# Entanglement source and quantum memory analysis for zero added-loss
multiplexing

Jeffrey H. Shapiro<EMAIL_ADDRESS>Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 USA
Michael G. Raymer<EMAIL_ADDRESS>Department of Physics, Oregon Center for
Optical, Molecular, and Quantum Science, University of Oregon, Eugene, Oregon
97403 USA  Clark Embleton<EMAIL_ADDRESS>Department of Physics, Oregon
Center for Optical, Molecular, and Quantum Science, University of Oregon,
Eugene, Oregon 97403 USA  Franco N. C. Wong<EMAIL_ADDRESS>Research Laboratory
of Electronics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139 USA  Brian J. Smith<EMAIL_ADDRESS>Department of
Physics, Oregon Center for Optical, Molecular, and Quantum Science, University
of Oregon, Eugene, Oregon 97403 USA

###### Abstract

High-rate, high-fidelity entanglement distribution is essential to the
creation of a quantum internet, but recent achievements in fiber (248 km at
9/s rate) and satellite-based (1200 km at 1.1/s rate) entanglement
distribution fall far short of what is needed. Chen  _et al_. [Phys. Rev.
Appl. 19, 054209 (2023)] proposed a means for dramatically increasing
entanglement-distribution rates via a scheme they call zero added-loss
multiplexing (ZALM). ZALM’s quantum transmitter employs a pair of Sagnac-
configured spontaneous parametric downconverters (SPDCs), channelization via
dense wavelength-division multiplexing (DWDM) filtering, and partial Bell-
state measurements (BSMs) to realize a near-deterministic, heralded source of
frequency-multiplexed polarization-entangled biphotons. Each biphoton is
transmitted to Alice and Bob along with a classical message identifying its
frequency channel and whether a $\psi^{-}$ singlet or a $\psi^{+}$ triplet was
heralded. Alice and Bob’s quantum receivers then use DWDM filtering and
temporal-mode conversion to interface their received biphotons to intra-cavity
color-center quantum memories. This paper delves deeply into ZALM’s SPDCs,
partial-BSMs, and Duan-Kimble loading of Alice and Bob’s quantum memories. Its
principal results—the density operators for the SPDC sources and the quantum
memories—allow heralding probability, heralding efficiency, and fidelity to be
evaluated for both the polarization-entangled biphotons and the loaded quantum
memories, thus enabling exploration of the parameter space for optimizing
ZALM’s performance. Even without a comprehensive optimization analysis, the
paper’s examples already demonstrate two critical features of the ZALM
architecture: (1) the necessity of achieving a near-separable channelized
biphoton wave function to ensure the biphoton sent to Alice and Bob is of high
purity; and (2) the premium placed on Alice and Bob’s temporal-mode
converters’ enabling narrowband push-pull memory loading to ensure the
arriving biphoton’s state is faithfully transferred to the intra-cavity color
centers.

##  I Introduction

Entanglement, the quintessential manifestation of quantum physics, [1] enables
a host of new technological capabilities. Among them is quantum teleportation
[2], a communication primitive on which a quantum internet can be built [3, 4,
5, 6]. Consequently, long-distance high-rate entanglement distribution—over
optical fiber or via free-space propagation—is eagerly being sought. A 248 km
distance record was established in 2022 for a fiber link [7], but it provided
a minuscule 9/s rate owing to the 79 dB link loss. Quantum repeaters [6, 8, 9]
can multiply the distance over which entanglement can be distributed at high
rate over optical fiber, but they have yet to reach an advanced state of
development. Alternatively, a low earth orbit (LEO) satellite-based system has
distributed entanglement between ground stations separated by 1200 km [10].
Moreover, a LEO satellite’s motion, relative to the ground, affords a
distribution rate largely independent of the distance between ground stations,
cf. the quantum key distribution results from the same satellite [11].
However, the satellite-based entanglement-distribution rate (1.1/s) at 1200 km
ground-station separation is still far below that needed for a useful quantum
internet, and the satellite system’s secret-key rate (1.1 kbit/s) at that
separation is likewise far below that needed for one-time pad transmission of
large data files.

Recently, a potential means for dramatically increasing entanglement-
distribution rates—over fiber or via satellite—was proposed by Chen  _et al._
[12]. Their proposal uses spontaneous parametric downconverters (SPDCs) and
partial Bell-state measurements (BSMs) [13] to herald the generation of
polarization-entangled biphotons across a large number of frequency-
multiplexed signal channels that collectively span the downconverters’
$\sim$10 THz phase-matching bandwidth. Thus, multi-pair generation in any
particular frequency channel can be kept low while the overall generation rate
becomes nearly deterministic as the number of channels increases. Furthermore,
as explained in Ref [12], by transmitting the heralded biphotons to Alice and
Bob’s terminals—along with classical messages identifying their heralded
frequency channels and whether the heralded polarization-entangled Bell state
is a $\psi^{-}$ singlet or a $\psi^{+}$ triplet—Alice and Bob can use quantum
frequency conversion [14, 15] and time-lens bandwidth compression [16] to
efficiently couple the received biphotons to pairs of intra-cavity color-
center quantum memories.

Chen  _et al_. dubbed their approach zero-added-loss multiplexing (ZALM),
because it avoids the switching losses incurred in previously proposed
multiplexing schemes [17, 18]. Like Ref. [3]’s fiber-based entanglement-
distribution architecture, and Ref. [10]’s satellite-based entanglement-
distribution system, ZALM uses a source-in-the-middle configuration, which is
known [19] to offer an efficiency—hence a distribution rate—advantage over its
meet-in-the-middle and sender-receiver alternatives in both fiber and free-
space operation. The free-space advantage is further increased when
atmospheric turbulence is concentrated near Alice and Bob’s terminals, as it
is if the ZALM source is on a LEO satellite.

Reference [12] contains a broad assessment of ZALM performance, concentrating
on the entanglement-distribution rate and its tradeoff with color-center
entangled-state fidelity, and it includes simulation results for ZALM’s mode
conversion, i.e., the combination of quantum frequency conversion and
bandwidth compression needed to efficiently couple arriving biphotons to the
intra-cavity color centers, as well as other implementation considerations.

