Implementation of a Single-Shot Receiver for Quaternary Phase-Shift Keyed CoherentStates

M. T. DiMario,1 E. Carrasco,1 R. A. Jackson,1, 2 and F. E. Becerra1, ∗

1Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 871312Nanoscience and Microsystems Engineering, University of New Mexico, Albuquerque, New Mexico 87131

We experimentally investigate a strategy to discriminate between quaternary phase-shift keyedcoherent states based on single-shot measurements that is compatible with high-bandwidth com-munications. We extend previous theoretical work in single-shot measurements to include criticalexperimental parameters affecting the performance of practical implementations. Specifically, weinvestigate how the visibility of the optical displacement operations required in the strategy impactthe achievable discrimination error probability, and identify the experimental requirements to out-perform an ideal Heterodyne measurement. Our experimental implementation is optimized based onthe experimental parameters and allows for the investigation of realistic single-shot measurementsfor multistate discrimination.

I. INTRODUCTION

Discrimination of non-orthogonal states, such as coher-ent states, cannot be performed perfectly by any deter-ministic measurement [1]. Measurements that minimizethe probability of error are of fundamental interest, andhave applications for information processing as well asquantum and classical communications [2–6]. The useof multiple coherent states to encode information cangreatly increase the amount of information that can betransmitted in a single coherent state, and allow for com-munications with high spectral efficiency [7]. In addition,the non-orthogonality of coherent states makes them use-ful for secure communication via quantum key distribu-tion [8, 9]. However, this non-orthogonality is also detri-mental to optical communication because it causes errorin the measurements that decode the information [1], andlimits the achievable rate of information transfer. Mea-surements for the discrimination of coherent states withsensitivities beyond the limits of conventional technolo-gies have a large potential for optimizing some quantumcommunication protocols [10] and enhancing optical com-munication rates [11, 12].

Discrimination measurements for two coherent stateswith different phases that can achieve sensitivities be-yond conventional measurement limits have also beentheoretically investigated [13–15] and experimentally re-alized [10, 16, 17]. In a strategy proposed by Kennedy[15] for the discrimination of two coherent states withopposite phases {| − α〉, |α〉}, the two input states areunconditionally displaced in phase space to the states |0〉and |2α〉, respectively, and photon counting is used todiscriminate between these displaced states. This dis-crimination strategy achieves error probabilities belowthe optimal Gaussian measurement for two states, cor-responding to the Homodyne measurement [13, 18], fora certain range of input powers. Furthermore, the am-

∗Electronic address: fbecerra@unm.edu

plitude of the displacement operation in phase space canbe optimized to minimize the error probability [13, 16].This strategy, termed the optimized Kennedy strategy,achieves discrimination errors below the Homodyne Limitfor all input powers.

Strategies for the discrimination of the four coherentstates |αk〉 ∈ {|α〉, |iα〉, | − α〉, | − iα〉} have been pro-posed [19–23] and experimentally demonstrated [12, 23–25]. Based on displacement operations, photon count-ing, and fast feedback, these strategies achieve errors be-low the quantum noise limit (QNL) set by an ideal Het-erodyne measurement. On the other hand, single-shotmeasurement strategies for multiple states based on dis-placement operations and single photon counting withoutfeedback [26, 27] are in principle compatible with high-bandwidth communication systems, and may provide ad-vantages for high-speed quantum and classical communi-cations. One such strategy proposed by Izumi et. al. [26]uses single-shot measurements that perform multiple hy-pothesis testing simultaneously by multiple displacementoperations. This strategy is a generalization of the opti-mized Kennedy measurement for four states, where thetotal error probability is minimized with optimized am-plitudes of the displacement operations.

In this work, we experimentally demonstrate the strat-egy for the discrimination of four coherent states withsingle-shot measurements based on the work in Ref.[26].The strategy can also be seen as a “minimum resource”measurement as it does not require feedback or photonnumber resolution, and employs the minimum numberof simultaneous measurements. We utilize an inherentlyphase-stable polarization-based setup to simultaneouslytest multiple hypotheses for the input state. Moreover,here we extend the previous theoretical work to includenon-ideal visibility of the displacement operations, whichquantifies imperfections encountered in any realistic im-plementation. The non-ideal visibility is one of the mostcritical parameters affecting the performance of the statediscrimination process. Including this parameter, we ob-serve an excellent agreement between our experimentalobservations and the theoretical predictions. This new

