1. Introduction

Uncertainty relations are one of the hallmarks of quantum physics [1–3]. They are useful for a wide range of applications including quantum metrology [4–8], entanglement detection [9–13], quantum cryptography [14–17], quantum speed limit [18–20], and quantum control and measurements [21]. In the textbook literatures, uncertainty relations usually refer to the preparation uncertainty, which suggests that it is impossible to prepare an ensemble of quantum systems for which uncertainties for two non-commuting observables can be arbitrarily small [22–25]. In the previous investigations, uncertainty relations for two incompatible observables [26–30] and for multi-observables [31–33] have been widely investigated and experimentally tested in different physical systems, such as NMR [34,35], nitrogen-vacancy center in diamonds [36,37], neutrons [38–42] and photons [43–49].

On the other hand, a unitary operator is a fundamental tenet of quantum theory. Every evolution of a closed quantum system is governed by acting unitary operators on the state of the system and the evolution of an open system can be represented by acting unitary operators on an enlarged system consisting of the quantum system as a subsystem [50]. Thus, uncertainty relations of unitary operators are naturally of broad interest to investigate [51]. More importantly, it is proven that the unitary uncertainty relation is stronger than, and can be used to derive, the standard Heisenberg and Robertson-Schrödinger uncertainty relations [52]. The unitary uncertainty relation, therefore, implies the standard uncertainty relation for observables. In 2016, a variance-based uncertainty relation for two arbitrary unitary operators acting on physical states of any Hilbert space was derived by Bagchi and Pati [53] and has been tested experimentally with photonic qutrits in 2017 [54]. In 2018, Bong et al. [52] derived and experimentally investigated another variance-based uncertainty relation for any $n$ unitary operators. In 2019, Yu et al. [55] provided a set of uncertainty principles for unitary operators using a sequence of inequalities with the help of the geometric-arithmetic mean inequality. As these inequalities are fine-grained compared with the well-known Cauchy-Schwarz inequality, the framework naturally improves the results based on the latter, such as the one in [52]. As such, they claimed their unitary uncertainty relations outperform the best known bound introduced in [52]. As the improvement is due to replacement of the Cauchy-Schwarz inequality underlying all previous uncertainty principles, their method provides fundamentally better bounds.

In this paper, we experimentally test the uncertainty relations theoretically derived in [55] with single photons and interferometric networks. Unitary uncertainty relations work for $n$ operators and we test the one for two operators for example. It is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literature [52]. The lower bound of the unitary uncertainty relation can be even further strengthened by the symmetry of permutation [55]. Our setup allows for measurements directly leading to the values of all the terms in the relation and releases the requirement of state tomography. The behaviour we find agrees with the predictions of quantum theory and obeys the unitary uncertainty relation in [55].

2. Theoretical idea

For an arbitrary unitary operator $U$ and a quantum state $| {\psi }\rangle$, the variance of the unitary operator $U$ with respect to $| {\psi }\rangle$ is defined by [51,56]

The variance of the unitary operator quantifies the disturbance of states and is bounded as $0\leq \Delta U^2\leq 1$ as it reaches its minimum value for a nondisturbing rephasing of the state and its maximum value for the maximally disturbing case that $U$ transforms the state to an orthogonal one. The unitary uncertainty relation between two unitary operators $U$ and $V$ is given by

Based on the geometric-arithmetic mean inequality, Yu et al. derived a variance-based unitary uncertainty relation in the product form for two unitary operators [55]

Thus, the lower bound of the unitary uncertainty relation for two unitary operators in [52] is the weakest bound in this sequence.