The present paper takes a more narrow view, viz., a deep dive into the SPDCs
and the partial BSMs that yield heralded polarization-entangled biphotons
across a set of frequency channels, and the ensuing color-center entangled-
state fidelity between Alice and Bob’s Duan-Kimble [20] intra-cavity quantum
memories. Two major results are the density operators for the heralded
biphoton produced by the source and for the loaded quantum memories. These
operators quantify the need to input near-separable channelized biphoton
states from two SPDC sources to the partial BSMs in order to get a high-purity
heralded biphoton, and thereby a high color-center entangled-state fidelity.
Moreover, this need applies even for ideal partial-BSM apparatus, and for
intra-cavity color-center quantum memories with perfect—unit magnitude and
$\pi$-rad phase difference—state-dependent reflectivities in their Duan-Kimble
loading protocol. More importantly, our density operators lend themselves to
numerical evaluations that enable accounting for nonidealities in the ZALM
source and for memories with realistic state-dependent reflectivities. In this
regard it is important to note that we employ a rigorously derived and
improved “push-pull” version of the Duan-Kimble protocol in our memory-loading
analysis [21].

The remainder of the paper is organized as follows. We begin, in Sec. II, with
an overview of ZALM, focusing on the dual-Sagnac SPDC source, partial BSM, and
Duan-Kimble quantum memories. In Section III, we characterize a channelized,
pulse-pumped Sagnac-configured SPDC source [22] of unheralded polarization-
entangled biphotons, starting from the perturbation theory result for
arbitrary pump spectrum and phase-matching function, and then specializing to
a transform-limited Gaussian pump pulse and Gaussian phase-matching function.
Section III also presents the channelized biphoton wave function’s singular-
value decomposition (SVD), which is useful for theoretical calculations and
importantly offers a practical route to numerical evaluation of memory-loading
performance in broadband conditions. Next, in Sec. IV, we derive the density
operator of the heralded polarization-entangled biphoton produced by the
channelized partial BSM on the idler photons from a dual-Sagnac source. There
we evaluate the heralding probability, heralding efficiency, purity, and error
probability of the general heralded biphoton and report numerical results for
the all-Gaussian special case. In Sec. V, we derive the intra-cavity color-
center density operator for the Duan-Kimble quantum memories operated in push-
pull mode. This section reports the fidelity for ideal (lossless, $\pi$-rad
phase shift) state-dependent memory-cavity reflectivities and how that
fidelity may be realized—at the expense of a reduction in the entanglement-
distribution rate—in narrowband operation with the push-pull protocol. The
main text concludes, in Sec. VI, with a brief treatment of the tradeoff
between ZALM’s entanglement-distribution rate and its inter-channel
interference, followed by discussion of some possibilities for extending our
analysis to include equipment nonidealities, alternative SPDC sources,
alternative memory protocols, etc. Appendix A and Appendix B then supply
derivation details for Secs. IV and V, respectively.

##  II ZALM Overview

Figure 1 is a high-level cartoon of free-space entanglement distribution via
ZALM. The ZALM quantum transmitter (QTX) contains two coherently pulse-pumped
Sagnac-configured SPDC sources, which we assume to be identical, each arranged
to emit polarization-entangled singlet states, as shown in Fig. 2 and analyzed
in Sec. III. Their idler beams are routed to the spectrally channelized
partial-BSM setup shown in Fig. 3 and analyzed in Sec. IV. Specifically, these
idlers are combined on a 50–50 beam splitter whose outputs have their
horizontal ($H$) and vertical ($V$) polarization components separated by
polarizing beam splitters (PBSs). The PBSs’ outputs are then channelized by
dense wavelength-division multiplexing (DWDM) filters whose channelized
outputs illuminate single-photon detectors (SPDs).

Figure 1: High-level cartoon of free-space entanglement distribution via zero
added-loss multiplexing (ZALM). QTX: quantum transmitter. QRX: quantum
receiver.  Figure 2: Schematic of a Sagnac-configured SPDC source [22] of
singlet-state signal-idler biphotons suitable for use in ZALM. A periodically-
poled lithium niobate (PPLN) crystal [23] is bidirectionally pulse-pumped for
type-0 nondegenerate phase matching. $D$, $H$, and $V$: diagonal, horizontal,
and vertical polarizations. HR: high reflector. $\lambda$: wavelength. PBS:
polarizing beam splitter. HWP: half-wave plate.

A particular pump pulse results in an $n$th-channel partial-BSM success when
its combined idlers register one $H$ and one $V$ detection at the $n$th
channel’s SPDs; see Figure 3. When these detections come from the $I_{+H}$ and
$I_{-V}$ or the $I_{+V}$ and $I_{-H}$ SPDs, they herald a $\psi^{-}$
polarization-singlet state, commonly referred to as the $HV$$-$$VH$ state. On
the other hand, when they come from the $I_{+H}$ and $I_{+V}$ or the $I_{-H}$
and $I_{-V}$ SPDs, they herald a $\psi^{+}$ polarization-triplet state,
commonly called the $HV$+$VH$ state.

Figure 3: Schematic of ZALM’s partial Bell-state measurement for heralding
polarization-entangled photon pairs. $S_{k}$ and $I_{k}$ for $k=1,2$: signal
($S$) and idler ($I$) beams from the $k$th dual-Sagnac source. BS: 50–50 beam
splitter. $I_{\pm}$: the idler-beam outputs from the 50–50 beam splitter. PBS:
polarizing beam splitter. $I_{\pm P}$ for $P=H,V$: horizontally ($H$) and
vertically ($V$) polarized outputs from the PBS illuminated by $I_{\pm}$.
DWDMI: idler-beam dense wavelength-division multiplexing filter. SPD: single-
photon detector.

The heralded polarization-entangled signal-signal ($S_{1}$-$S_{2}$) biphotons
are then sent to Alice and Bob’s quantum receivers (QRXs) via free-space
propagation, accompanied by classical messages specifying the heralded states
and their frequency channels. (For fiber connections to Alice and Bob’s QRXs,
polarization entanglement would be converted to time-bin entanglement at the
ZALM QTX before transmission. See Sec. VI for a discussion of how our analysis
can be applied to time-bin entanglement distribution over fiber.)

The propagation-surviving signal photons arriving at Alice and Bob’s QRXs are
DWDM channelized and, when the $n$th channel has been heralded by the ZALM
QTX, Alice and Bob’s $n$th-channel DWDM outputs are frequency converted and
bandwidth compressed to prepare them for loading into Alice and Bob’s intra-
cavity color-center quantum memories. Following these transformations—which we
take to be ideal, so as to focus our attention on the fundamental limits of
ZALM—the $n$th-channel signal at each QRX is separated into its horizontally
and vertically polarized components by a PBS, with the former undergoing a
state-dependent reflectivity from the memory cavity and the latter being time
delayed, as shown in Fig. 4 and analyzed in Sec. V. After rotation to vertical
polarization, the cavity-reflected signal is interfered with the time-delayed
version of the PBS’s vertically-polarized output on Fig. 4’s 50–50 beam
splitter [24]. Entanglement of Alice and Bob’s memories is heralded when each
QRX detects a photon from a particular QTX herald. As shown in Sec. V, a
successful herald loads a $\psi^{-}$ singlet state into their memories with,
in general, its entangled-state fidelity depending on which combination of
detectors registered counts.

