Experimental optimal verification of three-dimensional entanglement on a silicon chip

High-dimensional entanglement is significant for the fundamental studies of quantum physics and offers unique advantages in various quantum information processing (QIP) tasks. Integrated quantum devices have recently emerged as a promising platform for creating, processing, and detecting complex high-dimensional entangled states. A crucial step towards practical quantum technologies is to verify that these devices work reliably with an optimal strategy. In this work, we experimentally implement an optimal quantum verification strategy on a three-dimensional maximally entangled state using local projective measurements on a silicon photonic chip. A 95% confidence is achieved from 1190 copies to verify the target quantum state. The obtained scaling of infidelity as a function of the number of copies is -0.5497+-0.0002, exceeding the standard quantum limit of -0.5 with 248 standard deviations. Our results indicate that quantum state verification could serve as an efficient tool for complex quantum measurement tasks.


Introduction
An entangled state is an essential resource in a variety of QIP tasks. High-dimensional entangled states are of especial interest, owing to their distinctive properties compared to qubit states. They enable larger channel capacity and better noise tolerance in quantum communication [1][2][3], higher resolution of quantum sensing [4], greater efficiency and flexibility in quantum computing [5,6], and a richer variety of quantum simulations [7,8].

Verification procedure and experimental setup
Here we introduce the quantum state verification protocol. Suppose a quantum device is expected to produce a target state |Ψ⟩, whereas it produces 1 , 2 ,… independently in N runs. The task of QSV is to distinguish either i = |Ψ⟩ for all i, or ⟨Ψ| i |Ψ⟩ > 1 − for all i, i.e., the produced states are far from the target state |Ψ⟩ by ε. Ideally, the optimal strategy is to project i to the space of target state |Ψ⟩ and its orthogonal space. However, this would require entangled measurements in general, especially when the target state is entangled. Protocols based on local projective measurements are more feasible and experiment-friendly in practice. In this work, binary-outcome measurements perform from a set of accessible measurements. Each two-outcome measurement { , 1 − } (j = 1, 2, 3, …) is specified by an operator with probability , satisfying |Ψ⟩ = |Ψ⟩, corresponding to pass the test. The maximal probability that i can pass the test is given by [34] ⟨ | | ⟩≤1− where = is a verification strategy, 2 (Ω) is the second largest eigenvalue of Ω and Δ is the rejection probability of a single test. After N runs, the incorrect state can pass the test with probability being at most [1 − [1 − 2 (Ω)] ] . To guarantee confidence 1 − , the minimum number of N must be Eq.
(2) implies that the optimal strategy within the set of accessible measureme nts is to minimize 2 ( ). Recently, Li et al. [38] and Zhu et al. [45] proposed the optimal protocols for maximally entangled states using complete sets of mutually unbiased bases (MUB  [38,45,52]. Practically, the produced states often have a limited fidelity to the target state, which can result in an occasional rejection within a few amounts measurements, incurring an incorrect conclusion of QSV. As shown in Fig. 1, we consider a more practical protocol called quantum fidelity estimation based on a sufficient amount of copies (N) and use the relative frequency of passing copies (m) to describe the device in a statistical way [37,42,43,52]. For the bipartite states i generated by the practical quantum device, the task is to distinguish the following two cases (also called hypotheses): Case 1: the average fidelity of the quantum device's produced states is larger than 1 − : Case 2: the average fidelity of the quantum device's produced states is less than 1 − : As shown in Fig.1, a conclusion can be drawn as the states are within (σ ∈ ̅ , Case 1) the ε-target circle on average with a certain confidence. Intuitively, when the number of passing tests m is larger than N(1 − ∆ ), the device belongs to Case 1 with high probability, otherwise it would be very unlikely to obtain the experimental data. Quantitatively, the confidence 1 − that the device belongs to Case 1 can be determined using the Chernoff bound [35, 42-44, 49, 50, 52]: where is Kullback-Leibler divergence. Notice that the value measures how unlikely the given data are if Case 2 is true. We generate many copies of i , and apply the optimal conjugate strategy sequentially according to the corresponding projectors. From the measured data, a single coincidence count can be obtained for each randomly chosen measurement setting and decide whether i passes the test or not. We increase the number of copies and obtain the passing probability m/N. The values of δ(ε) are obtained from Eq. (7) given certain ε(δ). . Alice randomly chooses a projective measurement S i (with i = 1, … , d + 1) from the complete set of MUB with probability d + 1, while Bob performs the projective measurement in the conjugate basis. Each measurement returns a binary outcome 1 or 0, associated with pass or fail of the test, respectively. After N runs, the protocol returns m passing outcomes, giving a passing probability of m/N.
Here we experimentally realize the generation and verification of a pair of entangled qutrits on a silicon chip (Fig.2a) encompassing three dual Mach-Zehnder-interferometer rings (DMZI-Rs) [53][54][55]. Signal (1549.5 nm) and idler (1558.3 nm) photons are generated by annihilating two pump photons (1552.1 nm) in non-degenerate spontaneous four wave mixing (SFWM) in the DMZI-R sources, and further separated by on-chip wavelength division multiplexers (WDMs). This produces a three-dimensional state of the form Ψ = ∑ | ⟩| ⟩ 2 =0 , the coefficient can be arbitrarily changed by adjusting pump power and phase over the sources. With a balanced generation rate for all three sources and zero relative phases of the pump, we generate a maximally entangled state of two qutrits |Ψ⟩ = 1 √3 (|00⟩ + |11⟩ + |22⟩), where |0⟩ , |1⟩, and |2⟩ represents the individual path states of single photons. Each qutrit can be locally manipulated by reconfigurable linear optical circuits for implementing arbitrary 3-D unitary operations via three-dimensional multiports (3D-MPs). 3D-MPs are composed of 28 phase shifters and 22 multimode interferometers (MMIs) to realize ( ) and ( ) rotations, respectively.
The full scheme of our experiment is shown in Fig.2b. A tunable picosecond fiber laser produces pulses with 7.8 ps duration, 60.2 MHz repetition rate, and -0.8dBm average power. A bandpass filter with 1.2 nm bandwidth then suppresses the unwanted amplified spontaneous emission (ASE) noise of the laser for ~ 40 dB. Before the light is coupled via transverse electric (TE) grating couplers into the chip, a polarization controller (PC) optimize the polarization of the pump for maximizing fiber-tochip. Off-chip single-channel WDMs filter entangled photons emerging from the chip to remove the residual pump. Six superconducting single photon detectors (SSPDs) detect photons filtered after WDMs with ~80% detection efficiency, ~100 Hz dark count rates, a ~ 50 ns dead time. A field-programmable gate array (FPGA)-based time tag device collects electrical signals of the detector. The losses for the photons in the path entanglement measurement add up to 18.73 dB~19.13 dB. A programmable current source controls all the heaters with a range of 0-20 mA and 16-bit resolution. The coincidence counts between path i (i = 1, 3, 5) and path j (j = 2, 4, 6) are extracted as the experimental results. Finally, we verify the qutrit entanglement by analyzing coincidence counts from the outputs.

