$, the two compatible observables $$\mathcal{C}$$ and $$\mathcal{D}$$ can be set as C_± = (I ± ασ_y)/2 and D_± = (I ± βσ_z)/2, where α² + β² = 1 is satisfied as a result of the requirement for unit rank of the POVMs. In our case, C_± and D_± are not directly measurable but are obtained from the POVM operators G_+,± = [I + ασ_y ± βσ_z]/4 and G_−,± = [I − ασ_y ± βσ_z]/4. As clarified below, by using the carrier transition and then making projective measurements on $$|\uparrow \rangle$$, we can achieve measurements of the observables A_±, B_±, and G_±,±.

In the operations presented below, we define α = sin (θ_in) and β = cos (θ_in) (Fig. 1B). For a state ρ, the error measure (23, 25) between $$\mathcal{A}$$ and $$\mathcal{C}$$ is estimated by the Wasserstein distance $${\mathrm{\Delta }}_{\mathrm{\rho }}(\mathcal{A},\mathcal{C}{)}^2$$ (see Materials and Methods), and similarly, we have $${\mathrm{\Delta }}_{\mathrm{\rho }}(\mathcal{B},\mathcal{D}{)}^2$$ for the difference between $$\mathcal{B}$$ and $$\mathcal{D}$$. Heisenberg’s EDR for the pair of incompatible observables is determined by maximizing the summation of the two Wasserstein distances over all the possible states of the system withwhere the second equality holds when the system is prepared in $$|\mathrm{\varphi }\rangle = \text{cos}({\mathrm{\theta }}_1/2)|\downarrow \rangle -\mathit{i} \text{sin}({\mathrm{\theta }}_1/2){\mathit{e}}^{\mathit{i}{\mathrm{\phi }}_1}|\uparrow \rangle$$, with θ₁ and φ₁ defined in Fig. 2B, and the state-independent lower bound $${\mathrm{\Delta }}_{\text{bl}}{(\mathcal{A},\mathcal{B})}^2:=2(\sqrt{2}-1)$$ of the uncertainty is reached at $$\mathrm{\alpha }=\mathrm{\beta }=1/\sqrt{2}$$. Equation 2 represents a worst-case estimate of the inaccuracy applicable to all possible states.

Under the rotating-wave approximation (see the Supplementary Materials), the Hamiltonian of our case in units of ħ = 1 is given by H_C = Ω(σ₊e^iφ + σ₋e^−iφ)/2 (26), where Ω is the Rabi frequency representing the laser-ion coupling strength, σ_± are the usual Pauli operators, and φ is the laser phase. As shown in Fig. 2B, the experiment starts from the state $$|\downarrow \rangle$$, and the system evolves under U_C(θ, φ), that is,with θ = Ωt determined by the evolution time.

To verify Heisenberg’s EDR, we vary α and β to reach the maximal Wasserstein distance as in Eq. 2. The first step is to prepare the state $$|\mathrm{\phi }\rangle$$. We fix the laser phase φ₁ and steer the system under U_C(θ₁, φ₁) toward $$|\mathrm{\varphi }\rangle$$, which is tuned with the change of α and β for an optimal value corresponding to the worst-case estimate of inaccuracy. The operation is executed by a 729-nm laser coupling $$|\uparrow \rangle$$ and $$|\downarrow \rangle$$ for 2 to 3 μs (see details in the Supplementary Materials). The second step is to measure the necessary observables A_±, B_±, and G_±,±, which is achieved by another evolution under U_C(θ₂, φ₂) and then a detection on the state $$|\uparrow \rangle$$. To this end, we first drive the $$|\downarrow \rangle \leftrightarrow |\uparrow \rangle$$ transition by the 729-nm laser following the scheme in Table 1. Detection is then made by reapplying the cooling lasers and counting the emitted photons for 4 ms by the photomultiplier tube.

