The uncertainty principle lies at the heart of quantum mechanics, which makes it different from classical theories of the physical world. It behaves as a fundamental limitation describing the precise outcomes of incompatible observables, and plays a significant role in quantum information science from quantum key distribution [1–4] to quantum random number generation [5, 6], and from quantum entanglement witness [7–9] to quantum steering [10, 11] and quantum metrology [12, 13] (also see Ref. [14] for the review of uncertainty relation and applications).

The seminal concept of uncertainty relation was proposed by Heisenberg in 1927 [15], in which he observed that the measurement of position x of an electron with error Δ(x) causes the disturbance Δ(p) on its momentum p. In particular, their product has a lower bound set by Planck constant, that is, Δ(x)Δ(p) ∼ ℏ. Later, Robertson generalized the Heisenberg’s uncertainty relation to two arbitrary observables by Δ A Δ B ≥ 1 2 | ⟨ [ A , B ] ⟩ | , with ΔA (ΔB) being the standard deviation of observable A (B), [A, B] = AB− BA being the commutator of A and B, and ⟨⋅⟩ being the expected value in a given state ρ [16]. Indeed, such an uncertainty relation has a state-dependent lower bound so that it fails to reveal the intrinsic incompatibility when A and B are noncommuting.

To address the issue of state-independence of Robertson’s uncertainty relation, the entropic uncertainty relation has been developed by Deutsch [17], Kraus [18], and Maassen and Uiffink [19]: Consider a quantum state ρ and two observables A and B; the eigenstates |a _i ⟩ and |b _i ⟩ of observable A and B constitute measurement bases A = { | a i 〉 } and B = { | b i 〉 } . The probability of measuring A on state ρ with ith outcome is p _i = Tr[ρ|a _i ⟩⟨a _i |], and the corresponding Shannon entropy of measurement outcomes is H(A) = −∑ _i p _i log₂ p _i . Then, H(A) + H(B) is lower bounded by H(A) + H(B) ≥− log₂ c with c = max _i,j |⟨a _i |b _j ⟩|² the maximal overlap between |a _i ⟩ and |b _j ⟩. According to the definition of Shannon entropy, H(A) quantifies the uncertainty or lack of information associated to a random variable, but does not indicate whether the uncertainty comes from classical or quantum parts. For instance, the measurement of Pauli observable Z on states | + 〉 = ( | 0 〉 + | 1 〉 ) / 2 and I/2 = (|0⟩⟨0| + |1⟩⟨1|)/2 both lead to H(Z) = 1.

It is natural to consider quantum coherence, which is one of the defining features of quantum mechanics, to quantify the quantum component in uncertainty [20–22]. Along with this, rigorous connections between quantum coherence and entropic uncertainty have been established [23, 24] based on the framework of coherence quantification [25], and the quantum uncertainty relations (QURs) have been theoretically constructed with various coherence measures [26]. On the experimental side, the QURs using relative entropy of coherence have been demonstrated to investigate the trade-off relation [27] and connection between entropic uncertainty and coherence uncertainty [28]. Still, there are several unexplored matters along the line of experimental investigations. First, although various QURs have been theoretically constructed with relative entropy of coherence, the experimental feasibility and comparison have not been tested. Second, the experimental realizations of QURs using other coherence measures beyond relative entropy of coherence are still lacking. Finally, the lower bounds in QURs are generally obtained with quantum state tomography (QST) [27, 28], which becomes a challenge when the dimension of quantum state increases.

In this study, we experimentally investigate QURs constructed with three coherence measures, relative entropy of coherence, l ₁ norm of coherence, and coherence of formation, on a family of single-photon states. The lower bound of the QURs is indicated with classical shadow (CS) algorithm [29]. We show that the tightness of coherence lower bounds depends on the reference bases and the purity of quantum state.

This article is organized as follows: In Section 2, we introduce the basic idea of QUR using quantum coherence measures. In Section 3, we briefly introduce the CS algorithm to detect the purity of a quantum state. In Sections 4 and 5, we present the experimental demonstration and results. Finally, we draw the conclusion in Section 6.

A functional C can be regarded as a coherence measure if it satisfies four postulates: nonnegativity, monotonicity, strong monotonicity, and convexity [25]. The different coherence measure plays different roles in quantum information processing. For instance, the relative entropy of coherence plays a crucial role in coherence distillation [30], coherence freezing [31, 32], and the secret key rate in quantum key distribution [33]. The coherence of formation represents the coherence cost, that is, the minimum rate of a maximally coherent pure state consumed to prepare the given state under incoherent and strictly incoherent operations [30]. The l ₁-norm of coherence is closely related to quantum multi-slit interference experiments [34] and is used to explore the superiority of quantum algorithms [35–37]. We refer to Ref. [38] for the review of resource theory of quantum coherence. In the following, we give a brief review of QURs constructed with coherence measures of relative entropy of coherence, l ₁-norm of coherence, and coherence of formation [26].

