Adaptive generalized measurement for unambiguous state discrimination of quaternary phase-shift-keying coherent states

Generalized quantum measurements identifying non-orthogonal states without ambiguity often play an indispensable role in various quantum applications. For such unambiguous state discrimination scenario, we have a finite probability of obtaining inconclusive results and minimizing the probability of inconclusive results is of particular importance. In this paper, we experimentally demonstrate an adaptive generalized measurement that can unambiguously discriminate the quaternary phase-shift-keying coherent states with a near-optimal performance. Our scheme is composed of displacement operations, single photon detections and adaptive control of the displacements dependent on a history of photon detection outcomes. Our experimental results show a clear improvement of both a probability of conclusive results and a ratio of erroneous decision caused by unavoidable experimental imperfections over conventional static generalized measurements.


I. INTRODUCTION
Quantum measurements are ubiquitous in quantum information science. Designing measurements to appropriately discriminate quantum states and efficiently reading out the information encoded in them is of general importance for quantum computation [1,2], sensing [3,4] and communication [5][6][7]. Quantum measurements in the framework of state discrimination can be classified into two major scenarios: minimum error discrimination (MED) and unambiguous state discrimination (USD). For MED, a measurement is designed to minimize the average error in discriminating quantum states [8,9], whereas in USD, the aim is to discriminate quantum states with error-free conclusive decisions but with a finite probability of having inconclusive results [10][11][12]. Measurements capable of unambiguously identifying nonorthogonal quantum states are desirable for applications where exact identification of the quantum states without ambiguity is required. Particular examples are quantum key distribution (QKD) [13][14][15][16] and quantum digital signatures [17][18][19].
The discrimination of coherent states is the principal example where non-trivial and carefully designed quantum measurements can enhance the performance over classical ones [8]. The MED of multiple coherent states has been well investigated, particularly for the binary phase-shift-keying (BPSK) [20][21][22][23][24][25][26] and the quaternary phase-shift-keying (QPSK) coherent state alphabets [27][28][29][30][31][32][33]. It is widely acknowledged that displacement operations combined with photon detection provides a discrimination error that beats the shot noise limit achievable with conventional quadrature detection [24][25][26]. Furthermore, by including feedback control of the displacement operation, adapted by the detection events, it is possible to reach optimal MED performance for BPSK [21] and near-optimal performance for QPSK signals [27][28][29] assuming that the feedback control is infinitely fast. Although such an adaptive measurement scheme is technically challenging, its clear advantage has been exper-imentally observed in several experiments [30][31][32][33][34][35]. As for the USD strategy, generalized measurements enabling USD for M -ary PSK coherent states are realizable with M displacement operations and photon detectors [16]. This simple strategy without the complicated adaptive displacements is known to accomplish the optimal USD performance for BPSK coherent states: It maximizes the probability of identifying the states; in other words, minimizes the probability of inconclusive results [36,37]. On the other hand, in the more general case with multiple coherent states, M > 2, there is a substantial gap between the optimal USD performance and the simple strategy [16,38,39].
In this paper, we propose, theoretically investigate and experimentally demonstrate a generalized measurement scheme that unambiguously discriminates QPSK coherent states with a near-optimal success probability based on photon counting and real-time adaptive control of a displacement operation. We show that, compared to conventional static protocols without adaptive control, our strategy is significantly enhancing the probability of successfully obtaining conclusive results while suppressing the ratio of erroneous decisions induced by unavoidable experimental imperfections.
We first introduce our proposed generalized measurement with adaptive displacement operations in Sec. II. In Sec. III, we present the results of our experimental demonstration and compare them with the conventional static schemes without adaptive control. We conclude the paper in Sec. IV.

