1. INTRODUCTION

Quantum devices that prepare and measure quantum states are indispensable for quantum information processing. Before implementing quantum schemes, these devices must be precisely calibrated to ensure that the prepared quantum states or executed measurements are sufficiently credible. Conventional methods such as quantum state tomography (QST) [1] and quantum detector tomography (QDT) [2] have to perform precise projections to different bases and reconstruct the model with the projection probabilities. However, all of these methods inevitably encounter a dilemma: to certify the intended quantum states requires a well-calibrated measurement device, and to certify the desired measurement requires accurate preparation of quantum states; thus, the experimenter eventually stumbles into an infinite regress.

A particular branch of quantum information science has been created to overcome this dilemma: full-device-independent (FDI) or semi-device-independent (SDI) methods. In FDI schemes, one can predict the quality of prepared entangled states given a (nearly) maximal Bell violation is observed by examining Bell-like correlations [3–6]. FDI protocols for bipartite entangled states or measurements are intensively investigated [7–13], and, more recently, a FDI protocol for general genuine multipartite entanglements has been formalized and demonstrated in experiment [14–16]. However, FDI protocols hinge on the observation of maximal violation of certain Bell inequalities, which incurs a penalty of weak robustness to experiment imperfections. Alternatively, with additional knowledge on the Hilbert space dimension, or extra assumptions such as energy bound and imperfection bound [17–21], SDI schemes can be applied to fundamental prepare-and-measurement (P&M) experiments, which are more feasible to implement and tolerant of realistic experimental imperfections [22–25]. To date, SDI protocols have been developed for qubit states and measurements [26], pairs of mutually unbiased bases, and symmetric informationally complete measurements [27,28].

Currently, SDI schemes are limited to only a few classes of quantum states. On the other hand, different quantum states are needed to achieve various QIP schemes, such as quantum random number generation (QRNG) [29,30], quantum key distribution (QKD) [31,32], quantum contextuality [33–35], abundant random algorithms [36,37], and fault-tolerant universal quantum computation [38–40]. In this sense, it calls for SDI full calibration of quantum devices, which renders an overall evaluation of the quantum devices to produce an arbitrary quantum state in a Hilbert space of knowing dimension, but without any extra knowledge about the preparation or measurement devices.

In this study, we propose and demonstrate an SDI scheme to evaluate a device’s ability to produce arbitrary states by calibrating $t$-designs that accommodate discrete and highly symmetric structures. We bridge the gap between continuous and discrete structures with a concept termed the “covering radius,” which describes the minimum radius it requires to cover the entire original sphere ${S^d}$ if one draws circles of the same radius centered at a group of points. This concept determines the upper bound of the intrinsic bias in producing an arbitrary state given a perfect $t$-design. The global error to prepare the $t$-design is quantified by the frame potential, which can be acquired via Tavakoli’s protocol [41]. In our scheme, we extrapolate the maximally possible error to generate single states in the $t$-design from the measured frame potential, and then we fuse this practical error and the covering radius to evaluate the overall bias to generate one arbitrary target state. We show that our experimental results for a qubit system could serve as SDI certificates for the secure key in QKD and authentic randomness in QRNG, and SDI proof for magic states distillability.

2. THEORETICAL FRAMEWORK

A. Full Calibration Scheme

Granted a quantum device to produce arbitrary states in a Hilbert space ${{\cal H}^d}$, full calibration aims to evaluate the maximum infidelity $\xi$ between a pure target state $\rho$ and its real production $\rho ^\prime $, which is calculated as $\xi = \mathop {\max}\nolimits_{\rho \in {{\cal H}^d}} 1 - F(\rho ^\prime ,\rho)$, and $F(\rho ^\prime ,\rho) = {\rm tr}\sqrt {{\rho ^{1/2}}\rho ^\prime {\rho ^{1/2}}}$ is the fidelity between $\rho$ and $\rho ^\prime $. Intuitively it is a formidable challenge to evaluate $\xi$ since ${{\cal H}^d}$ is a continuous set containing an infinite number of $\rho$. To circumvent this deadlock, one can instead certify the production of a finite set of states named $X$. Provided that this set $X$ is densely and evenly distributed, one can acquire an upper bound of the bias to produce arbitrary states with the same devices.

