Abstract
As a very fundamental principle in quantum physics, uncertainty principle 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 fivequbit 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.
Introduction
The uncertainty principle is one of the most important principle in quantum physics. It implies the impossibility of simultaneously determining the definite values of incompatible observables. The more precisely an observable is determined, the less precisely a complementary observable can be known. Based on the distributions of measurement outcomes, the quantum uncertainty relations can be described in various ways; see for instance refs. ^{1,2,3,4,5,6,7,8,9,10,11,12,13}
The uncertainty principle was first formulated via the standard deviation of a pair of complementary observables, known as the Heisenberg’s uncertainty principle^{1} ΔxΔp ≥ ħ/2 for the coordinate x and the momentum p in an infinite dimensional Hilbert space. Later the Robertson−Schrödinger uncertainty inequality^{2,3} presented an uncertainty relation for two arbitrary observables in a finite dimensional Hilbert space. Instead of the standard deviation of observables, the uncertainty principle can also be elegantly formulated in terms of entropies related to measurement bases. When a quantum system is projected onto a certain basis {i_{θ}〉 i = 1, 2, ⋯, d}, where θ labels the measured observable, the uncertainty of the measurement results has been characterized by the Shannon entropy \(H_\theta = \mathop {\sum}\nolimits_{i = 1}^d  p_i\log _2p_i\), where p_{i} is the probability to obtain the ith basis state i_{θ}〉. The larger the Shannon entropy H_{θ} is, the more uncertain the measurement results are. In terms of the Shannon entropies of the measurement results, the uncertainty principle can be formulated as \(H_\theta + H_\tau \ge \log _2\frac{1}{c}\).^{4} Here 1/c quantifies the degree of complementarity of two observables θ and τ.
The above uncertainty relations only concern a single quantum system. By taking the entanglement with a memory system into account,^{14} an entropic uncertainty relation in the presence of quantum memory has been investigated in ref. ^{15}. It has been shown that for a bipartite state ρ_{AB}, performing measurements on one of the subsystems A gives rise to the following relation
where S(AB) and S(θB) (S(τB)) denote, respectively, the conditional von Neumann entropies of the initial bipartite state ρ_{AB} and the final bipartite state ρ_{θB} (ρ_{τB}) after the measurement in the basis {i_{θ}〉} ({i_{τ}〉}). This uncertainty relation was further extended to the smooth entropy case.^{16} With considering the entangled quantum memory, these uncertainty relations have potential applications in entanglement witnessing and quantum key distributions (QKD).^{15,17,18} However, the above results all concern measurements only on two observables and are given in inequality forms.
In this work, we consider projective measurements based on mutually unbiased bases (MUBs).^{19,20,21,22,23,24} The MUB measurements are complementary to each other in the sense that any pair of bases are maximally unbiased. They are deeply connected to the Born’s principle of complementarity^{21} and closely related to the waveparticle duality.^{25,26} A complete set of MUBs consists of at most d + 1 observables, where d is the dimension of the state space. Comparing with the incomplete case, the advantage of a complete set of MUB measurements is informatively complete^{23,24} and meaningful in quantum information progressing.^{21} It is therefore not surprising that a complete set of MUB measurements is crucial in entanglement detection.^{27,28} It was also proved that using a complete set of MUBs is much better than using two observables in QKD.^{19}
In ref. ^{6} an entropic uncertainty relation involving d + 1 MUB measurements has been obtained in terms of von Neumann entropy. However, it only dealt with a single system (in this case the von Neumann entropy is just the Shannon entropy of the measurement probability distributions), and the uncertainty relation is given by an inequality. In fact, the Shannon entropy is a natural measure of our ignorance regarding the properties of a classical system, because in classical measurements the observation removes our ignorance about the state by revealing the properties of the system which are considered to be preexisting and independent of the observation. In contrast to classical measurements, one cannot say that quantum measurements reveal a preexisting property of a quantum system. Therefore, the Shannon entropy could be thought of as “conceptually” inadequate in quantum physics.^{29} In ref. ^{26} the authors proposed a new measure of quantum information, which takes into account that the only features of quantum systems known before a measurement are the probabilities for various events to occur. It has significant physical meaning and various applications in quantum information processing such as quantum randomness, quantum state estimation, quantum teleportation and quantum metrology.^{30,31,32,33,34,35,36} Moreover, a series of works have been shown, together with many applications, that in single quantum system, the sum of the individual measures of information for MUBs is invariant under the choice of the particular set of complementary observations and conserved if there is no information exchange with environments.^{26,29,30,31,32,33,34,37,38,39,40,41,42}
In this article, we adopt the information measure proposed in ref. ^{26} and consider the uncertainty relation in the presence of quantum memory. Interestingly, we find that if we take a complete set of MUB measurements into account, we can obtain an uncertainty equality that the sum of measurement uncertainties over all MUBs on a subsystem in the presence of quantum memory is equal to a fixed quantity determined by the initial state. It gives rise to a kind of conservation relation of the uncertainties related to these MUB measurements. We further show the elegant applications of our result in quantum guessing game and quantum random number generation. We also experimentally verify this uncertainty equality by directly measuring the uncertainties on a nuclear spin system. Our method avoids the tomography process and allows one to perform verification experiments in large quantum systems.