From Section 2, it is obvious that the purity P of ρ is the key ingredient in the experimental testing of various QURs. The purity P can be calculated by reconstructing the density matrix of ρ with QST, which is very costly as the Hilbert space of ρ increases. Another protocol employs two copies of ρ for the detection of P , that is, P = Tr [ Π ρ ⊗ ρ ] , with Π being the local swap operator of two copies of the state [40, 41].

Very recently, the CS algorithm has been theoretically proposed for efficient quantum state detection [29], and has been experimentally realized in the detection of purity of unknown quantum states [42, 43]. In CS algorithm, a randomly selected single-qubit Clifford unitary U is applied on ρ, and then the rotated state UρU ^† is measured in the Pauli-Z basis, that is, Z = { | z 0 〉 = | 0 〉 , | z 1 〉 = | 1 〉 } . With the outcome of |z _i ⟩, the estimator ρ ̂ is constructed by ρ ̂ = 3 U † | z i 〉 〈 z i | U − I . It is equivalent to measure J = U ^† ZU ( J = { U | 0 〉 , U | 1 〉 } ) on ρ, and the measurement basis J is randomly selected from the Pauli observable basis set J ∈ { X , Y , Z } , with a uniform probability K ( J ) = 1 / 3 . The estimator ρ ̂ can be rewritten as ρ ̂ = 3 | k 〉 〈 k | − I , where |k⟩ ∈ {|x ₀⟩, |x ₁⟩, |y ₀⟩, |y ₁⟩, |z ₀⟩, |z ₁⟩}. In particular, | x 0 〉 = | + 〉 = ( | 0 〉 + | 1 〉 ) / 2 and | x 1 〉 = | − 〉 = ( | 0 〉 − | 1 〉 ) / 2 are the eigenvectors of Pauli observable X and | y 0 〉 = | L 〉 = ( | 0 〉 + i | 1 〉 ) / 2 and | y 1 〉 = | R 〉 = ( | 0 〉 − i | 1 〉 ) / 2 are the eigenvectors of Pauli observable Y. It is worth noting that the construction of estimator ρ ̂ only requires one sample. In our demonstrations, one sample is one two-photon coincidence. For a set of estimators { ρ ̂   i } constructed with N _s samples, the purity of state ρ can be estimated by two randomly selected independent ρ ̂   i and ρ ̂   j , that is, P ̂ = ∑ i ≠ j Tr [ Π ρ ̂   i ⊗ ρ ̂   j ] / N s ( N s − 1 ) .

To test the aforementioned QURs of various coherence measures, we consider the following single-qubit state: ρ τ = τ | + 〉 〈 + | + 1 − τ I 2 , (10) with 0 ≤ τ ≤ 1. Note that τ = 1 corresponds to the pure state | + ⟩ and τ = 0 corresponds to the maximally mixed state I/2. The experimental setup to generate state in Eq. 10 is shown in Figure 1A. Two photons are generated on a periodically poled potassium titanyl phosphate (PPKTP) crystal pumped by an ultraviolet CW laser diode. The generated two photons are with orthogonal polarization denoted as |HV⟩, where |H⟩ and |V⟩ denote the horizontal and vertical polarization, respectively. Two photons are separated on a polarizing beam splitter (PBS), which transmits |H⟩ and reflects |V⟩. The reflected photon is detected to herald the existence of transmitted photon in state |H⟩, which is then converted to | + 〉 = ( | H 〉 + | V 〉 ) / 2 by a half-wave plate (HWP) set at 22.5°. We sent the heralded photon into a 50:50 beam splitter (BS₁), which transmits (reflects) the single photon with a probability of 50%. The photons in transmitted and reflected mode are denoted as |t⟩ and |r⟩, respectively. Two tunable attenuators are set at modes |t⟩ and |r⟩ to realize the ratio of transmission probability in |t⟩ and |r⟩ of τ 1 − τ . The photon in |r⟩ passes through an unbalanced Mach–Zehnder interferometer (MZI) consisting of two PBS and two mirrors, which acts as a completely dephasing channel in polarization degree of freedom (DOF), that is, | + 〉 〈 + | → I / 2 . Finally, the two beams are incoherently mixed on BS₂ to erase the information of path DOF, which leads to the state ρ(τ) in both output ports. A step-by-step calculation detailing the evolution of the single-photon state through this setup is given in Eq. 11: | H 〉 → HWP @ 22.5 ° | + 〉 = 1 2 | H 〉 + | V 〉 → BS 1 | + 〉 ⊗ 1 2 | t 〉 + | r 〉 → at  | t 〉  and  | r 〉 two attenuators | + 〉 ⊗ τ | t 〉 + 1 − τ | r 〉 → at  | r 〉 unbalanced MZI τ | + 〉 〈 + | ⊗ | t 〉 〈 t | + 1 − τ I / 2 ⊗ | r 〉 〈 r | → incoherently combined BS 2 τ | + 〉 〈 + | + 1 − τ I / 2 . (11)

In our experiment, we set the parameter τ = 0 to τ = 1, with an increment of 0.1, and totally generated 11 states. For each generated state, we detect the QURs with the setup shown in Figure 1B. The lower bound in QURs related to purity P CS is measured with CS algorithm. C RE J is detected with projective measurement on basis J , along with the measured purity. C l 1 J ( C f J ) is calculated with reconstructed ρ(τ). All the measurement bases are realized with a HWP, a quarter-wave plate (QWP), and a PBS.