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FIG. 1: Theoretical Scheme: (a) Input states are preparedin one of four possible states |αk〉 ∈ {|α〉, |iα〉, |−α〉, |− iα〉}.(b) The power of the input state is split into three arms withratios R1, R2, and R3, and each arm is designed to test adifferent hypothesis of the input state by displacement oper-ations and photon counting. Arms 1, 2, and 3 are set to teststates |α〉, |iα〉, and | − α〉, respectively, with optimized dis-

placements D(βi) that minimize the probability of error. Thedetection outcomes of three single photon detectors (SPD)for zero (di = 0) or non-zero (di = 1) photons provide infor-mation about the possible input state. These outcomes areused to obtain a decision rule based on maximum a posterioriprobability criterion, which is shown in (c).

model can be used as a guide for implementations of thissingle-shot multi-state discrimination strategy with cur-rent technologies. In Sec. 2 we describe the theoreticalbackground for the strategy and our analysis includingimperfect experimental parameters. In Sec. 3 we de-scribe the implementation of the measurement and pro-cedures, in Sec. 4 we present our experimental results,and we include our conclusions in Sec. 5.

II. THEORETICAL BACKGROUND

Figure 1 shows the schematic for the discrimina-tion strategy of the four quadrature phase-shift-keyed(QPSK) coherent states with different phases |αk〉 =||α|eiφ〉 (where φ = kπ/2 and k ∈ {0, 1, 2, 3}), as shownin Fig. 1(a), based on multiple hypothesis testing fromRef. [26]. The input state |αk〉 is split into three sepa-rate detection arms as shown in Fig. 1(b), each of which

tests a different hypothesis from the set {|α〉, |iα〉, |−α〉}by displacing the input state and photon counting. Thedisplacement operation D(βi) in arm “i” has a phasethat is equal to the phase of the hypothesis being testedand magnitude set to an optimized value that minimizesthe probability of error. The displacement of the in-put state in each arm D(βi)|αk〉 is followed by photoncounting with single photon detectors (SPD) with anon/off detection scheme [26]. The set of the three SPDoutputs provide eight possible joint detection outcomes:d = {d1, d2, d3} ∈ {(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1),(0, 1, 1), (0, 1, 0), (0, 0, 1), (0, 0, 0)}, where di = 0 cor-responds to detecting zero photons in arm “i” and di = 1corresponds to detecting one or more photons. Based onthe joint detection outcome d, the measurement uses themaximum a posteriori probability criterion as a decisionrule (see Fig. 1(c)). The state with the highest jointposterior probability P (αk|d):

P (αk|d) = P (αk|d1, β1)P (αk|d2, β2)P (αk|d3, β3) (1)

is assumed to be the correct input state. Here P (αk|d)is the conditional posterior probability that the inputstate was |αk〉 given that the detection record d wasobserved. P (αk|di, βi) is the conditional probability forstate |αk〉 given the detection result of di and a displace-ment amplitude βi in arm “i”, calculated through Bayes’theorem: P (αk|di, βi)P (di) = P (di|αk, βi)P (αk). HereP (di) is the probability of observing d and P (αk) arethe initial probabilities of the input states {|αk〉}, whichare assumed to be equiprobable P (αk) = 1

4 . The con-ditional probability of photon detection P (di|αk, βi) isgiven by the Poissonian probabilities:

P (di|αk, βi) =

{e−|αk−βi|2 , di = 0

1− e−|αk−βi|2 , di = 1(2)

The total probability of error for this strategy is [26]:

PE = 1− 1

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P (αk|αk, β1, β2, β3) (3)

where P (αk|αk, β1, β2, β3) is the probability of correctdiscrimination given the input state |αk〉 was displacedby an amount β1, β2, and β3 in arm 1, arm 2, and arm3, respectively. Optimization of the displacement fieldamplitudes |βi| allows for the minimization of the proba-bility of error in Eq. (3) given experimental imperfectionsand splitting ratios R1, R2, and R3 (see Fig. 1b), whichprovide each arm “i” with an optimal fraction Ri|α|2 ofthe total input power |α|2 [26].