Figure 4: Notional schematic for Duan-Kimble loading of Alice and Bob’s intra-
cavity color-center quantum memories, assuming the partial BSM heralds an
$n$th-channel biphoton. $\tilde{S}_{k}$: the frequency-converted, bandwidth-
compressed $n$th-channel signal photon arriving at Alice’s memory ($k=1$) and
Bob’s memory ($k=2$). PBS: polarizing beam splitter. $\tilde{S}_{k_{P}}$: the
horizontally ($P=H$) and vertically ($P=V$) polarized components of
$\tilde{S}_{k}$. CIRC: optical circulator. QM: quantum memory. OC and HR:
optical coupler and high-reflector of a single-ended Fabry-Pérot (FP) cavity
[25]. SiV: silicon-vacancy color center [26] contained in the FP cavity.
$\tilde{S}^{\prime}_{k_{H}}$: $H$-polarized signal photon after reflection
from the QM. TD: $T$-sec-long time delay. M: mirror. HWP: half-wave plate.
$\tilde{S}^{\prime}_{k_{V}}$: $V$-polarized signal photon obtained by passing
$\tilde{S}^{\prime}_{k_{H}}$ through the HWP. BS: 50–50 beam splitter.
$\tilde{S}^{\prime}_{K\pm}$: outputs from the 50–50 beam splitter. SPD±:
single-photon detectors on the beam splitter’s $\pm$ output ports.

The original version of the Duan-Kimble quantum memory [20], and related
experiments with SiV-based memories [27, 28, 29], used an on-off protocol. In
a simplified four-state model, in which the qubit is stored in two ground
states each of which couples to a different excited state, on-off operation
has the optical frequency _on_ resonance with the cavity and one of the
ground-to-excited-state transitions and far _off_ resonance for the other
transition. Raymer  _et al_. [21] introduced the push-pull protocol, in which
the optical frequency is on resonance with the cavity but midway between the
two ground-to-excited-state transitions, as shown in Fig. 5, and their
analysis showed that push-pull operation offers several advantages over on-off
operation. Hence we adopt the push-pull scheme for this study, although our
memory analysis can be modified to apply to on-off operation.

Figure 5: Schematic of the SiV memory’s four-state model with push-pull
operation. $\tilde{\omega}_{S_{0}}$: center frequency of the frequency-
shifted, bandwidth-compressed signal photons. $\omega_{c}$: resonance
frequency of the SiV-containing cavity. $|g_{1}\rangle$ and $|g_{2}\rangle$:
ground states of the two independent SiV transitions in which the qubit will
be stored. $|e_{1}\rangle$ and $|e_{2}\rangle$: excited states of the two
independent SiV transitions. $\Delta_{12}$: blue detuning of transition 2 from
transition 1. In push-pull operation, $\tilde{\omega}_{S_{0}}=\omega_{c}$ is
blue (red) detuned by $\Delta_{12}/2$ from the first (second) transition.

In summary, ZALM’s goal is to spectrally multiplex heralded biphotons at its
QTX thereby increasing the entanglement-distribution rate to remote locations
while minimizing the occurrence of multiple biphotons from a given pump pulse
in a given DWDM channel. For example, with 100 channels, each with a 1%
probability of containing a biphoton from a particular pump pulse, the overall
probability of there being a biphoton for distribution to Alice and Bob from
that pump pulse is $\sim$1, whereas the probability of a heralded channel
containing two biphotons is $\sim$0.01%. ZALM’s advantageous scaling, in
comparison with its entanglement-distribution competitors, was demonstrated
theoretically in Chen  _et al_. [12].

We are now prepared to dive into analyzing ZALM’s source, partial-BSM, and
memory loading. Before doing so, however, it is worth warding off a possible
misunderstanding about heralding. Aside from losses, our analyses will assume
ideal equipment, e.g., the detectors will not have dark counts, the beam
splitters (polarizing and 50–50) will be perfect, etc. Nevertheless, we will
see that a partial BSM which heralds a $\psi^{\mp}$ state need _not_ result in
a $\psi^{\mp}$ state being sent to Alice and Bob. Indeed, there is, in
general, a nonzero probability that a $\psi^{\mp}$ herald from the partial-BSM
results in a $\psi^{\pm}$ state being transmitted to Alice and Bob, thus
degrading the ZALM scheme’s performance. The culprit here is the ZALM QTX’s
having Sagnac-configured SPDC sources that emit _nonseparable_ channelized
biphotons. To the best of our knowledge, our work is the first to explicitly
treat that error probability and its effect on heralded-biphoton purity and
consequent color-center entangled-state fidelity. See Ou  _et al_. [30],
however, for an early treatment of quantum interference between the outputs
from a pair of SPDC sources.

##  III Sagnac-Configured SPDC Source

Here we begin our paper’s journey—from biphoton source, through partial BSM,
to loaded intra-cavity color-center quantum memories—by analyzing the Sagnac-
configured SPDC source shown in Fig. 2, focusing on its ensuing behavior when
its signal and idler are channelized by complementary DWDM filters.

###  III.1 Perturbation theory for the SPDC source

In Fig. 2, a horizontally ($H$) polarized pulse with center frequency
$\omega_{P}$ pumps a type-0 phase matched PPLN crystal in both the counter-
clockwise (ccw) and clockwise (cw) directions. Operation is nondegenerate in
frequency, with the SPDC’s ccw-propagating and cw-propagating signal and idler
having center frequencies $\omega_{S_{0}}$ and $\omega_{I_{0}}$ satisfying
$\omega_{S_{0}}+\omega_{I_{0}}=\omega_{P}$ and
$|\omega_{S_{0}}-\omega_{I_{0}}|\gg\Omega_{\rm PM}$, where $\Omega_{\rm PM}$
is the crystal’s phase-matching bandwidth. Type-0 phase matching results in
the ccw- and cw-propagating signal and idler being $H$ polarized as they exit
the crystal. Consequently, Fig. 2’s half-wave plate (HWP), triple-wavelength
PBS and dichroic mirrors are able to separate the pump, signal, and idler
after they exit the Sagnac loop.

