# Demonstration of Optimal Non-Projective Measurement of Binary Coherent
States with Photon Counting

M. T. DiMario  Center for Quantum Information and Control, Department of
Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131
Joint Quantum Institute, National Institute of Standards and Technology and
the University of Maryland, College Park, Maryland 20742  F. E. Becerra
<EMAIL_ADDRESS>Center for Quantum Information and Control, Department of
Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131

###### Abstract

Quantum state discrimination is a central problem in quantum measurement
theory, with applications spanning from quantum communication to computation.
Typical measurement paradigms for state discrimination involve minimum
probability of error or unambiguous discrimination with minimum probability of
inconclusive results. Alternatively, an optimal inconclusive measurement, a
non-projective measurement, achieves minimal error for a given inconclusive
probability. This more general measurement encompasses the standard
measurement paradigms for state discrimination and provides a much more
powerful tool for quantum information and communication. Here, we
experimentally demonstrate the optimal inconclusive measurement for the
discrimination of binary coherent states using linear optics and single photon
detection. Our demonstration uses coherent displacement operations based on
interference, single photon detection, and fast feedback to prepare the
optimal feedback policy for the optimal non-projective quantum measurement
with high fidelity. This generalized measurement allows us to transition among
standard measurement paradigms in an optimal way from minimum error to
unambiguous measurements for binary coherent states. As a particular case, we
use this general measurement to implement the optimal minimum error
measurement for phase coherent states, which is the optimal modulation for
communications under the average power constraint. Moreover, we propose a
hybrid measurement that leverages the binary optimal inconclusive measurement
in conjunction with sequential, unambiguous state elimination to realize
higher dimensional inconclusive measurements of coherent states.

##  I Introduction

Quantum measurement theory provides a fundamental understanding of the limits
on the achievable sensitivity for distinguishing quantum states [1, 2, 3].
Physically realizable strategies that attain, or even approach, the ultimate
sensitivity limits for distinguishing nonorthogonal coherent states have a
wide range of applications in optical communication [4, 5, 6, 7, 8, 9],
cryptography [10, 11, 12, 13, 14, 15, 16, 17], and quantum information
processing [18, 19, 20]. A central problem in quantum measurement theory and
quantum information processing is the discrimination between two quantum
states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ with a certain optimal
measurement given an optimality criterion, depending on the specific
application [2, 21, 22].

Two fundamental measurement paradigms for quantum state discrimination involve
either minimum error or unambiguous state discrimination. Minimum-error state
discrimination (MESD) aims to achieve minimal probability of error
$P_{\mathrm{E}}$ [23, 24, 25, 26, 27, 28, 29, 1, 31, 32, 33, 34]. The Helstrom
bound [24] gives the ultimate limit for $P_{\mathrm{E}}$, which is achieved by
projective measurements onto complex superpositions of quantum states.
Notably, the optimal MESD measurement for binary coherent states can be
realized with linear optics, single photon detection, and fast feedback [35,
36]. In contrast, unambiguous state discrimination (USD) allows for perfect
discrimination with $P_{\mathrm{E}}=0$, but requires a non-zero probability of
inconclusive results $P_{\mathrm{I}}\neq 0$. Such a non-projective measurement
is described by a positive operator-valued measure (POVM) with three elements
[37, 2, 38], and aims to achieve the smallest possible $P_{\mathrm{I}}$ [39,
40, 41, 42, 43, 44, 45, 46, 47]. The realization of optimal USD of binary
coherent states does not require feedback [48, 49, 12], allowing for simpler
implementations [50, 45] compared to optimal MESD.

While optimal projective measurements exist for certain binary discrimination
tasks [24, 2, 51, 52], quantum measurement theory allows for a broader class
of generalized quantum measurements which are not projective. These
generalized measurements provide a more powerful tool for quantum information
processing and communications [2]. Among these general quantum measurements,
the optimal inconclusive measurement achieves the smallest possible error
probability for a fixed probability of inconclusive results [53, 37]. This
measurement is a non-projective measurement, and thus described by a non-
projective POVM, that encompasses MESD and USD measurement paradigms.
Moreover, non-projective quantum measurements allow for more exotic
discrimination tasks such as quantum state elimination [54], state comparison
[55, 56, 57], and discrimination with a fixed error margin [58]. Furthermore,
understanding optimal inconclusive measurements for binary states may provide
a path for realizing arbitrary non-projective POVMs in a two dimensional
Hilbert space [59, 60].

Figure 1: (a) Schematic of the generalized receiver for the optimal
inconclusive measurement. The input states are displaced in phase space using
an optimal waveform $u(t)$ for the LO field and followed by a single photon
detector (SPD) and feedback operations. (b) Optimal waveform magnitudes
$|u(t)|$ for different mean photon numbers (top panel). The bottom panel shows
an example of the waveform $u(t)$ for a particular measurement record where
the phase switches between $0$ and $\pi$ every photon detection (Photon
Detections, middle panel). The circles along the $x-$axis show the current
hypothesis for the input state as the measurement progresses. (c) Error
probability $P_{\mathrm{E}}$ for the optimal inconclusive measurement as a
function of specified probability of inconclusive results $P_{\mathrm{I}}$ for
$|\alpha|^{2}$=0.2, 0.4, and 0.6. The colored circles along the $y$-axis
($P_{\mathrm{I}}=0$) correspond to smallest possible $P_{\mathrm{E}}$ in the
paradigm of MESD, and the colored squares along the $x$-axis
($P_{\mathrm{E}}=0$) correspond to the smallest possible $P_{\mathrm{I}}$ in
the paradigm of USD. (d) Experimental setup used for demonstrating the optimal
inconclusive measurements of binary coherent states.

Theoretical work on quantum measurement theory has shown that it is possible
to realize an optimal inconclusive measurement for a broad class of quantum
states based on local operations and classical communications [58, 61, 62].
However, the corresponding measurement operators for the discrimination of
optical coherent states do not necessarily have a feasible physical
realization. While suboptimal inconclusive measurements of coherent states can
be realized based on linear optics and single photon detection [63], their
performance falls short of the performance of the optimal inconclusive
measurement. Recent work in Ref. [4] proposed a physical realization of a
strategy for the optimal inconclusive measurement of binary coherent states.
It was shown that such a non-projective measurement can be realized using
displacement operations, single photon detection, and feedback [4], which are
the same physical elements needed for implementing arbitrary binary projective
measurements [65, 52].

In this work, we experimentally demonstrate the optimal inconclusive
measurement for binary coherent states [4]. The measurement splits the energy
of the input state into two temporal modes. It performs a MESD measurement in
the first mode providing conclusive results with a certain probability of
error, and an optimal inconclusive measurement in the single-state domain in
the second mode determining whether the measurement result is inconclusive.
Our demonstration uses low noise, high bandwidth real-time feedback
conditioned on single photon detections to prepare the optimal displacement
operations required for the optimal inconclusive measurement. We further use
this generalized optimal measurement to realize the optimal MESD for phase
coherent states, which is the optimal modulation for optical communications
under the average power constraint, thus demonstrating the optimal quantum
receiver for coherent optical communications. Lastly, we show that the binary
optimal inconclusive measurement enables the realization of inconclusive
discrimination of three coherent states when used together with measurements
for unambiguous state elimination based on with hypothesis testing. This
proposed method can in principle be extended to high dimensional inconclusive
measurement strategies of coherent states.

##  II Results

###  II.1 Optimal Inconclusive Measurement

The optimal inconclusive measurement is a non-projective quantum measurement
that encompasses the MESD and USD paradigms and optimizes the tradeoff between
errors and inconclusive results [53, 37]. By construction, the optimal
inconclusive measurement achieves the minimal error probability
$P_{\mathrm{E}}$ for a specified inconclusive probability $P_{\mathrm{I}}$ [4,
2]. A feasible realization of the POVM for the optimal inconclusive
measurement $\\{\hat{\Pi}_{1},\hat{\Pi}_{2},\hat{\Pi}_{?}\\}$ for binary
coherent states was recently proposed in [4]. Notably, this optimal non-
projective measurement can in principle be realized by a generalization of the
optimal receiver for MESD, called the Dolinar receiver. This optimal MESD
receiver is based on displacement operations in phase space implemented by
interfering the input state with a local oscillator (LO) field, single photon
detection, and feedback with an optimal feedback policy. The displacement has
a magnitude with an optimal waveform and a phase conditioned to photon
detection [35, 36, 66].