Fig. 2(a) shows the numerically calculated optimal dis-placement ratios |βopt,i|2/Ri|α|2 for the three detectionarms. Arm 2 (thin dashed blue line) has a higher opti-mal displacement ratio compared to arms 1 and 3 (solidred line). However, the optimal displacement ratios for

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all three arms converge to a 1:1 ratio as the input meanphoton number 〈n〉 = |α|2 increases. The inset (i) inFig. 2(a) shows the optimal splitting ratio for each armas a function of mean photon number. These ratios startat {R1, R2, R3} = {0.33, 0.33, 0.33} but quickly convergeto about {0.40, 0.20, 0.40}. We note that the state thatarm 2 is testing for (|αk〉 = |iα〉) has the highest overlapwith the other states being tested (|α〉, |−α〉). This maybe why the optimal splitting ratio results in a reducedamount of power that converges to 20% of the total in-put power.

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FIG. 2: Theoretical Optimal Parameters andError Probabilities. (a) Optimal displacement ratios(|βopt,i|2/Ri|α|2) for the three detection arms that minimizethe probability of error. Arms 1 and 3 in a red solid lineand arm 2 in a blue dashed line. The inset shows theoptimal power ratios R1, R2, and R3 in arms 1, 2 and 3,respectively. (b) Error probabilities for different values ofdetection efficiency (η), dark counts (ν) and visibility (ξ)of the displacement operations for fixed splitting ratios of{R1, R2, R3} = {0.40, 0.20, 0.40} in arms 1, 2 and 3, respec-tively. Cases shown: detection efficiency of η = 1.0, darkcount rate of ν = 10−6, and visibility of ξ =0.995 (red solidline); η = 0.80, ν = 10−6, with ξ = 1.0 (green solid line); andη = 1.0, ν = 10−6, with ξ = 1.0 (blue solid line). Thin dot-ted lines of the same color represent zero dark counts ν = 0.The Helstrom bound, corresponding to the ultimate quan-tum limit [1], and Heterodyne Limit (QNL) are also plottedin thick dashed lines.

Following the theoretical work in Ref. [26], we ana-lyzed the discrimination strategy including realistic, non-

ideal detection efficiency (η) and non-zero dark counts(ν). Furthermore, we extend the work in [26] to incor-porate imperfections in the displacement operations byallowing the interference of the two fields (|αk〉, |βi〉) tohave non-ideal visibility (ξ), resulting from mode mis-match and other imperfections encountered in realisticimplementations [12, 23, 24]. In our analysis we considerthree detection arms, since no further improvements inthe strategy are expected by increasing the number ofarms [27]. We only considered fixed values for the split-ting ratios R1, R2, and R3, which are set to the optimalsplitting ratios of {0.40, 0.20, 0.40} in arms 1, 2, and 3,respectively. These splitting ratios are near optimal formean photon numbers larger than 3, which is the regimein which this single-shot strategy is expected to show ad-vantages over the QNL.

Fig. 2(b) shows the probability of error as a functionof mean photon number |α|2 for different values of thevisibility of the displacement operations (ξ), SPD detec-tion efficiency (η), and with or without dark counts (ν)at a rate of ν = 10−6 in each arm for fixed splitting ra-tios of {0.40, 0.20, 0.40}. As discussed in Ref. [26], thissingle-shot strategy with an ideal detection efficiency andvisibility outperforms a Heterodyne measurement at theQNL for |α|2 > 3. Non-zero dark counts reduce the per-formance of this strategy at high |α|2 until it encountersa “noise floor” set by the dark count rate, which appearsto be persistent for situations with reduced detection ef-ficiency (see case with η=0.80 in Fig. 2b).

However, our analysis shows that the visibility of thedisplacement operations is a critical parameter, and evenvery small deviations from ideal visibility severely modifythe performance of the strategy. We observe that evenwith perfect detection efficiency η=1.0, a reduced visi-bility of ξ=0.995 prevents the probability of error frombeing below the QNL. Moreover, reduction in visibilityimposes a stricter “floor” on the probability of error thanthe one from non-zero dark counts. The error comesdown to this floor and then increases slightly with |α|2.For reference at |α|2 = 15, a visibility of ξ = 0.99998, andperfect detection efficiency, the floor of the error proba-bility will still be slightly above the QNL. Other valuesof visibility and detection efficiency define regions in theparameter space where this strategy is expected to out-perform the ideal Heterodyne measurement, as describedin Sec. 4.