Using standard perturbation theory [12, 22], justified by the inefficient
nature of spontaneous parametric downconversion, a pump pulse will, to first
order, result in the following ccw-propagating and cw-propagating unnormalized
(vacuum plus biphoton) states entering Fig. 2’s PBS:

$\displaystyle|\tilde{\psi}\rangle_{SI}^{\rm ccw}=\sqrt{P_{0}}\,|{\bf
0}\rangle_{SI}$ $\displaystyle\,\,-\sqrt{P_{1}}\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{SI}(\omega_{S},\omega_{I})|\omega_{S}\rangle_{S_{V}}|\omega_{I}\rangle_{I_{V}},$
(1) $\displaystyle|\tilde{\psi}\rangle_{SI}^{\rm cw}=\sqrt{P_{0}}\,|{\bf
0}\rangle_{SI}$ $\displaystyle\,\,+\sqrt{P_{1}}\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{SI}(\omega_{S},\omega_{I})|\omega_{S}\rangle_{S_{H}}|\omega_{I}\rangle_{I_{H}},$
(2)

where $P_{0}\sim 1$ is the probability that the ccw (cw) pump pulse does not
produce a ccw (cw) biphoton, $P_{1}\ll P_{0}$ is the probability that it does,
and the sign difference between the two biphotons is due to the HWP. In these
equations: $|{\bf 0}\rangle_{SI}$ is the signal-idler vacuum state;
$\Psi_{SI}(\omega_{S},\omega_{I})$ is the normalized,

$\int\\!\frac{{\rm d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{SI}(\omega_{S},\omega_{I})|^{2}=1,$ (3)

frequency-domain biphoton wave function; the $|\omega_{K}\rangle_{K_{P}}$ for
$K=S,I$ and $P=H,V$, denote horizontally ($P=H$) and vertically ($P=V$)
polarized, frequency-$\omega_{K}$, single-photon states [31] of the signal
($K=S$) and idler ($K=I$); and integrals without limits are taken to be from
$-\infty$ to $\infty$ because $\Omega_{\rm PM}\ll\omega_{K_{0}}$. So, because
the ccw and cw pumps are coherent with each other, the preceding ccw and cw
states give us the following unnormalized (vacuum plus biphoton) state leaving
the Sagnac source:

$|\tilde{\psi}\rangle_{SI}=P_{0}|\,{\bf
0}\rangle_{SI}+\sqrt{2P_{0}P_{1}}\,|\psi\rangle_{SI},$ (4)

where

$|\psi\rangle_{SI}\equiv\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\\!\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{SI}(\omega_{S},\omega_{I})|\psi^{-}(\omega_{S},\omega_{I})\rangle_{SI},$
(5)

is the Sagnac source’s post-selected, normalized biphoton state, with
$|\psi^{-}(\omega_{S},\omega_{I})\rangle_{SI}$ being the polarization-
entangled singlet state,

$|\psi^{-}(\omega_{S},\omega_{I})\rangle_{SI}\equiv(|\omega_{S}\rangle_{S_{H}}|\omega_{I}\rangle_{I_{V}}-|\omega_{S}\rangle_{S_{V}}|\omega_{I}\rangle_{I_{H}})/\sqrt{2}.$
(6)

The biphoton wave function plays a pivotal role in what follows. It is
properly normalized and proportional to
$\mathcal{E}_{P}(\omega_{S}+\omega_{I})\Phi_{\rm PM}(\omega_{S},\omega_{I})$
[12, 22], where $\mathcal{E}_{P}(\omega)$ is the spectrum of the pump pulse’s
positive-frequency field and $\Phi_{\rm PM}(\omega_{S},\omega_{I})$ is the
PPLN crystal’s phase-matching function. The next subsection introduces the
channelized version of $\Psi_{SI}(\omega_{S},\omega_{I})$, and uses its all-
Gaussian special case to illustrate a fundamental tradeoff between heralding
efficiency and purity.

###  III.2 Channelized SPDC source

Anticipating the partial-BSM’s DWDM channelizing of the idlers from the ZALM
QTX’s pair of Sagnac SPDCs, and the complementary DWDM channelizing of the
signals arriving at Alice and Bob’s QRXs, let us examine their effects on the
$|\psi\rangle_{SI}$ biphoton from Eq. (5). To do so, we will employ idealized
$N$-channel DWDM filters—DWDMS for the signal and DWDMI for the idler—with
$\delta B$ Hz bandwidth brickwall channels and $\Delta B$ Hz channel spacing,
where $\Delta B>\delta B$ provides inter-channel guard bands. Their $n$th-
channel frequency responses are

$H_{S_{n}}(\omega_{S})=\left\\{\begin{array}[]{ll}1,&\mbox{for
$|\omega_{S}-\omega_{S_{0}}+2\pi n\Delta B|\leq\pi\delta B$}\\\\[3.61371pt]
0,&\mbox{otherwise},\end{array}\right.$ (7)

and

$H_{I_{n}}(\omega_{I})=\left\\{\begin{array}[]{ll}1,&\mbox{for
$|\omega_{I}-\omega_{I_{0}}-2\pi n\Delta B|\leq\pi\delta B$}\\\\[3.61371pt]
0,&\mbox{otherwise},\end{array}\right.$ (8)

for $n=-(N-1)/2,-(N-3)/2,\ldots,(N-1)/2$, with $N$ being an odd integer.

For a biphoton produced with monochromatic SPDC pumping, energy conservation
dictates that an idler photon at frequency
$\omega_{I}=\omega_{I_{0}}+\delta\omega$ is accompanied by a signal photon at
frequency $\omega_{S}=\omega_{S_{0}}-\delta\omega$. For pulsed pumping,
however, this need not be the case, because there is a range of energy-
conserving frequencies for the signal-photon companion of an idler at
frequency $\omega_{I}=\omega_{I_{0}}+\delta\omega$.

To illustrate this behavior—and for future use throughout the partial-BSM and
memory-loading analyses—we introduce the all-Gaussian biphoton wave function
[32, 33],

$\displaystyle\Psi_{SI}(\omega_{S},\omega_{I})$
$\displaystyle=\sqrt{8\pi\sigma_{P}/\Omega_{\rm
PM}}\,e^{-(\Delta\omega_{S}+\Delta\omega_{I})^{2}\sigma^{2}_{P}/16}$
$\displaystyle\,\,\times
e^{-4(\Delta\omega_{S}-\Delta\omega_{I})^{2}/\Omega_{\rm PM}^{2}},$ (9)

with $\Delta\omega_{K}\equiv\omega_{K}-\omega_{K_{0}}$, for $K=S,I$, being the
signal and idler’s detunings from their respective center frequencies. The
first exponential term in Eq. (9) comes from
$\mathcal{E}_{P}(\omega_{S}+\omega_{I})$ for a transform-limited Gaussian pump
pulse with time-domain complex envelope proportional to
$e^{-4t^{2}/\sigma_{P}^{2}}$, and the second exponential term comes from
$\Phi_{\rm PM}(\omega_{S},\omega_{I})$ for a PPLN crystal engineered to have a
Gaussian phase-matching function [32, 34, 35, 36].