To acquire $\xi$ in a realistic full calibration scheme, we introduce two classical concepts, “covering radius” and “covering angle” [42], which are widely applied in error-correction code [43]. With the knowledge of covering radius, the vectors originating from the center of ${S^d}$ and ending at such a covering circle cut a cone, and the cone angle is defined as the covering angle ${\delta _c}$. We can transfer the covering radius to a quantum version in the sense that all quantum states form a sphere in the Hilbert space ${{\cal H}^d}$, and the distance can be characterized by the infidelity between states. In this case, for the qubit system, the sphere in Fig. 1(a) can be viewed as the Bloch sphere, and the covering radius/angle of a set of pure states, namely, $X = \{{\rho _i}\}$, can be defined as

 

Fig. 1. (a) Diagram of the ideal full calibration scheme. On the surface of the sphere, the states in the set $X$ (green dots) are assigned as the centers of the orange circles, which cover the entire surface with the minimum possible radius. For any state $\rho$ (the red dot), there always exists a state ${\rho _i} \in X$ (the big green dot) whose distance from $\rho$ is within the covering radius ${r_c}$ (i.e., within one orange circle). When producing the target state $\rho$, the device can always turn to the nearest ${\rho _i}$ as substitution, with the bias $\xi$ below ${r_c}$. (b) Full calibration scheme considering practical errors. The red vector represents one arbitrary target state $\rho$. The orange cone’s cone angle is covering angle ${\delta _c}$, and one can always find a state vector ${\rho _i}$ (green vector) in $X$ within the orange cone as a substitution of $\rho$. However, the realistic ${\rho _i}$ may deviate to an error state ${\rho _i^\prime}$ (blue vector), and the green cone represents the maximum possible error with the cone angle denoted as ${\delta _{{\rm err}}}$. The overall bias with a realistic $X$ should take both ${\delta _c}$ and ${\delta _{{\rm err}}}$ into account.

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In ideal cases when the device can perfectly produce all the states in $X$, we have a covering radius as the natural bound for $\xi$, i.e., $\xi \le {r_c}$. In other words, premised that $X$ is well calibrated, for arbitrary target state $\rho$, we always turn to produce the nearest state ${\rho _i}$ in $X$, and the maximum infidelity due to this proximate substitution is just ${r_c}$.

However, inevitable imperfections give birth to an error in producing ${\rho _i}$ and result in an error state ${\rho ^\prime _i}$. In this case, the bias regarding the ${r_c}$ of an ideal set X should be replaced by an overall bias. As diagrammed in Fig. 1(b), the maximum angle between the target state and the nearest ${\rho _i} \in X$ is ${\delta _c}$, and the maximum angle between ${\rho _i}$ and its actual production ${\rho _i^\prime}$ is ${\delta _{{\rm err}}}$. With the knowledge of ${\delta _c}$ and ${\delta _{{\rm err}}}$, the maximum angle between $\rho$ and ${\rho ^\prime _i}$ can be calculated as ${\delta _c} + {\delta _{{\rm err}}}$. Correspondingly, the overall infidelity $\xi$ with imperfect $X$ is upper bounded by the following inequation:

In Eq. (2), ${\delta _c}$ is determined by the specific $X$, and ${\delta _{{\rm err}}}$ is estimated from the calibration of $X$. For an ideal scenario when ${\delta _{{\rm err}}}$ is zero, the right-hand side of Eq. (2) becomes ${r_c}$. To perform an accurate full calibration, one must primarily decrease ${\delta _c}$ by selecting a set $X$ accommodating a sufficiently dense and symmetric structure and then prepare this $X$ with a smaller ${\delta _{{\rm err}}}$. In the following sections, we will show that quantum $t$-designs are satisfying candidates for full calibration, and the preparation error can be estimated in an SDI way.

B. SDI Full Calibration with $t$-Designs

Quantum $t$-designs are sets of states that provide the same averaged value of a $t$-degree polynomial, while only a small number of states have to be involved in measurement. Due to these highly symmetric structures, quantum $t$-design can be applied to QST [44], quantum key distribution [31,32], Bell experiments [13], and entanglement detection [45,46]. As special paradigms, both MUB and SIC sets are quantum 2-design [47]. Also, as an important element of fault-tolerant universal quantum computation, quantum magic states serve as 3-design [38,48,49]. Three numbers can uniquely define a $t$-design $X(N,d,t)$: the set volume $N$, the Hilbert space dimension $d$, and the design number $t$. (For details and generation of $t$-designs, see Sections 1 and 2 in Supplement 1) A higher $t$ implies a higher degree of symmetry [50], and $t$-design is also a ($t - 1$)-design. For example, 1-design spherical codes mean the center of the sphere, and by adding to a new symmetric rule, one can obtain 2-design [50].