To investigate the accuracy of estimated purity P CS with CS algorithms, we also calculate the purity P QST with reconstructed density matrix of ρ(τ) from QST with N _S = 2000. The results of | P QST − P CS | are shown in Figure 2A. The more the samples used in CS algorithm, the smaller | P QST − P CS | is. We observe | P QST − P CS | < 0.1 when N _s ≥ 600. Especially, | P QST − P CS | = 0.0036 when N _s = 2000. In Figure 2B, we show the results of P CS and P QST with N _S = 2000 on 11 prepared ρ(τ), in which the experimental results of P CS and P QST have good agreements with the theoretical predictions. In the following, all the results with CS algorithm are obtained with 2000 samples. We also compare the accuracy of estimated purity P from CS algorithm and QST with the same N _s (see Supplementary Material for the results).

We first focus on the lower bounds in QURs using relative entropy of coherence, that is, Eqs 2–5. We calculate the lower bounds in Eqs 2–5 with the estimated P CS on ρ(τ = 1), ρ(τ = 0.894), ρ(τ = 0.688), and ρ(τ = 0.291), respectively. As shown in Figure 3A, we observe that the lower bounds in Eqs 4, 5 have the same value and outperform others when A and B are mutually unbiased (c = 0.5). When c becomes larger, lower bounds in Eqs 2, 3 are stricter than those in 4 and Eq. 5. However, the situation is quite different when the purity becomes smaller. As shown in Figure 3B–D, the values of lower bounds in Eqs 3, 4 are negative (we denote them as 0) when c is larger than certain values, which means that the lower bounds are loosened as C RE A ( ρ ) + C RE B ( ρ ) > 0 for all ρ.

To investigate the tightness of various lower bounds, we measure C RE A ( ρ ) + C RE B ( ρ ) in different reference bases. We select observables A and B from set J(θ) = cos θZ + sin θX. Specifically, we fix A = J (0°) and choose B = J (90°), J (66.42°), and J (36.86°), which correspond to c = 0.5, 0.7, and 0.9. For each observable J(θ), we perform the projective measurement on basis J ( θ ) , and calculate the Shannon entropy of measurement outcomes H (J(θ)). Thus, we obtain C RE J ( θ ) ( ρ ( τ ) ) = H ( J ( θ ) ) − S VN ( ρ ( τ ) ) , where S _VN(ρ(τ)) can be calculated from P CS . The results of QURs using relative entropy of coherence are shown in Figure 4. As shown in Figure 4A, the lower bounds in Eqs 4, 5 have the same values as C RE A ( ρ ) + C RE B ( ρ ) is lower bounded by 1 − S _VN(ρ), when c = 0.5 according to the definitions in Eqs 4, 5. When c is larger, the lower bound in Eq. 2 is stricter than others as reflected in Figure 4B and Figure 4C.

Next, we investigate the QURs using l ₁-norm of coherence and coherence of formation as described in Eqs 7–9. We choose observables A = J (0°) = Z and B = J (90°) = X in the coherence measure, which corresponds to c = 0.5. The C l 1 Z ( ρ ) and C l 1 X ( ρ ) are calculated according to Eq. 7 with the reconstructed density matrix of ρ(τ). Thus, C f Z ( ρ ) and C f X ( ρ ) can be calculated with C l 1 Z ( ρ ) and C l 1 X ( ρ ) as C f ( ρ ) = h 1 + 1 − C l 1 ( ρ ) 2 [26]. The results of QURs using l ₁ norm of coherence and coherence of formation are shown in Figure 5A and Figure 5B, respectively, in which the measured coherence is well bounded by the measured lower bounds.

In this study, we experimentally investigate quantum uncertainty relations using various coherence measures. The lower bounds in quantum uncertainty relations are detected with the classical shadow algorithm, in which the measurement cost is quite small and independent of the dimension of quantum states. For the quantum uncertainty relation using relative entropy of coherence, we show that the tightness of lower bounds is highly related to the reference basis and purity of quantum state. Moreover, we test the quantum uncertainty relation using l ₁ norm of coherence and coherence of formation.

Our results confirm that the tightness of lower bound in quantum uncertainty relations is related to the purity of quantum states and the reference bases, which can benefit the choice of quantum uncertainty relations when considering the experimental imperfections in practice. For instance, the imperfections in state preparation and measurement apparatus correspond to the purity and reference bases in the lower bound, respectively. More importantly, our method can be generalized to multipartite states while it keeps its efficiency. The multipartite coherence could be efficiently estimated using the stabilizer theory [44, 45] and the classical shadow algorithm to detect that the purity of multipartite state is efficient as well [43].