To test the strong unitary uncertainty relation (“strong” here means the unitary uncertainty relation in [55] is stronger than that given in the previous literature [52]), we consider a pure state $| {\psi _\theta }\rangle=\cos \theta | {0}\rangle-\sin \theta | {d-1}\rangle$ on a $d$-dimensional Hilbert space and two unitary operators

3. Experimental test

Our experiment uses polarizing and spatial degrees of freedom of single photons and an interferometric network to realize controllable unitary transformations. We can determine the value of $\langle U\rangle$, $\langle V\rangle$, $\langle U^\dagger \rangle$ and $\langle U^\dagger V\rangle$ for an input state from the average output photon numbers. As shown in Fig. 1, the experimental setup involves three stages: the specific state preparation, state evolution under either of $U$, $V$, $U^\dagger$ and $U^\dagger V$, interference-based measurement on the input state and the evolved state [57–60]. A pair of photons is generated via the spontaneous parametric down conversion in the periodically poled potassium titanyl phosphate crystal (PPKTP), with one serving as a trigger and a signal photon filtered out by an interference filter which restricts the bandwidth of photons to $3$nm. It then undergoes through an interferometric network [61,62], composed of wave plates and beam displacers (BDs) [57,63], and is finally detected by avalanche photodiodes (APDs), in coincidence with the trigger photon.

 

Fig. 1. Experimental demonstration of strong unitary uncertainty relations. A pair of photons is generated via the spontaneous parametric down conversion in the periodically poled potassium titanyl phosphate crystal (PPKTP), with one serving as a trigger and the other projected into the interferometric network as the single photon. After passing through a polarizing beam splitter (PBS) and a half-wave plate (HWP), the polarization of the signal photon is initialized. For a higher-dimensional system, quantum states are encoded in both polarizing and spatial degrees of freedom and prepared by a beam displacer (BD) following by the PBS and HWP. It then undergoes an interferometric network, composed of wave plates and BDs, and is projected into the basis states. The optical axes of BD$_i$ ($i=1,2,4,5,6$) are cut so that vertically polarized light is directly transmitted and horizontal light undergoes a $3$mm lateral displacement into a neighboring mode. BD$_3$ and BD$_7$ separate light with different polarizations $6$mm. Finally the signal photon is detected by avalanche photodiodes (APDs), in coincidence with the trigger photon.

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In a $2$-dimensional case, the input state is represented by polarizations of the heralded single photons. The basis states $| {0}\rangle$ and $| {1}\rangle$ are encoded by the horizontal and vertical polarizations of the photons, respectively. The input state $| {\psi _\theta }\rangle$ is prepared by letting heralded single photons to pass through a half-wave plate (HWP) at $-\theta /2$. We choose $17$ values of $\theta \in \left [0,\pi /2\right ]$. The terms of the inequality, i.e., $\Delta U^2\Delta V^2$ and $|\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle |^2$, can be calculated by the expected values of the certain unitary operators. The expected values of the unitary operators can be obtained by projecting the evolved states $| {\psi _A}\rangle=A| {\psi _\theta }\rangle$ to the input state $| {\psi _\theta }\rangle$, where $A=U, V, U^\dagger , U^\dagger V$ is the corresponding unitary operator. For the input state $| {\psi _\theta }\rangle$, the evolution $U$ ($U^\dagger$) adds a phase on $| {1}\rangle$ and keeps $| {0}\rangle$ unchanged. Whereas, $V$ ($U^\dagger V$) is a flip operation. First, we reproduce the polarization states of the photons in either arm of an interferometer in the input state $| {\psi _\theta }\rangle$ via the HWPs (H$_1$, H$_2$ and H$_3$) and BD$_1$. The interferometer consists two BDs (BD$_1$ and BD$_2$) and several wave plates. The unitary operator $A$ is realized by rotating the HWP (H$_4$) in one (upper) arm of the interferometer, while the identity operator $\mathbb {I}_2$ is realized in the other (lower) arm. Some wave plates are positioned in order to keep the path-length difference less than the coherence length. We vary the unitary operators pairwise in steps such that all the terms of the left- and right-hand sides of the inequality (3) is sampled and at each setting, where the state of the photons passing each arm is either of $| {\psi _\theta }\rangle$ or $| {\psi _A}\rangle$.