$t$-designs with sufficiently large $t$ and $N$ are adequate candidates for full calibration due to the high symmetry and large volume. To be specific, if in the neighborhood of any target state, we could always find a member in $X[N,d,t]$ within an admissible bias; then the full calibration can be done via the certification of this $t$-design. As shown in Figs. 2(a) and (b), for a certain dimension $d$, the set volume $N$ grows with design number $t$ as $N \sim O({t^d})$ [53]; correspondingly, the covering angle decreases, resulting in a more negligible bias in full calibration.

 

Fig. 2. (a) Dependence of the set volume $N$ on the design number $t$ for known $t$-designs X $(N,2,t)$ [51,52]. Although an analytic description cannot be extrapolated, roughly $N$ is exponentially growing with $t$. (b) Dependence of the covering angle ${\delta _c}$ on the set volume $N$ for known $t$-designs X $(N,2,t)$, which is calculated via Eq. (1) with the error as low as 0.01 deg (see Section IV in Supplement 1 for details). Qualitatively, larger volume implies higher symmetric structure and leads to a smaller covering angle.

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To execute SDI full calibration with $t$-designs, one must know the covering angle ${\delta _c}$ and the error angle ${\delta _{{\rm err}}}$. The covering angle for a special $t$-design can be identified from Eq. (1) via the Monte Carlo method, and the results for known $t$-designs with $t \le 21$ are shown in Fig. 2(b). The remaining question is how to estimate ${\delta _{{\rm err}}}$ in an SDI way. In the following analysis, we address this question by upper bounding ${\delta _{{\rm err}}}$ via the frame potential of $t$-designs, which can be measured through an SDI way [41]. Tavakoli proposed an SDI method to calibrate $t$-design states by accessing the frame potential ${F_t}$ [41], and for a set with $N$ member states in a dimension not exceeding $d$, the frame potential ${F_t}$ can be calculated as

From Eq. (3), to measure ${F_t}$, one has to acquire all possible $|\langle {\psi _i}|{\psi _j}\rangle |$, which can be obtained from two-state unambiguous state discrimination (USD). When the USD is subject to an error rate not exceeding $\epsilon$ the measured value of ${F_t}$ is denoted as $F_t^ \epsilon$, which necessarily satisfies $F_t^ \epsilon \ge {F_t} \ge {J_t}$. (See Section 3 in Supplement 1 for the details of measuring $F_t^ \epsilon$.) We define the differential frame potential as $F_t^ \epsilon - {J_t}$, which serves as a natural verifier of the produced $t$-design and can be used to upper bound the error angle ${\delta _{{\rm err}}}$.

Provided a measured value of $F_t^ \epsilon - {J_t}$, the worst case for full calibration is that only one $t$-design state ${\rho _0} = |{\psi _0}\rangle \langle {\psi _0}|$ suffers errors, while the other $N - 1$ states are perfect. Suppose the unique error state is ${\rho _0^\prime} = {\rho _0} + \Delta$; the infidelity ${\xi _{{\rm err}}}$ between ${\rho _0^\prime}$ and ${\rho _0}$ is upper bounded by the following inequality:

The differential frame potential is calculated as

To implement our SDI full calibration scheme, we have to utilize nonantipodal designs of which $\epsilon$ is necessarily smaller than the so-called max-error margin (see Section 5 of Supplement 1).

3. EXPERIMENT RESULTS

The SDI scheme to certify $t$-designs can be performed by the setup shown in Fig. 3. Alice handles a setup to produce $t$-designs by encoding the states into the polarization of the heralded photons generated by a nonlinear crystal. Then she sends it to Bob, who implements the USD to diagnose how the produced states are close to an ideal $t$-design (see Section 6 of Supplement 1). Both the preparation and measurement are executed according to a preestablished SDI protocol in the Theoretical Framework.

 

Fig. 3. Experimental Setup. Alice uses a 405.4 nm diode laser to pump a periodically poled KTiOPO4 (PPKTP) nonlinear crystal, where orthogonally polarized photon pairs are created with degenerate wavelength at 810.8 nm. A band-pass filter (BPF) at 810 nm blocks the pump light, while permitting a high transmission of down-conversed photon pairs. The vertically polarized photon is reflected at the polarized beam splitter (PBS) and then detected by a single photon detector (SPD) to produce heralding signals. The transmitted horizontally polarized photon is used to proceed $t$-design states preparation and certification. Granted the random number generator (RNG) outputs a number $x$, a corresponding state ${\psi _x}$ is encoded to the transmitted photon by a motorized half-wave plate (HWP) and a motorized quarter-wave plate (QWP). The verifier named Bob generates two random numbers $i,j$ from another RNG, and then performs USDs for ${\psi _i}$ and ${\psi _j}$ if he has $x = i$ or $x = j$. The USD apparatus is composed of three beam displacers (BDs), four motorized wave plates, and two HWPs with fixed angles. Bob eventually obtains three outcomes, respectively attributed to definite conclusions of receiving ${\psi _i}$ and ${\psi _j}$, and a third inconclusive outcome. The success rate $P_s^{i,j}$ to discriminate ${\psi _i}$ and ${\psi _j}$ can then be accessed for all possible ($i,j$) in a $t$-design.