Figure 1(a) shows the concept of the optimal inconclusive measurement. The
input state $|\pm\alpha\rangle$ and strong LO field interfere on a high
transmittance beam splitter to implement a displacement operation
$\hat{D}(u(t))$. The receiver implements the optimal displacement waveform
$u(t)$, where the phase of the LO switches between 0 and $\pi$ based on the
photon detection outcomes from the single photon detector (SPD) during the
measurement time. In the proposal for the optimal inconclusive discrimination
strategy [4], the generalized receiver performs optimal measurements in two
temporal modes during the measuring time $0\leq t\leq 1$ using displacement
operations, single photon detection, and feedback. In the first temporal mode
($0\leq t\leq t_{1}$), the receiver performs an optimal MESD measurement to
discriminate between $\\{|\pm\alpha\rangle\\}$ with minimal error using the
optimal displacement waveform [35, 36, 66, 4]. In the second temporal mode
($t_{1}<t\leq 1$), the receiver performs an optimal inconclusive measurement
in the so-called single-state domain, where the measurement becomes a
projective measurement such that the POVM element for the least probable is
zero, e.g. $\hat{\Pi}_{2}=0$ [4, 58]. Without loss of generality, the most
probable state after the first mode is $|\alpha\rangle$, and the non-zero POVM
elements are $\hat{\Pi}_{1}$ and $\hat{\Pi}_{?}$. Therefore, the receiver in
the second temporal mode attempts to determine whether the measurement result
is inconclusive, i.e. the receiver realizes a MESD measurement between the
correct and inconclusive outcomes. Reference [4] shows that this projective
measurement in the second temporal mode (single-state domain) can be realized
by a Dolinar-like receiver with a different optimal displacement waveform,
which is the key element that we leverage for demonstrating the optimal
inconclusive measurement. The total displacement waveform $u(t)$ that
implements the optimal inconclusive measurement is given by [4]:

$u(t)=\begin{cases}\frac{(-1)^{N_{1}(t)}\alpha}{\sqrt{1-4p(1-p)K^{2t}}}&0\leq
t\leq t_{1}\\\
\frac{(-1)^{N_{2}(t)+N_{0}}\alpha}{\sqrt{1-4v(1-v)K^{2(t-t_{1})}}}&t_{1}<t\leq
1.\end{cases}$ (1)

Where $N_{1}(t)$ and $N_{2}(t)$ are the total number of photons detected up to
time $t$ for the first and second temporal mode, respectively, and
$N_{0}\in\\{0,1\\}$ based on $|\alpha|^{2}$, $P_{\mathrm{I}}$, and $p$ [4].
The total optimal waveform $u(t)$ is comprised of $u_{1}(t)$ and $u_{2}(t)$,
each of which are optimal for the two temporal modes (see Methods Section I
for details). The magnitude of $u(t)$ is predetermined based on the values of
$|\alpha|^{2}$, $P_{\mathrm{I}}$, and $p$, but the sign of $u(t)$ (phase of
the LO) is adaptively switches between positive and negative (LO phase of 0
and $\pi$) each photon detection due to the $(-1)^{N_{1}(t)}$ and
$(-1)^{N_{2}(t)+N_{0}}$ terms.

Figure 2: (a) Experimental results for the optimal inconclusive measurement
for $|\alpha|^{2}$=0.2, 0.4, and 0.6, in blue, orange, and yellow,
respectively. Each point corresponds to the measured values of
$\\{P^{\mathrm{exp}}_{\mathrm{I}}(1),P^{\mathrm{exp}}_{\mathrm{E}}(1)\\}$ and
the error bars represent one standard deviation from five sets of $5\times
10^{4}$ individual experiments each. The solid lines show the expected results
and the dashed gray lines show the ideal performance for each $|\alpha|^{2}$.
The colored circles and squares on the $y$-axis and $x$-axis show the optimal
$P_{\mathrm{E}}$ and $P_{\mathrm{I}}$ for ideal MESD and USD, respectively.
Inset (i): evolution of $P^{\mathrm{exp}}_{\mathrm{E}}$,
$P^{\mathrm{exp}}_{\mathrm{C}}$, and $P^{\mathrm{exp}}_{\mathrm{I}}$ for
$|\alpha|^{2}$=0.2 and and $P^{\mathrm{exp}}_{\mathrm{I}}$=0.31. (b)
Experimental results (blue points) of an optimal MESD measurement, the Dolinar
receiver, for phase coherent states $\\{|\pm\alpha\rangle\\}$. The gray and
red solid lines show the Helstrom bound for $\eta=1.0$ and $\eta=0.72$,
respectively, and the dashed lines show the corresponding error for a homodyne
measurement.

The top panel of Fig. 1(b) shows the displacement magnitude $|u(t)|$ for the
optimal inconclusive strategy with inconclusive probability
$P_{\mathrm{I}}=0.19$ for $|\alpha|^{2}$ = 0.2, 0.4, and 0.6. The discrete
jumps in $|u(t)|$ for each $|\alpha|^{2}$ correspond to the time $t_{1}$ when
the receiver switches between the measurements two temporal modes. The
receiver implements a minimum-error measurement with a Dolinar receiver during
$0\leq t\leq t_{1}$ with $u_{1}(t)$ between $|\pm\alpha\rangle$. The receiver
then implements the optimal inconclusive measurement in the single-state
domain $\\{\hat{\Pi}_{1},\hat{\Pi}_{?}\\}$ with a Dolinar-like receiver during
$t_{1}<t\leq 1$ with $u_{2}(t)$. The final outcome of the measurement is
either an inconclusive result with probability $P_{\mathrm{I}}$, a correct
discrimination result with probability $P_{\mathrm{C}}$, or an error with
probability $P_{\mathrm{E}}=1-P_{\mathrm{C}}-P_{\mathrm{I}}$ [4]. The bottom
panel of Fig. 1(b) shows the displacement amplitude $u(t)$ for an example
measurement record. The provisional hypothesis (circles) and the phase of the
waveform change each time a photon is detected. The red dashed line
($t_{1}\approx 0.70$) shows where the receiver switches from MESD of the two
input states in the first temporal mode, to MESD between the more likely state
given the current detection record and the inconclusive outcome in the second
temporal mode.

Figure 1(c) shows the resulting probabilities
$\\{P_{\mathrm{I}},P_{\mathrm{E}}\\}$ of the optimal inconclusive measurement
for equiprobable coherent states $\\{|\pm\alpha\rangle\\}$ with $|\alpha|^{2}$
= 0.2, 0.4, and 0.6, in blue, orange, and yellow, respectively. The colored
circles along the $y$-axis ($P_{\mathrm{I}}=0$) correspond to the smallest
possible $P_{\mathrm{E}}$ in the paradigm of MESD (Helstrom Bound), and the
colored squares along the $x$-axis ($P_{\mathrm{E}}=0$) correspond to the
smallest possible $P_{\mathrm{I}}$ in the paradigm of USD (sometimes referred
to as the IDP Bound [39, 40, 41]). Thus, the optimal inconclusive measurement
is the generalization of MESD and USD and interpolates between these
measurement paradigms in an optimal way using more general non-projective
measurements. In general, optimal inconclusive measurement for the
discrimination of two general quantum states
$\\{|\psi_{1}\rangle,|\psi_{2}\rangle\\}$ is represented by three POVM
elements $\\{\hat{\Pi}_{1},\hat{\Pi}_{2},\hat{\Pi}_{?}\\}$ where a positive
outcome of $\\{\hat{\Pi}_{1,2}\\}$ indicates that the state
$|\psi_{1,2}\rangle$ is present, and
$\hat{\Pi}_{?}=\hat{I}-\hat{\Pi}_{1}-\hat{\Pi}_{2}$ corresponds to an
inconclusive result. Optimality indicates that this non-projective measurement
achieves the minimum error for a fixed probability of inconclusive outcomes.

While the proposed implementation of this non-projective quantum measurement
in Ref. [4] is in principle feasible, its demonstration requires a high degree
of control for the preparation of optimal waveforms with high fidelity, and
the ability to realize feedback measurements with high bandwidth and low noise
(see Supplementary Note I). Moreover, the validation of optimal performance
requires absolute power measurements at the single photon level. In our
experimental demonstration we address the issues to satisfy these stringent
requirements, which allows us to demonstrate experimentally this complex
quantum measurement with high fidelity. Figure 1(d) shows our experimental
setup for the demonstration of the optimal inconclusive measurement for binary
coherent states. We use an interferometric setup to generate the input states
and local oscillator field, a single photon detector (SPD), and an FPGA
(Altera Cyclone IV, 50 MHz base clock) connected to a digital-to-analog
converter (DAC) to implement the required optimal displacement waveform $u(t)$
for the optimal inconclusive measurement using fiber-coupled amplitude (AM)
and phase (PM) modulators (see Methods Sections B, C, and E for details). We
actively stabilize our interferometer using a second 780 nm laser and a
feedback loop to maintain a well-defined relative phase (see Methods Section C
for details). Our implementation achieves an overall detection efficiency
$\eta=0.72(1)$ ($\eta=\eta_{\mathrm{SPD}}\eta_{\mathrm{sys}}$ where
$\eta_{\mathrm{SPD}}=0.82(1)$ is the SPD efficiency and
$\eta_{\mathrm{sys}}=0.88(1)$ is the system transmittance), interference
visibility $\xi=0.998(1)$, and dark counts $\nu=0.03(1)$ per pulse. The
experiment operates at a 4 kHz repetition rate, alternating between
experimental trials (1024 time bins, 160 ns each) and interferometer
stabilization with a $\approx 66\%$ duty cycle. We have also realized
numerical investigations of the effects of realistic imperfections described
in Supplementary Note I. Based on these studies, we observe that reduced
detection efficiency degrades the achievable performance for all error and
inconclusive probabilities. Reduced interference visibility and increased dark
counts mainly degrades the performance of strategies where the desired
$P_{\mathrm{E}}$ or $P_{\mathrm{I}}$ are small, i.e. near the MESD and USD
regimes.

###  II.2 Demonstration of the Optimal Inconclusive Measurement

We implement the optimal inconclusive measurement for equiprobable coherent
states. In our experimental demonstration, we obtain the time evolution of the
error $P^{\mathrm{exp}}_{\mathrm{E}}(t)$, correct
$P^{\mathrm{exp}}_{\mathrm{C}}(t)$, and inconclusive
$P^{\mathrm{exp}}_{\mathrm{I}}(t)$ probabilities by reconstructing the results
in post-processing, and compare them to the expected probabilities
$\\{P_{\mathrm{E}}(t),P_{\mathrm{C}}(t),P_{\mathrm{I}}(t)\\}$ [4]. The final
inconclusive $P^{\mathrm{exp}}_{\mathrm{I}}(t=1)$ and error
$P^{\mathrm{exp}}_{\mathrm{E}}(t=1)$ probabilities for a given $|\alpha|^{2}$
correspond to a single realization of the optimal inconclusive measurement.
Figure 2(a) shows the experimental results for $|\alpha|^{2}$ = 0.2, 0.4, and
0.6 in blue, orange, and yellow, respectively. The points show the
experimental data
$\\{P^{\mathrm{exp}}_{\mathrm{I}}(1),P^{\mathrm{exp}}_{\mathrm{E}}(1)\\}$ and
the error bars represent one standard deviation from five experimental runs of
$5\times 10^{4}$ independent experiments each. The black lines show the
theoretical expectation from Monte-Carlo simulations of the experiment
incorporating experimental imperfections (See Supplementary Note IV). We note
that we obtain the expected performance of our demonstration by directly
simulating the experiment including experimental imperfections and effects
without requiring ant fitting procedures. The dashed gray lines show the ideal
($\eta=1$) performance for each mean photon number. The colored circles and
squares on the $y$-axis and $x$-axis show the optimal $P_{\mathrm{E}}$ and
$P_{\mathrm{I}}$ for ideal MESD and USD, respectively, for each
$|\alpha|^{2}$.