To read out the values of $\langle A\rangle$, we need to measure the inner product of $| {\psi _\theta }\rangle$ and $| {\psi _A}\rangle$ [57–60]. We then apply projective measurements in the basis $\{| {0}\rangle,| {1}\rangle,(| {0}\rangle+| {1}\rangle)/\sqrt {2},(| {0}\rangle+i| {1}\rangle)/\sqrt {2}\}$, which can be realized by a polarizing beam splitter (PBS) and wave plates. We can determine the values of $\langle A\rangle$ for the input state by noting that the average output photon number is given by the clicks of the corresponding APDs. The average output photon number is related to the inner product of two states in the two arms of the interferometer and then one can obtain an interference pattern to determine the values of $\langle U\rangle$, $\langle V\rangle$, $\langle U^\dagger \rangle$ and $\langle U^\dagger V\rangle$.

To obtain the values of $I_k$, we need to extract $|u_i|=|\langle \phi _i|\delta U| {\psi _\theta }\rangle|$ and $|v_i|=|\langle \phi _i|\delta V| {\psi _\theta }\rangle|$. However, $\delta U$ and $\delta V$ are not unitary and we can not then repeat the above procedure by simply replacing the input state as $(A-\langle A\rangle )| {\psi _\theta }\rangle$ ($A=U,V$). We reproduce the polarization states of the photons in either of the arms of the interferometer in the input state $| {\psi _\theta }\rangle$. At the same time, we use the HWP (H$_1$) in front of BD$_1$ to adjust the ratio between the number of photons in the upper arm of the interferometer and that in the lower arm to be $1:|\langle A\rangle |$, where $\langle A\rangle$ is obtained in the previous procedure. We realize $A$ in the upper arm of the interferometer via the HWP ($H_4$), and $e^{i\varphi }\mathbb {I}_2$ in the other arm via the QWP-HWP-QWP (Q$_1$-H$_6$-Q$_2$). We assume that $\langle A\rangle =|\langle A\rangle | e^{i\varphi }$. We vary $A$ in steps and, at each setting, where the state of the photons passing each arm is either of $\langle A\rangle | {\psi _\theta }\rangle$ or $| {\psi _A}\rangle$. We assume $\langle A\rangle | {\psi _\theta }\rangle=\alpha | {0}\rangle+\beta | {1}\rangle$ and $| {\psi _A}\rangle=a| {0}\rangle+b| {1}\rangle$ (not normalized). By varying the setting angles of H$_5$ and H$_7$, the output state is either $a| {0}\rangle+\alpha | {1}\rangle$ or $b| {0}\rangle+\beta | {1}\rangle$ (not normalized). Thus, the values of $|a_i|$ can be represented by $|a_1|\propto \sqrt {|a|^2-a\alpha ^*-a^*\alpha -|\alpha |^2}$ and $|a_2|\propto \sqrt {|b|^2-b\beta ^*-b^*\beta -|\beta |^2}$, where $a_i=u_i, v_i$. Finally, we perform the projective measurements on either of the output states in the basis $\{| {0}\rangle,| {1}\rangle,(| {0}\rangle+| {1}\rangle)/\sqrt {2},(| {0}\rangle+i| {1}\rangle)/\sqrt {2}\}$ and reconstruct the coefficients of the output states ($\{|a|^2,a\alpha ^*, a^*\alpha ,|\alpha |^2\}$ or $\{|b|^2,b\beta ^*,b^*\beta ,|\beta |^2\}$) from the outcomes of the projective measurements. The values of $|u_i|$ and $|v_i|$ can be calculated from the reconstructed coefficients of the output states.

For a $3$-dimensional case, a qutrit is represented by three modes of the heralded single photons. The basis states $| {0}\rangle$, $| {1}\rangle$ and $| {2}\rangle$ are encoded by the horizontal polarization of the photon in the lower mode, the horizontal polarization of the photon in the upper mode and the vertical polarization in the upper mode, respectively. The signal photons pass though a PBS and a HWP and are split by a BD (BD$_3$) into upper and lower modes to generate the input state $| {\psi _\theta }\rangle=\cos \theta | {0}\rangle-\sin \theta | {2}\rangle$.

