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Abstract
Certifying quantum measurements is increasingly important for foundational insights in quantum information science. Here, we report an experimental certification of unknown quantum measurements in a semi-device-independent setting. For the first time, we experimentally demonstrate that genuine three-outcome positive operator-valued measures (POVMs) can be certified under the assumption of a limited overlap between the prepared quantum states. The generalized quantum measurements are realized through discrete-time quantum walk and our experimental results clearly show that three-outcome POVMs can be certified even in the presence of noise. Finally, we experimentally investigate that optimal POVMs for performing unambiguous state discrimination can be self-tested. Our work opens new avenues for robust certification of quantum systems in the prepare-and-measure scenario.
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1. Introduction
Quantum measurements are crucial for quantum technologies and they reveal some of the most startling non-classical features of quantum theory. Projective measurements in quantum theory are represented by complete sets of orthogonal projectors. However, the most general description of a quantum measurement is given by positive-operator-valued measures (POVMs) which is collection of positive operators summing up to identity. In fact, we can consider projective measurements as instances of POVMs but not all POVMs are projective measurements. Moreover, these nonprojective measurements not only have good applications in quantum information processing, but also have specific advantages over projective measurements in practical tasks, such as quantum cryptography [1,2], randomness generation and certification [3,4], quantum computing [5], quantum tomography [6] and quantum state discrimination [7,8].
With the increasing complexity of quantum systems, there is a growing need for certification and verification of nonprojective measurements. Due to noise and technical imperfections present in quantum devices, and limited trust to them, a trend of certifying quantum states and measurements in a device-independent (DI) manner has emerged [9–11], which is referred to as "self-testing". Under the fully DI approach, the quantum device is prepared as a black box and the quantum system is characterized only by the measurement results, without making assumptions about the internal construction of the involved devices. DI certification of nonprojective measurements has also been realized based on the violation of the Bell inequality [12,13]. Although a number of experiments have been implemented in the DI manner, this elegant approach faces many obstacles in experiments. The main difficulty is that fully DI certification methods require a loophole-free Bell inequality violation.
Motivated by finding a compromise between the idea of DI and ease of implementation in the experiment, more and more attention has been directed at the semi-device-independent (SDI) approach [14–16] which benefits from the fact that prepare-and-measure (PAM) experiments are more experimentally appealing, while also allowing for realistic experimental imperfections [17–22], the price to pay for this simplification is that an additional assumption on the system is required in order to get meaningful results. The most common assumption is a constraint on the dimension of quantum systems and minimal scheme for certifying three-outcome qubit measurements has also been introduced in the prepare-and-measure scenario [23]. Recently, Armin Tavakoli et al. [24] investigated the problem of self-testing nonprojective measurements in a PAM scenario under the assumption of bounded dimension for the first time.
Compared to the previous work, it would be interesting to investigate the certification of nonprojective measurements using different assumptions and we follow a different approach by considering the assumption of a limited overlap between the prepared quantum states instead of bounded dimension. First, dimension is not a directly measurable quantity and the assumption of bounded dimension is not straightforward to justify. This motivates the study of different approaches in the SDI setting with different types of assumptions. Alternative approaches based on new assumptions have been put forward in the cases of bounds on entropy of quantum information [25], energy of quantum states [26], and overlap between prepared quantum states [3]. In particular, we are interested in certifying quantum measurements using the assumption of a limited overlap between the prepared quantum states. The advantage of this approach is that the dimension of the encoded system is no longer required and the use of inner product information is sufficient to inscribe the non-orthogonality of the encoded states. This type of approach is particularly useful for analysing the performance of quantum communication and randomness generation [3], which draws a direct connection between the distinguishability of quantum states and quantum correlations. What’s more, the overlap between the prepared quantum states can be used as a bounded fidelity with respect to a set of target states and quantum correlations can be investigated with bounded distrust [27]. Second, the new assumption opens new avenues for robust certification of genuine three-outcome POVMs in the prepare-and-measure scenario.