Results
Measurement of uncertainty relations
Let (p_{1}, p_{2}, …, p_{d}) be the probabilities for the d measurement outcomes. The lack of information about the jth outcome with respect to a single experimental trial is given by p_{j}(1 − p_{j}). The total lack of information regarding all d possible experimental outcomes is then given by \(\mathop {\sum}\nolimits_{j = 1}^d {p_j} (1  p_j) = 1  \mathop {\sum}\nolimits_{j = 1}^d {p_j^2}\), which is minimal if one probability is equal to a unity and maximal if all the probabilities are equal. In fact, \(1  \mathop {\sum}\nolimits_{j = 1}^d {p_j^2}\) is nothing but 1 − Tr(ρ^{2}), where ρ is the state after a quantum (projective) measurement, the linear entropy of the measured state. Therefore, the lack of information regarding all d possible experimental outcomes can be described by the linear entropy of a dlevel quantum state ρ, S_{L}(ρ) = 1 − Tr(ρ^{2}). S_{L}(ρ) ranges from 0 (when ρ is a pure state) to (d − 1)/d (when ρ is maximally mixed). Unlike that in ref. ^{26}, here we do not introduce a normalization factor to have a range between 0 and log_{2} d, so the measure of uncertainty in terms of linear entropy does not have the unit of a “bit”. However, it quantifies uncertainty in a natural way: an uncertainty of 0 means the outcome is 100% certain while an uncertainty approaching 1 means the outcome is almost random.
For a bipartite state ρ_{AB} in a d × D (D ≥ d) dimensional composite Hilbert space, if system A is projected on to the basis {i_{θ}〉 i = 1, 2, ⋯, d}, the overall state of the composite system after the nonselective measurement^{43,44} on A is given as
We can introduce the conditional linear entropy
as a measure of the uncertainty about Alice’s measurement result given Bob’s state, where the reduced state ρ_{B} = Tr_{A}(ρ_{θB}) = Tr_{A}(ρ_{AB}) is independent of the measurement basis. It is straightforward to show that the conditional linear entropy S_{L}(θB) is always nonnegative. As an example, suppose ρ_{AB} is a maximally entangled pure state. Alice can perform a measurement on her system in any basis. The possible resulting states of Bob’s system are orthogonal to each other, and each possible resulting state is in onetoone correspondence to Alice’s resulting state. Therefore, given Bob’s state, Alice’s measurement result can be determined with certainty. In this case S_{L}(θB) vanishes. If \(\rho _{AB} = \mathop {\sum}\nolimits_{i = 1}^d {\sqrt {\lambda _i} } \left {ii} \right\rangle\) is a partially entangled state written in its Schmidt bases, after Alice measures her system in the Schmidt basis, Bob’s possible resulting states are orthogonal to each other and the Alice’s measurement result is completely determined without uncertainty when Bob’s state is given. This is also confirmed by the vanishing conditional entropy as \(S_L(\rho _{\theta B}) = S_L(\rho _B) = \mathop {\sum}\nolimits_i {\lambda _i^2}\). However, if Alice performs a measurement on a basis that is not the Schmidt basis, the possible resulting states of Bob’s system are not orthogonal and cannot be distinguished with certainty, and thus uncertainty of Alice’s measurement result exists even when Bob’s state is known. This fact is again confirmed by the observation that the conditional linear entropy is strictly greater than zero in this case. The conditional linear entropy is thus a good measure of the uncertainty about Alice’s measurement result given Bob’s state. It depends on the basis in which the measurement is performed in general. When Alice tries to find a basis to perform the measurement on her system so that Bob will know her result with minimum uncertainty, then using another MUB to perform the measurement will result in Bob having a large uncertainty about Alice’s result. However, the whole uncertainty running over all possible MUB measurements is fixed. This uncertainty relation is formulated in the following theorem (the proof involves subtle mathematical techniques, see Method A).
Theorem For any density matrix ρ_{AB} on a composite Hilbert space H_{A} ⊗ H_{B} of dimension d × D, we have the following uncertainty equality
when a complete set of d + 1 MUBs exists for the ddimensional Hilbert space H_{A}.