We observe that our demonstration of the optimal inconclusive measurement with
$\eta=0.72$ for $|\alpha|^{2}$ = 0.2, 0.4, and 0.6 reaches errors below the
ideal Helstrom bound when $P_{\mathrm{I}}\gtrapprox 0.18$. This shows that a
non-ideal implementation of the optimal inconclusive measurement can surpass
the ideal Helstrom bound at the expense of having inconclusive results111We
note that while the Helstrom bound is the minimum discrimination error that
can be achieved by a deterministic measurement, this bound is not the lowest
error for a general quantum measurement that allows for inconclusive results
[53]. As such, the optimal inconclusive measurement then allows for errors
below the Helstrom bound for $P_{\mathrm{I}}\neq 0$, and achieves zero error
at a rate of inconclusive results given by the IDP bound [39, 40, 41].. The
inset of Figure 2(a) shows an example of the evolution of $P_{\mathrm{E}}(t)$
(blue), $P_{\mathrm{C}}(t)$ (orange), and $P_{\mathrm{I}}(t)$ (yellow) as the
measurement progresses for $|\alpha|^{2}=0.2$ and $P_{\mathrm{I}}\approx
0.31$. The solid lines show the theoretical expectation including experimental
imperfections and the points show the experimental results for
$P^{\mathrm{exp}}_{\mathrm{E}}(t)$, $P^{\mathrm{exp}}_{\mathrm{C}}(t)$, and
$P^{\mathrm{exp}}_{\mathrm{I}}(t)$ every 50 time bin steps. Note that the
measurement switches from a MESD measurement to an optimal inconclusive
measurement in the single-state domain at $t_{1}\approx 0.57$.

###  II.3 Optimal MESD of Binary Phase States

The optimal inconclusive measurement generalizes MESD and USD [4], and can be
used to demonstrate the optimal MESD measurement, the Dolinar receiver [35],
by setting $P_{\mathrm{I}}=0$. Previous work [36] demonstrated a Dolinar
receiver for intensity-modulated coherent states
$\\{|0\rangle,|\alpha\rangle\\}$ and achieved performance below the shot noise
after correcting for system losses and detection efficiency. However, phase
encoded coherent states $\\{|\pm\alpha\rangle\\}$ are the optimal modulation
for binary coherent communications under the energy constraint. This is
because this alphabet has the smallest overlap, and therefore the highest
distinguishability, for a fixed average energy of the states [67, 68, 48]. To
this end, we use the optimal inconclusive measurement to demonstrate a Dolinar
receiver for the phase encoded binary coherent states, and thus demonstrate
the optimal quantum receiver for coherent optical communications.

Figure 2(b) shows the experimental results (blue points) and the expected
error probability (solid black) for the optimal MESD measurement, the Dolinar
receiver, for phase coherent states together with the Helstrom (solid) and
homodyne limits (dashed). The Helstrom bound and the homodyne limits corrected
to our overall efficiency ($\eta=0.72$) are included for reference. We observe
that our demonstration of the Dolinar receiver approaches the corrected
Helstrom bound and shows an excellent agreement with the theoretical
predictions (solid black line). We note that our implementation of the optimal
MESD measurement for BPSK states with overall efficiency of $\eta=0.72$
achieves $P_{\mathrm{E}}=0.18$ for $|\alpha|^{2}=0.2$, which is below the
ideal homodyne limit that corresponds to the optimal Gaussian measurement for
BPSK [67]. This error rate is similar to the one achieved by a sub-optimal
receiver without feedback in Ref. [27] using a high efficiency ($\eta=0.99$)
superconducting detector resulting on an overall system efficiency of
$\eta=0.91$. Thus, we conclude the strategy demonstrated here based on complex
adaptive measurements can potentially provide overall higher sensitivities
than the sub-optimal strategy under the same loss and realistic experimental
noise and imperfections. In principle the optimal inconclusive measurement
also allows for construction of the optimal USD measurement where
$P_{\mathrm{E}}=0$. However, the experimental imperfections such as dark
counts and non-ideal interference visibility prevent the receiver from
achieving $P_{\mathrm{E}}=0$, see Fig. 2(a). Nevertheless, the above framework
allows for finding the optimal waveform to implement this optimal measurement.

###  II.4 Higher Dimensional Inconclusive Strategies

Figure 3: Inconclusive discrimination of the ternary phase-shift keyed (TPSK)
states coherent states using the binary optimal inconclusive measurement. (a)
The proposed inconclusive measurement for TPSK states uses sequential
unambiguous state elimination, followed by the optimal binary inconclusive
measurement (See main text for details). (b) Conditional error probability
$P_{\mathrm{E}}/(1-P_{\mathrm{I}})$ as a function of inconclusive probability
$P_{\mathrm{I}}$ for $|\alpha|^{2}=$ 0.2 (blue), 0.4 (orange), and 0.6
(yellow). We compare the proposed measurement with the parameter $f$ = 0.66
(solid) and $f$ = 0.90 (dashed) to the performance of ideal heterodyne
detection (dotted). See text for details.

We investigate how to leverage the optimal inconclusive measurement of binary
coherent states to enable inconclusive state discrimination of higher
dimensional encodings. We propose a hybrid measurement that utilizes binary
optimal inconclusive measurements in conjunction with unambiguous state
elimination, which can realize such a non-projective inconclusive measurement
of ternary phase-shift keyed (TPSK) $\\{|\alpha\rangle,|\alpha
e^{i2\pi/3}\rangle,|\alpha e^{i4\pi/3}\rangle\\}$, and can be extended to
higher dimensions. This measurement first aims to eliminate all but two
possible input states via hypothesis testing, and then utilize the optimal
inconclusive measurement in the reminding binary states. Figure 3(a) shows the
measurement operations conditioned to single photon detection, realized by the
proposed high-dimensional inconclusive measurement. The receiver realizes an
elimination measurement based on hypothesis testing (red region, Fig.
3(a-$i$)) for the state $|\alpha\rangle$ on a fraction $f/3$ of the total
input state. This state elimination measurement is based on displacement
operation of $|\alpha\rangle$ to the vacuum state $|0\rangle$ and single
photon detection, such that detection of a photon unambiguously eliminates
$|\alpha\rangle$ as a possible input state. If a photon is detected in the
first stage (Stage 1), the receiver then performs an optimal inconclusive
measurement (blue region, Fig. 3(a-$i$)) to optimally discriminate between
$|\alpha e^{i2\pi/3}\rangle$ and $|\alpha e^{i4\pi/3}\rangle$ using the
remaining fraction $1-f/3$ of the input energy. If no photons are detected
during the first stage, the receiver then realizes a state elimination
measurement for the input state $|\alpha e^{i2\pi/3}\rangle$ also using a
fraction $f/3$ of the total input power (Fig. 3(a-$ii$)) . Now if a photon is
detected in the second stage (Stage 2), an optimal inconclusive measurement
discriminates between the remaining two possible input states using a fraction
$1-2f/3$ of the input power, where the factor of 2 comes from the first stage.
If no photons are detected in the second stage, then the receiver tests for
the state $|\alpha e^{i4\pi/3}\rangle$ in Stage 3, also using a fraction $f/3$
of the total input state (Fig. 3(a-$iii$)) . If a photon is detected, an
optimal inconclusive measurement discriminates between the remaining two
states now using a fraction $1-3f/3$ of the input power. If no photons are
detected in the third hypothesis test for unambiguous state elimination, we
define the measurement outcome to be inconclusive.

Figure 3(b) shows the simulation results for the proposed inconclusive
measurement of TPSK states based on the optimal inconclusive measurement for
binary states for mean photon numbers $|\alpha|^{2}=$ 0.2, 0.4, and 0.6. The
$x$-axis corresponds to the inconclusive probability and the $y$-axis
corresponds to the conditional error probability
$P_{\mathrm{E}}/(1-P_{\mathrm{I}})$, i.e. the error probability given that a
conclusive outcome was obtained. The solid and dashed lines show the proposed
inconclusive measurement of three coherent states with $f=0.66$ and $f=0.90$,
respectively. The dotted lines show the result
$P^{\mathrm{Het}}_{\mathrm{E}}/(1-P^{\mathrm{Het}}_{\mathrm{I}})$ for using
ideal heterodyne detection, where measurement outcomes with the largest error
probability are designated as inconclusive until the desired inconclusive
probability is achieved as in [45]. The optimal limits for USD and MESD for
three states are represented with squares and circles in the x-axis
($P_{\mathrm{E}}=0$) and y-axis ($P_{\mathrm{I}}=0$), respectively [69, 49].