III. EXPERIMENT

A. Experimental Setup

Figure 3 shows the diagram of the experimental setup.Coherent states are prepared in 5.42 µs long pulses usinga 633nm power-stabilized HeNe laser together with anacousto-optic modulator (AOM) at a rate of 11.7 kHz.The light pulses are coupled into a single mode fiber(SMF) and then directed to the polarization-based in-

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HypothesisPBS ER 2.03(2)x10-4 3.13(2)x10-4 1.45(1)x10-4

FIG. 3: Experimental Configuration. The coherent statepulses are generated by a 633nm HeNe laser and an acousto-optic modulator (AOM) and then coupled into the experi-mental set-up using a single mode fiber (SMF). A variableattenuator (VA) defines the mean photon number of the in-put state. A half-wave plate (HWP0) and quarter-wave plate(QWP0) define the four possible states in the vertical po-larization which are then split into the three arms with asplitting ratio of {0.40, 0.20, 0.40} with two beam splitters.The input state in the vertical polarization co-propagates withthe strong displacement field (LO) in the horizontal polariza-tion. The displacements of the input state in each arm areperformed by a HWP and a polarizing beam splitter (PBS)

and the photons in the displaced state D(βi)|αk〉 are countedwith SPD’s. The detection record d is collected by an field-programmable gate array (FPGA) which transfers the datato the computer for analysis of the discrimination strategy.The extinction ratio (ER) for each PBSi is shown and for apower ratio of the input state to LO of 1/100 this leads to theobserved visibilities of {0.991(1), 0.990(1), 0.993(1)} in arms1, 2, and 3, respectively.

terferometric set-up. The horizontal polarization is de-fined as horizontal to the optical table. A half-wave plate(HWP0) along with a quarter wave plate (QWP0) areused to prepare the four possible input states |αk〉 ∈{|α〉, |iα〉, |−α〉, |−iα〉} with phases φ ∈ {0, π/2, π, 3π/2},respectively in the vertical (V) polarization. The stronglocal oscillator field |LO〉 is defined in the horizontal po-larization (H). HWP0 is set at an angle relative to thehorizontal so that 1% of the total power is contained inthe input state |αk〉 in the vertical polarization (V) [28]with a mean photon number 〈n〉 = |α|2 that is calibratedwith a transfer-standard calibrated detector [29].

After the input state is prepared, the state is splitinto three detection arms with a 60/40 beam splitter

and a 50/50 beam splitter. In order to obtain the de-sired splitting ratios of {0.40, 0.20, 0.40} in arms 1, 2,and 3, respectively, the beam splitters are rotated tochange the incident angles, which allows for some de-gree of tunability of the splitting ratios, such that the50/50 beam splitter behaves as a 66/33 beam splitter. Byusing this technique, we achieve splitting ratios for thevertical polarization of: {0.400(1), 0.200(1), 0.400(2)}.We implement the optimized displacement operationsD(β1), D(β2), and D(β3) in each arm via polarizationinterferometry by using a HWP and a polarizing beamsplitter (PBS) in each arm [28]. Additional phase plates,not shown, are used in each arm to compensate for polar-ization rotation and birefringence. In arm “i”, the halfwave plate HWPi is set to an angle to produce inter-ference between the input state |αk〉 and the displace-ment field βi(〈n〉) on a polarizing beam splitter PBSiwith a power ratio between these fields being Si(〈n〉) =|βi(〈n〉)|2/Ri|α|2, which correspond to the optimal dis-placements that minimize the probability of error PE inEq. (3). Three SPDs detect the photons in the displaced

states D(βi)|αk〉 in each arm and a field-programmablegate array (FPGA) collects and transfers the joint de-tection result d = {d1, d2, d3} to a computer for anal-ysis of the strategy as discussed in Sec. 2. The detec-tion efficiency, visibility and dark counts for each arm areshown in Fig. 3. The extinction ratios for each PBSi are{2.03(2)x10−4, 3.13(2)x10−4, 1.45(1)x10−4}, and for apower ratio of the input state to LO of 1/100 this leads tothe observed visibilities of {0.991(1), 0.990(1), 0.993(1)}in arms 1, 2, and 3, respectively.