The theorem shows that the total uncertainty related to the measurements over all d + 1 MUBs of a subsystem is exactly given by a fixed quantity, \(d{\mathrm{Tr}}(\rho _B^2)  {\mathrm{Tr}}(\rho _{AB}^2)\), which is determined only by the initial bipartite state. This quantity is always nonnegative and can be viewed as the total measurement uncertainty of a subsystem, given the state of the other subsystem. Different from the uncertainty inequality (1) based on the von Neumann entropy, here we obtain the equality (4). Note that this equality (4) is also completely different from the one given in ref. ^{45} which is based on only ONE positive operatorvalued measure consisting of uniformly all the measurement operators of d + 1 MUBs (see Remark in Method A). In general, when there are only M MUBs available or when we are only interested in certain M MUBs, we always have the following uncertainty inequality,
With each additional MUB, the lower bound of total uncertainty is increased by a fixed amount \({\mathrm{Tr}}(\rho _B^2)  \frac{1}{d}{\mathrm{Tr}}(\rho _{AB}^2)\).
To illustrate the implications of the theorem, let us consider that Alice and Bob are both users of quantum technology. In order to make a hard decision on whether she should accept Bob’s invitation to see a film, Alice asks Bob to send her a qubit A. Alice can measure the qubit with the three (Pauli) observables σ_{x}, σ_{y} and σ_{z} at her choice. After the measurement, Alice announces her choice of the observable, and Bob is supposed to guess the Alice’s measurement results. Alice would accept (deny) Bob’s request if 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 stage. From the theorem, we know that the sum of uncertainties (of Bob’s guess at Alice’s measurement results given the state of B) in three different cases is equal to the quantity \(Q = 2{\mathrm{Tr}}(\rho _B^2)  {\mathrm{Tr}}(\rho _{AB}^2)\) that is completely determined by the initial state ρ_{AB}. Bob can minimize the quantity Q by preparing an EPR state, thus win Alice’s acceptance with certainty, a result that cannot be obtained from an uncertainty inequality like the one based on Shannon entropy (see Fig. 1).
On the practical side, the theorem also provides possible applications in quantum random number generation, especially semiselftesting quantum random number generators (QRNGs) which are more robust to device imperfections. In a typical setup of a semiselftesting QRNG,^{46,47,48} Alice and Bob share a quantum state ρ_{AB}, e.g., an EPR pair. If both parties are trusted, the measurement outcome of one party will be random to the other party when Alice measures in the computational basis and Bob measures in the diagonal basis. However, if one of the parties is corrupted, e.g., due to device imperfections, this scheme is broken. To show this, consider that one party switches to the same basis as the other party. A common solution is that each party randomly uses multiple basis, such as σ_{x}, σ_{y}, or σ_{z} basis.^{46,47,48} Now, we consider the following semiselftesting scenario. Alice first chooses a reference frame, and randomly performs measurements 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 choose her measurement basis, i.e., the basis choice may be manipulated by an adversary who wishes to corrupt Alice’s randomness, such as Bob. Hence, the entropy of Alice’s random outcomes with respect to Bob is the smallest entropy S_{L}(θB) among all measurement choices. To maximize this quantity, the theorem shows that S_{L}(θB) should be equal for all θs. Thus, the maximum entropy of a semiselftesting QRNG is \([d{\mathrm{Tr}}(\rho _B^2)  {\mathrm{Tr}}(\rho _{AB}^2)]/(d + 1)\). This limit on semiselftesting QRNGs also cannot be obtained from an uncertainty inequality. Finally, note that entanglementbased QRNGs considered here have a higher randomness generation rate compared to prepareandmeasure QRNGs,^{49} and cannot be analyzed by using the tools developed by Brukner and Zeilinger.^{26}
Experimental verification
To experimentally investigate the uncertainty conservation, a twoqubit system ρ_{AB}, chosen as the test system, is prepared in the following states:
where ψ_{α}〉 = cos(α/2)01〉 − sin(α/2)10〉. These states are mixed states composed of one pure state with weight x and the maximal mixed state with weight (1 − x)/4. The parameters α characterizes the entanglement of the pure part and x characterizes the purity of the state. When α = π/2 and x = 1, the bipartite state is one of the Bell states. The other three Bell states can be obtained by local unitary operations while the linear entropy remains invariant under such transformations.
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 largesize quantum systems. Since the linear entropy is directly related to the purity Tr(ρ^{2}) that can be directly obtained by Tr(ρ^{2}) = Tr(V_{2}ρ ⊗ ρ)^{53} 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 V_{2} is the SWAP operation, i.e., V_{2}ψ_{1}ψ_{2}〉 = ψ_{2}ψ_{1}〉, 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 1bromo2,4,5trifluorobenzene as a fivequbit NMR quantum system which consists of two ^{1}H spins and three ^{19}F spins, dissolved in the liquidcrystal N(4methoxybenzylidene)4butylaniline (MBBA). Spins H_{3} and H_{4} are labeled as the bipartite system ρ_{AB}, spins F_{1} and F_{2} as the copy system ρ_{A′B′}, and spin F_{5} as the probe qubit ρ_{probe}. The effective Hamiltonian of the fivequbit system in double rotating frame is
where σ_{z} is the Pauli operator, ν_{i} is the chemical shift of spini and J_{jk} + 2D_{jk} is the effective coupling constant of spinj and spink. The molecular structure is shown in Fig. 2b, and the relevant parameters are shown in Method B.