In this work, we consider the problem of experimentally certifying non-projective measurements following a SDI protocol [28] by making assumptions on the lower bound of the overlap between prepared quantum states. In our experiment, we implement genuine three-outcome POVMs certification in the PAM scenario based on quantum walk. Our results not only allow us to certify genuine three-outcome POVMs in a SDI manner, but also experimentally confirmed that the certification is robust to noise. Furthermore, we also experimentally demonstrate that the larger the overlap between the quantum states is, the more noise the certification can tolerate. Finally, we experimentally show that optimal POVMs for performing unambiguous state discrimination (USD) can be self-tested.
2. Theory
The PAM scenario consists of preparation and measurement devices as shown in Fig. 1(a). When pressing button in preparation, the preparation device prepares a binary input $x \in \{0,1\}$ and emits state $\rho _{x}$. When the button in measurement device is pressed, the device performs an unknown POVM $M_{b}$ on the incoming state and the measurement produces an outcome $b \in \{0,1,2\}$, then the probability distribution is $p(b|x)=\operatorname {Tr}\left [\rho _{x} M_{b}\right ]$.
Fig. 1. (a) Certification of nonprojective quantum measurements in the prepare-and-measure scenario (b) Schematic representation of the three-outcome POVMs with symmetrical Bloch vectors in x-z plane.
In this work, we characterize the unknown POVM in a SDI scenario based on the assumption of a lower bound on the indistinguishability of the two quantum states. The measurement device gets information about the input $x$ only from the quantum state $\rho _{x}$ given by the preparation device. We use the fidelity [29,30] to define the overlap of two quantum states $\rho _{0}$ and $\rho _{1}$. For mixed states the fidelity is
Noticed that genuine three-outcome POVMs cannot be decomposed into a convex combination of two-outcome POVMs, the corresponding elements of POVMs $M_{b}$ can be recorded as
where $j=0, 1, 2$, then the measurements can be divided into two classes: (i) when $\{M^{j}_{b}\}_{b=0,1,2}$ is a valid POVM with $M_{j}^{j}=0$, then we say that $M_{b}$ is not a genuine three-outcome POVM; (ii) when $M_{j}^{j}\neq 0$, $M_{b}$ is a genuine three-outcome POVM. In order to characterize the boundary of the two-outcome POVM, we define $X=\frac {1}{2}[p(0|0)+p(1|1)]$ and $Y=\frac {1}{2}[p(0|1)+p(1|0)]$, which are the average probability of success or failure in guessing correctly the initial state $\left |\Phi _{0}\right \rangle$ and $\left |\Phi _{1}\right \rangle$ under different measurements. According to Eq. (2), there are three two-outcome strategies $\left \{0, K_{1}, I-K_{1}\right \}$, $\left \{K_{2}, 0, I-K_{2}\right \}$, $\left \{K_{3}, I-K_{3}, 0\right \}$ to characterize the boundary of the behaviors achievable by two-outcome POVMs [28]. Where $K_{i}$ represents one of the elements of two-outcome measurements. Hence, after the numerical calculation [28], the set of behaviors achievable by two-outcome POVMs (defined as $\mathrm {P}_{2}(\delta )$) can be obtained through the convex hull of the point (0,0), ellipse Eq. (3) and Eq. (4), which means the behaviors do not belong to the convex hull can certify genuine three-outcome POVMs.3. Experiment
In our experiment, the wavelength of laser is 405nm, we use a polarization beam splitter (PBS) with a half-wave plate (HWP) to control the polarization of the pump beam. The beam is first focused in a 10 mm periodically polarized KTiOPO$_{4}$ (PPKTP) crystal by a lens ($f=20cm$), and then turned into parallel light by another lens with the same parameters. After the pump beam light enters the PPKTP crystal, phase matching is performed to produce single photon pairs that are coupled into the single-mode fiber separately, and an interference filter (IF, $\Delta f=10nm$) is required to remove the pump beam light. One single photon in the pair is detected to herald the single photon and the other photon is injected into the optical quantum walk. When the power of pump beam is 2.67 mW, total coincidence counts collected by the single-photon detector are about $3 \times 10^{4}$ in one second.