II. ADAPTIVE GENERALIZED MEASUREMENT FOR QPSK SIGNALS
We consider the unambiguous discrimination of four coherent states defined as |α m = |α| e (2m+1)iπ/4 where m = 0, . . . , 3 and |α| represents the magnitude of the signal state. This set of coherent states is illustrated in phase space in Fig. 1 ≥ 0. A figure of merit of the USD is the total probability of obtaining conclusive results, where p m is the a priori probability for |α m which throughout this paper is assumed to be identical for the four states, p m = 1/4, and p a|m = α m |Π USD a |α m is the probability of obtaining the (potentially inconclusive) result a given an incoming state |α m . In Fig. 1(b), we show a transition diagram for |α 0 . The state is successfully identified without ambiguity if measurement results corresponding to the POVM elementΠ USD 0 are obtained (the red arrow). Some measurement results will be inconclusive. These are represented by the solid black arrow. In an ideal USD measurement, the decisions k = 1, 2, 3 represented by the dashed arrows will never occur, that is, p k|m = 0 for k = m. In practice, however, experimental imperfections will inevitably induce a certain amount of ambiguity in the measurement, resulting in non-zero p k|m . The error probability will be our other figure of merit and is defined as the ratio of these erroneous conclusions to the total conclusive result probability: Our strategy for unambiguous discrimination of the non-orthogonal QPSK signals consists of beam splitters, displacement operations and single photon detectors (SPDs). A schematic of this protocol is depicted in Fig. 1(c). An input coherent state |α m is equally split by the beam splitters into M states |γ m with γ m = α m / √ M . Each of the states are then displaced and detected by an SPD. The displacement operations, , are implemented such that one of the QPSK states is displaced to the vacuum state. The SPD is capable of discriminating whether there exists at least one photon ("on") or not ("off"). These two outcomes are described by the POVM {Π off = e −ν ∞ n=0 (1 − η) n |n n| ,Π on =Î −Π off }, where ν is the dark count rate and η is the detection efficiency, assumed to be the same for all SPDs. Therefore, the probability of having an "off" outcome when the displacementD(−γ i ) is performed on a state |γ m is = exp −ν − 2η |α| 2 M (1 − ξ cos((m − i)π/2)) . For "on" it is P (on|m; i) = 1 − P (off|m; i). Here, we introduced the visibility of the displacement operation denoted as ξ. The dark count and the visibility imperfection both lead to false conclusive results since we may obtain an "on" event even when the incoming state is displaced to the vacuum state, i.e. i = m. A simple static USD strategy for the QPSK states can be performed with 4 stages (M = 4), where the displacement operations are performed with four different phases. This strategy corresponds to the shaded area in Fig. 1(c). Conclusive results are obtainable if any three of the SPDs give the outcome "on"; otherwise the result is inconclusive. The probability of having conclusive results for the USD measurement with M = 4 can be analytically obtained from eq. (3) and the decision scheme to be where P s = P (off|m; i) when i = m ± s mod 4, that is P s = e −ν−(1−(1−s)ξ)η|α| 2 /2 . While this simple strategy enables us to unambiguously discriminate the QPSK coherent states (assuming no dark counts ν = 0 and perfect visibility ξ = 100%), even for an ideal detection efficiency (η = 100%), the probability of having such conclusive results is significantly lower than what is achievable by the optimal USD measurement [38] (See Fig. 2(a) for M = 4).
We now turn to our proposed scheme for improving the probability of conclusive results. Here, we increase the number of splittings to some M > 4 and maintain the static structure outlined above for the first four modes. For the remaining modes, though, the choice of displacement phase should now be dynamically adapted to the outcomes of all the previous SPD measurements. This introduces a temporal ordering of the modes which we can therefore consider as different stages of the receiver. See Fig. 1(c). The displacement operation is set to displace the hypothetical state |γ m to the vacuum state in cyclic order of m = 0 → 2 → 1 → 3 → 0 → · · · . Detecting a photon eliminates the possibility of receiving the hypothetical state and the displacement is subsequently set to test for other hypothetical states in the following stages. Hence, the displacement condition at the j'th stage is determined according to the counting history up to the j − 1'th stage. For example, the hypothetical state to be tested at the fifth stage, m 5 is set to be the first of the states in the cycle which hasn't yet been ruled out. Finally, results are conclusive if and only if any three out of the M SPDs detect photons. Our adaptive strategy continues the displacement with a fixed hypothetical condition even after three "on" events are obtained and the results are regarded as inconclusive if a fourth "on" event occurs due to the visibility imperfection or the dark count. It is worth noting that we may be able to improve the probability of conclusive results by making the final conclusion based on Bayesian inference of the received state instead of regarding some cases as inconclusive results.