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Fig. 4. (a) Normalized frame potentials ${\tilde F^ \epsilon}_{{t^\prime}}$ for tested $t$-designs. Each line connects the points representing the measured ${\tilde F^ \epsilon}_{{t^\prime}}$ for a prepared $t$-design. When $t^\prime \le t$, approximately the tested set can be regarded as a $t^\prime $-design, and ${\tilde F_{{t^\prime}}}$ is close to its ideal value 1, as labeled by the triangles. Otherwise, the tested set cannot mimic a $t^\prime $-design if $t^\prime \gt t$, and the corresponding circles climb evidently indicating that a $t$-design is incapable to accommodate a structure possessing overhigh symmetry. (b) Upper bound of infidelity in full calibration with tested $t$-designs. Here, the total height of each column is given by inequality Eq. (7), and the blue bar is quantified as ${r_c}$. Their difference is attributed to the practical errors in the P&M experiment, which indicates the high robustness of this SDI full calibration.

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The error rate of USD is tested in the experiment, which is as low as 0.002 (see Section 7 in Supplement 1). To keep the max-error margin greater than the error rate, we test $X(4,2,2)$, $X(7,2,2)$, $X(9,2,2)$, $X(10,2,3)$, $X(15,2,3)$, $X(19,2,4)$, $X(16,2,5)$, and $X(24,2,7)$ in the experiment [52].

For each set investigated, we record the success rate of USD for all possible pairs in the set and calculate $F_{{t^\prime}}^ \epsilon$ with $t^\prime $ ranging from 1 to 7. To characterize the quality of the produced set, we define $\tilde F_{{t^\prime}}^ \epsilon$ as the normalized frame potential, which is calculated as ${\tilde F_{{t^\prime}}} = F_{{t^\prime}}^ \epsilon /{J_{{t^\prime}}}$. This figure of merit is impervious to the set volume, and its minimum value, i.e., $1$, can only be attained by perfect preparation and measurement. In other words, any errors in the P&M experiment necessarily lead to an $\tilde F_k^ \epsilon$ exceeding 1. The results of $\tilde F_{{t^\prime}}^ \epsilon$ for eight tested sets are shown in Fig. 4(a).

By measuring the frame potential of a certain set with a P&M apparatus, we can estimate the upper bound of the infidelity $\xi$ to produce an arbitrary state on the Bloch sphere. The upper bounds are calculated from Eq. (7) and shown as columns in Fig. 4(b). Each column comprises two bars, with the blue bar representing ${r_c}$ of an ideal $t$-design. The orange bar is the error amplitude in the P&M experiment to calibrate the realistic $t$-design. Generally, the blue bars dominate the overall bias for these tested sets and decrease with increasing $t$ and $N$. The orange bars weakly fluctuate since an unaltered P&M apparatus produces the tested sets. Lower columns indicate more negligible overall bias and more accurate full calibration. Specifically, $X(24,2,7)$ is the best candidate for full calibration among the eight tested sets. The corresponding infidelity to produce arbitrary states is below 8.75%, which is evaluated via an SDI method. It is worth noting that this upper bound is obtained by assuming an ultimate scenario, while normally each state in the set suffers identical errors and the orange bar in Fig. 4(b) could be approximately divided by $n$. In particular, when the target states are just the ones in the set, the blue bar vanishes and only the practical errors need to be taken into account.

Knowing the ability of a certain quantum device to generate arbitrary qubit states, we can further acquire SDI certificates for the secure key of QKD and the authentic randomness of QRNG that can be generated from this device. For QKD, Alice requires two sets, $\mathbb{X}$ and $\mathbb{Z}$, with the preparation quality defined as $q = - \log \mathop {\max}\nolimits_{x,z} |\langle {\psi _x}|{\psi _z}\rangle |$ [54]. This value reaches 1 for perfect BB84 or Six States protocols. Assuming the state preparation setup can produce an arbitrary state with infidelity less than $\xi$, $q$ is degraded such that $q \ge 1 - \log [1 + 4(1 - \xi)\sqrt {\xi (1 - \xi)}]$. If a potential eavesdropper exploits this state imperfection, the upper bound of the asymptotic key rate (when communicating infinite bits) becomes $R \le 1 - \log [1 + 4(1 - \xi)\sqrt {\xi (1 - \xi)}]$ [55,56]. This extension does not rely on assumptions about the measurement or the channel and can be applied to all kinds of QKD protocols (see Section 8 in Supplement 1 for details).