B. Calibration

The implementation of the single-shot discriminationstrategy requires precise calibration of the optimized dis-placement operations in the polarization basis. This cal-ibration involves determination of the visibility of thedisplacement operation, and the angle of HWPi thatprovides the specific power ratio between the input fieldRi|α|2 and the displacement field |βi|2 corresponding tothe optimal displacements in arm “i”. As a first step,we determine the nulling angle ψi of HWPi at whichdisplacement to vacuum is achieved by total destructiveinterference at the output of the PBSi for |α〉 in arm 1,|iα〉 in arm 2, and | − α〉 in arm 3. We then find theHWP angle that provides the level of interference corre-sponding to the optimal power ratio Si(〈n〉) for each armfor different photon numbers. Appendix A describes thecalibration procedure.

Figure 4 shows an example for the calibration of theoptimal displacements in arm 1, which is testing the hy-pothesis that the input state is |α〉. The four curvesshow the normalized intensity after PBS1 for a rangeof angles of HWP1 for each input state with phasesφ ∈ {0, π/2, π, 3π/2}. Here, the red curve correspondsto the case when the input state is |α〉 (φ = 0). A fit

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FIG. 4: Interference in Arm 1. Normalized intensityafter the PBS1 as a function of HWP1 angle in arm 1 test-ing the hypothesis that the input state is |α〉 for the fourpossible input states {|α〉, |iα〉, | − α〉, | − iα〉} with phasesφ = {0, π/2, π, 3π/2}, respectively. Point A corresponds tothe maximum nulling at the nulling angle ψ1 and point D cor-responds to the intensity at the nulling angle ψ1 when sendingthe state with a π phase shift (φ = π). These two points arethe minimum (A) and the maximum (D) of an interferencefringe and are used to estimate the visibility ξ of the displace-ment operation in arm 1. The inset (i) shows the expectednormalized interference as a function of input phase φ withminimum and maximum corresponding to points A and D,respectively. Points B and C correspond to HWP1 being setto the nulling angle ψ1 and sending the states φ = 3π/2 andφ = π/2. These two points are used to check the quality ofstate preparation. Also shown (green squares) are examplesof the angles that correspond to the optimal displacement ra-tios Si(〈n〉) for 〈n〉 = 1, 2, 3. The angle δ1 corresponds to theHWP1 angle which allows only light with vertical polarizationto be transmitted through the PBS1.

of the normalized intensity given by Eq. (A1) to thiscurve provides the determination of the nulling angle ofψ1 ≈ 244.0◦ and the HWP offset of δ1 ≈ 247.2◦. The an-gle δ1 corresponds to the HWP angle which allows onlylight with vertical polarization to be transmitted throughthe PBS1. This information is used to calculate the an-gles θ1(〈n〉) of HWP1 that give the optimal displacementratio of S1(〈n〉) for each mean photon number 〈n〉 (seeAppendix A). The normalized intensities at the corre-sponding angles θ1(〈n〉) for the optimal displacement ra-tios for 〈n〉 = 1, 2, 3 are shown as examples with greensquares along the red curve (φ = 0, |αk〉 = |α〉).

Inset Fig. 4(i) shows the expected normalized interfer-ence in arm 1 after the PBS1 as a function of the phaseof the input state “φ” as a solid black line, and the mea-sured intensities for the actual phases of the input statesφ = 0, π/2, π, 3π/2 corresponding to the points (A, B, C,D) on the interference fringe, respectively, at the nullingangle for arm 1 (ψ1 ≈ 244.0◦). We estimate the visibil-

ity ξ for each arm by using the intensity from the nulledstate at point “A” and the state that is shifted by π,shown by point “D”. We use the difference in intensitybetween sending the input state with phase φ = π/2 ,point “C”, and the input state with φ = −π/2, point “B”,as an indicator to diagnose for imperfections and inaccu-racies in the state preparation by HWP0 and QWP0. Ourpolarization-based experimental setup provides excellentphase stability and high accuracy for the implementationof the optimized displacements with low noise.