We compare achievable probabilities of conclusive results in Fig. 2(a). Green and black solid curves represent the conclusive probabilities for the simple 4-stage strategy, with perfect visibility and no dark count, and the optimal ideal USD measurement [38], respectively. Our strategy offers a major improvement to the static scheme, potentially closing most of the gap towards the optimal USD as evidenced by the red solid curve for M = 10 and the black dashed curve for M = 100 in a perfect visibility and no dark count condition. These probabilities are evaluated by Monte Carlo simulations. Although the analysis of the performance in the asymptotic limit is not straightforward, our numerical analysis indicates that the probability of conclusive results is saturated around 100 stages. We further evaluate the probabilities of conclusive results for various visibility conditions, ξ = 99.8 (short dashed), 99.6 (long dashed) and 99.4% (dotted) with the finite dark count ν = 1.0 × 10 −3 . In the large mean photon regime, the performance of the adaptive strategy with M = 10 is degraded because of the visibility imperfection while it is not critical for M = 4. Since we continue the measurement after having three "on" events and conclude the result as inconclusive if a fourth "on" event is obtained, the probability of having more than three "on" events increases, which reduces the probability of conclusive results. A similar USD strategy relying on the adaptive displacement was discussed in [16], where an analytical expression of the conclusive probability was derived and the conclusive probability of its asymptotic limit M → ∞ shows a similar performance with our scheme with M = 100.
In Fig. 2(b), we show the error probabilities evaluated for the same values of M , ξ and ν as in Fig. 2(a). Ev-idently, the visibility imperfection is quite detrimental in terms of the error probability. However, the adaptive strategy, while not only improving the conclusive result probability, can also significantly reduce the error, as shown here for M = 10. This reduction can be explained with similar reasoning as above: Instead of deciding on conclusive results after three "on" events, we continue to perform the displacement operation with a fixed hypothetical condition for the rest of the stages for confirmation. This reduces the probability of erroneously having conclusive results. For very low mean photon numbers, the achievable error probability is mostly determined by the dark count rate.
Finally, we note that unambiguous discrimination can be emulated by heterodyne measurement if one is willing to accept the finite error of having conclusive results. We will discuss the performance of the heterodyne measurement strategy and compare it with the performance of the photon detection protocol including experimental imperfections later in this paper.

III. EXPERIMENT
In the previous section, we suggested to realize the adaptive measurement strategy by splitting the input state into multiple spatial modes. However, the measurement strategy can also be carried out in a temporal mode version where the state is virtually split into M time bins and the displacement is successively updated in time through real-time feedback [22]. For the experimental realization, we chose the latter strategy as it only requires a single spatial mode, a single displacement operation and a single photon counter. Our experimental setup is shown in Fig. 3. A continuous-wave laser at 1550 nm is split into two paths in order to prepare a signal state and a strong reference field for the displacement operation. The intensity and the phase of the QPSK signal state are controlled with a variable attenuator and a phase shifter consisting of a piezo transducer embedded in a circular mount with an optical fiber looped around. The signal state interferes with the reference field for the displacement operation at a 99:1 fiber coupler, which enables us to implement the displacement operation. By using an optical switch, the laser intensity is switched between high and low for the phase calibration and the data acquisition, respectively. The relative phase between the signal and the reference is set to one of the four phase conditions (2m+1)π/4 (m = 0−3) by monitoring an output of the 99:1 fiber coupler using a conventional photo detector. During the data acquisition period, the output of the 99:1 fiber coupler is guided to a superconducting nanowire single photon detector (SSPD) [40,41] and, instead of randomly preparing the QPSK coherent states, 300 identical signal states are measured after releasing the phase stabilization loop. This phase calibration and data acquisition procedure is repeated 20 times for each of the QPSK states. The voltage applied to the phase modulator, corresponding to the phase condition of the reference field, is controlled by a field programmable gate array (FPGA) dependent on the counting history of the SSPD. The FPGA implements the adaptive strategy using a hard-coded lookup-table. Due to the finite bandwidth of the digital-to-analog converter that generates a signal to control the phase modulator, we discard a time interval 0.3 µs between each time bin [32]. This corresponds to a 4.5(1.5)% discarding loss for M = 10(4) since the temporal width of the signal state is defined to be 60 µs in our experiment.
