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ARTICLE OPEN Uncertainty equality with quantum memory and its experimental verification

Hengyan Wang1,2,3,4, Zhihao Ma5, Shengjun Wu6, Wenqiang Zheng7, Zhu Cao8, Zhihua Chen7, Zhaokai Li1,3,4, Shao-Ming Fei9,10, Xinhua Peng 1,3,4, Vlatko Vedral11,12 and Jiangfeng Du1,3,4

As a very fundamental principle in quantum physics, has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five- spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations. npj (2019) 5:39 ; https://doi.org/10.1038/s41534-019-0153-z

INTRODUCTION Shannon entropies of the measurement results, the uncertainty 1 4 fi The uncertainty principle is one of the most important principle in principle can be formulated as Hθ þ Hτ  log2 c. Here 1/c quanti es quantum physics. It implies the impossibility of simultaneously the degree of complementarity of two observables θ and τ. determining the definite values of incompatible observables. The The above uncertainty relations only concern a single quantum more precisely an observable is determined, the less precisely a system. By taking the entanglement with a memory system into 14 complementary observable can be known. Based on the distribu- account, an entropic uncertainty relation in the presence of 15 tions of measurement outcomes, the quantum uncertainty relations quantum memory has been investigated in ref. . It has been 1–13 ρ can be described in various ways; see for instance refs. shown that for a bipartite state AB, performing measurements on The uncertainty principle was first formulated via the standard one of the subsystems A gives rise to the following relation deviation of a pair of complementary observables, known as the 1 ’ 1 Δ Δ ≥ ħ SðθjBÞþSðτjBÞlog þ SðAjBÞ; (1) Heisenberg s uncertainty principle x p /2 for the coordinate x 2 c and the momentum p in an infinite dimensional . Later where S(A|B) and S(θ|B)(S(τ|B)) denote, respectively, the condi- the Robertson−Schrödinger uncertainty inequality2,3 presented an tional von Neumann entropies of the initial bipartite state ρ and uncertainty relation for two arbitrary observables in a finite AB the final bipartite state ρθ (ρτ ) after the measurement in the basis dimensional Hilbert space. Instead of the standard deviation of B B {|iθ〉} ({|iτ〉}). This uncertainty relation was further extended to the observables, the uncertainty principle can also be elegantly smooth entropy case.16 With considering the entangled quantum formulated in terms of entropies related to measurement bases. memory, these uncertainty relations have potential applications in 〉 = When a quantum system is projected onto a certain basis {|iθ |i 1, entanglement witnessing and quantum key distributions ⋯ θ 2, , d}, where labels the measured observable, the uncertainty of (QKD).15,17,18 However, the above results all concern measure- the measurementP results has been characterized by the Shannon ments only on two observables and are given in inequality forms. d entropy Hθ ¼ i¼1 Àpi log2 pi,wherepi is the probability to obtain In this work, we consider projective measurements based on 19–24 the ith basis state |iθ〉. The larger the Shannon entropy Hθ is, the mutually unbiased bases (MUBs). The MUB measurements are more uncertain the measurement results are. In terms of the complementary to each other in the sense that any pair of bases

1Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, 230026 Hefei, China; 2Department of Physics, Zhejiang University of Science and Technology, 310023 Hangzhou, China; 3CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, 230026 Hefei, China; 4Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, 230026 Hefei, China; 5School of Mathematical Sciences, Shanghai Jiaotong University, 200240 Shanghai, China; 6Kuang Yaming Honors School, Nanjing University, 210023 Nanjing, Jiangsu, China; 7College of Science, Zhejiang University of Technology, 310023 Hangzhou, Zhejiang, China; 8Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, 100084 Beijing, China; 9School of Mathematical Sciences, Capital Normal University, 100048 Beijing, China; 10Max- Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany; 11Atomic and Laser Physics, University of Oxford, Clarendon Laboratory, Parks Road, OX1 3PU Oxford, United Kingdom and 12Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore Correspondence: Xinhua Peng ([email protected]) or Jiangfeng Du ([email protected]) These authors contributed equally: Hengyan Wang, Zhihao Ma, Shengjun Wu Received: 19 November 2018 Accepted: 11 April 2019

Published in partnership with The University of New South Wales H. Wang et al. 2 are maximally unbiased. They are deeply connected to the Born’s be described by the linear entropy of a d-level quantum state 21 2 principle of complementarity and closely related to the wave- ρ, SL(ρ) = 1 − Tr(ρ ). SL(ρ) ranges from 0 (when ρ is a pure state) to particle duality.25,26 A complete set of MUBs consists of at most d (d − 1)/d (when ρ is maximally mixed). Unlike that in ref. 26, here + 1 observables, where d is the dimension of the state space. we do not introduce a normalization factor to have a range Comparing with the incomplete case, the advantage of a between 0 and log2 d, so the measure of uncertainty in terms of complete set of MUB measurements is informatively complete23,24 linear entropy does not have the unit of a “bit”. However, it and meaningful in quantum information progressing.21 It is quantifies uncertainty in a natural way: an uncertainty of 0 means therefore not surprising that a complete set of MUB measure- the outcome is 100% certain while an uncertainty approaching 1 ments is crucial in entanglement detection.27,28 It was also proved means the outcome is almost random. that using a complete set of MUBs is much better than using two For a bipartite state ρAB in a d × D (D ≥ d) dimensional composite 19 observables in QKD. Hilbert space, if system A is projected on to the basis {|iθ〉 |i = 1, 2, In ref. 6 an entropic uncertainty relation involving d + 1 MUB ⋯, d}, the overall state of the composite system after the measurements has been obtained in terms of von Neumann nonselective measurement43,44 on A is given as entropy. However, it only dealt with a single system (in this case Xd the von Neumann entropy is just the Shannon entropy of the ρ ρ : θB ¼ jiiθ Ahiθj A hiθj ABjiiθ A (2) measurement probability distributions), and the uncertainty i¼1 relation is given by an inequality. In fact, the Shannon entropy is a natural measure of our ignorance regarding the properties of a We can introduce the conditional linear entropy classical system, because in classical measurements the observa- θ : ρ ρ ρ2 ρ2 SLð jBÞ ¼ SLð θBÞÀSLð BÞ¼Trð BÞÀTrð θBÞ (3) tion removes our ignorance about the state by revealing the properties of the system which are considered to be pre-existing as a measure of the uncertainty about Alice’s measurement result ’ ρ = ρ = ρ and independent of the observation. In contrast to classical given Bob s state, where the reduced state B TrA( θB) TrA( AB) measurements, one cannot say that quantum measurements is independent of the measurement basis. It is straightforward to θ reveal a pre-existing property of a quantum system. Therefore, the show that the conditional linear entropy SL( |B) is always ρ Shannon entropy could be thought of as “conceptually” inade- nonnegative. As an example, suppose AB is a maximally quate in quantum physics.29 In ref. 26 the authors proposed a new entangled pure state. Alice can perform a measurement on her measure of quantum information, which takes into account that system in any basis. The possible resulting states of Bob’s system