Our experimental setup is based on a suitable discrete-time quantum walk [31–33] and we realize the desired single-qubit POVM by measuring the position of the walker in the final step. Generally speaking, the discrete-time quantum walk consists of four elements: the walker, the coin carried by the walker, the coin tossing method and the walking rules. The evolution (U) of quantum walk consists of conditional translation operation (T) and coin operator (C): $U=TC$, where conditional translation operation is realized by beam displacer (BD), and the coin operators in different positions are realized by wave plates with different angles. A one-dimensional discrete-time quantum walk system has two degrees of freedom, the position of the particle $x$ and the coin state $c$. In our experiment, they are encoded in the longitudinal spatial mode and the polarization states $|H\rangle$ and $|V\rangle$ of the single photon respectively. The BD used in our experiment does not displace the vertical polarized photons ($|x, V\rangle \rightarrow |x-1, V\rangle$), but introduce a transverse displacement $|x, H\rangle \rightarrow |x+1, H\rangle$ for the horizontally polarized photon. The position-dependent coin operation operator $C_{x,n}$ for the $n$th step is
3.1 Certification of three-outcome POVM
The experimental setup is shown in Fig. 2. In our experiment, the two initial states are $\left |\Phi _{0,1}\right \rangle =\cos \theta |0\rangle \pm \sin \theta |1\rangle$ and different initial states can be prepared by rotating the angle of HWP1 in the setup. The qubit measurements ($M_{0}$, $M_{1}$, and $M_{2}$) are realized through quantum walk, which consists of three beam displacers and three half-wave plates (HWP2, HWP3, HWP4), by adjusting the angle of these half-wave plates in the optical path, all the input photons pass through the three output ports (0, 1, 2) respectively, and then the coupling efficiency of each port is adjusted so that the detection efficiency difference between them does not exceed $5\%$. In order to implement the certification with different overlaps, we choose three different overlaps in our experiment, namely $\delta =0.5$, $\delta =0.7$, and $\delta =0.9$, then we can obtain different initial states and measurement bases according to the different overlap $\delta$. For example, when $\delta =0.5$, it can be deduced that the two initial states are $\left |\Phi _{0}\right \rangle =\frac {\sqrt {3}}{2}|0\rangle +\frac {1}{2}|1\rangle$ and $\left |\Phi _{1}\right \rangle =\frac {\sqrt {3}}{2}|0\rangle -\frac {1}{2}|1\rangle$ respectively, and the angle of HWP1 is $15^{\circ }$ and $-15^{\circ }$. According to the setting of the coin operator $C$, the optical axes of BD must be aligned, that is, an interferometer is formed between them, and the interference extinction ratio is stable at 2000:1 in our experiment, the stabilization time is about 1h. Once aligned, we start setting the corresponding coin operator in each step. In order to implement the three-outcome POVM, the site-dependent coin rotations for the first three steps are
Fig. 2. Experimental setup. A periodically poled KTiOPO$_{4}$ (PPKTP) nonlinear crystal placed inside a phase-stable sagnac interferometer is pumped by a single mode 405 nm laser to produce single photon pairs at 810 nm. A polarized beam splitter (PBS) and a half-wave plate (HWP) are used to control the polarization mode of the pump beam. A dichroic mirror and a filter are used to remove the pump beam light. The single photon pairs are coupled into single-mode fiber separately, after that one photon in the pair is injected into the optical quantum walk. Each pair of single photons will be detected with single photon detectors (SPDs) and the coincidence is recorded and analyzed by an ID800 (ID Quantique). BD, beam displacer; PBS, polarized beam splitter; IF, interfering filter; Di, dichroic; HWP, half-wave plate.