We achieve a total transmittance of 91% from the 99:1 fiber coupler to the fiber right before the SSPD by splicing all optical fibers. The SSPD provides a detection efficiency around 73% and a dark count of about 27 Hz. Therefore, our system is able to accomplish a total system efficiency of about 66%. For calibration of the system detection efficiency η SE , a laser beam, propagating along the signal path, is divided into two paths, where one is used to monitor the laser power to estimate the power propagating along the other path. The laser power is attenuated to the photon level by inserting a cascade of well calibrated optical fiber filters such that the SSPD can detect photons without being saturated. The total system efficiency is evaluated by comparing the observed count rate η SE |α| 2 with the expected photon count rate |α| 2 , where the latter is estimated from the laser power and the filter attenuation. The total system detection efficiency can be estimated with around 1.5% uncertainty including the finite precision of calibrating the filters, the uncertainty in the splitting ratio of the 50:50 fiber coupler and other systematic errors.
Experimental results for the probability of having conclusive results are shown in Fig. 4(a) and (b). Filled circles are associated with the experimentally obtained results. We evaluate the mean values and the error bars of the experimental results from 5 independent measurement runs. Green and red represent measurements for the static case of M = 4 and the adaptive case for M = 10, respectively. The signal mean photon number |α| 2 is estimated by measuring the attenuated mean photon number, η SE |α| 2 , directly by the SSPD, and then compensating for the losses. We measure the attenuated mean photon number and the total detection efficiency multiple times before and after the data acquisition to evaluate the error bars on the mean photon number. The theoretically attained maximum probability of conclusive results using the optimal USD strategy is shown by the black solid line. Dashed lines represent the theoretical predictions for our measurement strategy assuming η SE = 66%, ξ = 99.4%, ν = 1.5 × 10 −3 and the discarding loss because of the finite bandwidth of the digitalto-analog converter. The experimental results show that our adaptive strategy is able to significantly enhance the probability of conclusive results. Moreover, the good agreement between the experimental results and the theory indicate that our system is well-controlled. We also investigate the performance of our measurement using an imaginary SPD with unit detection efficiency. Due to the finite transmittance of our system, the maximum achievable system efficiency is limited to η SE = 91%. The expected performance with such an ideal SPD is shown by open circles in Fig. 4(a) while the red and green solid lines represent the theoretical prediction with system efficiency of η SE = 100%, perfect visibility and no dark count. Indeed, state-of-the-art SPDs with detection efficiency of more than 93% at the telecom wavelength have been reported [42] and, by installing such SPDs, our measurement will be able to further reduce the gap between the optimal USD bound and the physically implementable USD measurement.
Furthermore, we evaluate error probabilities in Fig. 5(a) and (b). Experimental results and theoretical predictions under the experimental condition are shown by filled circles and dashed lines. A few discrepancies are observed between the theory and the experimental results which is caused by instabilities of the interference at the displacement operation. The shaded regions in Fig. 5(b) correspond to the predicted error probabilities when using a heterodyne measurement that is set to attain probabilities of conclusive results that are above the ones attained by the photon detection scheme corresponding to the filled circles in Fig. 4. For the heterodyne measurement, the USD is emulated by applying thresholding in phase space and subsequently post processing the obtained outcomes. The POVM for inconclusive results can be represented asΠ ? = x,p∈S |x x| ⊗ |p p| dxdp for some region S of phase space. There exists a tradeoff between the probabilities associated with errors and conclusive results and therefore the post processing is optimized such that the error probability is minimized while achieving a certain probability for a conclusive result. We adapt a linear thresholding approach for simplicity, where the POVM isΠ ? = x th −x th p th −p th |x x|⊗|p p| dxdp and x th = p th . More advanced post processing can be performed by numerically optimizing the region S, but the improvement over the linear thresholding strategy is very small [39,43]. Our adaptive measurement strategy shows a clear improvement of the error probability over the conventional simple protocol with M = 4 for all |α| 2 , and moreover, it beats the heterodyne strategy in the low photon number regime up to approximately |α| 2 = 2.5.