1234567890():,; the only features of quantum systems known before a measure- are orthogonal to each other, and each possible resulting state is ment are the probabilities for various events to occur. It has in one-to-one correspondence to Alice’s resulting state. Therefore, fi given Bob’s state, Alice’s measurement result can be determined signi cant physical meaning and various applications in quantum P pffiffiffiffi θ ρ d λ information processing such as quantum randomness, quantum with certainty. In this case SL( |B) vanishes. If AB ¼ i¼1 ijiii is state estimation, quantum teleportation and quantum metrol- a partially entangled state written in its Schmidt bases, after Alice – ogy.30 36 Moreover, a series of works have been shown, together measures her system in the Schmidt basis, Bob’s possible resulting with many applications, that in single quantum system, the sum of states are orthogonal to each other and the Alice’s measurement the individual measures of information for MUBs is invariant under result is completely determined without uncertainty when Bob’s fi the choice of the particular set of complementary observations state is given. This is also con Prmed by the vanishing conditional 2 and conserved if there is no information exchange with entropy as S ðρθ Þ¼S ðρ Þ¼ λ . However, if Alice performs a 26,29–34,37–42 L B L B i i environments. measurement on a basis that is not the Schmidt basis, the possible In this article, we adopt the information measure proposed in resulting states of Bob’s system are not orthogonal and cannot be 26 ref. and consider the uncertainty relation in the presence of distinguished with certainty, and thus uncertainty of Alice’s quantum memory. Interestingly, we find that if we take a measurement result exists even when Bob’s state is known. This complete set of MUB measurements into account, we can obtain fact is again confirmed by the observation that the conditional an uncertainty equality that the sum of measurement uncertain- linear entropy is strictly greater than zero in this case. The ties over all MUBs on a subsystem in the presence of quantum conditional linear entropy is thus a good measure of the memory is equal to a fixed quantity determined by the initial state. uncertainty about Alice’s measurement result given Bob’s state. It gives rise to a kind of conservation relation of the uncertainties It depends on the basis in which the measurement is performed in related to these MUB measurements. We further show the elegant general. When Alice tries to find a basis to perform the applications of our result in quantum guessing game and measurement on her system so that Bob will know her result quantum random number generation. We also experimentally with minimum uncertainty, then using another MUB to perform verify this uncertainty equality by directly measuring the the measurement will result in Bob having a large uncertainty uncertainties on a nuclear spin system. Our method avoids the about Alice’s result. However, the whole uncertainty running over tomography process and allows one to perform verification all possible MUB measurements is fixed. This uncertainty relation experiments in large quantum systems. is formulated in the following theorem (the proof involves subtle mathematical techniques, see Method A). Theorem For any density matrix ρ on a composite Hilbert RESULTS AB space HA ⊗ HB of dimension d × D, we have the following Measurement of uncertainty relations uncertainty equality Let (p1, p2, …, pd) be the probabilities for the d measurement  Xdþ1 1 outcomes. The lack of information about the jth outcome with S ðθjBÞ¼d Trðρ2ÞÀ Trðρ2 Þ (4) respect to a single experimental trial is given by p (1 − p ). The L B d AB j j θ¼1 total lack of informationP regarding all d possibleP experimental d d 2 when a complete set of d + 1 MUBs exists for the d-dimensional outcomes is then given by j¼1 pjð1 À pjÞ¼1 À j¼1 pj , which is Hilbert space HA. minimal if one probability is equal toP a unity and maximal if all the The theorem shows that the total uncertainty related to the d 2 − probabilities are equal. In fact, 1 À j¼1 pj is nothing but 1 Tr measurements over all d + 1 MUBs of a subsystem is exactly given ρ2 ρ fi ρ2 ρ2 ( ), where is the state after a quantum (projective) measure- by a xed quantity, dTrð BÞÀTrð ABÞ, which is determined only by ment, the linear entropy of the measured state. Therefore, the lack the initial bipartite state. This quantity is always nonnegative and of information regarding all d possible experimental outcomes can can be viewed as the total measurement uncertainty of a

npj Quantum Information (2019) 39 Published in partnership with The University of New South Wales H. Wang et al. 3 subsystem, given the state of the other subsystem. Different from party when Alice measures in the computational basis and Bob the uncertainty inequality (1) based on the von Neumann entropy, measures in the diagonal basis. However, if one of the parties is here we obtain the equality (4). Note that this equality (4) is also corrupted, e.g., due to device imperfections, this scheme is completely different from the one given in ref. 45 which is based broken. To show this, consider that one party switches to the same on only ONE positive operator-valued measure consisting of basis as the other party. A common solution is that each party 46–48 uniformly all the measurement operators of d + 1 MUBs (see randomly uses multiple basis, such as σx, σy,orσz basis. Now, Remark in Method A). In general, when there are only M MUBs we consider the following semi-self-testing scenario. Alice first available or when we are only interested in certain M MUBs, we chooses a reference frame, and randomly performs measurements always have the following uncertainty inequality, in one of the MUB basis. The reference frame is assumed to be  reliably chosen, but Alice does not have a free will to randomly XM 2 1 2 choose her measurement basis, i.e., the basis choice may be S ðθjBÞðM À 1Þ Trðρ ÞÀ Trðρ Þ : (5) L B d AB manipulated by an adversary who wishes to corrupt Alice’s θ¼1 randomness, such as Bob. Hence, the entropy of Alice’s random With each additional MUB, the lower bound of total uncertainty is outcomes with respect to Bob is the smallest entropy SL(θ|B) fi ρ2 1 ρ2 increased by a xed amount Trð BÞÀd Trð ABÞ. among all measurement choices. To maximize this quantity, the To illustrate the implications of the theorem, let us consider that theorem shows that SL(θ|B) should be equal for all θs. Thus, the Alice and Bob are both users of quantum technology. In order to maximum entropy of a semi-self-testing QRNG is make a hard decision on whether she should accept Bob’s ρ2 ρ2 = ½dTrð BÞÀTrð ABފ ðd þ 1Þ. This limit on semi-self-testing QRNGs invitation to see a film, Alice asks Bob to send her a qubit A. Alice also cannot be obtained from an uncertainty inequality. Finally, can measure the qubit with the three (Pauli) observables σx, σy and note that entanglement-based QRNGs considered here have a σz at her choice. After the measurement, Alice announces her higher randomness generation rate compared to prepare-and- choice of the observable, and Bob is supposed to guess the Alice’s measure QRNGs,49 and cannot be analyzed by using the tools measurement results. Alice would accept (deny) Bob’s request if developed by Brukner and Zeilinger.26 his guess is correct (wrong). Bob tries to gain Alice’s acceptance by entangling the qubit A with his local qubit B in the preparation Experimental verification stage. From the theorem, we know that the sum of uncertainties ’ ’ To experimentally investigate the uncertainty conservation, a two- (of Bob s guess at Alice s measurement results given the state of B) ρ ρ2 qubit system AB, chosen as the test system, is prepared in the in three different cases is equal to the quantity Q ¼ 2Trð BÞÀ ρ2 ρ following states: Trð ABÞ that is completely determined by the initial state AB. Bob can minimize the quantity Q by preparing an EPR state, thus win 1 À x ρ ðα; xÞ¼xjiψα hjψα þ I ; (6) Alice’s acceptance with certainty, a result that cannot be obtained AB 4 4 from an uncertainty inequality like the one based on Shannon where |ψα〉 = cos(α/2)|01〉 − sin(α/2)|10〉. These states are mixed entropy (see Fig. 1). states composed of one pure state with weight x and the maximal On the practical side, the theorem also provides possible mixed state with weight (1 − x)/4. The parameters α characterizes applications in quantum random number generation, especially the entanglement of the pure part and x characterizes the purity semi-self-testing quantum random number generators (QRNGs) of the state. When α = π/2 and x = 1, the bipartite state is one of which are more robust to device imperfections. In a typical setup the Bell states. The other three Bell states can be obtained by local 46–48 of a semi-self-testing QRNG, Alice and Bob share a quantum unitary operations while the linear entropy remains invariant ρ state AB, e.g., an EPR pair. If both parties are trusted, the under such transformations. measurement outcome of one party will be random to the other The key part of the experiments is to measure the system’s (conditional) linear entropy. Similar to the measurement of von Neumann entropy in previous experiments,50,51 linear entropy can be indirectly measured by full quantum state tomography.52 However, this is inefficient for large-size quantum systems. Since the linear entropy is directly related to the purity Tr(ρ2) that can be 2 53 directly obtained by Tr(ρ ) = Tr(V2ρ ⊗ ρ) with a copy of ρ, this allows us to employ an operational and direct way to experimentally verify the uncertainty conservation relation. Here the operator V2 is the SWAP operation, i.e., V2|ψ1ψ2〉 = |ψ2ψ1〉, that exchanges the states of two subsystems. By using one ancillary probe qubit to perform the interferometric measurement, we can directly obtain all the required information of the purities from the probe qubit, as shown in Fig. 2a.