In our experiment, eight different measurements are realized for a given overlap $\delta$. When $\delta =0.5$ and $\delta =0.9$, the angles of the HWP1 and HWP2 used to prepare the initial states and measurements can be found in Table 1, and the angle of HWP3 and HWP4 is fixed at $45^{\circ }$ and $22.5^{\circ }$. What’s more, we can see the measurement results (the value of $X$ and $Y$) directly in Fig. 3(a) and Fig. 3(c) when $\delta$ is 0.5 and 0.9. It can be intuitively found that the geometric regions of $\mathrm {P}_{2}(\delta )$ and $\mathrm {P}_{3}(\delta )$ are different for a given overlap, which means we can judge whether the measurement device implements genuine three-outcome POVMs directly from the experimental results. The specific experimental results for $\delta =0.5$ and $\delta =0.9$ are shown in Table 2, the experimental results are very close to the theoretical value and the average error is about 0.005. The errors in the experiments mainly come from imperfect wave plates and photon counting statistics.
Fig. 3. Experimental results for the certification of genuine three-outcome POVMs when $\delta =0.5$, $\delta =0.7$, and $\delta =0.9$. We show the geometrical regions of the sets of $P_2(\delta )$ and $P_3(\delta )$ when $\delta =0.5$, $\delta =0.7$, and $\delta =0.9$. The solid line represents the theoretical value and the square represents the experimental results. 3-POVM (2-POVM) stands for three-outcome POVM (two-outcome POVM). The error bars are all due to quantum statistics of photon counts.
Table 1. The angles of HWP used to prepare the initial states and the POVMs when $\delta$ is 0.5, 0.7 and 0.9. $\theta _0$ ($\theta _1$) is the angle of HWP1 used to prepare the initial state $\left |\Phi _{0}\right \rangle$ ($\left |\Phi _{1}\right \rangle$). We implement eight different POVMs for these overlaps and $\alpha$ are angles of HWP2 used to prepare the POVMs, while the angle of HWP3 and HWP4 is fixed at $45^{\circ }$ and $22.5^{\circ }$. 6mm
Table 2. Experimental values ($X$ and $Y$) for the certification of genuine three-outcome POVMs when $\delta =0.5$ and $\delta =0.9$. 4mm
As for the case of $\delta =0.7$, we can find some of the angles of the HWP in Table 1 and all the measurement results can be found in Fig. 3(b), 13 different measurements are used to perform the experiment. As shown in Fig. 3(d), experimental results that do not fall into the regions of $\mathrm {P}_{2}(\delta )$ can clearly demonstrate that our measurements are genuine three-outcome POVMs. The error bars in the experimental results are all due to quantum statistics of photon counts.
3.2 Robustness against noise
Noise can be considered the natural enemy of quantum information. In our experiment, we investigate the noise resistance of the certification under the assumption of a limited overlap between the prepared quantum states. For a given overlap $\delta$, we prepare the optimal value of $\phi$, which characterize the most robust measurement through numerical calculation. For example, when $\delta$ is 0.5 and 0.9, the optimal value of $\phi$ is $\phi =0.4048\pi$ and $\phi =0.378\pi$. At this time, the initial states and measurements are achieved by rotating the angle of HWP1 for $\pm 15^{\circ }$, $\pm 6.46^{\circ }$ and HWP2 for $23.79^{\circ }$, $21.23^{\circ }$.
Here, white noise is introduced in the form of background photons generated by sources of light simulating a realistic operational environment for a quantum communication system. Our method of adding white noise, namely by fixing a constant luminosity light source close to our detectors. As shown in Fig. 4, the most remarkable outcome of this study is the fact that we demonstrate an increased resistance to noise by increasing the overlaps between the prepared quantum states. In other words, the behaviors of the initial states with $\delta =0.5$ will first fall into the regions of two-outcome POVMs with the same noise (Fig. 4(a)), which means initial states with larger overlaps were able to overcome more noise to certify the genuine three-outcome POVMs (Fig. 4(b)).