Fig. 3
Experimental results on the variation of passing probability ( m/N ) versus the number of copies. The passing probability will reach a stable value 0.9568. The blue symbol is the experimental error bar, which is obtained by 300 trials of measurement.
We give the variation of extracted passing probability versus the number of copies in Fig. 3. The passing probability reaches a stable value 0.9568. The fidelity between σ and |Ψ⟩⟨Ψ|, which can be inferred from the asymptotic passing probability via Eq. (1), is about 94.24% assuming that the number of copies is large enough so that m/N approximates 1 − . This result indicates that the quantum device is stable. In Fig. 4 we present the results of applying QSV to estimate the average fidelity of the quantum states generated by the quantum device. In Fig. 4, δ is obtained by setting infidelity ε as 0.08 (Case 1). Fig. 4 shows that within N=1190 copies, the passing probability / = 0.9563 and δ approaches 0.05. We thus conclude that the generated states have an average fidelity larger than 0.92 with 95% confidence. The values of δ are obtained from Eq. (7) and plotted in log-scale in the insets of Fig. 4. For the estimation of confidence in verifying our target state in Case 1, these few amounts of copies required in our protocol are useful when only limited resources are available in practical quantum device.  4 Experimental results for the QSV. When infidelity ε is set to be 0.08, δ is plotted as a function of the number of copies, where 1-δ is the confidence of the device belonging to Case 1. δ approaches 0.05 within 1190 copies for Case 1. Insets show δ versus the number of copies in log-scale.

Fig. 5
The variation of infidelity parameter vs. the number of copies. The parameter ε is log-log plotted versus the number of copies with δ = 0.05. The blue symbol is the experimental error bar, which is obtained by 300 trials of measurements. The error of the slope is obtained by fitting 100 groups of ε-N data.
In Fig. 5, we set δ = 0.05 and calculate the infidelity ε as a function of the number of copies. In the log-log plot, ε drops fast at small number of copies and converges to 0.0576. The slope obtained from fitting the linear part is approximately -0.5497±0.0002. The error of the slope is obtained by fitting 100 groups of ε-N data. The non-optimal scaling of ε versus 1/N is due to the limited fidelity of the generated state. This scaling obviously exceeds the standard quantum bound of -0.5 by 248 standard deviations [42,43], which is obtained by subtracting the bound (-0.5) from the fitting value (-0.5497) and dividing it by the error (0.0002). This result indicates that QSV is an efficient quantum measurement tool for high-dimensional entangled states. It is worthy to note that the slope can approach the Heisenberg limit of −1 in quantum metrology if the fidelity is further improved [43].

Conclusion
In summary, we experimentally demonstrate the optimal verification of maximally entangled qutrit state generated on a silicon chip with local projective measurements. The variation of confidence and infidelity parameters with the number of copies are presented. We give a comprehensive analysis of the generated state and present a precise estimation of the reliability and stability of the device. The results provide a scaling parameter better than the value of the standard quantum limit. QSV represents a compelling technique for quantifying the prepared state with substantially lower complexity than the quantum state tomography. Measurement settings of chip-based verification protocol can be implemented with programmable and reconfigurable devices. The present procedure for chip-based states verification can be extended to higher-dimensional and multipartite on-chip entangled states, requiring good scalability in the integrated platform. Our work paves the way for the future realization of large-scale chip-based high-dimensional quantum states verification.