Physical system. To verify the equation in the experiments, we used the sample named 1-bromo-2,4,5-trifluorobenzene as a five- qubit NMR quantum system which consists of two 1H spins and 19 Alice Bob three F spins, dissolved in the liquid-crystal N-(4-methoxybenzy- lidene)-4-butylaniline (MBBA). Spins H3 and H4 are labeled as the bipartite system ρAB, spins F1 and F2 as the copy system ρA′B′, and spin F5 as the probe qubit ρprobe. The effective Hamiltonian of the fi Fig. 1 a Sketch of the proposal. b Illustration of implications of the ve-qubit system in double rotating frame is theorem. Alice chooses to measure one of the three X5 X π ÀÁ σ σ σ i j k x, y and z on qubit A, and then informs Bob her choice and H ¼ πν σ þ J þ 2D σ σ ; (7) NMR i z 2 jk jk z z requests him to guess her measurement outcome. In order to guess i¼1 1j

Published in partnership with The University of New South Wales npj Quantum Information (2019) 39 H. Wang et al. 4

Fig. 2 a Quantum circuit of directly measuring the purity. b Molecular structure for NMR quantum register. c Experimental schematic for verifying the measurement uncertainty relation. MUB measurements are performed on subsystem A to get ρθB, and meanwhile the same process is applied on the mirror subsystem A′. Then controlled-SWAP gates are performed for the measurements of the linear entropies SL(θ|B) in LHS of Eq. (4). The purity information on the original state ρAB, i.e., the RHS of Eq. (4), are obtained without MUB measurement in the dashed ρ2 ρ2 ′ lines. The purities of the bipartite system AB: Trð ABÞ and Trð θBÞ, are obtained by two controlled-SWAP gates Cswap applying on subsystems AA ′ ′ ρ2 ρ2 and BB while only one Cswap is operated on subsystem BB (denoted by the solid line) for the purity of subsystem Trð BÞ and Trð BjθÞ, where ρB|θ = TrA(ρθB)

The molecular structure is shown in Fig. 2b, and the relevant sequence. However, by some calculations, these undesired parameters are shown in Method B. evolutions will lead to an error less than 2[1 − cos(2Δθ)] ≈ 0.065 on the purity measurements of the subsystem A or A′, Experimental procedure. Figure 2c shows the experimental mainly determined by the different evolutions on F1,

schematic for verifying the measurement uncertainty relations. It H3 due to the different effective coupling constants JF1;F5 þ Δθ π can be divided into four parts. 2DF1;F5 and JH3;F5 þ 2DH3;F5 . Here ¼ ½ðJF1;F5 þ 2DF1;F5 ÞÀ ðJ ; þ 2D ; ފt =2 with the duration t of pulsed 1. Preparing initial state. From thermal equilibrium state, we H3 F5 H3 F5 Gz Gz magnetic field gradient G . To perform projective measure- first initialized the system to a labeled pseudo-pure state z ments of M and M on specific spins, we first selectively (LPPS) ρ ¼ 1 I þ εσprobe ji0000 h0000j with 1 2 LPPS 32 32 z ABA0B0 rotate the spins with [π/2]− or [π/2] rotations, then 54 ε ≈ −5 y x selective-transition method, where 10 is the polar- F2;H4 performs the projective measurement of M0, e.g., Px ¼ ization and I32 is the 32-dimension identity matrix. There is F ;H ; ; F ;H ; π= 2 4 F2 H4 F2 H4 π= 2 4 F2 H4 56 no dynamical and measurement effect on the part of ½ 2ŠÀy À Pz and Py ¼½ 2Šx À Pz . identity density matrix; thus in the following, we conven- 3. Measuring the purities. After the MUBs on the subsystem A tionally denote the state with the deviation density matrix,55 and A′, performing the quantum circuit in Fig. 2a will give ρ ignoring the identity matrix. Then the product state ρAB(α, x) the purity information on the resulting state θB after MUBs. ⊗ ρA′B′(α, x) was prepared from |0000〉ABA′B′ 〈0000|, where ρA′ For example, when two controlled-SWAP gates (Cswap) are ′ ′ B′ is the copy of ρAB. We vary the weight x by rotation with applied to both subsystems AA and BB , one gets the purity ρ2 different angles and a following nonunitary gradient pulse. of the bipartite system AB:Trð θBÞ; when only one Cswap is The details of initialization process are shown in Method C. applied to the subsystem BB′, one gets the purity of 2. Performing MUB measurements. The complete MUB mea- ρ2 ρ = ρ subsystem B:Trð BjθÞ with B|θ TrA( θB), as shown in Fig. 2c. surements were implemented on subsystem A. For a two- Likely, the related purities of the original state ρAB: Tr(ρAB) dimensional system, the simplest case of MUBs are ρ2 no and Trð BÞ, are obtained by the similar procedure without 0 1 0 1 M ¼fj0i; j1ig; M ¼ j iþjpffiffi i ; j iÀjpffiffi i ; the MUBs. It can be noted the initial state of the probe qubit 0 1 2 2 no (8) is different from the original method in Fig. 2a. We initialize j0iþffiffiij1i j0iÀffiffiij1i σprobe M2 ¼ p ; p : the probe qubit as z in our experiment. However, this 2 2 will not affect the measure of the purity by the quantum σ σ They are just the eigenvectors of Pauli operators x, y and circuit Uqc in Fig. 2a, i.e., σz. In NMR, such MUB projective measurements can be fi 56 σprobe σprobe ρ ρ y ρ ρ ρ2 : emulated using pulsed magnetic eld gradients. Without Tr½ z Uqcð z ÞUqcŠ¼2Tr½V2ð ފ ¼ 2Trð Þ ′ interfering the unselected systems (B and its copy B ), we σ fi realized the MUB measurements on subsystem A and its Moreover, since the direct observable in NMR is x, the nal copy A′ by the gradient echo technology,57 i.e., by selective Hadamard gate can be canceled out by the readout operation. π pulses to the other spins, this dephasing operation (i.e., Therefore, through integrating the NMR spectra of the probe spin F5, we directly measure the purities on the related states. By projective measurement of σz) can be selectively performed on some specific spins. It is in principle necessary to refocus calculating the linear entropy, both sides of Eq. (4) are obtained all the evolutions under the internal Hamiltonian during the without quantum state tomography. gradient echo. However, this sequence can be simplified when we only care about the purity of the crashed state. Experimental results. The experimental results are shown in Fig. 3. For example, the projective measurement of M on spin As expected, the sum of uncertainties decreases to zero when the 0 bipartite system is in maximally entangled state. With certain α, F2 (A) and H4 (A′) are accomplished by ; F ;H ;F F ;H ;F lower purity corresponds to higher uncertainty. From Fig. 3a, b, we F2 H4 π 1 3 5 π 1 3 5 Pz ¼ Gz À½ Šx À Gz À½ ŠÀx , where the evolu- can see that the experimental results are in accord with theoretical tions under internal effective coupling Hamiltonian related expectations and the uncertainty conversation relation holds with to three qubits F1,H3 and F5 are reserved during the pulse high precision. Figure 3c shows the final NMR spectra of the probe