The overall $P_{\mathrm{I}}$ achieved by this strategy contains two
contributions $P_{\mathrm{I}}=P^{(1)}_{\mathrm{I}}+P^{(2)}_{\mathrm{I}}$,
where $P^{(1)}_{\mathrm{I}}$ comes from the state elimination stage, and
$P^{(2)}_{\mathrm{I}}$ from the binary optimal inconclusive measurement. In
the state elimination stage, the detection of vacuum during all three
hypothesis tests results in an inconclusive outcome as each state is equally
likely. This produces a lower bound ($P^{(1)}_{\mathrm{I}}$) on the attainable
inconclusive probability $P_{\mathrm{I}}$ depending on the values of $f$ and
$|\alpha|^{2}$. In the binary optimal inconclusive stage, the proposed
measurement defines a “target” inconclusive probability
($P^{(2)}_{\mathrm{I}}$) which can be set for any $|\alpha|^{2}$ to achieve
the desired overall $P_{\mathrm{I}}$ of the strategy. We observe that the
proposed measurement with both values of the parameter $f$, which
parameterizes the unambiguous state elimination stages, can outperform
heterodyne detection. Moreover, we note that a smaller value of $f=0.66$
achieves a smaller error probability, but this also puts a limit on the
smallest attainable inconclusive probability of $P_{\mathrm{I}}\approx$0.76,
0.58, and 0.45 for $|\alpha|^{2}=$ 0.2, 0.4, and 0.6, respectively. On the
other hand, a larger value of $f=0.90$ (dashed lines) allows for a smaller
inconclusive probability but at the cost of a larger error probability
compared to $f=0.66$ (solid lines). This trade-off is due to the fact that a
larger value of $f$ results in a smaller contribution to $P_{\mathrm{I}}$ from
the inconclusive outcome during the state elimination stage (detecting vacuum
during all stages). However a larger value of $f$ results in a smaller
fraction of the total input energy of the state for the binary optimal
inconclusive measurement, which results in a larger error probability. Then,
the optimal choice of energy fraction $f$ will depend on the particular
application of this proposed measurement. For example, if we are willing to
tolerate more inconclusive results $P_{\mathrm{I}}$ to achieve a given small
target error threshold $P_{\mathrm{E}}$ such as for communications with error
detection, correction, and erasures [70], we should choose a small value of
$f$.

The proposed inconclusive measurement for three states can be extended to
higher dimensions. Using this technique, an inconclusive measurement of $M$
input coherent states can be realized through the implementation of $M-1$
hypothesis testing stages for unambiguous state elimination [49, 45], followed
by the binary optimal inconclusive measurement. Given that the binary optimal
inconclusive measurement can always achieve $P_{\mathrm{E}}=0$ for
$P^{(2)}_{\mathrm{I}}<1$, there is always a range of error probabilities for
which this strategy will outperform heterodyne detection (note
$P^{\mathrm{Het}}_{\mathrm{E}}=0$ only when
$P^{\mathrm{Het}}_{\mathrm{I}}=1$), i.e. there is always an error regime where
$P_{\mathrm{I}}<P^{\mathrm{Het}}_{\mathrm{I}}$. Ideally, this performance is
achieved at the smallest possible value of $P_{\mathrm{I}}$, which will depend
on the number of possible states (See Supplementary Note III).

We note that the proposed measurement uses similar techniques in the state
elimination stage as the Bondurant receiver for MESD of multiple states [71],
and the USD receiver based on feedback and state elimination [48]. However,
the proposed strategy in here makes use of the optimal inconclusive
measurement of two states for allowing to transition between the measurement
paradigms of MESD and USD to realize an optimized inconclusive measurement of
multiple coherent states. While the performance of the strategy for more
states will degrade due to the increased inconclusive probability in the state
elimination stage, we expect that this strategy will serve as the basis for
designing optimized inconclusive strategies in higher dimensions. A possible
example for inconclusive measurement strategies could use hybrid measurement
schemes combining Gaussian measurements, such as homodyne, with photon
counting [72, 73]. In these schemes, the Gaussian measurement can eliminate a
sub-set of the states, and photon counting would be used for state elimination
in a smaller sub-set of states followed by the binary optimal inconclusive
measurement.

##  III Discussion

Optimal inconclusive measurements are generalized quantum measurements that
encompass standard paradigms of state discrimination including MESD and USD.
These non-projective measurements allow for diverse state discrimination tasks
and provide a more powerful tool for classical and quantum information
processing [63, 17]. In optical communication, inconclusive measurement
results can be treated as an erasure channel, and optimal inconclusive
measurements can be leveraged to increase the amount of information transfer
by utilizing communication codes well suited for erasure channels [70]. These
optimal inconclusive measurements can also enable hybrid repeater schemes
where inconclusive measurements of coherent states are used to entangle remote
quantum memories [74, 75]. Recent advancements in quantum measurement theory
showed that such complex quantum measurements for binary coherent states can
be realized using single photon detection and local operations and classical
communication in a two-mode measurement [4]. This measurement strategy splits
the energy of the input state into two temporal modes. It performs a MESD
measurement of the input state in the first mode with a certain probability of
error, and an optimal inconclusive measurement in the single-state domain in
the second mode determining whether the measurement result is inconclusive.
The optimality of this measurement makes it possible to achieve minimal error
for a given inconclusive probability. Moreover, such generalized quantum
measurements can be realized with Dolinar-like optimal receivers for coherent
states.

Here, we experimentally demonstrate the optimal inconclusive measurement
proposed in [4]. Our demonstration uses coherent displacement operations,
single photon detection, and fast feedback to implement these general non-
projective quantum measurements with high fidelity in a real system. We
further use this measurement to demonstrate the optimal MESD for phase-encoded
binary coherent states, which is the optimal modulation for optical
communications under the average power constraint. While our proof of
principle demonstration of the optimal inconclusive measurement was realized
at moderate measurement rates, future implementations based on integrated
photonics with high-bandwidth optical modulation and processing [76] within
small footprints, together with advancements in high-bandwidth integrated
nanowire detectors will allow for demonstrations at GHz bandwidths. These
results show that Dolinar-like receivers can be used to perform a wide variety
of measurements within a two dimensional Hilbert space with current
technologies. Furthermore, we show how the binary optimal inconclusive
measurement can be leveraged to perform inconclusive measurements in higher
dimensions with hybrid measurements using sequential unambiguous state
elimination of multiple states. Our work contributes to our understanding of
the fundamental and practical limits of measurements based on single photon
detection, coherent displacement operations, and feedback, and can further our
understanding of quantum measurement theory [59]. Moreover, these measurement
techniques can potentially allow for implementations of more general non-
projective measurements in two dimensional spaces using linear optics and
single photon detection.

##  IV Methods

###  IV.1 Optimal Displacement Waveform

The optimal inconclusive measurement
$\\{\hat{\Pi}_{1},\hat{\Pi}_{2},\hat{\Pi}_{?}\\}$ for binary coherent states
can be realized with a generalized Dolinar receiver [4] . In this modified
strategy [4], the generalized receiver performs optimal measurements in two
temporal modes during the measuring time $0\leq t\leq 1$. In the first
temporal mode, $0\leq t\leq t_{1}$, the optimal inconclusive receiver performs
an optimal MESD measurement to discriminate between $\\{|\pm\alpha\rangle\\}$
with minimal error using the optimal displacement waveform [35, 36, 66, 4]:

$u_{1}(0\leq t\leq
t_{1})=\frac{(-1)^{N_{1}(t)}\alpha}{\sqrt{1-4p(1-p)K^{2t}}},$ (2)

Here $K^{2}=|\langle-\alpha|\alpha\rangle|^{2}=e^{-4|\alpha|^{2}}$, $p$ is the
prior probability of the most likely state, and $N_{1}(t)$ is the total number
of detected photons in the first mode up to time $t\leq t_{1}$, where
$N_{1}(0)=0$. Note that during the first temporal mode the phase of the LO
displacement field (sign of $u_{1}(t)$) switches between 0 and $\pi$ each time
a photon is detected, similar to the Dolinar receiver [77]. During the
measurement in the first temporal mode, the provisional hypothesis for the
input state at time $t$ is $|\alpha\rangle$ if $N_{1}(t)$ is even and
$|-\alpha\rangle$ if $N_{1}(t)$ is odd. The provisional probabilities for the
two input states after the first temporal mode are
$\\{P_{\mathrm{C}}^{(1)},1-P_{\mathrm{C}}^{(1)}\\}$ with:

$P_{\mathrm{C}}^{(1)}=\frac{1}{2}\Big{(}1-\sqrt{1-p(1-p)e^{-4t_{1}|\alpha|^{2}}}\Big{)},$
(3)

which corresponds to the Helstrom bound for the coherent states
$\\{|\pm\sqrt{t_{1}}\alpha\rangle\\}$. The optimal waveform $u_{1}(t)$ in Eq.
(2) and the evolution of t he probability of correct detection
$P_{\mathrm{C}}(t)$ at $t$ can be obtained using Bayesian updating [78, 79, 4]
or optimal control [66].

In the second temporal mode ($t_{1}<t\leq 1$), the receiver performs an
optimal inconclusive measurement in the so-called single-state domain, where
$P_{\mathrm{I}}$, $P_{\mathrm{C}}^{(1)}$, and $(1-t_{1})|\alpha|^{2}$ are such
that $\hat{\Pi}_{2}=0$ [4, 58]. Without loss of generality, the most probable
state is $|\alpha\rangle$, and the non-zero POVM elements are $\hat{\Pi}_{1}$
and $\hat{\Pi}_{?}$. This optimal inconclusive projective measurement in the
single-state domain can be realized by a Dolinar-like receiver with optimal
displacement waveform [4]:

$u_{2}(t_{1}<t\leq
1)=\frac{(-1)^{N_{2}(t)+N_{0}}\alpha}{\sqrt{1-4v(1-v)K^{2t}}},$ (4)

where $p$ in Eq. (2) is replaced by the quantity $v$, which depends on
$P_{\mathrm{I}}$, $p$, and $|\alpha|^{2}$ [4]. $N_{2}(t)$ is the number of
photons detected in the second mode with $N_{2}(t_{1})=0$, and $N_{0}$
determines the phase of the LO at $t_{1}$: $N_{0}=0$ if $v>0.5$ and $N_{0}=1$
otherwise.