A faithful observation requires a clear understanding of the operational imperfections. From an effective period of Rabi oscillation, we estimate the error of the initial-state preparation to be 3(1)% and the imperfection in the detection to be 0.35(2)% (the numbers in parentheses are the SEM). The radial thermal phonons cause a dephasing-like behavior that yields an accumulative deviation in the evolution. All these errors are experimentally determined, and the induced deviation can be corrected. Hence, the Rabi oscillation under a π/2 pulse of U_C can reach a fidelity of 99.8(1)% (see the Supplementary Materials), and thus, the observed data of $$\mathcal{A}$$, $$\mathcal{B}$$, $$\mathcal{C}$$, and $$\mathcal{D}$$ demonstrate an excellent agreement with the theoretical prediction. Errors, reflecting the fluctuation due to unstable laser power and magnetic field, are calculated and included in the SD.

Typical experimental data sets of $$\langle {\mathit{A}}_{\pm }\rangle$$, $$\langle {\mathit{B}}_{\pm }\rangle$$, $$\langle {\mathit{C}}_{\pm }\rangle$$, and $$\langle {\mathit{D}}_{\pm }\rangle$$ are depicted in Fig. 3, which clearly demonstrate no possibility of good approximations of $$\mathcal{C}$$ to $$\mathcal{A}$$ and $$\mathcal{D}$$ to $$\mathcal{B}$$, simultaneously. Provided θ_in → π/2, we have $$\mathcal{C}\to {\mathrm{\sigma }}_{\mathit{y}}$$, indicating the nearly perfect case for $$\mathcal{C}$$ approaching $$\mathcal{A}$$. However, in this case, we have $$\mathcal{D}\to \mathit{I}/2$$, implying that we cannot obtain any information about $$\mathcal{B}$$. With θ_in away from π/2, $$\mathcal{D}$$ approaches $$\mathcal{B}$$, and meanwhile, $$\mathcal{C}$$ turns out to be much different from $$\mathcal{A}$$. The sum of their differences, reflecting the balance between the two differences, reaches the minimum at θ_in = π/4 (see the inset of Fig. 4).

The error–trade-off relation is witnessed in Fig. 4 by the Wasserstein distances $$\mathrm{\Delta }(\mathcal{A},\mathcal{C}{)}^2$$ and $$\mathrm{\Delta }(\mathcal{B},\mathcal{D}{)}^2$$, which are calculated by the experimental data in Fig. 3. The observation fits the theoretical prediction “when one is more precisely measured, the other is more disturbed” very well. One cannot predict both outcomes of two incompatible measurements to arbitrary precision. Because it results from the maximal Wasserstein distances over all the possible states in the system, the observed error–trade-off relation represents the state-independent inaccuracy and reflects the essence of Heisenberg’s EDR.

Because $$\mathcal{A}$$ and $$\mathcal{B}$$ in our case are maximally incompatible, the lower bound of the uncertainty can reach $$2(\sqrt{2}-1)$$, the minimum in Eq. 2. We plot this lower bound in Fig. 4 by the dashed line tangent to the state-independent curve of the error–trade-off relation. The tangent point implies $$\mathrm{\alpha }=\mathrm{\beta }=1/\sqrt{2}$$. It is worth noting that the error bars, which dominantly resulted from the statistical deviation (due to quantum projection noise), represent the largest valid range of the experimental observation, rather than the true values allowed to be below the theoretically predicted lower bound of the uncertainty. Besides, the error bars here are four times as long as those in Fig. 3, reflecting the maximum possible deviation of statistics in measuring four variables (see Materials and Methods). More measurements can shrink the error bars but could not present new physics with respect to the 40,000 measurements performed here. Moreover, by fixing α and β, but varying the state $$|\mathrm{\varphi }\rangle$$, we can obtain tangent curves below the state-independent curve, which represent the error–trade-off relation with the state dependence. This example, with $$\mathrm{\alpha }=\mathrm{\beta }=1/\sqrt{2}$$, is illustrated in the Supplementary Materials. Furthermore, extending our implementation to other Pauli operators, for example, $$\mathcal{A}={\mathrm{\sigma }}_{\mathit{x}}$$ and $$\mathcal{B}={\mathrm{\sigma }}_{\mathit{z}}$$, is straightforward and will result in the same EDR as simply verified in Fig. 5. For this case, we performed operations for optimal state preparation and measurement largely different from those for $$\mathcal{A}={\mathrm{\sigma }}_{\mathit{y}}$$ and $$\mathcal{B}={\mathrm{\sigma }}_{\mathit{z}}$$ (see the Supplementary Materials) but obtained similar EDRs. The similarity indicates the universality of Heisenberg’s uncertainty relation.