npj Quantum Information (2019) 39 Published in partnership with The University of New South Wales H. Wang et al. 5

Fig. 3 Experimental results of verifying the measurement uncertainty with the input states ρAB(α,1)(a) and ρAB(π/2, x)(b). Red and blue dots represent the measured values of LHS and RHS of Eq. (4), respectively. The experimental data are rescaled by extracting the decoherence effect, shown by filled squares. The original data are shown in Method E. The dark gray curves are theoretical expectations. The bars are plotted from the infidelity of readout process. c Experimental NMR spectra of probe spin (F5)forρAB(π/4, 1). From bottom to up, the related resulting states after measurement are, respectively, ρAB, ρxB, ρyB, ρzB, ρB, ρB|x, ρB|y, ρB|z. d Measured purities of the resulting states shown in (c). Each purity is obtained from the sum of the integral values of the corresponding peak in (c) qubit for one certain initial state ρAB(π/4, 1). The sum of the of conservation of measurement uncertainties, in the sense that integral values of all the peaks is read as the purity of the related (4) is invariant under the transformation of MUBs. state in our experiment, as shown in Fig. 3d. Our theorem gives an uncertainty equality relation for arbitrary In our experiments, to avoid the error accumulation and dimensional bipartite systems consisting of a quantum system alleviate the influence of the decoherence, we used high-fidelity and a coupled quantum memory. It should be emphasized that, engineered quantum control pulses, which exploit the gradient even for single-partite systems, it is already quite difficult to obtain ascent pulse engineering (GRAPE) algorithm,58 to implement the an uncertainty equality relation for high-dimensional case. The quantum circuit in the experiments. The experiments for pure high-dimensional bipartite case is much more complex than the states contain eight GRAPE pulses with the total duration of about case of single-partite one.26,29–34,37–42 Therefore, as one sees in 74 ms, while for mixed states, we used 9~11 GRAPE pulses with Method A, it is not surprising that the derivation of our uncertainty total durations of 67−85 ms. We numerically optimized all GRAPE relations needs subtle mathematical techniques. pulses with considering 5% ratio frequency (rf) field inhomogene- With the help of one mirror system of the measured system and ity, so that they are more robust in experiments. All GRAPE pulses one additional probe qubit, we have provided the first experi- used in the experiments have theoretical fidelities above 99.3%. mental verification of this measurement uncertainty relation in an Numerical simulations show that the imperfection of GRAPE NMR quantum processor, where the experimental data of pulses causes infidelity of 2−4% in the final states. Due to the uncertainty quantities have been directly obtained by measuring short relaxation times of the liquid-crystal sample, the experi- the involved entropies without quantum state tomography. This ments suffer severe decoherence effect. We numerically simulated method allows one to perform verification experiments in large the dynamical process and estimated the attenuation factors quantum systems, and deal with the experimental data by caused by decoherence effect in the experiments.59,60 Then we standard statistical and information-theoretical methods. These rescaled the experimental results. The details can be also found in results may give rise to significant applications in quantum Method E. In the numerical simulations, we found that transverse information processing such as quantum metrology. For closed relaxation time T2 plays a leading role in the decoherence process, systems, it is well known that uncertainty relation determines the while the longitude relaxation time T1 has little influence. The precision limit of quantum metrology based on complementary imperfection in preparing the labeled PPS also causes some errors. basis. For open systems, with B the environment and A the system The highest unexpected peak in the labeled PPS NMR spectrum of to be measured, our equality presents a complete characterization spin F5 is about 3% intensity of the only expected peak. of the precision limit for measuring the system under MUBs. Such precision limit or accuracy determined by uncertainty relations also appear in quantum computing when quantum gates like DISCUSSION CNOT are physically implemented. Hence our results may high- In conclusion, we have derived a novel entropic measurement light further studies on both fundamental problems in quantum uncertainty relation in bipartite systems with a quantum memory. mechanics and the applications. It has been shown that after a complete set of MUB measurements on one partite, the total uncertainty on the other partite is exactly given by the purities of the initial system and the memory. METHODS Substantially different from the previous uncertainty relations with A: Proof of the theorem inequalities, we presented an equality of uncertainty relation for In a d-dimensional Hilbert space H, let {|iθ〉|i = 1,⋯, d} denote a basis θ the case with a quantum memory, which implies direct applica- labeled by . A set of M such bases is called mutually unbiased if tions to quantum random number generation and quantum 2 1 jhijiθjjτ ¼ (9) guessing games. Moreover, the relation (4) is independent of the d choices of the MUBs. Therefore, the relation (4) gives rise to a kind for any i, j = 1, 2, ⋯, d, and θ, τ = 1, ⋯, M with θ ≠ τ. When d is a power of a