 

Fig. 5. (a) SDI certificates for QKD and QRNG. With bounded error $\xi$ in state preparation, the SDI certificates for the generating rates of shared keys in QKD and authentic randomness in QRNG are plotted as solid curves. The stars represent the results using arbitrary qubits states calibrated via $X(24,2,7)$ in full calibration, and the diamonds represent the results with adequate states just in $X(15,2,3)$.

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For QRNG, Alice produces states from $\mathbb{X}$, and Bob measures them to $\mathbb{Z}$. The authentic randomness encoded in the states from a potential eavesdropper $E$ can be expressed as the min-entropy ${H_{{\min}}}(Z|E)$ [54], the lower bound of which can be calculated as ${H_{{\min}}}(Z|E) \ge 1 - {h_{{\max}}}(\xi)$, where ${h_{{\max}}}$ is the binary max-entropy [57,58]. Consequently, the extracted rate of the authentic randomness is bounded by $R \le 1 - {h_{{\max}}}(\xi)$ (see Section 9 of Supplement 1 for details).

The states to implement QKD or QRNG can be either selected as ordinary ones calibrated via SDI full calibration, or just the member states in a certain $t$-design. Utilizing the results for $X(24,2,7)$ shown in Fig. 4, the SDI certificates for the generating rates of secure keys in QKD and authentic randomness in QRNG are identified to be 0.35 and 0.05, respectively, granted the maximum value of $\xi$ is 0.0875 (see Fig. 5). There are also $t$-designs that exactly include states required for QKD and QRNG, and, thus, solely the practical errors in P&M need to be taken into account and $\xi$ is replaced by ${\xi _{{\rm err}}}$. $X(15,2,3)$ is such a set, and the corresponding rate certificates are 0.85 and 0.71 for QKD and QRNG (see Fig. 5), when ${\xi _{{\rm err}}}$ is smaller than 0.003 as confirmed in Fig. 4.

Furthermore, our full calibration method has applications in fault-tolerant universal quantum computation, which relies on T-states (magic states) in addition to Clifford gates [38]. According to Bravyi and Kitaev’s threshold for distillation with noisy magic states of $\frac{1}{2}(1 - \sqrt {3/7}) \approx 0.173$ [48], our experiments, generating arbitrary states with $\xi \le 0.0875$, confirm the distillability of T-states from the calibrated device.

4. DISCUSSION

Actually, our full calibration scheme could employ any discrete set of states rather than the $t$-designs; however, $t$-designs are intuitively appealing since they provide highly symmetric structures in the Hilbert space. Granted the same number of states, a more symmetric structure is likely to reach a smaller covering angle, which renders better accuracy to produce arbitrary states. Another benefit of using $t$-designs is that an SDI method has been proposed to certify $t$-designs by Tavakoli [41], which renders an SDI method to evaluate the global error to prepare the $t$-design, in terms of the frame potential. To execute the full calibration scheme, first we have to extrapolate the error bound to generate single states in the $t$-design via the measured frame potential, and then fuse this error with the covering angle to figure out the overall bias. In this sense, our work is not merely a variant of Tavakoli’s protocol but employs it as a basic tool to realize SDI full calibration.

In Tavakoli’s protocol, to evaluate the frame potential one needs a number of measurements that scale quadratically in the number of states. Therefore, it is clearly calling for efficient SDI protocols to certify $t$-designs with large state volumes. Once this efficient method emerges, it can be immediately incorporated into our protocol to tackle high-dimensional systems. Nevertheless, our work introduces the possibility of SDI certification for arbitrary states and demonstrates this new protocol for the most used qubit systems.

To conclude, we experimentally demonstrate an SDI full calibration via the certification of $t$-designs. With uncharacterized P&M devices, the overall bias in the infidelity of producing an arbitrary target state can be related to the frame potential of a discrete $t$-design set. The inherent bias due to the discrete structure and the practical errors from the P&M devices jointly determined this overall bias. The nontrivial robustness renders this SDI method a useful benchmarking tool to advance quantum technologies.