In our experiment, we demonstrated variation of the observables with respect to α and β. Our operations included steering toward the state $$|\mathrm{\varphi }\rangle$$ by U_C(θ₁, φ₁) and realizing the observables by U_C(θ₂, φ₂), followed by the detection on the state $$|\uparrow \rangle$$. The step can be mathematically written aswhere U = U_C(θ₂, φ₂)U_C(θ₁, φ₁) and K = A, B, G.

The trace distance for a pair of observables E and F, with $${\mathit{E}}_{+}=({\mathit{e}}_0\mathit{I}+\vec{\mathit{e}}\cdot \vec{\mathrm{\sigma }})/2$$ (E₋ = I − E₊) and $${\mathit{F}}_{+}=({\mathit{f}}_0\mathit{I}+\vec{\mathit{f}}\cdot \vec{\mathrm{\sigma }})/2$$ (F₋ = I − F₊) applied on the state $$\mathrm{\rho }=(\mathit{I}+\vec{\mathit{r}}\cdot \vec{\mathrm{\sigma }})/2$$, is given bywhere $${\mathit{P}}_{\mathit{i}}^{\mathcal{J}}=\text{Tr}[\mathrm{\rho }{\mathit{J}}_{\mathit{i}}]$$ (J = E, F) is the probability distribution. In the qubit case, the trace distance was actually the Wasserstein distance defined in Busch et al. (23) for inaccuracies. Equation 5 shows that the SD of the trace distance was four times that of the p_i observed.

In our case, $${\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{A},\mathcal{C})}^2=2(1-\mathrm{\alpha })|{\mathit{r}}_{\mathit{y}}|$$ and $${\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{B},\mathcal{D})}^2=2(1-\mathrm{\beta })|{\mathit{r}}_{\mathit{z}}|,$$ where r_y = sin θ₁ cos φ₁ and r_z = − cos θ₁ (see the Supplementary Materials). The probability distribution for the observables in the main text can be measured as in Eq. 4. Heisenberg’s EDR is $$\mathrm{\Delta }{(\mathcal{A},\mathcal{B})}^2:=\underset{\mathrm{\rho }}{\text{max}}[{\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{A},\mathcal{C})}^2+{\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{B},\mathcal{D})}^2]$$, where the maximum is reached by a state $$|\mathrm{\varphi }\rangle$$ prepared at $${\mathrm{\theta }}_1= \text{arcsin}[(1-\mathrm{\alpha })/\sqrt{{(1-\mathrm{\alpha })}^2+{(1-\mathrm{\beta })}^2}]$$ and φ₁ = 0. Here, we have to mention that the $$\mathrm{\Delta }(\mathcal{A},\mathcal{B}{)}^2$$ we defined was slightly different from that in Busch et al. (23), which used the summation of the respective maximum of $${\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{A},\mathcal{C})}^2$$ and $${\mathrm{\Delta }}_{\mathrm{\rho }}{(\mathcal{B},\mathcal{D})}^2$$. We preferred to work with our formulation because of the convenience in experimental implementation.

Numerical simulation was performed to fit the experimental observation and assess the imperfection of experimental execution. The main deviations in our experiment came from two aspects: thermal phonons from the radial direction yielding offsets of Rabi oscillations and imperfection in qubit detection. For the former, an effective mean deviation at different moments could be estimated by means of a fitting method (36, 37). This kind of mean deviation is nearly constant and thus easily corrected. The detection error yielded a mean deviation of 0.35(2)% and could be calibrated by a practical method (38). The statistical error in our experiment was calculated by the Monte Carlo simulation with a peak value of 0.025. The decay and dephasing times of the qubit were 1.1 s and 2 ms, respectively, whose detrimental effects were negligible during the short periods (~ 8 μs) of our operations. Other possible imperfections could also lead to small errors, such as fluctuations of AC Stack shift due to power instability of the 729-nm laser and the fluctuating magnetic field, which were assessed to be less than 2% from the Rabi oscillation and included in the SD represented by the error bars.