Published in partnership with The University of New South Wales npj Quantum Information (2019) 39 H. Wang et al. 6 , a complete set of d + 1 MUBs exists. When d is an arbitrary written via a partial trace over system C P ÈÉÂà integer number, the maximal number of MUBs is unknown. For example, ρ ρ when d = 6, only three MUBs have yet been found. However, for any d ≥ 2, Fð CBÞ¼ TrC jiiθ Ahjθj jijθ Chiθj CB ij there exist at least three MUBs. For the purpose of this paper, we only P ρ (18) assume that M MUBs are available in a d-dimensional Hilbert space, where ¼ jiiθ Ahjθj hiθj CBjijθ ij M is less than or equal to the maximal number of MUBs that can exist. ¼ ρ In a composite Hilbert space H H of two qubits, we construct the AB following M(d − 1) + 1 states, for any θ ∈ {1, ⋯, M}. Similarly, ρθB can be written as ()"# Xd 1 Xd Φ ffiffiffi Ã; ρ θ θ θ θ ρ : j i¼p ji1iA jii1 B (10) θB ¼ TrC jii Ahi j jii Chi j CB (19) d i¼1 i¼1 Hence 1 Xd () ϕ ffiffiffi ωkðiÀ1Þ Ã j θ;k i¼p jiθiA jiiθ B (11) PM PM P d 1 ρ ρ 1 ρ i¼1 ðd AB À θBÞ¼ TrC d jiiθ Ahjθj jijθ Chiθj CB θ θ 2πi/d * ¼1 ¼1 ij with ω = e , k = 1, …, d − 1 and θ = 1, 2, …, M. Here |iθ〉 denotes the  PM P complex conjugate of |iθ〉 with respect to the computational basis (which Tr iθ iθ iθ iθ ρ (20) fi 〉 À C jiAh j jiCh j CB can be chosen as the rst basis {|i1 } without loss of generality). It is θ¼1 i − + fi () straightforward to show that the M(d 1) 1 bipartite states de ned in PM dPÀ1 TC Eqs. (10) and (11) are normalized and orthogonal to each other. ϕ ϕ ρ : ¼ÀTrC θ;k ACh θ;k j CB Therefore, these M(d − 1) + 1 states can be used for constructing a basis θ¼1 k¼1 in the composite Hilbert space H H. Since there are at most d2 2 − + ≤ 2 We have used Eq. (14) to obtain the last equality. From Eqs. (13) and (18) orthogonal states in a d -dimensional space, one has M(d 1) 1 d , θ = ≤ + + with 1, we also have which implies M d 1, i.e., there are at most d 1 MUBs for a d- no dimensional Hilbert space. When M = d + 1, i.e., a complete set of d + 1 1 ρ ¼ Tr ðjiΦ hΦjÞTC ρ : (21) MUBs in a d-dimensional space is available, the states defined in Eqs. (10) d AB C AC CB and (11) constitute a complete basis for the composite Hilbert space From Eqs. (20) and (21) and the obvious relation I ⊗ ρ = Tr [(I ⊗ I ) . On the other hand, when M < d + 1, one can complete a basis of A B C a C H H ρ ], we can rewrite Γ as the composite Hilbert space by adding p = (d − 1)(d + 1 − M) additional CB AB orthonormal states {|ϕα〉 |α = 1, ⋯, p}. The projector onto the subspace PM Γ I ρ 1 ρ 1 ρ ρ spanned by these additional states is denoted by P, i.e., AB ¼ A B À d AB þ ðd AB À θBÞ ÈÉθ¼1 (22) XM XdÀ1 Xp ¼ Tr PTC ρ : Φ Φ ϕ ϕ φ φ : C AC CB P¼I I À jihjÀ θ;k h θ;k j¼ jihα αj (12) θ¼1 k¼1 α¼1 Here the operator PAC is the projector defined on HA HC according to (12), i.e., It is obvious that P¼0 when M = d + 1. Let T denote the partial transpose with respect to the computational PM dPÀ1 2 Φ Φ ϕ ϕ basis of the second Hilbert space. One immediately has PAC ¼ I I À jihjÀ θ;k h θ;kj θ ¼1 k¼1 (23) 1 Xd Pp T2 φ φ : ðjihΦ ΦjÞ ¼ jihi1 j1j jihj1 i1j: (13) ¼ jiα ACh αj d α 1 i;j¼1 ¼ P = + = ϕ 〉 dÀ1 ωkðiÀjÞ − ≠ − = fi When M d 1, then p 0, the states | α AC in HA HC do not exist, As k¼1 equals to 1 for i j andd 1 when i j, it is not dif cult both PAC and ΓAB vanish. When M ≤ d, we have to show  p ÀÁ ! P T dÀ1 T2 d d Γ φ φ C ρ X X X AB ¼ TrC jiα ACh αj CB 1 α ϕθ; hϕθ; j ¼ jihiθ iθj jihiθ iθjÀ jihiθ jθj jihjθ iθj: (14) ¼1 k k d no k 1 i¼1 i;j¼1 Pp ÀÁ ¼ φ φ TC ρ ¼ TrC jiα ACh αj CB α Suppose ρAB is a bipartite state on the composite Hilbert space HA HB ¼1 p noÀÁÀÁ of dimension d × D, and suppose {|iθ〉 } is the θth MUB in the d-dimensional P A φ TC ρ φ TC : Hilbert space , after system A is projected onto the θth MUB the overall ¼ TrC jiα AC CB h αjAC HA α¼1 bipartite state is written as The last equality is due to the fact that operators on HC ÀÁcan have cyclicffiffiffiffiffiffiffi Xd Θ φ TC pρ permutations under the partial trace over C. Let α  jiα AC CB, ρθ ¼ jiiθ hiθj A hiθjρ jiiθ : (15) B A AB A which are operators on HA HC HB. Then we have i¼1 Xp no Given a bipartite state ρ and a set of M MUBs in H ,wedefine an y AB A ΓAB ¼ TrC ΘαðΘαÞ : (24) operator α¼1 XM y M À 1 Since ΘαðΘαÞ are always nonnegative-denofinite operators on Γ  I ρ þ ρ À ρθ (16) AB A B d AB B Θ Θ y fi θ¼1 HA HC HB, the operators TrC αð αÞ are nonnegative-de nite operators on H H , so is their sum. Therefore Γ ≥ 0. This completes on HA HB. This operator is Hermitian, and it has the following nice A B AB property. the proof of the proposition. Γ fi Proposition When M = d + 1, the operator Γ vanishes: Γ = 0. When According to the proposition, AB is a nonnegative-de nite operator AB AB ≤ = + M ≤ d, it is nonnegative-definite when M d, and it vanishes when M d 1. Hence, for any nonnegative- definite operator ΠAB on HA HB, ΓAB  0: (17) TrABðΓABΠABÞ0 (25) In order to prove the proposition, we introduce an additional Hilbert ≤ = + space H of dimension d, and introduce a linear map F that maps when M d, and the inequality becomes an equality when M d 1. C Π = ρ = Let AB AB, Eq. (25) yields operators on HC to operators on HA, such that Fðjii1 Chj1jÞ ¼ jii1 Ahj1j (i, j 1,⋯, d). One can easily show that iθ jθ iθ jθ for θ = 1, …, M. Let Γ ρ FðjiCh jÞ ¼ jiAh j 0  TrABð AB ABÞ F À1 denote the inverse map, and let ρ FÀ1ðρ Þ denote the CB AB PM (26) corresponding state on with respect to the state ρ on ρ2 MÀ1 ρ2 ρ2 : HC HB AB ¼ Trð BÞþ d Trð ABÞÀ Trð θBÞ : ρ ρ θ HA HB. Therefore, the map F CB !Fð CBÞ can also be conveniently ¼1