Fig. 4. Certifying three-outcome POVMs in the presence of noise. (a) The square represents the behavior of $X$ and $Y$ for $\delta =0.5$ without noise, we use the blue circles to show the experimental results with noise. When increasing the intensity of the ambient light in the lab, the behaviors fall into the regions of two-outcome POVMs. (b) When increasing the intensity of the ambient light, the behaviors of the initial states with $\delta =0.9$ still stay in the regions of three-outcome POVMs. To avoid noise, single-photon detectors are placed in black boxes and the average dark counts in each path are about 346/s and the dark coincidence counts are 0/s. When we add white noise, the average dark counts are about 173936/s in each path and the dark coincidence counts are about 230/s.
3.3 Self-testing of the optimal POVMs for USD
Optimal discrimination between two nonorthogonal quantum states is an important task for quantum-information processing. Unambiguous state discrimination allows two nonorthogonal states to be discriminated without error, but with a finite probability of getting a inconclusive result [34–36]. In our experiment, we experimentally show that optimal POVMs for performing unambiguous state discrimination can be self-tested. To achieve unambiguous state discrimination, the elements of the POVMs must be orthogonal to the states. It is calculated that when $\phi = 2\theta$ in Eq. (5), $M_{0}$, $M_{1}$, and $M_{2}$ are exactly the three-outcome POVMs of unambiguous quantum state discrimination. The two nonorthogonal states are prepared with equal probability: $\left |\Phi _{0,1}\right \rangle =\cos \theta |0\rangle \pm \sin \theta |1\rangle$, the site-dependent coin rotations for the first three steps are
In our experiment, to distinguish the two nonorthogonal states, five different overlaps of initial states are selected, namely $\delta =0.1$, $\delta =0.3$, $\delta =0.5$, $\delta =0.7$ and $\delta =0.9$. The corresponding measurement basis are given in Eq. (5) with $\phi = 2\theta$ and the angles of half-wave plates are given in Table 3. We can directly determine which kind of POVM is implemented by the measurement device only by the final experimental data in Fig. 5, then the optimal POVMs for performing unambiguous state discrimination can be self-tested when the maximum success probability $p_{max}=1-\delta$ of USD is achieved in our experiment. Finally, based on our experimental results, we can clearly find that USD can be treated as a general method to certify three-outcome POVMs, which is a good application for the unambiguous state discrimination.
Table 3. The angles of wave plates (HWP1 and HWP2) used to prepare the intial states and perform the unambiguous state discrimination with three-outcome POVMs and two-outcome POVMs, while the angle of HWP3 and HWP4 is fixed at $45^{\circ }$ and $22.5^{\circ }$. 5mm
Fig. 5. Experimental results for the certification of genuine three-outcome POVMs with USD. For a given overlap, if the success probability of USD is larger than $P_{succ,2}$, then we can confirm that the measurement device implements genuine three-outcome POVMs.
4. Conclusion
We investigate the problem of experimentally certifying non-projective measurements in a SDI scenario by making assumptions on the lower bound of the overlap between prepared quantum states. In our experiment, we implement the certification of genuine three-outcome POVMs through quantum walk. Our results not only allow us to certify genuine three-outcome POVMs, but also experimentally confirmed that the certification is robust to noise. We also experimentally demonstrate that the larger the overlap between the quantum states is, the more noise the certification can tolerate. Finally, we show that optimal POVMs for performing unambiguous state discrimination can be self-tested. We believe that our work greatly extend the robust certification of quantum systems in the prepare-and-measure scenario.
Funding
National Natural Science Foundation of China (62105086); Scientific Research Foundation for Scholars of HZNU (4085C50221204030).
Acknowledgments
The authors thank Weixu Shi for the helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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