The total displacement waveform for the optimal inconclusive receiver is thus
a combination of $u_{1}(t)$ in Eq. (2) and $u_{2}(t)$ in Eq. (4) resulting on
the total optimal displacement in Eq. (1) in main text. This strategy
therefore implements a standard Dolinar receiver during the first mode $0\leq
t\leq t_{1}$, and then a Dolinar-like receiver during $t_{1}<t\leq 1$ assuming
the input states at $t=t_{1}$ have prior probabilities $\\{v,1-v\\}$ [4].

###  IV.2 Experimental Setup Details

In our experimental demonstration, optical pulses are generated from a Helium-
Neon laser and a pulsed acousto-optic modulator (AOM), and then split into the
signal arm (upper) and LO arm (lower), as shown in Figure 1(d). The input
states are prepared with an attenuator (Att.) and a phase modulator (PM). The
LO field is prepared by a PM with a multiplexer (MUX), and an amplitude
modulator (AM) with a digital-to-analog converter (DAC). The input state and
the LO field interfere on a 99/1 beam splitter (BS) to implement the optimal
displacement waveforms $\hat{D}(u(t))$ conditioned on photon detection events
using a single photon detector (SPD). A field programmable gate array (FPGA)
stores the magnitude of the optimal waveform $|u(t)|$ in Eq. (1) in memory,
prepares the amplitude and phase of the LO conditioned on $N_{1}(t)$,
$N_{2}(t)$, and $N_{0}$, and implements the strategy for the optimal
inconclusive measurement. We discretize time $t$ into 1024 time bins of 160 ns
each where a photon can be detected to approximate a continuous measurement.
Our implementation achieves a feedback bandwidth of about 6 MHz, which is
limited by the APD output latency, electronic bandwidth of controllers,
switches and FPGA, accounting for about 50 ns and optical delays in the
interferometric setup (100 ns). The FPGA processes and stores the photon
detections during these time bins and sends the detection histories to a
computer. We reconstruct the measurement probabilities in post-processing. The
optimal inconclusive measurement requires very large values of the ratio
between mean photon numbers of the displacement field and the input state,
$R=|u(t)|^{2}/|\alpha|^{2}$. However, experimentally there is a maximum ratio
$R$ that can be reliably implemented. In our demonstration, we set the maximum
of this ratio to $R=50$, which is limited by the extinction ratio ($\approx
20$ dB) of the AM in the LO arm of the setup. The impact of finite values for
$R$ and other experimental imperfections are discussed in the Supplementary
Notes I & II.

###  IV.3 FPGA Implementation

We use an Opal Kelly ZEM4310 to control the experiment, which is based on an
Altera Cyclone IV FPGA and has a base clock rate of 50 MHz. We discretize the
discrimination measurements into 1024 time bins of 160 ns each such that a
single shot of the experiment corresponds to a pulse that is 163.8 $\mu$s
long. The magnitude of the LO waveform for each of the 1024 time bins is pre-
calculated for each $|\alpha|^{2}$ and inconclusive probability
$P_{\mathrm{I}}$ and stored in a look-up table within the FPGA as an 8-bit
value. The phase of the LO flips between 0 and $\pi$ each time a photon is
detected. This method for preparing the optimal LO waveform given by Eq. (1)
allows us to efficiently implement the desired optimal inconclusive
measurement in our demonstration.

The optimal inconclusive strategy requires precise and fast control of the LO
phase. We control the phase of the LO by changing the voltage applied to the
phase modulator between two values which correspond to a phase of 0 and a
phase of $\pi$. Each time the phase of the LO changes, we ignore the output of
the APD for 160 ns to avoid accidental photon detections. This “blanking time”
is obtained by noting that the combined electrical and optical delay time
between changing the modulation voltage and observing the corresponding
photons at the APD is approximately 150 ns. This also has the benefit of
reducing the effective after-pulsing probability to close to zero. Typically,
the probability of detecting an after-pulse is at its maximum immediately
after the dead-time of the APD ($\approx 40$ ns for our implementation), but
this probability quickly decays with time. We note that without any blanking,
the cumulative after-pulsing probability of our APD is $P_{\mathrm{AP}}\approx
0.015$ and $P_{\mathrm{AP}}<0.001$ with 100 ns of blanking

###  IV.4 Interferometer Stabilization

In order to maintain a well-defined relative phase between the signal and LO
fields, we actively stabilize the interferometer. We run the experiments with
a 4 kHz repetition rate to give an experimental duty cycle of $\approx 66\%$
(experiment time of $\approx$ 165 $\mu$s, locking time of $\approx$ 91
$\mu$s). During the part of the experimental duty cycle when the experiment is
not taking place, the relative phase between the two arms of the
interferometric setup actively stabilized with a feedback loop using a PID
controller and a piezo on the back of a mirror in the signal arm, see Fig. 2.
We obtain the error signal for stabilization of the interferometer using a
narrow-band laser at 780 nm, which is actively stabilized in frequency to an
atomic line in rubidium using saturated absorption spectroscopy. Light from
this laser propagates in an opposite direction through the interferometer
compared to the light at 633 nm, and is detected with a differential detector
to measure the phase fluctuations. Slightly before the discrimination
measurement begins, the feedback loop is paused and voltage to the piezo is
fixed at its current value. The stabilization feedback loop resumes after the
discrimination measurement is completed.

###  IV.5 Optical Modulators

The setup uses fiber coupled Lithium-Niobate, amplitude and phase electro-
optic modulators (AM and PM), with a 3 dB bandwidth of $\approx 1$ GHz. The
phase modulators (PMs) have a $\pi$-voltage of $V_{\pi}=1.5$ V and the
amplitude modulator (AM) has a $\pi$-voltage of $V_{\pi}=750$ mV with an
extinction ratio of $\approx 20$ dB. The amplitude and phase of the LO and the
phase of the signal fields are adjusted using three 8-bit digital-to-analog
converters (DAC), voltage-controlled gain circuits, and summing amplifiers.

ACKNOWLEDGEMENTS
This work was supported by the National Science Foundation (NSF)
(PHY-1653670).

AUTHOR CONTRIBUTIONS
F.E.B. supervised the work. M.T.D. designed the experimental implementation
and performed the measurements. All authors contributed to the analysis of the
theoretical and experimental results, conceived the idea of generalizing
inconclusive measurement to higher dimensions, and contributed to writing the
manuscript.

COMPETING INTERESTS
The authors declare that there are no competing interests.

DATA AVAILABILITY
The data that support the findings of this study are available from the
authors upon request.

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Supplementary Material: Demonstration of Optimal Non-Projective Measurements
of Binary Coherent States with Photon Counting

## Supplementary Note I: Impact of Experimental Imperfections

We investigate the effect of non-ideal experimental parameters on the
performance of the optimal inconclusive measurement and the Dolinar receiver.
A critical experimental parameter is the ratio
$R=\mathrm{max}(|u(t)|^{2})/|\alpha|^{2}$ between the maximum local oscillator
(LO) energy $\mathrm{max}(|u(t)|^{2})$ and the energy of the input state
$|\alpha|^{2}$ that can be reliably implemented in the experiment. In
principle, the optimal inconclusive measurement requires very large values of
$R$ (e.g. $R=\infty$ at $t=0$ for equiprobable states). However, having a
finite value of $R$ that can be applied limits our ability to implement the
desired optimal displacement $u(t)$ perfectly, which impacts the performance
of the strategy. Supplementary Figure 4 shows the effect of a finite value of
$R=50$ on the optimal inconclusive measurement for $|\alpha|^{2}$=0.2, 0.4,
and 0.6. The dashed lines show the performance for when $R=\infty$, for
reference. We obtain each data point by solving the evolution equations for
$P_{\mathrm{I}}$ and $P_{\mathrm{E}}$ using the bounded optimal waveform
$u^{\prime}(t)=\mathrm{min}[u(t),\sqrt{R}|\alpha|\mathrm{sign}(u(t))]$ with a
finite $R$, and then plotting the obtained values of $P_{\mathrm{E}}$ and
$P_{\mathrm{I}}$. We observe that the main effect of finite $R$ is a gap $g$
in the achievable $P_{\mathrm{I}}-P_{\mathrm{E}}$ curve. We define this gap as
$g^{2}=(\Delta P_{\mathrm{E}})^{2}+(\Delta P_{\mathrm{I}})^{2}$, where the
dashed gray lines show $\Delta P_{\mathrm{E}}$ and $\Delta P_{\mathrm{I}}$ for
$|\alpha|^{2}=0.2$. The inset shows $g^{2}$ as a function of $R$ for each
value of $|\alpha|^{2}$ on a log-log scale, showing a $1/R$ scaling.

Intrinsic imperfections such as non-ideal interference visibility and non-zero
dark counts can produce unexpected photon detections, which cause errors in
the discrimination process. Supplementary Figure 5(a-c) shows the performance
of the optimal inconclusive measurement for $|\alpha|^{2}=0.2$ for different
levels of experimental imperfections (solid colored lines) compared to the
ideal case (dashed line). Non-ideal values of (a) the detection efficiency
$\eta$, (b) the interference visibility $\xi$, and (c) the dark count rate
$\nu$ all increase the attainable error probability compared to the ideal
case. A reduction in $\eta$ increases the error probability by approximately
the same amount across the entire range of values of $P_{\mathrm{I}}$. On the
other hand, higher dark counts have a larger impact closer to the MESD
($P_{\mathrm{I}}\approx 0$) and USD ($P_{\mathrm{E}}\approx 0$) regimes.
Supplementary Figure 5(d-f) shows the performance of the optimal MESD
measurement, the Dolinar receiver, for different levels of experimental
imperfections (solid colored lines) compared to the ideal case (dashed black
line) and homodyne limit (dashed red line). We observe that for the Dolinar
receiver, non-ideal interference visibility (e) has the largest detrimental
effect, significantly increasing the error probability at larger values of
$|\alpha|^{2}$.