npj Quantum Information (2019) 39 Published in partnership with The University of New South Wales H. Wang et al. 7 Therefore, coherence terms that were eliminated by a following field gradient pulse ρ XM Gz. Accordingly, the LPPS LPPS was prepared. 2 2 M À 1 2 In the following we only consider the deviation part À Trðρθ ÞÀTrðρ ÞÀ Trðρ Þ: (27) B B d AB σprobe θ¼1 z ji0000 ABA0B0h0000j. The composite system was prepared into the ρ2 state Adding MTrð BÞ to the above inequality, we immediately have  XM 2 2 2 1 2 σprobe ρ α; ρ α; σprobe ψ ψ 1Àx ðTrðρ ÞÀTrðρ ÞÞðM À 1Þ Trðρ ÞÀ Trðρ Þ : (28) z ABð xÞ A0B0ð xÞ¼ z ðxj αih αjþ 4 I4Þ B θB B d AB θ¼1 ψ ψ 1Àx ; ðxj αih αjþ 4 I4Þ The inequality becomes an equality when M = d + 1. Thus, the theorem in the main text has been proved. probe from σ j0000ih0000j, where |ψα〉 = cos(α/2)|01〉 − sin(α/2)|10〉. In the Remark: The approach we admitted here is methodologically similar to z case of x = 1, the bipartite systems are entangled where the rotation angle the one used in ref. 26 (see also ref. 29). In ref. 45 an elegant uncertainty α varies from 0 to π/2 for producing different entanglement. α = 0 equality has also been presented based on ONE positive operator-valued α = π measure consisting uniformly all the measurement operators of d + 1 corresponds to separable state and /2 corresponds to the maximally α = π MUBs. As we take into account all the d + 1 MUB projection measurements entangled state. Setting /2 and scanning x from 0 to 1 with interval individually, our uncertainty relation is completely different from the one 0.25, we prepared the states with different mixedness. In preparing given in ref. 45 In fact, the uncertainty relation in ref. 45 cannot be applied to our QRNG self-testing scenario, because the equality derived in ref. 45 holds only when the system is measured in one of the MUBs with uniformly random probability. While our equality holds no matter whether the MUB is uniformly chosen. Such a property is crucial in practical QRNG application, as one needs to restrict the input randomness to choose a (a) mutually unbiased basis. -37477 1915.7 63.5 342.5 144.4 240ms 0.8s B: Quantum register 2492 -12.8 1556.6 64.5 310ms 1.5s We used the sample named 1-bromo-2,4,5-trifluorobenzene as a five-qubit 2479 129.2 1887.4 313ms 1.5s 1 19 NMR quantum system which consists of two H spins and three F spins, -44888 -59.4 174ms 0.6s dissolved in the liquid-crystal N-(4-methoxybenzylidene)-4-butylaniline. Unit: Hz The structure of the molecule is shown in Fig. 4. Due to the partial average -47340 180ms 0.8s effect in the liquid-crystal solution, the direct dipole−dipole interaction will be scaled down by the order parameter.61 In our sample, the partial (b) 19F average effect makes the direct dipole−dipole couplings of homonuclear spectrum F5 F F spins much smaller than the difference between the chemical shifts of 2 1 related nuclear spins, all the dipole−dipole Hamiltonians are reduced to σi σj fi the form of z z. Therefore, the effective Hamiltonian of the ve-qubit system in rotating frame is X5 X π ÀÁ

NMR signal (arb. Units) NMR signal −100 −110 −120 H ¼ πν σi þ J þ 2D σj σk ; (29) [ppm] NMR i z 2 jk jk z z i¼1 1j

C: Initial state preparation We first initialized the quantum register into a labeled pseudo-pure state 10 8 6 4 [ppm] ρ εσprobe = (arb. Units) NMR signal (LPPS) LPPS ¼ z ji0000 ABA0B0h0000jþ1 32I32 from the equilibrium state where ε is the polarization about 10−5. We just neglected the identity part because unitary and nonunitary operations all have no influence on it. β Fig. 4 a Characteristics of the 1-bromo-2,4,5-trifluorobenzene We applied 30 unitary operators with the form U eÀi ij Vij to redistribute ¼ molecule. The chemical shifts and effective coupling constants (in the populations between energy levels i and j. Here is the single Vij Hz) are on and above the diagonal in the table, respectively. The last quantum transition operator between levels i and j, that is, a 32 × 32 matrix two columns show the transversal relaxation times and the whose elements are zero except for two elements i; j j; i 1=2. Vijð Þ¼Vijð Þ¼ longitudinal relaxation times of each nucleus. b, c The thermal According to the desired distribution, the angle β were numerically ij equilibrium 19F and 1H spectra calculated. In this produce, the unitary operators generated undesired

σprobe ρ α; ρ α; α Fig. 5 a Quantum circuit of preparing z ABð 1Þ A0B0ð 1Þ from LPPS. The rectangles represent single-qubit rotations with angles , π π fi σprobe ρ π= ; ρ π= ; /2 and along y-axis. Gz is the eld gradient pulse along z-axis. b Pulse sequences for preparing z ABð 2 xÞ A0B0ð 2 xÞ with the 2 2 temporal averaging method. The rotation angles δ1,2,3,4 are arccos[(1 − x) ], arccos[x(1 − x)], arccos[x(1 − x)] and arccos(x ), which are used to adjust the coefficients of four parts in Eq. (30)

Published in partnership with The University of New South Wales npj Quantum Information (2019) 39 H. Wang et al. 8

Fig. 6 Experimental pulse sequence. The LPPS preparation contains one GRAPE pulse and one z-axis gradient pulse. The GRAPE pulse is used to redistribute the populations and the field gradient pulse is used to eliminate the undesired coherences. The operation V(α, x) is to prepare σprobe ρ α; ρ α; fi π − the initial state z ABð xÞ A0B0ð xÞ. Its speci c form is shown in Fig. 5. The solid blue rectangle is /2 rotation along ( y)-axis in x- measurement and along x-axis in y-measurement. The dashed blue rectangles denote π/2 rotations with the opposite phases to those in solid blue rectangles. No need to operate them in z-measurement. The readout part contains two Cswap gates applying on subsystems AA′ and BB′. The dashed Cswap(AA′) is not required in reading out the purity of subsystem B

Fig. 7 a The original results without rescaling of verifying the measurement uncertainty relation. The bars are plotted from the infidelity of readout process. b The directly measured purities. The hollow dots are the expected purities. The blue asterisks are the directly measured purities and the red ones are the rescaled results. The left five figures correspond to purities when the initial bipartite state is ρAB(α, 1) where α = 0, π/8, π/4, 3π/8, π/2 from top to bottom. And the right five figures correspond to ρAB(π/2, x) where x = 0, 0.25, 0.5, 0.75, 1 from top to fi ρ2 ρ2 ρ2 ρ2 ρ2 ρ 2 ρ 2 ρ 2 ρ θ = bottom. In each gure, from left to right, they are Trð ABÞ, Trð xBÞ, Trð yBÞ, Trð zBÞ, Trð BÞ, Tr½ð BjxÞ Š, Tr½ð BjyÞ Š, Tr½ð BjzÞ Š, where θB ( x, y, z) are the resulting state after the complete MUB measurements on subsystem A, and ρB|θ = TrA(ρθB)