IV. RESULTS AND DISCUSSION

Figure 5 shows the experimental results for the imple-mentation of the single-shot discrimination measurementfor the four non-orthogonal states {|α〉, |iα〉, | − α〉, | −iα〉} with optimized displacements implemented with ourpolarization-based setup. Included in the plot are thetheoretical predictions, the Helstrom bound [1], the Het-erodyne Limit (QNL) for ideal detection efficiency, andthe Heterodyne Limit with the same detection efficiencyas our experimental implementation ηavg = 0.778(1),which includes optical losses in the set-up as well as de-tector efficiency in each arm. We observe that our resultslie slightly above the adjusted QNL (77.8% detection effi-

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FIG. 5: Experimental Results. Experimental results forthe error probability as a function of mean photon numberof the input state 〈n〉 = |α|2. Included are the HelstromBound (Green) and Heterodyne Limit (QNL) (Black) as wellas the QNL adjusted for the overall experimental detectionefficiency of 77.8% (Grey). The effect of non-ideal visibil-ity can be seen by the floor that is imposed by the reducedvisibility resulting in an achieved PE of 3.6x10−2 at a meanphoton number of |α|2 ≈ 10. The experimental results are invery good agreement with theoretical predictions (Red). Ournumerical study can provide a guide for future implemen-tations of these single-shot strategies regarding the requiredexperimental parameters needed to outperform a Heterodyne(QNL) measurement.

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FIG. 6: Error probability for non-ideal parameters. Comparison of error probability PE for the non-Gaussian measure-ment with the Heterodyne limit Phet in a logarithmic scale log10(PE/Phet) as a function of visibility and mean photon numberwith (a) detection efficiency η = 1 and (b) η = 90 for ν = 10−6. Negative values of log10(PE/Phet) indicate improvementover Heterodyne, i.e. PE/Phet < 1. Empty areas above red boundaries correspond to values of log10(PE/Phet) > 1. (c) RatioPE/Phet in the parameter space for visibility and detection efficiency for 〈n〉 = 6. Note that it is possible to attain improvementover Heterodyne, PE/Phet < 1, with non-ideal detection efficiency and visibility. The black circle corresponds to a receiverwith η = 98% and ξ = 99.98%.

ciency) for low mean photon numbers and then encounterthe floor which is set by the visibility in each arm as dis-cussed in Sec. 2. Our experimental observations are invery good agreement with the theoretical predictions thatinclude the critical parameters for state discrimination:detection efficiencies, SPD dark counts, and visibilitiesof the displacement operations. This indicates good fi-delities in state preparation and implementations of op-timized displacements in the separate detection arms.

This experimental setup allows for the robust im-plementation of the optimized displacements basedon polarization-based interferometers, and complementsprevious experimental demonstrations with optimizeddisplacements in the context of BPSK state discrimina-tion [10, 16]. Our experimental setup can be used forstudies of more sophisticated strategies with photon num-ber resolving detection [27], which may provide highersensitivities and further robustness for realistic imple-mentations, and could allow for performing polarizationand intensity noise tracking [30]. Furthermore, single-shot discrimination strategies of multiple states with po-larization multiplexing can be used to improve the per-formance of different communication protocols based onmulti-state measurements of coherent states [31–33], andcould potentially be compatible with high-rate and high-spectral-efficiency communications.

Our results validate experimentally the single-shotmulti-state discrimination strategy in Ref. [26] and pro-vide an extension to incorporate the experimental im-perfections resulting in the reduction of the visibility ofthe optimal displacements. As predicted and observedin the experimental results, the visibility of the displace-ment operation significantly affects the performance ofthe strategy. Moreover, our analysis allows us to de-termine what visibility is required to outperform a Het-erodyne (QNL) measurement with detectors that have areduced efficiency and dark counts.

Figs. 6(a) and 6(b) show our investigation of the ex-

pected ratio of the error probability PE for the non-Gaussian measurements to the Heterodyne measurementPhet in logarithmic scale log10(PE/Phet) as a function ofvisibility ξ and mean photon number for different detec-tion efficiencies η with dark counts of ν = 10−6. Blueregions for which log10(PE/Phet) < 0 indicate improve-ment over Heterodyne with PE/Phet < 1, showing thatnon-Gaussian receivers with non-ideal visibility, detec-tion efficiency, and dark counts can provide a real ad-vantage over the ideal Heterodyne. Fig. 6(c) shows theratio PE/Phet in the parameter space of visibility ξ anddetection efficiency η for 〈n〉 = 6 and ν = 10−6. Weobserve that there is a trade-off between ξ and η allow-ing different system configurations for receivers to reachthe regime for which PE < Phet. These studies allow fordetermining the requirements for implementations of re-ceivers with non-ideal detectors and optical componentsto surpass the Heterodyne limit, and can be used to guidefuture demonstrations of these strategies.