Supplementary Figure 4: Impact of a finite value of
$R=\mathrm{max}(|u(t)|^{2})/|\alpha|^{2}$ on the optimal inconclusive
measurement. The blue, orange, and yellow solid lines show the simulated
results for $|\alpha|^{2}$=0.2, 0.4, and 0.6, respectively, for a strategy
with $R=50$, and the dashed lines show the results for $R=\infty$. A value of
$R<\infty$ produces a gap $g^{2}=(\Delta P_{\mathrm{E}})^{2}+(\Delta
P_{\mathrm{I}})^{2}$ in the $P_{\mathrm{I}}-P_{\mathrm{E}}$ curve. The dashed
gray lines show the ranges $\Delta P_{\mathrm{I}}$ and $\Delta P_{\mathrm{E}}$
for $|\alpha|^{2}$=0.2 that cannot be accessed by the measurement. The inset
shows the scaling of the gap $g$ as a function of $R$, which follows $~{}1/R$
scaling for each value of $|\alpha|^{2}$. Supplementary Figure 5: Impact of
imperfections on the optimal inconclusive measurement (top row) and the
optimal MESD measurement, the Dolinar receiver (bottom row). Colored lines
show the effect of different levels of detection efficiency $\eta$ (a,d),
interference visibility $\xi$ (b,e), and dark counts $\nu$ (c,f) on the
attainable performances of the corresponding measurements for
$|\alpha|^{2}=0.2$.

## Supplementary Note II: Phase Correction Calibration

The LO amplitude modulator (AM) also incidentally modulates the LO phase
depending on the depth of the amplitude modulation, which is related to the
parameter $R$. The relationship between the applied amplitude modulation and
the resulting extra phase modulation is typically linear for low modulation
depths. However, the implementation of the optimal inconclusive measurement
requires modulation as large as experimentally possible (in principle
infinitely large at $t=0$ for equal prior probabilities) to implement the
necessary displacement waveform for a particular strategy. Thus, to correct
the phase modulation caused by the AM we apply a phase correction to the phase
modulator (PM) in the LO arm, such that any extra phase modulation from the AM
will be exactly canceled out once the light passes through both modulators.
This correction allows us to experimentally implement the displacement
waveform which implements the desired optimal inconclusive measurement. A
critical aspect of the strategy for the optimal inconclusive measurement is
that the optimal waveform for the LO magnitude has a deterministic shape, i.e.
takes on values which implement a predetermined displacement magnitude. Since
the LO magnitude is predetermined, the necessary phase correction is
predetermined as well. Thus, in our implementation of this strategy in the
FPGA, the correction for the unwanted phase shift introduced by the amplitude
modulator is included as an extra factor for the phase of the LO in the FPGA.
This modification allows us to realize feedforward operations to the PM of the
LO in real time for implementing the phase correction of this undesirable
effect.

We calibrate this phase correction by measuring the voltage required to
correct the phase shift from the AM as a function of the bit-value
(proportional to voltage via the DAC) applied to the AM. To measure the
necessary amount of phase correction, we use an interferometric measurement
where the Signal and LO fields interfere, while the interferometer is
stabilized. We apply two copies of a waveform for phase calibration to the PM
for the Signal field back-to-back within a single light pulse. The phase
calibration waveform implements the phase pattern [0, $\pi/2$, 0, $\pi$, 0,
$3\pi/2$, 0], where each value has the same time duration [1]. During the
first copy, the phase and amplitude modulation of the LO is turned off, and
during the second copy we enable the AM of the LO with a particular bit-value.

We calibrate the phase shift induced by the AM of the LO by observing the
intensity of the interference pattern after the Signal and LO interfere.
Comparing the measured intensity for relative phases $\pi/2$ and $3\pi/2$
allows us correct the phase shift induced by the AM by adjusting the PM of the
LO to achieve the expected interference for these two phases. These two phases
should correspond to the same intensity, but the phase error from the AM
causes them to be uneven and we simply adjust the phase correction until they
are balanced. We repeat this process to obtain the phase correction voltage
for different values of amplitude modulation to obtain a phase correction
calibration curve. This calibration curve corresponds to the extra voltage we
must apply to the LO PM to cancel out the phase error from amplitude
modulation in the measurement. During the experimental runs, this phase
correction is simply added to the current LO phase, which depends on the
random photon detections in the measurement, and the strength of the
correction depends on the value sent to the AM of the LO, which is coded in
the FPGA as a look-up table.

Supplementary Figure 6: Example correction curve for the phase error imparted
by the LO AM. The points show the phase modulator voltage required to correct
the incidental phase shift from amplitude modulation. At low modulation depth
(large bit-value due to our configuration), the correction is relatively
linear as expected. However at large modulation (small bit-value) the needed
phase correction becomes highly non-linear. The blue line shows the data fit
using a smoothing spline and the orange line shows the corresponding bit-value
added to the phase of the LO to correct the phase error.

Supplementary Figure 6 shows an example of the measured phase correction
curve. The dots show the measured voltage applied to the PM required to
correct the phase error from amplitude modulation. Different points on the
x-axis (Amplitude Modulation Bit-Value) correspond to different levels of
amplitude modulation depth which implement different displacement magnitudes.
The blue line shows a fit to the measured points using a smoothing spline. The
orange curve shows the bit-value that needs to be applied to the phase
modulator in order to correct the phase error, which appears to be piece-wise
constant due to the finite precision in digitization (8-bits) within the FPGA.
For example, an amplitude modulation bit-value of 150 will require the FPGA to
add a bit-value of 19 to the LO phase to cancel out any phase errors, and an
amplitude modulation value of 50 requires a phase correction bit value of 31
be added to the LO phase. This correction ensures that there is no incidental
phase shifts caused by amplitude modulation, allowing us to precisely
implement the correct displacement waveform needed for a particular optimal
inconclusive measurement.

## Supplementary Note III: Extending to High Dimensional Inconclusive
Measurements

We show how to construct inconclusive measurements in high dimensions using
the combination of hypothesis testing followed by the binary optimal
inconclusive measurement. We extend the strategy proposed in Section III to
M-PSK coherent state alphabets by again dividing the single measurement into
two stages. In the first stage, the receiver implements hypothesis tests until
there are only two possible remaining states. There are on the order of
$2^{M}$ possible detection histories for the hypothesis testing stage, and
most of these outcomes will result in an inconclusive result, which yields
$P^{(1)}_{\mathrm{I}}$. The binary optimal inconclusive measurement in the
second stage can then achieve arbitrarily small error probabilities
$P_{\mathrm{E}}$ at the expense of increasing the total inconclusive
probability $P_{\mathrm{I}}$ by a relatively small amount given by
$P^{(2)}_{\mathrm{I}}$. In contrast, heterodyne detection requires that
$P^{\mathrm{Het}}_{\mathrm{I}}$ asymptotically approaches
$P^{\mathrm{Het}}_{\mathrm{I}}=1$ in order to reach asymptotically small error
probabilities $P^{\mathrm{Het}}_{\mathrm{E}}$ (though
$P^{\mathrm{Het}}_{\mathrm{E}}=0$ only at $P^{\mathrm{Het}}_{\mathrm{I}}=1$).
Thus, examining the minimum possible inconclusive probability
$P^{(1)}_{\mathrm{I}}$ allows us to examine the scaling in the performance of
the proposed hybrid approach as the number of states $M$ increases, since
lower error rates can always be achieved by a small increase in the total
$P_{\mathrm{I}}=P^{(1)}_{\mathrm{I}}+P^{(2)}_{\mathrm{I}}$ where
$P_{\mathrm{I}}<1$ for $P_{\mathrm{E}}=0$.

The minimum inconclusive probability $P^{(1)}_{\mathrm{I}}$ is given by the
total probability that the hypothesis testing stage results in a detection
history where there are three or more possible states remaining. A given state
$s_{j}\in\\{s_{1}...s_{M}\\}$ is eliminated as a possible state if the
receiver detects a photon when testing for that state. The entire hypothesis
testing stage consists of testing for each possible input state until either
only two states remain, or all states have been tested. We denote the
detection history of the hypothesis testing stage as $\textbf{d}_{i}$ and the
set of all detection histories which result in an inconclusive result as
$\textbf{D}_{\mathrm{inc}}$. We then obtain the minimum inconclusive
probability $P^{(1)}_{\mathrm{I}}$ as:

$\displaystyle P^{(1)}_{\mathrm{I}}$
$\displaystyle=\sum_{\textbf{d}_{i}\in\textbf{D}_{\mathrm{inc}}}P(\textbf{d}_{i})$
(5)
$\displaystyle=\sum_{\textbf{d}_{i}\in\textbf{D}_{\mathrm{inc}}}\sum_{j=1}^{M}P(\textbf{d}_{i}|s^{*}=s_{j})P(s^{*}=s_{j})$
(6)
$\displaystyle=\sum_{\textbf{d}_{i}\in\textbf{D}_{\mathrm{inc}}}\sum_{j=1}^{M}\prod_{k=1}^{M}\mathcal{L}(d_{i,k}|\hat{s}=s_{k},s^{*}=s_{j})\times\frac{1}{M}$
(7)

where $P(\textbf{d}_{i})$ is the probability of obtaining detection history
$\textbf{d}_{i}$ from the hypothesis testing stage,
$P(\textbf{d}_{i}|s^{*}=s_{j})$ is the probability of obtaining the detection
history $\textbf{d}_{i}$ given the actual input state $s^{*}$ is $s_{j}$, and
$P(s_{j})=1/M$ is the prior probability. Also,
$\mathcal{L}(d_{i,k}|\hat{s}=s_{k},s^{*}=s_{j})$ is the photon counting
likelihood of detecting $d_{i,k}$ from a single hypothesis test given that the
receiver tests the state $\hat{s}=s_{k}$ and the actual input state is
$s^{*}=s_{j}$. We note that $\mathcal{L}(1|s_{j},s_{j})=0$ and this enforces
that some detection histories never actually occur. Given that
$P^{(1)}_{\mathrm{I}}\rightarrow 1$ as $M\rightarrow\infty$, we obtain the
maximum conclusive probability as
$P^{\mathrm{max}}_{\mathrm{conc}}=1-P^{(1)}_{\mathrm{I}}$.