Similar to the procedure for the $2$-dimensional case, we reproduce the polarization of the photons in either arm of an interferometer in the input state $| {\psi _\theta }\rangle$ with a BD (BD$_4$). The unitary operators $U$, $V$, $U^\dagger$ and $U^\dagger V$ are realized by an interferometer (BD$_5$ and BD$_6$) and wave plates and the expected values of the unitary operators are determined by the interference pattern. With the experimental result of $\langle U \rangle$ and $\langle V\rangle$, we can then extract $I_k$. The projective measurements in the basis $\{| {1}\rangle,| {2}\rangle,(| {1}\rangle+| {2}\rangle)/\sqrt {2},(| {1}\rangle+i| {2}\rangle)/\sqrt {2}\}$ on the output states are realized by a BD (BD$_7$), a PBS and wave plates. The outcomes of the measurements are registered by the clicks of the corresponding APDs. We can obtain the values of each term in the unitary uncertainty relation (3) for the $3$-dimensional case.

For a $4$-dimensional case, the encoding and experimental demonstration are similar to those for the $3$-dimensional case. Thus, we can test the strong unitary uncertainty relations for $d=2,3,4$ cases.

4. Experimental results

In Fig. 2, we show the experimental results of testing the uncertainty relation for two unitary operators in (3) for the $d=2,3,4$ cases. The solid lines with different colors represent the theoretical predictions of the left- and right-hand sides of the relation and $I_k$, respectively. The symbols represent the experimental results of the inequality with $17$ input states $| {\psi _\theta }\rangle$. The experimental results agree remarkably well with the theoretical predictions. For the $2$-dimensional case, the uncertainty relations in (2) and (3) are equivalent and saturated for all pure states. However, the unitary uncertainty relation in (3) outperforms that in (2) for $d>2$, as shown by the experimental results.

 

Fig. 2. Experimental results of testing the unitary uncertainty relation (3). The unitary operators in both arms of the interferometer remain fixed and we input a family of states with a varying parameter $\theta$. (a) Experimental results for a $2$-dimensional case. The solid blue curve describes the theoretical predictions of left-hand side $\Delta U^2 \Delta V^2$ and right-hand side $\left |\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle \right |^2$ of the uncertainty relation (3) and $I_2$. (b) Experimental results for a $3$-dimensional case. The solid blue and red curves characterize the theoretical predictions of $\Delta U^2 \Delta V^2$ and $I_2$, respectively. The solid green curve represents the theoretical predictions of $I_2$ and $\left |\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle \right |^2$. (c) Experimental results for a $4$-dimensional case. The solid blue curve represents the theoretical prediction of $\Delta U^2 \Delta V^2$. The solid red curve represents the theoretical predictions of $I_2$ and $I_3$. The solid green curve represents the theoretical predictions of $I_4$ and $\left |\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle \right |^2$. The data points represent the experimental results of corresponding terms of the uncertainty relations. Error bar indicate the statistical uncertainties, which are obtained based on assuming Poissonian statistics of photons.

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Remarkably, the bounds $I_k$ can be further strengthened by symmetry of permutations [55]. Taking the $3$-dimensional case as an example, we consider the same input state $| {\psi _\theta }\rangle=\cos \theta | {0}\rangle-\sin \theta | {2}\rangle$ and apply the symmetric group $S_3$ on the set $\{1,2,3\}$ naturally by permutation. This is used to strengthen the lower bound of the unitary uncertainty relation. For any two permutations $\pi _1,\pi _2\in S_3$, the induced action of $S_3\times S_3$ on $I_k$ ($k=2,3$) is given by

In Fig. 3, we show the experimental results of the verification of the strengthened uncertainty relation for two unitary operators in (7) for the $3$-dimensional case. Four solid curves represent the theoretical predictions of $\Delta U^2\Delta V^2$, $\max _{\pi _{1},\pi _2\in S_{3}}(\pi _1,\pi _2)I_2$, $\max _{\pi _{1},\pi _2\in S_{3}}(\pi _1,\pi _2)I_3$, $I_2$, $I_3$ and $|\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle |^2$, respectively. The symbols represent the corresponding experimental results. We obtain $u_i$ and $v_i$ ($i=1,2,3$) for each input state, and maximize $(\pi _1,\pi _2)I_2$ and $(\pi _1,\pi _2)I_3$ over permutations $\pi _1,\pi _2\in S_3$. Thus, we experimentally demonstrate that the new uncertainty bound of the inequality (3) for two unitary operators [55] outperforms that of previous uncertainty relation (2) based on the Cauchy-Schwarz inequality [26,53] or the positive semi-definiteness of the Gram matrix [52] in the whole range. Furthermore, a strengthened unitary uncertainty relation theoretically derived in [55] is verified experimentally.