σprobe ρ π= ; ρ π= ; σprobe ρ π= ; ρ π= ; z ABð 2 xÞ A0B0ð 2 xÞ, we divided it into 4 parts, i.e., z ABð 2 xÞ A0B0ð 2 xÞ. The pulse sequences for the state preparation are shown in Fig. 5a, b. σprobe 1Àx 1Àx ; z ð 4 I2Þ ð 4 I2Þ σprobe 1Àx ψ ψ ; z ð I2Þ ðxj π=2ih π=2jÞ 4 (30) σprobe ψ ψ 1Àx ; D: Experimental procedure z ðxj π=2ih π=2jÞ ð 4 I2Þ probe The total experimental procedure is shown in Fig. 6, including the LPPS σ ðxjψπ= ihψπ= jÞ ðxjψπ= ihψπ= jÞ; z 2 2 2 2 preparation, the initial state preparation, MUB measurements and the and adopted the temporal averaging method62 to realize them. By readout of the purities. When the measurement is along y-axis, the first and summing the four output states, we obtained the desired state last rotations in the MUB measurement part should be substituted by

npj Quantum Information (2019) 39 Published in partnership with The University of New South Wales H. Wang et al. 9 rotations along x and −x, and the angles remain also π/2. In the z-axis REFERENCES measurement case, these two rotations are omitted. 1. Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kine- matik und Mechanik. Z. Phys. 43, 172–198 (1927). E: The attenuation factors caused by decoherence effect and 2. Robertson, H. P. The uncertainty principle. Phys. Rev. 34, 163 (1929). experimental measured purities 3. Schrödinger, E. About Heisenberg uncertainty relation. Sitz. Preuss. Akad. Wiss. Phys. Math. Kl. 19, 296 (1930). fi Even through the high- delity GRAPE pulses in experiments, the 4. Maassen, H. & Uffink, J. B. M. Generalized entropic uncertainty relations. Phys. Rev. experimental results are still severely affected due to the decoherence Lett. 60, 1103 (1988). due to the long running times comparing to the T2 relaxation times of 5. Ozawa, M. Uncertainty relations for noise and disturbance in generalized quan- nuclear spins. tum measurements. Ann. Phys. 311, 350 (2004). In our experiments, the environmental noises are suitably modeled as 6. Wu, S., Yu, S. & Mølmer, K. Entropic uncertainty relation for mutually unbiased Markovian, and the evolution of the system is given by the Lindblad bases. Phys. Rev. A 79, 022104 (2009). master equation 7. Branciard, C. Error-tradeoff and error-disturbance relations for incompatible X ÀÁ quantum measurements. Proc. Natl Acad. Sci. USA 110, 6742 (2013). ρ_ ; ρ ρ y y ρ ρ y ; ¼Ài½HS þ HCðtÞ Šþ 2Lα Lα À LαLα À LαLα (31) 8. Friedland, S., Gheorghiu, V. & Gour, G. Universal uncertainty relations. Phys. Rev. α Lett. 111, 230401 (2013). ł Ł Ż where HS is the system Hamiltonian, HC(t) is the time-dependent external 9. Pucha a, Z., Rudnicki, .& yczkowski, K. Majorization entropic uncertainty rela- control Hamiltonian, and the operators Lα are Lindblad operators tions. J. Phys. A: Math. Theor. 46, 272002 (2013). representing the coupling with the environment. The experimentsqffiffiffiffi are 10. Vallone, G., Marangon, D., Tomasin, M. & Villoresi, P. Quantum randomness cer- γ tified by the uncertainty principle. Phys. Rev. A 90, 052327 (2014). ασα; α mainly affected by the dephasing process, and therefore Lα ¼ 2 z ð ¼ 11. Maccone, L. & Pati, A. K. Stronger uncertainty relations for all incompatible 1; :::; 5Þ and γα = 1/T2α and observables. Phys. Rev. Lett. 113, 260401 (2014). X ÀÁ 12. Busch, P., Lahti, P. & Werner, R. F. Quantum root-mean-square error and mea- ρ_ ; ρ γ σαρσα ρ : ¼Ài½HS þ HCðtÞ Šþ α z z À (32) surement uncertainty relations. Rev. Mod. Phys. 86, 1261 (2014). α 13. Kaniewski, J., Tomamichel, M. & Wehner, S. Entropic uncertainty from effective In the experiments, the high-fidelity GRAPE pulses we used are kinds of anticommutators. Phys. Rev. A 90, 012332 (2014). 14. Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entangle- shaped pulses consisting of thousands of slices with a constant HC(ti) the duration of each slice, where δt is 2−25 μs. Consequently, for each slice, ment. Rev. Mod. Phys. 81, 865 (2009). the state of the system can be approximately given by 15. Berta, M., Christandl, M., Colbeck, R., Renes, J. M. & Renner, R. The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659 (2010). X5 X1 16. Tomamichel, M. & Renner, R. The uncertainty relation for smooth entropies. Phys. α δ δ α ρ ρ δ Ài½HSþHCðtiþ1 ފ tρ i½HSþHCðtiþ1ފ t y; ðtiþ1Þ¼ ðti þ tÞ¼ Ei e ðtiÞe Ei (33) Rev. Lett. 106, 110506 (2011). α¼1 i¼0 17. Tomamichel, M., Lim, C. C. W., Gisin, N. & Renner, R. Tight finite-key analysis for . Nat. Commun. 3, 634 (2012). where the Kraus operations pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 18. Gehring, T. et al. Implementation of continuous-variable quantum key distribu- α α α tion with composable and one-sided-device-independent security against E ¼ λα1; E ¼ 1 À λασ (34) 0 1 z coherent attacks. Nat. Commun. 6, 8795 (2015).