Using the all-Gaussian wave function with brickwall filtering and parameter
values from Table 1, Fig. 6 plots, versus channel number, the SPDC source’s
heralding probability, i.e., the probability that the idler photon will pass
through the DWDMI filter’s $n$th channel,

$\Pr(I_{n})=\int\\!\frac{{\rm d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{SI_{n}}(\omega_{S},\omega_{I})|^{2},$ (10)

where
$\Psi_{SI_{n}}(\omega_{S},\omega_{I})\equiv\Psi_{SI}(\omega_{S},\omega_{I})H_{I_{n}}(\omega_{I})$.
Figure 7 then plots, versus channel number, the heralding efficiency, viz.,
the probability that the signal photon will pass through the DWDMS filter’s
$n$th channel given that its idler companion passed through the DWDMI filter’s
$n$th channel,

$\Pr(S_{n}\mid I_{n})=\frac{\displaystyle\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})|^{2}}{\displaystyle\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{SI_{n}}(\omega_{S},\omega_{I})|^{2}},$ (11)

where
$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})\equiv\Psi_{SI}(\omega_{S},\omega_{I})H_{S_{n}}(\omega_{S})H_{I_{n}}(\omega_{I})$
is the dual-Sagnac source’s channelized biphoton wave function.

Figures 6 and 7 reveal some interesting behaviors. First, $\Pr(I_{n})$ is
virtually independent of the pump pulse’s duration and rolls off, with
increasing $|n|$, owing to the Gaussian nature of the phase-matching function.
Second, $\Pr(S_{n}\mid I_{n})$ is constant across DWDM channels—hence largely
independent of the phase-matching bandwidth—but it is strongly affected by the
pump pulse’s duration. Indeed, as expected, when the pump pulse’s bandwidth,
$\sim$$1/\sigma_{P}$ is significantly less than the channel bandwidth, $\delta
B$, as it is in Case 1, we get $\Pr(S_{n}\mid I_{n})\approx 0.94$, but when
$1/\sigma_{P}=2.5\,\delta B$, as it is in Case 2, $\Pr(S_{n}\mid I_{n})$ drops
to 0.44.

Figure 6: Heralding probability, $\Pr(I_{n})$ from Eq. (10), plotted versus channel number assuming Eq. (9)’s all-Gaussian biphoton wave function with brickwall filtering and parameter values from Table 1. The Case 1 and Case 2 curves are indistinguishable.  Figure 7: Heralding efficiency, $\Pr(S_{n}\mid I_{n})$ from Eq. (11), plotted versus channel number assuming Eq. (9)’s all-Gaussian biphoton wave function with brickwall filtering and parameter values from Table 1.  Parameter | Symbol | Case 1 | Case 2
---|---|---|---
pump-pulse duration | $\sigma_{P}$ | 160 ps | 16 ps
phase-matching bandwidth | $\Omega_{\rm PM}/2\pi$ | 6.37 THz | 6.37 THz
channel bandwidth | $\delta B$ | 25 GHz | 25 GHz
DWDM channel spacing | $\Delta B$ | 30 GHz | 30 GHz
# of DWDM channels | $N$ | 81 | 81
Table 1: Parameter values used in Figs. 6–14. $\Omega_{\rm PM}/2\pi=6.37$ THz
corresponds to a $\sigma_{\rm cor}=0.1$ ps biphoton correlation time in Ref.
[33]’s time-domain wave function.

It might seem that long pump pulses would be desirable, because they ensure
high values of the Sagnac source’s channelized heralding efficiency, i.e.,
$\Pr(S_{n}\mid I_{n})$. Heralding efficiency, however, is _not_ the only
figure of merit worthy of consideration. Both ZALM’s partial BSMs and its
intra-cavity color-center quantum memories rely, for their proper functioning,
on high-quality quantum interference, e.g., between $\tilde{S}^{\prime}_{kV}$
and $\tilde{S}_{kV}$ in Fig. 4 for the quantum memories. High-quality quantum
interference requires co-polarized photons of high purity. To assess what it
takes to achieve that high purity, consider the the unnormalized density
operator of an $n$th-channel biphoton from a single Sagnac source,

$\displaystyle\tilde{\rho}_{S_{n},I_{n}}$ $\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{I}}{4\pi^{2}}\,\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})\Psi_{S_{n}I_{n}}^{*}(\omega^{\prime}_{S},\omega_{I}^{\prime})$
$\displaystyle\,\,\times|\psi^{-}(\omega_{S},\omega_{I})\rangle_{SI}\,{}_{SI}\langle\psi^{-}(\omega_{S}^{\prime},\omega_{I}^{\prime})|,$
(12)

where, for $K=S,I$,
${\boldsymbol{\omega}}_{K}\equiv(\omega_{K},\omega^{\prime}_{K})$ and ${\rm
d}^{2}{\boldsymbol{\omega}}_{K}\equiv{\rm d}\omega_{K}\,{\rm
d}\omega_{K}^{\prime}$. By tracing out the idler we obtain an unnormalized
version of the conditional density operator given by

$\displaystyle\tilde{\rho}_{S_{n}\mid I_{n}}$ $\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})\Psi_{S_{n}I_{n}}^{*}(\omega^{\prime}_{S},\omega_{I})$
$\displaystyle\,\,\times\left[|\omega_{S}\rangle_{S_{H}}\,{}_{S_{H}}\langle\omega^{\prime}_{S}|+|\omega_{S}\rangle_{S_{V}}\,{}_{S_{V}}\langle\omega^{\prime}_{S}|\right]/2,$
(13)

whose normalization constant is

$\displaystyle{\rm Tr}(\tilde{\rho}_{S_{n}\mid I_{n}})$
$\displaystyle=\int\\!\frac{{\rm d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})|^{2}$
$\displaystyle=\Pr(S_{n},I_{n}),$ (14)

where $\Pr(S_{n},I_{n})$ is the joint probability that the signal and idler
photons both pass through their DWDM filter’s $n$th channels.