As a concrete example, we note that current develop-ments in superconducting detectors have demonstrateddetection efficiencies η ≥ 98% and negligible dark countsν ≈ 0 [34]. In addition, off-the-shelf high extinction ra-tio polarizers can reach extinction ratios of 106:1 [35]or higher [36], resulting in displacement visibilities ofξ ≥ 99.98%. The black circle in Fig. 6(c) correspondsto a non-Gaussian receiver based on these technologieswith η = 98% and ξ = 99.98%. This receiver has anexpected advantage over the Heterodyne of about 20%,with PE/Phet = 0.80, at 〈n〉 = 6. This shows that it ispossible to obtain advantages over the Heterodyne mea-surement based on single-shot measurements with cur-rent technologies.

7

V. CONCLUSION

We experimentally demonstrate a single-shot discrim-ination measurement for multiple non-orthogonal coher-ent states based on the work in Ref. [26], which uses theminimum number of simultaneous measurements and thesimplest photon detection method. Our setup based onpolarization multiplexing provides high phase stabilityand readily allows for arbitrary displacements to be im-plemented with high accuracy using only wave-plates andbeam splitters, and these new techniques can have appli-cations in high-bandwidth communication protocols.

Our setup allows us to investigate the critical exper-imental parameters that affect the probability of error,such as the non-ideal visibility of the displacement op-erations and non-ideal detectors. Our numerical studycan provide a guide for future implementations of thesestrategies to perform multi-state discrimination withsingle-shot measurements below the QNL in realistic sce-narios. We expect that our work will motivate furtherresearch on investigating and demonstrating robust dis-crimination strategies compatible with high-bandwidthfree space quantum communication with multiple statesunder realistic conditions of noise.

VI. ACKNOWLEDGEMENTS

This work was supported by NSF Grant PHY-1653670and PHY-1521016.

Appendix A: Calculation of HWP Angles forOptimal Displacements

The calibration of the HWP angles that perform theoptimal displacements in each arm is achieved by first ob-serving the interference after the PBS while scanning theHWP, and then finding the intensities that correspondto the optimal displacement ratios Si(〈n〉) = |βopt,i|/|αi|in arm “i”. Here |αi| =

√Ri|α|cos(2∆) is the signal am-

plitude after the PBS, ∆ is the HWP angle, and Ri|α|2is the signal power in that arm. The intensity after thePBS for arm “i” as a function of HWPi rotation angle isgiven by:

Ii(α, β, φ,∆) = Ri|α|2cos2(2∆) + |βi|2sin2(2∆) (A1)

−2√Ri|α||βi|sin(2∆)cos(2∆)cos(γ)

Here γ is the relative phase between the input signal|αk〉 and the local oscillator |βi〉 in arm “i”, and ∆ = θ−δicorresponds to the HWP rotation angle θ being shiftedby an offset δi. This offset δi corresponds to the angleof the HWPi that rotates the polarization such that onlythe signal in the V polarization is transmitted throughthe PBSi, which can be seen in Fig. (4) as the pointwhere the intensity is the same for all input states. Theintensity given by Eq. (A1) can be normalized to thepower of the input state Ri|α|2 in the V polarizationbefore the PBSi in arm “i” such that Inormi = Ii/Ri|α|2:

Inormi (α, β, φ,∆) = cos(2∆)2 + f2i sin(2∆)2 (A2)

−2fisin(2∆)cos(2∆)cos(γ)

where |βi|2 is the power of the LO in the H polarizationbefore the PBS in arm “i” and fi is the ratio of LO am-plitude to input amplitude fi = |βi|/

√Ri|α| in arm “i”

before the PBS. Given the parameters fi and δi, the ex-pected intensity Ii after PBSi in arm “i”, and the targetoptimal ratio Si(〈n〉) = |βopt,i|/|αi| of the LO to the in-put field after the PBS for a given 〈n〉, we find the HWPangle θi(〈n〉) that implements the optimal displacementin arm “i” as:

Si(〈n〉) =|βopt,i(θ, δi)||α′i(θ, δi)|

(A3)

=|βi|sin[2(θi(〈n〉)− δi)]√Ri|α|cos[2(θi(〈n〉)− δi)]

= fitan[2(θi(〈n〉)− δi)]

This procedure results is a list of angles θi(〈n〉) forthe HWP in each arm corresponding to the optimal ratioSi(〈n〉) of the two fields for different 〈n〉.

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