Àγαδt 19. Bradler, K., Mirhosseini, M., Fickler, R., Broadbent, A. & Boyd, R. Finite-key security with λα ¼ð1 þ e Þ=2. Therefore, we numerically simulated the experiments, extracted the analysis for multilevel . New J. Phys. 18, 073030 (2016). attenuation factors caused by decoherence effect and then rescaled the 20. Wootters, W. K. & Fields, B. D. Optimal state-determination by mutually unbiased experiment results for verifying the uncertainty conversation relation in measurements. Ann. Phys. (NY) 191, 363 (1989). Fig. 3a, b. The un-rescaled original results are shown in Fig. 7a. From this 21. Klappenecker, A. & Rötteler, M. Mutually unbiased bases are complex projective we can see that even though the right and left sides of the uncertainty 2-designs. In Proc. 2005 IEEE International Symposium on Information Theory (ISIT – equation are all reduced, they almost equal each other. Figure 7b shows 2005) 1740 1744. (IEEE Press, New York, 2005). Ż the directly measured purities of different states required in the 22. Durt, T., Englert, B. G., Bengtsson, I. & yczkowski, K. On mutually unbiased bases. – uncertainty conversation relation. Int. J. Quant. Inf. 8, 535 640 (2010). 23. Kalev, A. & Gour, G. Mutually unbiased measurements in finite dimensions. New J. Phys. 16, 053038 (2014). DATA AVAILABILITY 24. Zhu, H. Mutually unbiased bases as minimal Clifford covariant 2-designs. Phys. Rev. A 91, 060301 (2015). fi The main data supporting the nding of this study are available within the article. 25. Englert, B. G. Fringe visibility and which-way information: an inequality. Phys. Rev. Additional data can be provided upon request. Lett. 77, 2154 (1996). 26. Brukner, Č. & Zeilinger, A. Operationally Invariant Information in quantum mea- surements. Phys. Rev. Lett. 83, 3354 (1999). ACKNOWLEDGEMENTS 27. Spengler, C., Huber, M., Brierley, S., Adaktylos, T. & Hiesmayr, B. C. Entanglement We thank Marcin Pawłowski, Simone Severini and Dawei Lu for helpfull discussions. detection via mutually unbiased bases. Phys. Rev. A 86, 022311 (2012). This work was supported by the National Key R&D Program of China (Grant No. 28. Lu, D. et al. Tomography is necessary for universal entanglement detection with 2018YFA0306600, 2016YFA0301801, 2017YFA0303703), National Natural Science single-copy observables. Phys. Rev. Lett. 116, 230501 (2016). Foundation of China (Grants No. 11425523, 11661161018, 11675113, 11605153, 29. Brukner, Č. & Zeilinger, A. Conceptual inadequacy of the Shannon information in 11571313, 11575173), Natural Science Foundation of Zhejiang province (Grants No. quantum measurements. Phys. Rev. A 63, 022113 (2001). LQ19A050001), Anhui Initiative in Quantum Information Technologies (Grant No. 30. Zeilinger, A. A foundational principle for quantum mechanics. Found. Phys. 29, AHY050000), and the Key Project of Beijing Municipal Commission of Education 631 (1999). under No. KZ201810028042. 31. Řeháček, J. & Hradil, Z. Invariant information and quantum state estimation. Phys. Rev. Lett. 88, 130401 (2002). 32. Lee, J., Kim, M. S. & Brukner, Č. Operationally invariant measure of the distance AUTHOR CONTRIBUTIONS between quantum states by complementary measurements. Phys. Rev. Lett. 91, 087902 (2003). All authors researched, collated, and wrote this paper. 33. Kofler, J. & Zeilinger, A. Quantum information and randomness. Eur. Rev. 18, 469 (2010). 34. Madhok, V., Riofro, C. A., Ghose, S. & Deutsch, I. H. Information gain in tomo- ADDITIONAL INFORMATION graphy—a quantum signature of chaos. Phys. Rev. Lett. 112, 014102 (2014). Competing interests: The authors declare no competing interests. 35. Lee, J. & Kim, M. S. Entanglement teleportation via Werner states. Phys. Rev. Lett. 84, 4236 (2000). Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims 36. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum-enhanced measurements: in published maps and institutional affiliations. beating the standard quantum limit. Science 306, 1330–1336 (2004).

Published in partnership with The University of New South Wales npj Quantum Information (2019) 39 H. Wang et al. 10 37. Brukner, Č. & Zeilinger, A. Young’s experiment and the finiteness of information. 53. Ekert, A. K., Alves, C. M. & Oi, D. K. L. Direct estimations of linear and nonlinear Philos. Trans. R. Soc. Lond. A 360, 1061 (2002). functionals of a quantum state. Phys. Rev. Lett. 88, 21 (2002). 38. Luis, A. Complementarity and duality relations for finite-dimensional systems. 54. Peng, X. et al. Preparation of pseudo-pure states by line-selective pulses in Phys. Rev. A 67, 032108 (2003). nuclear magnetic resonance. Chem. Phys. Lett. 340, 509–516 (2001). 39. Song, W. & Chen, Z.-B. Invariant information and complementarity in high- 55. Knill, E., Laflamme, R., Martinez, R. & Tseng, C.-H. An algorithmic benchmark for dimensional states. Phys. Rev. A 76, 014307 (2007). quantum information processing. Nature 404, 368–370 (2000). 40. Brukner, Č. & Zeilinger, A. Information invariance and quantum probabilities. 56. Teklemariam, G., Fortunato, E. M., Pravia, M. A., Havel, T. F. & Cory, D. G. NMR Found. Phys. 39, 631 (2009). analog of the quantum disentanglement eraser. Phys. Rev. Lett. 86, 5845 (2001). 41. Rastegin, A. E. On the Brukner−Zeilinger approach to information in quantum 57. Levitt, M. Spin Dynamics (Wiley: West Sussex, England, 2001). measurements. Proc. R. Soc. A 471, 20150435 (2015). 58. Khaneja, N., Reiss, T., Kehlet, C., Schulte-Herbruggen, T. & Glaser, S. J. Optimal 42. Khrennikov, A. Reflections on Zeilinger−Brukner information interpretation of control of coupled spin dynamics: design of NMR pulse sequences by gradient quantum mechanics. Found. Phys. 46, 836 (2016). ascent algorithms. J. Magn. Reson. 172, 296 (2005). 43. Pechen, A., Il’in, N., Shuang, F. & Rabitz, H. Quantum control by von Neumann 59. Vandersypen, L. M. K. et al. Experimental realization of Shor’s quantum factoring measurements. Phys. Rev. A 74, 052102 (2006). algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001). 44. Kalev, A., Mann, A. & Revzen, M. Choice of measurement as the signal. Phys. Rev. 60. Vandersypen, L. M. K. Experimental quantum computation with nuclear spins in Lett. 110, 260502 (2013). liquid solution. Preprint at https://arxiv.org/abs/quant-ph/0205193 (2001). 45. Berta, M., Coles, P. J. & Wehner, S. Entanglement-assisted guessing of com- 61. Dong, R. Y. Nuclear Magnetic Resonance of Liquid Crystals (Springer, New York, plementary measurement outcomes. Phys. Rev. A 90, 062127 (2014). 1997). 46. Li, H.-W., Pawłowski, M., Yin, Z.-Q., Guo, G.-C. & Han, Z.-F. Semi-device- 62. Knill, E., Chuang, I. & Laflamme, R. Effective pure states for bulk quantum com- independent randomness certification using n → 1 quantum random access putation. Phys. Rev. A. 57, 3348 (1998). codes. Phys. Rev. A 85, 052308 (2012). 47. Cao, Z., Zhou, H. & Ma, X. Loss-tolerant measurement-device-independent quantum random number generation. New J. Phys. 17, 125011 (2015). Open Access This article is licensed under a Creative Commons 48. Cao, Z., Zhou, H., Yuan, X. & Ma, X. Source-independent quantum random Attribution 4.0 International License, which permits use, sharing, number generation. Phys. Rev. X 6, 011020 (2016). adaptation, distribution and reproduction in any medium or format, as long as you give 49. Pawłowski, M. & Żukowski, M. Entanglement-assisted random access codes. Phys. appropriate credit to the original author(s) and the source, provide a link to the Creative Rev. A 81, 042326 (2010). Commons license, and indicate if changes were made. The images or other third party 50. Li, C.-F., Xu, J.-S., Xu, X.-Y., Li, K. & Guo, G.-C. Experimental investigation of the material in this article are included in the article’s Creative Commons license, unless entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752–756 indicated otherwise in a credit line to the material. If material is not included in the (2011). article’s Creative Commons license and your intended use is not permitted by statutory 51. Prevedel, R., Hamel, D. R., Colbeck, R., Fisher, K. & Resch, K. J. Experimental regulation or exceeds the permitted use, you will need to obtain permission directly investigation of the uncertainty principle in the presence of quantum memory. from the copyright holder. To view a copy of this license, visit http://creativecommons. Nat. Phys. 7, 757–761 (2011). org/licenses/by/4.0/. 52. Paris, M. G. A. & Řeháček, J. (eds) Quantum State Estimation. Lecture Notes in Physics, Vol. 649 (Springer, New York, 2004). © The Author(s) 2019

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