Note that the quantity defined as

$\hat{I}({\boldsymbol{\omega}}_{S})\equiv\left[|\omega_{S}\rangle_{S_{H}}\,{}_{S_{H}}\langle\omega^{\prime}_{S}|+|\omega_{S}\rangle_{S_{V}}\,{}_{S_{V}}\langle\omega^{\prime}_{S}|\right]/2$
(15)

indicates that Eq. (13) is a randomly-polarized state, as expected from the
$S_{n}$-$I_{n}$ biphoton’s having maximal polarization entanglement. Because
of that random polarization, we have that

$\hat{I}({\boldsymbol{\omega}}_{S})\equiv\left[|\omega_{S}\rangle_{S_{P}}\,{}_{S_{P}}\langle\omega^{\prime}_{S}|+|\omega_{S}\rangle_{S_{P_{\perp}}}\,{}_{S_{P_{\perp}}}\langle\omega^{\prime}_{S}|\right]/2,$
(16)

for $(P,P_{\perp})$ being an _arbitrary_ polarization basis, and so our quest
for high-quality quantum interference in ZALM’s partial BSMs and quantum
memories can focus on the purity of $S_{n_{P}}$.

Projecting onto Eq. (13)’s $P$-polarized component, we get

$\displaystyle\tilde{\rho}_{S_{n_{P}}\mid I_{n}}$
$\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})\Psi_{S_{n}I_{n}}^{*}(\omega^{\prime}_{S},\omega_{I})$
$\displaystyle\,\,\times\,|\omega_{S}\rangle_{S_{P}}\,{}_{S_{P}}\langle\omega^{\prime}_{S}|/2,$
(17)

whose normalization constant is $\Pr(S_{n},I_{n})/2$. The square of the
normalized version of this density operator is therefore

$\displaystyle\hat{\rho}^{2}_{S_{n}\mid I_{n}}=$ (18)
$\displaystyle\frac{\displaystyle\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}\tilde{\omega}_{S}}{2\pi}\,\Phi_{n}(\omega_{S},\omega^{\prime}_{S})\Phi_{n}(\omega_{S}^{\prime},\tilde{\omega}_{S})|\omega_{S}\rangle_{S_{P}}\,{}_{S_{P}}\langle\tilde{\omega}_{S}|}{\displaystyle\left(\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})|^{2}\right)^{2}}\\!,$

with

$\Phi_{n}(\omega_{S},\omega^{\prime}_{S})\equiv\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{S_{n},I_{n}}(\omega_{S},\omega_{I})\Psi^{*}_{S_{n}I_{n}}(\omega^{\prime}_{S},\omega_{I}),$
(19)

from which we get that the purity of $S_{n_{P}}$ is

$\displaystyle{\rm Tr}(\hat{\rho}^{2}_{S_{n_{P}}\mid I_{n}})=$ (20)
$\displaystyle\frac{\displaystyle\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\left|\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})\Psi^{*}_{S_{n}I_{n}}(\omega^{\prime}_{S},\omega_{I})\right|^{2}}{\displaystyle\left(\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})|^{2}\right)^{2}}.$

Applying the Schwarz inequality to Eq. (20)’s numerator verifies that ${\rm
Tr}(\hat{\rho}^{2}_{S_{n_{P}}\mid I_{n}})\leq 1$, and it is seen that equality
occurs when the channelized biphoton wave function is separable, i.e.,
$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})=\Psi_{S_{n}}(\omega_{S})\Psi_{I_{n}}(\omega_{I})$.
In the next subsection we prove that separability is the only way to obtain
100% purity.

Figure 8 plots the single source’s single-polarization purity, ${\rm
Tr}(\hat{\rho}_{S_{n_{P}}\mid I_{n}}^{2})$, versus channel number assuming Eq.
(9)’s all-Gaussian biphoton wave function and brickwall filtering with
parameter values from Table 1. Like what we found for the heralding
efficiency, we see that the phase-matching bandwidth has no effect on the
purity while the pump pulse’s bandwidth does impact the purity. Unlike what we
found for the heralding efficiency, however, Case 2 is now preferable to Case
1, because the former’s low time-bandwidth product, $\sigma_{P}\delta B=0.4$,
ensures a nearly single temporal mode (100% purity) signal photon, whereas the
latter’s $\sigma_{P}\delta B=4$ yields a low-purity, ${\rm
Tr}(\hat{\rho}_{S_{n_{P}}\mid I_{n}}^{2})=0.27$, signal photon.

Figure 8: Single source’s single-polarization purity, ${\rm
Tr}(\hat{\rho}_{S_{n_{P}}\mid I_{n}}^{2})$ from Eq. (20), versus channel
number for the $n$th-channel signal photon resulting from an $n$th-channel
idler herald assuming Eq. (9)’s all-Gaussian biphoton wave function with
brickwall filtering and parameter values from Table 1.

###  III.3 Singular-value decomposition of
$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})$

The channelized biphoton wave function,
$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})$, has a singular-value decomposition
that we will write in the form

$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})=\sum_{\ell=1}^{\infty}\lambda_{\ell}\,\phi_{\ell}(\omega_{S})\psi_{\ell}(\omega_{I}),$
(21)

where $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{\ell}\geq\cdots\geq 0$
are its singular values, and the $\\{\phi_{\ell}(\omega_{S})\\}$ and
$\\{\psi_{\ell}(\omega_{I})\\}$ are, respectively, complete orthonormal (CON)
function sets on the passbands of $H_{S_{n}}(\omega_{S})$ and
$H_{I_{n}}(\omega_{I})$ [37]. This SVD will play important roles going
forward, beginning in this subsection where it is applied to Sec. III.2’s
formulas for $\Pr(S_{n},I_{n})$, $\Phi_{n}(\omega_{S},\omega^{\prime}_{S})$,
and ${\rm Tr}(\hat{\rho}^{2}_{S_{n_{P}}\mid I_{n}})$.