Experimental procedure
Figure 2c shows the experimental schematic for verifying the measurement uncertainty relations. It can be divided into four parts.

1.
Preparing initial state. From thermal equilibrium state, we first initialized the system to a labeled pseudopure state (LPPS) \(\rho _{{\mathrm {LPPS}}} = \frac{1}{{32}}I_{32} + \varepsilon \sigma _z^{{\mathrm {probe}}} \otimes \left {0000} \right\rangle _{ABA{\prime}B{\prime}}\langle 0000\) with selectivetransition method,^{54} where ε ≈ 10^{−5} is the polarization and I_{32} is the 32dimension identity matrix. There is no dynamical and measurement effect on the part of identity density matrix; thus in the following, we conventionally denote the state with the deviation density matrix,^{55} ignoring the identity matrix. Then the product state ρ_{AB}(α, x) ⊗ ρ_{A′B′}(α, x) was prepared from 0000〉_{ABA′B′} 〈0000, where ρ_{A′B′} is the copy of ρ_{AB}. We vary the weight x by rotation with different angles and a following nonunitary gradient pulse. The details of initialization process are shown in Method C.

2.
Performing MUB measurements. The complete MUB measurements were implemented on subsystem A. For a twodimensional system, the simplest case of MUBs are
$$\begin{array}{*{20}{l}} {M_0 = \{ 0\rangle ,1\rangle \} ,M_1 = \left\{ \frac{{0\rangle + 1\rangle }}{{\sqrt 2 }},\frac{{0\rangle  1\rangle }}{{\sqrt 2 }}\right\} ,} \hfill \\ {M_2 = \left\{ \frac{{0\rangle + i1\rangle }}{{\sqrt 2 }},\frac{{0\rangle  i1\rangle }}{{\sqrt 2 }}\right\} .} \hfill \end{array}$$(8)They are just the eigenvectors of Pauli operators σ_{x},σ_{y} and σ_{z}. In NMR, such MUB projective measurements can be emulated using pulsed magnetic field gradients.^{56} Without interfering the unselected systems (B and its copy B′), we realized the MUB measurements on subsystem A and its copy A′ by the gradient echo technology,^{57} i.e., by selective π pulses to the other spins, this dephasing operation (i.e., projective measurement of σ_{z}) can be selectively performed on some specific spins. It is in principle necessary to refocus all the evolutions under the internal Hamiltonian during the gradient echo. However, this sequence can be simplified when we only care about the purity of the crashed state. For example, the projective measurement of M_{0} on spin F_{2} (A) and H_{4} (A′) are accomplished by \(P_z^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}} = G_z  [\pi ]_x^{{\mathrm{F}}_{\mathrm{1}},{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}}  G_z  [\pi ]_{  x}^{{\mathrm{F}}_{\mathrm{1}},{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}}\), where the evolutions under internal effective coupling Hamiltonian related to three qubits F_{1}, H_{3} and F_{5} are reserved during the pulse sequence. However, by some calculations, these undesired evolutions will lead to an error less than 2[1 − cos(2Δθ)] ≈ 0.065 on the purity measurements of the subsystem A or A′, mainly determined by the different evolutions on qubits F_{1}, H_{3} due to the different effective coupling constants \(J_{{\mathrm{F}}_{\mathrm{1}},{\mathrm{F}}_{\mathrm{5}}} + 2D_{{\mathrm{F}}_{\mathrm{1}},{\mathrm{F}}_{\mathrm{5}}}\) and \(J_{{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}} + 2D_{{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}}\). Here \(\Delta \theta = \pi [(J_{{\mathrm{F}}_{\mathrm{1}},{\mathrm{F}}_{\mathrm{5}}} + 2D_{{\mathrm{F}}_{\mathrm{1}},{\mathrm{F}}_{\mathrm{5}}})  (J_{{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}} + 2D_{{\mathrm{H}}_{\mathrm{3}},{\mathrm{F}}_{\mathrm{5}}})]t_{G_z}/2\) with the duration \(t_{G_z}\) of pulsed magnetic field gradient G_{z}. To perform projective measurements of M_{1} and M_{2} on specific spins, we first selectively rotate the spins with [π/2]_{−y} or [π/2]_{x} rotations, then performs the projective measurement of M_{0}, e.g., \(P_x^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}} = [\pi /2]_{  y}^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}}  P_z^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}}\) and \(P_y^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}} = [\pi /2]_x^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}}  P_z^{{\mathrm{F}}_{\mathrm{2}},{\mathrm{H}}_{\mathrm{4}}}\).^{56}

3.