 

Fig. 3. Experimental results of testing the strengthened unitary uncertainty relation (7) for a $3$-dimensional case. The strengthened bound v.s. the bound $I_k$ for qutrit pure states. The solid blue curve describes the theoretical predictions of $\Delta U^2\Delta V^2$ and $\max _{\pi _{1},\pi _2\in S_{3}}(\pi _1,\pi _2)I_2$. The solid black and red curve represent the theoretical predictions of $\max _{\pi _{1},\pi _2\in S_{3}}(\pi _1,\pi _2)I_3$ and $I_2$, respectively. The solid green curve describes the theoretical predictions of $I_3$ and $|\langle U^\dagger V\rangle -\langle U^\dagger \rangle \langle V\rangle |^2$. The symbols represent the corresponding experimental results. Error bar indicate the statistical uncertainties, which are obtained based on assuming Poissonian statistics of photons.

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5. Discussion and conclusions

Uncertainty relations are one of the most important foundations of physics, defining the limits on what is possible in a quantum world. Especially, unitary uncertainty relations are more fundamental as it is proven that the unitary uncertainty relation is stronger than, and can be used to derive, the standard Heisenberg and Robertson-Schrödinger uncertainty relations. Thus, uncertainty relations of unitary operators are naturally of broad interest to investigate. In this paper, we test a very powerful and simple uncertainty relation for unitary operators, which significantly strengthens previous results. Unitary uncertainty relations work for $n$ operators and we test the one for two operators for example. It is saturated by any pure qubit state. For higher-dimensional states, it is stronger than the best known bound introduced in the previous literatures [26,30,32,33,52–54].

We test the unitary uncertainty relation experimentally with single photons and interferometric networks. From an operational point of view, we show that the uncertainty in the unitary operator is related to the visibility of quantum interference in an interferometer where one arm of the interferometer is affected by a unitary operator. Moreover, our setup allows for measurements directly leading to the values of all the terms in the relation and releases the requirements of state tomography.

Compared to the previous experiment [52], the unitary uncertainty relations tested here are stronger and the lower bounds can be further strengthened by the symmetry of permutation. More importantly, we improve the experimental setup. We measure the transition probabilities (or the square root of the transition probabilities) of the states instead of interference visibility, which reduces the complexity of measurements. Thus, we are able to test the uncertainty relations for high-dimensional unitary operators ($3$-dimensional and $4$-dimensional), while only the simplest case—the uncertainty relations for $2$-dimensional unitary operators was tested in [52]. On the other hand, as we measure the transition probabilities, the unitary uncertainty relations are directly related to some inequalities for the transition probabilities of the states.

Compared to the first experimental test of the unitary uncertainty relations of a strictly pure state [54], no prior knowledge of the state is required in our experiment. Moreover, the unitary uncertainty relations here do not assume or require pure states and all the results can be extended to mixed states $\rho$ by the definition of the variance of the unitary operator $A$ with respect to the input mixed state $\Delta A^2=\textrm {Tr}(\rho \delta A^\dagger \delta A)$ [55]. Thus, the unitary uncertainty relation becomes a general and powerful tool for testing quantumness of the physical system. Our experimental results show uncertainty bounds for arbitrary two unitary operators outperform those of the previous unitary uncertainty relation due to the Cauchy-Schwarz inequality or the positive semi-definiteness of the Gram matrix in the whole range. Furthermore, we experimentally prove that the bounds of the unitary uncertainty relation can be even further strengthened by symmetry of permutations.