Substituting Eq. (21) into Eqs. (14), (19), and (20), we can use the CON
natures of the mode sets $\\{\phi_{\ell}(\omega_{S})\\}$ and
$\\{\psi_{\ell}(\omega_{I})\\}$ to obtain their SVD forms:

$\displaystyle\Pr(S_{n},I_{n})$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\lambda_{\ell}\lambda_{\ell^{\prime}}\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\,\phi_{\ell}(\omega_{S})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S})$
$\displaystyle\,\,\times\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\psi_{\ell}(\omega_{I})\psi^{*}_{\ell^{\prime}}(\omega_{I})=\sum_{\ell=1}^{\infty}\lambda_{\ell}^{2},$
(22) $\displaystyle\Phi_{n}(\omega_{S},\omega^{\prime}_{S})$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\lambda_{\ell}\lambda_{\ell^{\prime}}\,\phi_{\ell}(\omega_{S})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S})$
$\displaystyle\,\,\times\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\psi_{\ell}(\omega_{I})\psi^{*}_{\ell^{\prime}}(\omega_{I})$
(23)
$\displaystyle=\sum_{\ell=1}^{\infty}\lambda_{\ell}^{2}\,\phi_{\ell}(\omega_{S})\phi^{*}_{\ell}(\omega^{\prime}_{S}),$
(24)

and

$\displaystyle{\rm Tr}(\hat{\rho}_{S_{n_{P}}\mid I_{n}}^{2})$
$\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\left|\sum_{\ell=1}^{\infty}\tilde{\lambda}_{\ell}^{2}\,\phi_{\ell}(\omega_{S})\phi^{*}_{\ell}(\omega^{\prime}_{S})\right|^{2}$
(25)
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\tilde{\lambda}^{2}_{\ell}\tilde{\lambda}^{2}_{\ell^{\prime}}\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\,\phi_{\ell}(\omega_{S})\phi^{*}_{\ell^{\prime}}(\omega_{S})$
$\displaystyle\,\,\times\int\\!\frac{{\rm
d}\omega^{\prime}_{S}}{2\pi}\,\phi^{*}_{\ell}(\omega^{\prime}_{S})\phi_{\ell^{\prime}}(\omega^{\prime}_{S})=\sum_{\ell=1}^{\infty}\tilde{\lambda}^{4}_{\ell},$
(26)

where the $\\{\tilde{\lambda}_{\ell}\\}$, given by
$\tilde{\lambda}_{\ell}\equiv\lambda_{\ell}/\sqrt{\sum_{\ell=1}^{\infty}\lambda^{2}_{\ell}}$,
are $\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})$’s normalized
($\sum_{\ell=1}^{\infty}\tilde{\lambda}^{2}_{\ell}=1$) singular values.

Because of the $\\{\tilde{\lambda}_{\ell}\\}$’s foregoing normalization, it is
obvious that ${\rm Tr}(\hat{\rho}_{S_{n_{P}}\mid I_{n}}^{2})\leq 1$, with
equality if and only if $\tilde{\lambda}_{1}=1$ and $\tilde{\lambda}_{\ell}=0$
for $\ell=2,3,\ldots$ In other words, ${\rm Tr}(\hat{\rho}_{S_{n_{P}}\mid
I_{n}}^{2})=1$ if and only if the channelized wave function is separable,

$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})=\sqrt{\Pr(S_{n},I_{n})}\,\phi_{1}(\omega_{S})\psi_{1}(\omega_{I}),$
(27)

as claimed in Sec. III.2.

##  IV Partial Bell-State Measurement

This section’s principal goals are to derive: (1) the dual-Sagnac source’s
normalized heralding probability, i.e., the probability that an ideal (unit
efficiency) $n$th-channel partial BSM will herald a $\psi^{\mp}$ biphoton
[38]; (2) the dual-Sagnac source’s heralding efficiency, viz., the probability
that the photons sent to Alice and Bob’s QRXs by the ZALM QTX pass through
those QRX’s $n$th DWDMS channels, given that the QTX’s partial BSM heralds an
$n$th-channel $\psi^{\mp}$ state; (3) the normalized density operator,
$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}$, of the biphoton sent to
Alice and Bob, given that the $\\{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}\\}$
event has occurred; and (4) the purity, ${\rm
Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})$, of the biphoton
sent to Alice and Bob, given that the
$\\{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}\\}$ event has occurred. In pursuing
these goals we assume ideal equipment, except for symmetric losses that we
coalesce into identical efficiencies, $0<\eta_{\rm qtx}<1$, assigned to Fig.
3’s SPDs but ignored for now as they are the normalization constant for the
first goal and they do not affect the others’ post-selected events.

The partial-BSM’s normalized heralding probability, which we denote
$\Pr(\psi^{\mp}_{n})$, is obtained as follows. Assuming ideal (unit
efficiency) equipment, we have that $\Pr(\psi^{\mp}_{n})$ is the probability
that both idler photons pass through the DWDMI filter’s $n$th channel _and_
result in one $H$ and one $V$ detection from that channel’s SPDs. Because the
two idlers are equally likely to be in any of the four polarization Bell
states, we get

$\displaystyle\Pr(\psi^{\mp}_{n})$ $\displaystyle=[\Pr(I_{n})]^{2}/4$ (28)
$\displaystyle=\frac{1}{4}\\!\left(\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{SI_{n}}(\omega_{S},\omega_{I})|^{2}\right)^{2}.$
(29)

Finding the other quantities of interest will require greater effort.

###  IV.1 Derivations of $\Pr(S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n})$ and
$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}$

Assuming that neither of its coherently-pumped, identical, Sagnac SPDC sources
makes a multi-pair emission, ZALM’s QTX produces, with probability
$(2P_{0}P_{1})^{2}$, the post-selected joint state
$|\psi\rangle_{S_{1}I_{1}}\otimes|\psi\rangle_{S_{2}I_{2}}$, where

$|\psi\rangle_{S_{k}I_{k}}\equiv\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\\!\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,\Psi_{SI}(\omega_{S},\omega_{I})|\psi^{-}(\omega_{S},\omega_{I})\rangle_{S_{k}I_{k}},$
(30)

for $k=1,2$, with

$\displaystyle|\psi^{-}(\omega_{S},\omega_{I})\rangle_{S_{k}I_{k}}\equiv$ (31)
$\displaystyle(|\omega_{S}\rangle_{S_{k_{H}}}|\omega_{I}\rangle_{I_{k_{V}}}-|\omega_{S}\rangle_{S_{k_{V}}}|\omega_{I}\rangle_{I_{k_{H}}})/\sqrt{2}.$

Note that $P_{1}\ll P_{0}$ suffices, to first order, for ignoring a multi-pair
emission from a single Sagnac-configured SPDC, but it does _not_ suffice for
ignoring multi-pair emissions from _both_ sources. That insufficiency arises
because using the first-order expressions from Eq. (30) in
$|\psi\rangle_{S_{1}I_{1}}\otimes|\psi\rangle_{S_{2}I_{2}}$ yields a second-
order expression. Thus terms involving source 1 emitting two pairs while none
are emitted by source 2 and vice versa should be included to get a joint
$S_{1}$-$S_{2}$ state that is correct to second order.