Measuring the purities. After the MUBs on the subsystem A and A′, performing the quantum circuit in Fig. 2a will give the purity information on the resulting state ρ_{θB} after MUBs. For example, when two controlledSWAP gates (C_{swap}) are applied to both subsystems AA′ and BB′, one gets the purity of the bipartite system AB: \({\mathrm{Tr}}(\rho _{\theta B}^2)\); when only one C_{swap} is applied to the subsystem BB′, one gets the purity of subsystem B: \({\mathrm{Tr}}(\rho _{B\theta }^2)\) with ρ_{Bθ} = Tr_{A}(ρ_{θB}), as shown in Fig. 2c. Likely, the related purities of the original state ρ_{AB}: Tr(ρ_{AB}) and \({\mathrm{Tr}}(\rho _B^2)\), are obtained by the similar procedure without the MUBs. It can be noted the initial state of the probe qubit is different from the original method in Fig. 2a. We initialize the probe qubit as \(\sigma _z^{{\mathrm {probe}}}\) in our experiment. However, this will not affect the measure of the purity by the quantum circuit U_{qc} in Fig. 2a, i.e.,
Moreover, since the direct observable in NMR is σ_{x}, the final Hadamard gate can be canceled out by the readout operation. Therefore, through integrating the NMR spectra of the probe spin F_{5}, we directly measure the purities on the related states. By calculating the linear entropy, both sides of Eq. (4) are obtained without quantum state tomography.
Experimental results
The experimental results are shown in Fig. 3. As expected, the sum of uncertainties decreases to zero when the bipartite system is in maximally entangled state. With certain α, lower purity corresponds to higher uncertainty. From Fig. 3a, b, we can see that the experimental results are in accord with theoretical expectations and the uncertainty conversation relation holds with high precision. Figure 3c shows the final NMR spectra of the probe qubit for one certain initial state ρ_{AB}(π/4, 1). The sum of the integral values of all the peaks is read as the purity of the related state in our experiment, as shown in Fig. 3d.
In our experiments, to avoid the error accumulation and alleviate the influence of the decoherence, we used highfidelity engineered quantum control pulses, which exploit the gradient ascent pulse engineering (GRAPE) algorithm,^{58} to implement the quantum circuit in the experiments. The experiments for pure states contain eight GRAPE pulses with the total duration of about 74 ms, while for mixed states, we used 9~11 GRAPE pulses with total durations of 67−85 ms. We numerically optimized all GRAPE pulses with considering 5% ratio frequency (rf) field inhomogeneity, so that they are more robust in experiments. All GRAPE pulses used in the experiments have theoretical fidelities above 99.3%. Numerical simulations show that the imperfection of GRAPE pulses causes infidelity of 2−4% in the final states. Due to the short relaxation times of the liquidcrystal sample, the experiments suffer severe decoherence effect. We numerically simulated the dynamical process and estimated the attenuation factors caused by decoherence effect in the experiments.^{59,60} Then we rescaled the experimental results. The details can be also found in Method E. In the numerical simulations, we found that transverse relaxation time T_{2} plays a leading role in the decoherence process, while the longitude relaxation time T_{1} has little influence. The imperfection in preparing the labeled PPS also causes some errors. The highest unexpected peak in the labeled PPS NMR spectrum of spin F_{5} is about 3% intensity of the only expected peak.
Discussion
In conclusion, we have derived a novel entropic measurement uncertainty relation in bipartite systems with a quantum memory. 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. Substantially different from the previous uncertainty relations with inequalities, we presented an equality of uncertainty relation for the case with a quantum memory, which implies direct applications to quantum random number generation and quantum guessing games. Moreover, the relation (4) is independent of the choices of the MUBs. Therefore, the relation (4) gives rise to a kind of conservation of measurement uncertainties, in the sense that (4) is invariant under the transformation of MUBs.
Our theorem gives an uncertainty equality relation for arbitrary dimensional bipartite systems consisting of a quantum system and a coupled quantum memory. It should be emphasized that, even for singlepartite systems, it is already quite difficult to obtain an uncertainty equality relation for highdimensional case. The highdimensional bipartite case is much more complex than the case of singlepartite one.^{26,29,30,31,32,33,34,37,38,39,40,41,42} Therefore, as one sees in Method A, it is not surprising that the derivation of our uncertainty relations needs subtle mathematical techniques.