As an example, the minimum inconclusive probability $P^{(1)}_{\mathrm{I}}$ for
$M=3$ as in Sec. III of the main text is given by the probability of detecting
zero photons in each hypothesis test, i.e. obtaining
$\textbf{d}_{0}=\\{0,0,0\\}$. The histories $\textbf{d}_{1}=\\{0,0,1\\}$,
$\textbf{d}_{2}=\\{0,1,0\\}$, and $\textbf{d}_{4}=\\{1,0,0\\}$ result in
performing the binary optimal inconclusive measurement on the remaining two
states, and the other detection histories never occur. Thus,
$\textbf{D}_{\mathrm{inc}}=\\{\textbf{d}_{0}\\}$ and we obtain:

$\displaystyle P^{(1)}_{\mathrm{I}}$ $\displaystyle=P(\textbf{d}_{0})$ (8)
$\displaystyle=\sum_{j=1}^{3}\prod_{k=1}^{3}\mathcal{L}(0|s_{j},s_{k})/3$
$\displaystyle=(1)\Bigg{(}e^{-2\frac{f_{3}|\alpha|^{2}}{3}\big{[}1-\mathrm{cos}(\frac{2\pi}{3})\big{]}}\Bigg{)}\Bigg{(}e^{-2\frac{f_{3}|\alpha|^{2}}{3}\big{[}1-\mathrm{cos}(\frac{4\pi}{3})\big{]}}\Bigg{)}.$

Which yields $P_{\mathrm{I}}^{(1)}\approx\\{0.766,0.587,0.449\\}$ for
$|\alpha|^{2}=\\{0.2,0.4,0.6\\}$ respectively with $f_{3}=2/3$ as in Sec. III
of the main text.

Supplementary Figure 7 shows the scaling of the maximum conclusive probability
$\mathrm{log}_{10}(P^{\mathrm{max}}_{\mathrm{conc}})=\mathrm{log}_{10}(1-P^{(1)}_{\mathrm{I}})$
as the number of states $M$ in the alphabet increases. The blue, orange,
yellow, purple, and green points show the results for a mean photon number per
bit of $|\alpha|^{2}/\mathrm{log}_{2}(M)=$ 0.2, 0.4, 0.6, 0.8, and 1.0. The
dashed lines are to guide the eye. We note that here the fraction $f$ of total
energy $|\alpha|^{2}$ used to perform the hypothesis testing changes for each
alphabet as $f_{M}=(M-1)/M$ such that here $M=3$ (inset) corresponds to the
results shown in Supplementary Fig. (3) in the main text. We find that the
quantity $\mathrm{log}_{10}(1-P^{(1)}_{\mathrm{I}})$ roughly scales
proportionally to $M^{2}$ for all total mean photon numbers, suggesting that
$P^{\mathrm{max}}_{\mathrm{conc}}\propto 10^{c_{\alpha}\times M^{2}}$ where
the coefficient $c_{\alpha}<0$ is specific to a particular value of
$|\alpha|^{2}$. We note that this scaling may change with a different choice
of $f_{M}$. Furthermore, for $M\geq 4$, a Bayesian approach to hypothesis
testing may provide better performance as opposed to randomly choosing between
the remaining untested states, such as in Refs. [1, 2, 3].

Supplementary Figure 7: Scaling of the maximum conclusive probability for
inconclusive measurements based on hypothesis testing for high dimensional
M-PSK alphabets. Colored points show the maximum conclusive probability
$P^{\mathrm{max}}_{\mathrm{conc}}=1-P^{(1)}_{\mathrm{I}}$, where a larger
$P^{\mathrm{max}}_{\mathrm{conc}}$ is better, for different numbers of states
in the PSK alphabet. The value of $P^{\mathrm{max}}_{\mathrm{conc}}$ increases
as the mean photon number per bit $|\alpha|^{2}/\mathrm{log}_{2}(M)$ increases
and $\mathrm{log}_{10}(1-P^{(1)}_{\mathrm{I}})$ scales approximately
proportional to $M^{2}$.

## Supplementary Note IV: Optimal Inconclusive Measurement Simulation Details

We simulate the optimal inconclusive measurement using two methods, one based
on evolving equations derived in Ref. [4] and the other based on Monte-Carlo
sampling. Both methods allow for inclusion of experimental imperfections such
as detection efficiency $\eta$, interference visibility $\xi$, dark count rate
$\nu$, finite maximum displacement power $R$, and finite digitization
precision from the digital-to-analog conversion within the experiment. A
particular optimal inconclusive measurement is fixed by first setting the
input energy $|\alpha|^{2}$, desired inconclusive probability
$P_{\mathrm{I}}$, and prior probabilities of the two states $\\{p,1-p\\}$.
These parameters allow for calculation of the switching time
$t_{1}=t_{1}(|\alpha|^{2},P_{\mathrm{I}},p)$ as well as the value of
$v=v(|\alpha|^{2},P_{\mathrm{I}},p)$, which determines the phase of the
displacement field at the beginning of the second temporal mode [4].

Algorithm 1 Optimal Inconclusive Simulation: Equations

1:Fix: $|\alpha|^{2}$, $P_{\mathrm{I}}$, and $p$ $\triangleright$ Energy,
inconclusive probability, and prior probability of more likely input state

2:Fix: $\eta$, $\xi$, $\nu$, $R$ $\triangleright$ Experimental Imperfections

3:

4:$t_{1}\leftarrow t_{1}(|\alpha|^{2},P_{\mathrm{I}},p)$ $\triangleright$ Eq.
13 in Ref. [4]

5:$v\leftarrow v(|\alpha|^{2},P_{\mathrm{I}},p)$ $\triangleright$ Eq. 18 in
Ref. [4]

6:

7:Get $|u(t)|$ from Eq. (1)

8:$|u_{\mathrm{exp}}(t)|=\mathrm{min}(\sqrt{R},|u(t)|)$

9:Transform $|u_{\mathrm{exp}}(t)|$ to nearest bit-value

10:Transform $|u_{\mathrm{exp}}(t)|$ back to amplitude

11:

12:Initial: $P_{\mathrm{C}}(0)=p$, $P_{\mathrm{I}}(0)=0$, $t=0$

13:$P_{\mathrm{E}}(0)=1-P_{\mathrm{C}}(0)-P_{\mathrm{I}}(0)$

14:while $0\leq t<t_{1}$ do $\triangleright$ Evolve for first temporal mode

15: $|\beta|\leftarrow|u_{\mathrm{exp}}(t)|$

16: $\langle\hat{n}\rangle_{\pm}\leftarrow\eta(|\alpha|^{2}+|\beta|^{2}\pm
2\xi|\beta||\alpha|)+\nu$

17: $P_{\mathrm{C}}(t+\delta t)\leftarrow P_{\mathrm{C}}(t)\big{(}1-\delta
t\langle\hat{n}\rangle_{-}\big{)}+\big{(}1-P_{\mathrm{C}}(t)\big{)}\delta
t\langle\hat{n}\rangle_{+}$

18: $P_{\mathrm{E}}(t+\delta t)\leftarrow 1-P_{\mathrm{C}}(t+\delta t)$

19: $P_{\mathrm{I}}(t+\delta t)\leftarrow 1-P_{\mathrm{C}}(t+\delta
t)-P_{\mathrm{E}}(t+\delta t)$

20: $t\leftarrow t+\delta t$

21:end while$\triangleright$ End of first temporal mode

22:

23:if $v\leq 0.5$ then

24: $P_{\mathrm{C}}(t_{1})\leftarrow 0$

25: $P_{\mathrm{I}}(t_{1})\leftarrow 1$

26: $P_{\mathrm{E}}(t_{1})\leftarrow
1-P_{\mathrm{C}}(t_{1})-P_{\mathrm{I}}(t_{1})$

27:else if $v>0.5$ then

28: $P_{\mathrm{C}}(t_{1})\leftarrow P_{\mathrm{C}}(t_{1})$

29: $P_{\mathrm{I}}(t_{1})\leftarrow P_{\mathrm{I}}(t_{1})$

30: $P_{\mathrm{E}}(t_{1})\leftarrow
1-P_{\mathrm{C}}(t_{1})-P_{\mathrm{I}}(t_{1})$

31:end if

32:

33:$p^{\prime}\leftarrow\frac{1}{2}\big{(}1-\sqrt{1-4p(1-p)e^{-4t_{1}|\alpha|^{2}}}\big{)}$
$\triangleright$ Helstrom Bound for states $|\sqrt{t_{1}}|\alpha|\rangle$

34:while $t_{1}\leq t<1$ do $\triangleright$ Evolve for second temporal mode

35: $|\beta|\leftarrow|u_{\mathrm{exp}}(t)|$

36: $\langle\hat{n}\rangle_{\pm}\leftarrow\eta(|\alpha|^{2}+|\beta|^{2}\pm
2\xi|\beta||\alpha|)+\nu$

37: $P_{\mathrm{C}}(t+\delta t)\leftarrow P_{\mathrm{C}}(t)\big{(}1-\delta
t\langle\hat{n}\rangle_{-}\big{)}+\big{(}p^{\prime}-P_{\mathrm{C}}(t)\big{)}\delta
t\langle\hat{n}\rangle_{+}$