To demonstrate the improvement in performance when increasing the number of detection stages, we show in Fig. 6 the results of the conclusive (a) and the error (b) probabilities for detectors with varying number of stages. We performed the measurements for two different fixed mean photon numbers, |α| 2 = 1.5 (red) and |α| 2 = 3.0 (green). Theoretical predictions under the given experimental condition with η = 66%, ξ = 99.55%, ν = 1.5 × 10 −3 as well as the discarding loss for delay compensation, are represented by crosses. Shaded areas correspond to the error probabilities for the hetero- dyne measurement under the condition that the probabilities of conclusive results are larger than the experimentally obtained probabilities shown by the filled circles in Fig. 6(a). Red and green colors are associated with the mean photon number |α| 2 = 1.5 and |α| 2 = 3.0, respectively. Both the probabilities of conclusive results and of errors can be improved by increasing the number of stages but the improvement saturates (or even slightly degrades) for a large number of stages since the discarding loss becomes dominant as the number of stages increases.

IV. CONCLUSIONS
We proposed and experimentally realized an adaptive generalized quantum measurement that unambigu-ously discriminate quaternary phase-shift-keying coherent states with a near-optimal performance. Our strategy consists of a displacement operation, a single photon detector and real-time adaptive control of the phase space displacements that depend on the history of single photon detection outcomes. We demonstrated the adaptive generalized measurement and, while the performance is degraded due to the finite efficiency of our system, we observed a clear improvement of the probability of having conclusive results in comparison with a simple scheme using a non-adaptive approach. Furthermore, we evaluated the probability of erroneously obtaining conclusive results which is caused by the non-perfect interference contrast of the displacement operation as well as the dark counts. By increasing the number of detection stages, the error probability can be suppressed, yielding better performance than a heterodyne measurement designed to reach a comparable probability of conclusive results for a wide range of signal mean photon numbers.
Since adaptive phase space displacements based on photon detections provide near-optimal performance for the minimum error discrimination of multiple coherent states in addition to the unambiguous state discrimination, it is expected to serve as a novel receiver technique in applications associated with classical coherent communication [44] as well as quantum communication, in particular quantum key distribution [45]. By performing the adaptive strategy, we observe the improvement of the probability of conclusive results while suppressing the error probability due to the dark count and visibility imperfection. However, a heterodyne measurement designed to approach the comparable conclusive results probability yields a better error probability in the large mean photon number regime. Here, we analyze the error probability achievable by our strategy with an SPD under various detection efficiency conditions and compare them with the heterodyne detection strategy.
Since our system has a finite transmittance of 91% from the signal preparation point to the SSPD, the maximum achievable total system efficiency is limited to 91%. In Fig. 7(a), we plot the ratio of the error probabilities of our strategy and the heterodyne measurement as a function of the quantum efficiency of the SPD for the mean photon numbers |α| 2 = 1.5 (filled circles) and |α| 2 = 3.0 (open circles). Green and red colors indicate the results for M = 4 and M = 10, respectively. The actual experimental condition corresponds to the detector efficiency of 73%, which gives a total system efficiency of 66%. To obtain Fig. 7(a), we first normalize the experimentally measured mean photon number with the detector efficiency of our SSPD (73%) and multiply by the detector efficiency of the imaginary SPD, and then choose the mean photon number closest to the |α| 2 = 1.5 and |α| 2 = 3.0 from a list of re-scaled mean photon numbers. Using the mean photon number and corresponding probability of conclusive results, we calculate the error probability for the heterodyne detection strategy in each detector effi-ciency condition. For |α| 2 = 1.5, a detector efficiency of more than 70% is sufficient to beat the heterodyne limit by using the adaptive strategy with M = 10. On the other hand, for the simple static strategy with M = 4, a detection efficiency of more than 90% is required to surpass the heterodyne limit. For |α| 2 = 3.0, the static protocol cannot beat the heterodyne limit even if we employ an ideal SPD with unit quantum efficiency. On the other hand, by using a state-of-the-art SPD with a detector efficiency of more than 80%, our adaptive detection strategy is able to outperform the ideal heterodyne detector.
Furthermore, we investigate the ratio of the error prob-abilities as a function of the mean photon number, both using the actual experimental condition with a detector efficiency of 73% (filled circles) and the ideal condition of a unit detector efficiency (open circles). These results are shown in Fig. 7(b). Our analysis indicates that our adaptive measurement strategy provides an error probability that is lower than the error probability of the heterodyne detector in a wide range of mean photon numbers if an SPD with high efficiency is available. On the other hand, it is clear that the simple static strategy is not able to beat the heterodyne measurement limit for large mean photon numbers.