With the help of one mirror system of the measured system and one additional probe qubit, we have provided the first experimental verification of this measurement uncertainty relation in an NMR quantum processor, where the experimental data of uncertainty quantities have been directly obtained by measuring the involved entropies without quantum state tomography. This method allows one to perform verification experiments in large quantum systems, and deal with the experimental data by standard statistical and informationtheoretical methods. These results may give rise to significant applications in quantum information processing such as quantum metrology. For closed systems, it is well known that uncertainty relation determines the precision limit of quantum metrology based on complementary basis. For open systems, with B the environment and A the system to be measured, our equality presents a complete characterization 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 CNOT are physically implemented. Hence our results may highlight further studies on both fundamental problems in quantum mechanics and the applications.
Methods
A: Proof of the theorem
In a ddimensional Hilbert space \({\cal{H}}\), let {i_{θ}〉i = 1,⋯, d} denote a basis labeled by θ. A set of M such bases is called mutually unbiased if
for any i, j = 1, 2, ⋯, d, and θ, τ = 1, ⋯, M with θ ≠ τ. When d is a power of a prime number, a complete set of d + 1 MUBs exists. When d is an arbitrary integer number, the maximal number of MUBs is unknown. For example, when d = 6, only three MUBs have yet been found. However, for any d ≥ 2, there exist at least three MUBs. For the purpose of this paper, we only assume that M MUBs are available in a ddimensional Hilbert space, where M is less than or equal to the maximal number of MUBs that can exist.
In a composite Hilbert space \({\cal{H}} \otimes {\cal{H}}\) of two qubits, we construct the following M(d − 1) + 1 states,
with ω = e^{2πi/d}, k = 1, …, d − 1 and θ = 1, 2, …, M. Here i_{θ}〉^{*} denotes the complex conjugate of i_{θ}〉 with respect to the computational basis (which can be chosen as the first basis {i_{1}〉} without loss of generality). It is straightforward to show that the M(d − 1) + 1 bipartite states defined in Eqs. (10) and (11) are normalized and orthogonal to each other.
Therefore, these M(d − 1) + 1 states can be used for constructing a basis in the composite Hilbert space \({\cal{H}} \otimes {\cal{H}}\). Since there are at most d^{2} orthogonal states in a d^{2}dimensional space, one has M(d − 1) + 1 ≤ d^{2}, which implies M ≤ d + 1, i.e., there are at most d + 1 MUBs for a ddimensional Hilbert space. When M = d + 1, i.e., a complete set of d + 1 MUBs in a ddimensional space is available, the states defined in Eqs. (10) and (11) constitute a complete basis for the composite Hilbert space \({\cal{H}} \otimes {\cal{H}}\). On the other hand, when M < d + 1, one can complete a basis of the composite Hilbert space by adding p = (d − 1)(d + 1 − M) additional orthonormal states {ϕ_{α}〉 α = 1, ⋯, p}. The projector onto the subspace spanned by these additional states is denoted by \({\cal{P}}\), i.e.,
It is obvious that \({\cal{P}} = 0\) when M = d + 1.
Let T_{2} denote the partial transpose with respect to the computational basis of the second Hilbert space. One immediately has
As \(\mathop {\sum}\nolimits_{k = 1}^{d  1} {\omega ^{k(i  j)}}\) equals to −1 for i ≠ j and d − 1 when i = j, it is not difficult to show
Suppose ρ_{AB} is a bipartite state on the composite Hilbert space \({\cal{H}}_A \otimes {\cal{H}}_B\) of dimension d × D, and suppose {i_{θ}〉_{A}} is the θth MUB in the ddimensional Hilbert space \({\cal{H}}_A\), after system A is projected onto the θth MUB the overall bipartite state is written as
Given a bipartite state ρ_{AB} and a set of M MUBs in \({\cal{H}}_A\), we define an operator
on \({\cal{H}}_A \otimes {\cal{H}}_B\). This operator is Hermitian, and it has the following nice property.