38: $P_{\mathrm{E}}(t+\delta t)\leftarrow P_{\mathrm{E}}(t)\big{(}1-\delta
t\langle\hat{n}\rangle_{+}\big{)}+\big{(}1-p^{\prime}-P_{\mathrm{E}}(t)\big{)}\delta
t\langle\hat{n}\rangle_{-}$

39: $P_{\mathrm{I}}(t+\delta t)\leftarrow 1-P_{\mathrm{C}}(t+\delta
t)-P_{\mathrm{I}}(t+\delta t)$

40: $t\leftarrow t+\delta t$

41:end while$\triangleright$ End of second temporal mode

Algorithm 1 shows the pseudo-code for the first simulation method where we
directly obtain the probability of correct detection $P_{\mathrm{C}}(t)$,
error probability $P_{\mathrm{E}}(t)$, and inconclusive probability
$P_{\mathrm{I}}(t)$ as a function of time elapsed over the course of the
measurement. This method is based on evolving these probabilities in time
using Eq. (9) and Eq. (25) from Ref. [4]. Lines 1 through 10 allow for
obtaining the magnitude of the experimental displacement waveform including
experimental imperfections. Line 8 enforces the maximum displacement power $R$
as discussed above and Line 9-10 enforces the finite precision (8-bits for our
implementation) when performing digital-to-analog conversion from the FPGA.
Lines 14-21 evolve $P_{\mathrm{C}}(t)$, $P_{\mathrm{E}}(t)$, and
$P_{\mathrm{I}}(t)$ according to Eq. (9) of Ref. [4] to simulate the first
temporal mode of the measurement. Lines 23-31 determine the value of the
probabilities at the start of the second temporal mode based on
$|\alpha|^{2}$, $P_{\mathrm{I}}$, and $p$ (See Ref. [4] for details). Lines
33-40 evolve the probabilities through the second temporal mode to give the
final values of the error, correct, and inconclusive probabilities.

Algorithm 2 shows the pseudo-code for the second simulation method based on
Monte-Carlo sampling. Instead of the probabilities, this method simulates the
measurement directly, i.e. as it occurs in the lab based on random sampling
from the photon counting distribution. Lines 1-5 fix the particular optimal
inconclusive measurement as well as the switching time and value of $v$. Lines
7-50 repeats the simulation of a single measurement (indexed by $i$)
$M_{\mathrm{MC}}$ number of times. During a single measurement, Lines 8-12
randomly choose the sign of the true input state $s_{i}=\pm 1$, corresponding
to $|\pm\alpha\rangle$. Lines 14-17 obtain the displacement magnitude waveform
including experimental imperfections. Lines 20-28 simulate the first temporal
mode. Line 21 determines the displacement amplitude based on the current total
number of photons detected $N$. Line 22 calculates the mean photon number
$\langle\hat{n}\rangle$ of the displaced input state detected by the single
photon detector including experimental imperfections. Line 23 samples a single
detection from a Poisson distribution with mean $\langle\hat{n}\rangle\delta
t$ where $\delta t=1/1024$ is the duration of a single time step (i.e. we
discretize the measurement into 1024 discrete time steps). Line 24 enforces
on/off detection since for each time step the detection result can only be
either vacuum or not-vacuum. Line 27 updates the provisional hypothesis for
the input state conditioned on the total number of photons detected thus far.
The first temporal mode continues until the time $t_{1}$ is reached. Lines
30-34 determine the displacement phase for the start of the second temporal
mode.

Algorithm 2 Optimal Inconclusive Simulation: Monte-Carlo

1:Fix: $|\alpha|^{2}$, $P_{\mathrm{I}}$, and $p$ $\triangleright$ Energy,
inconclusive probability, and prior probability of more likely input state

2:Fix: $\eta$, $\xi$, $\nu$, $R$ $\triangleright$ Experimental Imperfections

3:

4:$t_{1}\leftarrow t_{1}(|\alpha|^{2},P_{\mathrm{I}},p)$ $\triangleright$ Eq.
13 in Ref. [4]

5:$v\leftarrow v(|\alpha|^{2},P_{\mathrm{I}},p)$ $\triangleright$ Eq. 18 in
Ref. [4]

6:Fix: $M_{\mathrm{MC}}$ $\triangleright$ Number of Monte-Carlo samples

7:for $i\leftarrow 1$ to $M_{\mathrm{MC}}$ do

8: if $\mathrm{rand}>p$ then $\triangleright$ Randomly choose actual input
state

9: $s_{i}=-1$,

10: else

11: $s_{i}=1$

12: end if

13:

14: Get $|u(t)|$ from Eq. (1)

15: $|u_{\mathrm{exp}}(t)|=\mathrm{min}(\sqrt{R},|u(t)|)$

16: Transform $|u_{\mathrm{exp}}(t)|$ to nearest bit-value

17: Transform $|u_{\mathrm{exp}}(t)|$ back to amplitude

18:

19: Initial: $N=0$, $t=0$, $\hat{s}_{i}(0)=1$ $\triangleright$ Assume $p\geq
0.5$

20: while $0\leq t<t_{1}$ do $\triangleright$ Evolve for first temporal mode

21: $\beta\leftarrow(-1)^{N}|u_{\mathrm{exp}}(t)|$

22:
$\langle\hat{n}\rangle\leftarrow\eta\big{(}|\alpha|^{2}+|\beta|^{2}-2\xi\beta|\alpha|s_{i}\big{)}+\nu$

23: $d\sim\mathrm{Poisson}(n|\langle\hat{n}\rangle\delta t)$$\triangleright$
Vacuum/photon detection

24: If $d>1$, set $d=1$

25: $N\leftarrow N+d$ $\triangleright$ Total number of detected photons

26: $t\leftarrow t+\delta t$

27: $\hat{s}_{i}(t)=(-1)^{N}$ $\triangleright$ Current hypothesis for input

28: end while$\triangleright$ End of first temporal mode

29:

30: if $v\leq 0.5$ then

31: $N_{0}=1$

32: else if $v>0.5$ then

33: $N_{0}=0$

34: end if

35:

36: $N\leftarrow N_{0}$

37: while $t_{1}\leq t<1$ do $\triangleright$ Evolve for second temporal mode

38: $\beta\leftarrow\hat{s}_{i}(t_{1})\times(-1)^{N}|u_{\mathrm{exp}}(t)|$

39:
$\langle\hat{n}\rangle\leftarrow\eta\big{(}|\alpha|^{2}+|\beta|^{2}-2\xi\beta|\alpha|s_{i}\big{)}+\nu$

40: $d\sim\mathrm{Poisson}(n|\langle\hat{n}\rangle\delta t)$$\triangleright$
Vacuum/photon detection

41: If $d>1$, set $d=1$

42: $N\leftarrow N+d$ $\triangleright$ Total number of detected photons

43: $t\leftarrow t+\delta t$

44: if $\mathrm{mod}(N,2)=0$ then

45: $\hat{s}_{i}(t)=\hat{s}_{i}(t_{1})$ $\triangleright$ Hypothesis:
$|\pm\alpha\rangle$

46: else

47: $\hat{s}_{i}(t)=0$ $\triangleright$ Hypothesis: Inconclusive

48: end if

49: end while$\triangleright$ End of second temporal mode

50:end for

51:$P_{\mathrm{C}}(0)\leftarrow p$, $P_{\mathrm{I}}(0)\leftarrow 0$,
$P_{\mathrm{E}}(0)\leftarrow 1-p$

52:$t\leftarrow 0$ $\triangleright$ Reset time

53:while $0\leq t<1$ do $\triangleright$ Calculate sampled probabilities

54: $t\leftarrow t+\delta t$

55:
$P_{\mathrm{C}}(t)=\sum_{i}\big{(}\hat{s}_{i}(t)==s_{i}\big{)}/M_{\mathrm{MC}}$
$\triangleright$ Correct

56:
$P_{\mathrm{I}}(t)=\sum_{i}\big{(}\hat{s}_{i}(t)==0\big{)}/M_{\mathrm{MC}}$
$\triangleright$ Inconclusive

57: $P_{\mathrm{E}}(t)=1-P_{\mathrm{C}}(t)-P_{\mathrm{I}}(t)$ $\triangleright$
Error

58:end while

Lines 37-50 simulate the second temporal mode in a similar fashion to the
first mode, except now the receiver is attempting to discriminate between the
hypothesis after the first temporal mode $\hat{s}_{i}(t_{1})$ and the
inconclusive outcome denoted by $\hat{s}_{i}(t)=0$. Lines 38-41 set the
displacement field and sample the photon detection for the time step. Lines
44-48 update the provisional hypothesis $\hat{s}_{i}(t)$, which now switches
between the hypothesis after the first mode and the inconclusive outcome. This
behavior is in contrast to the first temporal mode where the provisional
hypothesis switches between the two possible input states. Lines 53-58
calculate the measured values of the correct, error, and inconclusive
probabilities based on the value of the hypothesis $\hat{s}_{i}(t)$ and the
true value $s_{i}$ across all $M_{\mathrm{MC}}$ Monte-Carlo samples.

## References

  * Becerra  _et al._ [2013] F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, and A. Migdall, Experimental demonstration of a receiver beating the standard quantum limit for multiple nonorthogonal state discrimination, Nature Photonics 7, 147 (2013).
  * Ferdinand  _et al._ [2017] A. R. Ferdinand, M. T. DiMario, and F. E. Becerra, Multi-state discrimination below the quantum noise limit at the single-photon level, npj Quantum Information 3, 43 (2017).
  * Burenkov  _et al._ [2020] I. Burenkov, M. Jabir, A. Battou, and S. Polyakov, Time-resolving quantum measurement enables energy-efficient, large-alphabet communication, PRX Quantum 1, 010308 (2020).
  * Nakahira and Usuda [2012] K. Nakahira and T. S. Usuda, Optimal receiver for discrimination of two coherent states with inconclusive results, Physical Review A 86, 052323 (2012).