Proposition When M = d + 1, the operator Γ_{AB} vanishes: Γ_{AB} = 0. When M ≤ d, it is nonnegativedefinite
In order to prove the proposition, we introduce an additional Hilbert space \({\cal{H}}_C\) of dimension d, and introduce a linear map \({\cal{F}}\) that maps operators on \({\cal{H}}_C\) to operators on \({\cal{H}}_A\), such that \({\cal{F}}(\left {i_1} \right\rangle _C\langle j_1) = \left {i_1} \right\rangle _A\langle j_1\) (i, j = 1,⋯, d). One can easily show that \({\cal{F}}(\left {i_\theta } \right\rangle _C\langle j_\theta ) = \left {i_\theta } \right\rangle _A\langle j_\theta \) for θ = 1, …, M. Let \({\cal{F}}^{  1}\) denote the inverse map, and let \(\rho _{CB} \equiv {\cal{F}}^{  1}(\rho _{AB})\) denote the corresponding state on \({\cal{H}}_C \otimes {\cal{H}}_B\) with respect to the state ρ_{AB} on \({\cal{H}}_A \otimes {\cal{H}}_B\). Therefore, the map \({\cal{F}}:\rho _{CB} \to {\cal{F}}(\rho _{CB})\) can also be conveniently written via a partial trace over system C
for any θ ∈ {1, ⋯, M}. Similarly, ρ_{θB} can be written as
Hence
We have used Eq. (14) to obtain the last equality. From Eqs. (13) and (18) with θ = 1, we also have
From Eqs. (20) and (21) and the obvious relation I_{A} ⊗ ρ_{B} = Tr_{C}[(I_{a} ⊗ I_{C})ρ_{CB}], we can rewrite Γ_{AB} as
Here the operator \({\cal{P}}_{AC}\) is the projector defined on \({\cal{H}}_A \otimes {\cal{H}}_C\) according to (12), i.e.,
When M = d + 1, then p = 0, the states ϕ_{α}〉_{AC} in \({\cal{H}}_A \otimes {\cal{H}}_C\) do not exist, both \({\cal{P}}_{AC}\) and Γ_{AB} vanish. When M ≤ d, we have
The last equality is due to the fact that operators on \({\cal{H}}_C\) can have cyclic permutations under the partial trace over C. Let \(\Theta _\alpha \equiv \left( {\left {\varphi _\alpha } \right\rangle _{AC}} \right)^{T_C}\sqrt {\rho _{CB}}\), which are operators on \({\cal{H}}_A \otimes {\cal{H}}_C \otimes {\cal{H}}_B\). Then we have
Since \(\Theta _\alpha (\Theta _\alpha )^\dagger\) are always nonnegativedefinite operators on \({\cal{H}}_A \otimes {\cal{H}}_C \otimes {\cal{H}}_B\), the operators \({\mathrm{Tr}}_C\left\{ {\Theta _\alpha (\Theta _\alpha )^\dagger } \right\}\) are nonnegativedefinite operators on \({\cal{H}}_A \otimes {\cal{H}}_B\), so is their sum. Therefore Γ_{AB} ≥ 0. This completes the proof of the proposition.
According to the proposition, Γ_{AB} is a nonnegativedefinite operator when M ≤ d, and it vanishes when M = d + 1. Hence, for any nonnegativedefinite operator Π_{AB} on \({\cal{H}}_A \otimes {\cal{H}}_B\),
when M ≤ d, and the inequality becomes an equality when M = d + 1.
Let Π_{AB} = ρ_{AB}, Eq. (25) yields
Therefore,
Adding \(M{\mathrm{Tr}}(\rho _B^2)\) to the above inequality, we immediately have
The inequality becomes an equality when M = d + 1. Thus, the theorem in the main text has been proved.
Remark: The approach we admitted here is methodologically similar to the one used in ref. ^{26} (see also ref. ^{29}). In ref. ^{45} an elegant uncertainty equality has also been presented based on ONE positive operatorvalued measure consisting uniformly all the measurement operators of d + 1 MUBs. As we take into account all the d + 1 MUB projection measurements individually, our uncertainty relation is completely different from the one given in ref. ^{45} In fact, the uncertainty relation in ref. ^{45} cannot be applied to our QRNG selftesting 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 mutually unbiased basis.
B: Quantum register
We used the sample named 1bromo2,4,5trifluorobenzene as a fivequbit NMR quantum system which consists of two ^{1}H spins and three ^{19}F spins, dissolved in the liquidcrystal N(4methoxybenzylidene)4butylaniline. The structure of the molecule is shown in Fig. 4. Due to the partial average effect in the liquidcrystal solution, the direct dipole−dipole interaction will be scaled down by the order parameter.^{61} In our sample, the partial average effect makes the direct dipole−dipole couplings of homonuclear spins much smaller than the difference between the chemical shifts of related nuclear spins, all the dipole−dipole Hamiltonians are reduced to the form of \(\sigma _z^i\sigma _z^j\). Therefore, the effective Hamiltonian of the fivequbit system in rotating frame is
where σ_{z} is the Pauli operator, ν_{i} is the chemical shift of spini and J_{jk} + 2D_{jk} is the effective coupling constant of spinj and spink. The relevant parameters are shown in Fig. 4.
C: Initial state preparation
We first initialized the quantum register into a labeled pseudopure state (LPPS) \(\rho _{{\mathrm {LPPS}}} = \varepsilon \sigma _z^{{\mathrm {probe}}} \otimes \left {0000} \right\rangle _{ABA{\prime}B{\prime}}\langle 0000 + 1/32I_{32}\) 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. We applied 30 unitary operators with the form \(U = e^{  i\beta _{ij}{\cal{V}}_{ij}}\) to redistribute the populations between energy levels i and j. Here \({\cal{V}}_{ij}\) is the single quantum transition operator between levels i and j, that is, a 32 × 32 matrix whose elements are zero except for two elements \({\cal{V}}_{ij}(i,j) = {\cal{V}}_{ij}(j,i) = 1/2\). According to the desired distribution, the angle β_{ij} were numerically calculated. In this produce, the unitary operators generated undesired coherence terms that were eliminated by a following field gradient pulse Gz. Accordingly, the LPPS ρ_{LPPS} was prepared.
In the following we only consider the deviation part \(\sigma _z^{{\mathrm {probe}}} \otimes \left {0000} \right\rangle _{ABA{\prime}B{\prime}}\langle 0000\). The composite system was prepared into the state
from \(\sigma _z^{{\mathrm {probe}}} \otimes 0000\rangle \langle 0000\), where ψ_{α}〉 = cos(α/2)01〉 − sin(α/2)10〉. In the case of x = 1, the bipartite systems are entangled where the rotation angle α varies from 0 to π/2 for producing different entanglement. α = 0 corresponds to separable state and α = π/2 corresponds to the maximally entangled state. Setting α = π/2 and scanning x from 0 to 1 with interval 0.25, we prepared the states with different mixedness. In preparing \(\sigma _z^{{\mathrm {probe}}} \otimes \rho _{AB}(\pi /2,x) \otimes \rho _{A{\prime}B{\prime}}(\pi /2,x)\), we divided it into 4 parts, i.e.,
and adopted the temporal averaging method^{62} to realize them. By summing the four output states, we obtained the desired state \(\sigma _z^{{\mathrm {probe}}} \otimes \rho _{AB}(\pi /2,x) \otimes \rho _{A{\prime}B{\prime}}(\pi /2,x)\). The pulse sequences for the state preparation are shown in Fig. 5a, b.
D: Experimental procedure
The total experimental procedure is shown in Fig. 6, including the LPPS preparation, the initial state preparation, MUB measurements and the readout of the purities. When the measurement is along yaxis, the first and last rotations in the MUB measurement part should be substituted by rotations along x and −x, and the angles remain also π/2. In the zaxis measurement case, these two rotations are omitted.
E: The attenuation factors caused by decoherence effect and experimental measured purities
Even through the highfidelity GRAPE pulses in experiments, the experimental results are still severely affected due to the decoherence due to the long running times comparing to the T_{2} relaxation times of nuclear spins.
In our experiments, the environmental noises are suitably modeled as Markovian, and the evolution of the system is given by the Lindblad master equation
where H_{S} is the system Hamiltonian, H_{C}(t) is the timedependent external control Hamiltonian, and the operators L_{α} are Lindblad operators representing the coupling with the environment. The experiments are mainly affected by the dephasing process, and therefore \(L_\alpha = \sqrt {\frac{{\gamma _\alpha }}{2}} \sigma _z^\alpha ,(\alpha = 1,...,5)\) and γ_{α} = 1/T_{2α} and
In the experiments, the highfidelity GRAPE pulses we used are kinds of shaped pulses consisting of thousands of slices with a constant H_{C}(t_{i}) the duration of each slice, where δt is 2−25 μs. Consequently, for each slice, the state of the system can be approximately given by
where the Kraus operations
with \(\lambda _\alpha = (1 + e^{  \gamma _\alpha \delta t})/2\).
Therefore, we numerically simulated the experiments, extracted the attenuation factors caused by decoherence effect and then rescaled the experiment results for verifying the uncertainty conversation relation in Fig. 3a, b. The unrescaled original results are shown in Fig. 7a. From this we can see that even though the right and left sides of the uncertainty equation are all reduced, they almost equal each other. Figure 7b shows the directly measured purities of different states required in the uncertainty conversation relation.
Data availability
The main data supporting the finding of this study are available within the article. Additional data can be provided upon request.
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Acknowledgements
We thank Marcin Pawłowski, Simone Severini and Dawei Lu for helpfull discussions. This work was supported by the National Key R&D Program of China (Grant No. 2018YFA0306600, 2016YFA0301801, 2017YFA0303703), National Natural Science Foundation of China (Grants No. 11425523, 11661161018, 11675113, 11605153, 11571313, 11575173), Natural Science Foundation of Zhejiang province (Grants No. LQ19A050001), Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000), and the Key Project of Beijing Municipal Commission of Education under No. KZ201810028042.
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Wang, H., Ma, Z., Wu, S. et al. Uncertainty equality with quantum memory and its experimental verification. npj Quantum Inf 5, 39 (2019). https://doi.org/10.1038/s415340190153z
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