We will not include the unwanted second-order terms in our analysis, because
they do not send photons to _both_ Alice _and_ Bob. Hence, when background
light and dark counts at Alice and Bob’s QRXs can be ignored, as we shall
assume, the unwanted second-order terms cannot lead to memory loads at both
receivers and we can safely proceed without including them.

Let $|\tilde{\psi}\rangle_{{\bf S}_{n}{\bf
I}_{n}}\equiv|\tilde{\psi}\rangle_{S_{1_{n}}I_{1_{n}}}\otimes|\tilde{\psi}\rangle_{S_{2_{n}}I_{2_{n}}}$
be the unnormalized state that results from restricting
$|\psi\rangle_{S_{1}I_{1}}\otimes|\psi\rangle_{S_{2}I_{2}}$ to the $n$th DWDMS
and DWDMI channels for the signal and idler, respectively. Unnormalized
density operators for $S_{1_{n}},S_{2_{n}},\psi^{\mp}$ events are found by
tracing over their heralding detections, giving us

$\displaystyle\tilde{\rho}_{S_{1_{n}}S_{2_{n}},\psi^{-}_{n}}$
$\displaystyle\equiv\int\\!\frac{{\rm d}\omega_{I_{+}}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I_{-}}}{2\pi}\,({}_{I_{+H}}\\!\langle\omega_{I_{+}}|\,{}_{I_{-V}}\\!\langle\omega_{I_{-}}|)|\tilde{\psi}\rangle_{{\bf
S}_{n}{\bf I}_{n}}\,{}_{{\bf S}_{n}{\bf
I}_{n}}\\!\langle\tilde{\psi}|(|\omega_{I_{-}}\rangle_{I_{-V}}\,|\omega_{I_{+}}\rangle_{I_{+H}})$
$\displaystyle\,\,+\int\\!\frac{{\rm d}\omega_{I_{+}}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I_{-}}}{2\pi}\,({}_{I_{+V}}\\!\langle\omega_{I_{+}}|\,{}_{I_{-H}}\\!\langle\omega_{I_{-}}|)|\tilde{\psi}\rangle_{{\bf
S}_{n}{\bf I}_{n}}\,{}_{{\bf S}_{n}{\bf
I}_{n}}\\!\langle\tilde{\psi}|(|\omega_{I_{-}}\rangle_{I_{-H}}\,|\omega_{I_{+}}\rangle_{I_{+V}}),$
(32)

and

$\displaystyle\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\,\psi^{+}_{n}}$
$\displaystyle\equiv\int\\!\frac{{\rm d}\omega_{I_{+}}}{2\pi}\int\\!\frac{{\rm
d}\omega^{\prime}_{I_{+}}}{2\pi}\,({}_{I_{+H}}\\!\langle\omega_{I_{+}}|\,{}_{I_{+V}}\\!\langle\omega^{\prime}_{I_{+}}|)|\tilde{\psi}\rangle_{{\bf
S}_{n}{\bf I}_{n}}\,{}_{{\bf S}_{n}{\bf
I}_{n}}\\!\langle\tilde{\psi}|(|\omega^{\prime}_{I_{+}}\rangle_{I_{+V}}\,|\omega_{I_{+}}\rangle_{I_{+H}})$
$\displaystyle\,\,+\int\\!\frac{{\rm d}\omega_{I_{-}}}{2\pi}\int\\!\frac{{\rm
d}\omega^{\prime}_{I_{-}}}{2\pi}\,({}_{I_{-H}}\langle\omega_{I_{-}}|\,{}_{I_{-V}}\\!\langle\omega^{\prime}_{I_{-}}|)|\tilde{\psi}\rangle_{{\bf
S}_{n}{\bf I}_{n}}\,{}_{{\bf S}_{n}{\bf
I}_{n}}\\!\langle\tilde{\psi}|(|\omega^{\prime}_{I_{-}}\rangle_{I_{-V}}\,|\omega_{I_{-}}\rangle_{I_{-H}}).$
(33)

Now, using the beam-splitter relations

$|\omega_{I}\rangle_{I_{1P}}=\frac{|\omega_{I}\rangle_{I_{+P}}+|\omega_{I}\rangle_{I_{-P}}}{\sqrt{2}},\mbox{
and
}|\omega_{I}\rangle_{I_{2P}}=\frac{|\omega_{I}\rangle_{I_{+P}}-|\omega_{I}\rangle_{I_{-P}}}{\sqrt{2}},\mbox{
for $P=H,V$},$ (34)

in $|\tilde{\psi}\rangle_{{\bf S}_{n}{\bf I}_{n}}$, and their adjoints in
${}_{{\bf S}_{n}{\bf I}_{n}}\langle\tilde{\psi}|$, Appendix A shows that

$\displaystyle\tilde{\rho}_{S_{1_{n}},S_{2_{n}},\psi^{\mp}_{n}}$
$\displaystyle=\frac{1}{8}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})|\psi^{\mp}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{\mp}({\boldsymbol{\omega}}^{\prime}_{S})|$
$\displaystyle\,\,+\frac{1}{8}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})|\psi^{\pm}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{\pm}({\boldsymbol{\omega}}^{\prime}_{S})|,$
(35)

where the kernels are

$\displaystyle
K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
$\displaystyle\equiv\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{1}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{2}})$
$\displaystyle\,\,+\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{2}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{1}}),$
(36)

and

$\displaystyle
K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
$\displaystyle\equiv\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{1}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{2}})$
$\displaystyle\,\,-\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{2}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{1}}),$
(37)

with ${\boldsymbol{\omega}}_{S}$ and ${\rm d}^{2}{\boldsymbol{\omega}}_{S}$
here and henceforth denoting $(\omega_{S_{1}},\omega_{S_{2}})$ and ${\rm
d}\omega_{S_{1}}{\rm d}\omega_{S_{2}}$.

In Eqs. (36) and (37), the superscripts $(c)$ and $(e)$ denote “correct” and
“error”, respectively, because the
$K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
kernel is associated with the term for which a $\psi^{\mp}_{n}$ herald results
in a $\psi^{\mp}_{n}$ biphoton being sent to Alice and Bob, while the
$K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
kernel is associated with the term for which a $\psi^{\mp}_{n}$ herald results
in a $\psi^{\pm}_{n}$ biphoton being sent to Alice and Bob.

Tracing $\tilde{\rho}_{S_{1_{n}},S_{2_{n}},\psi^{\mp}_{n}}$ in the Bell basis
we obtain the normalization constant,

$\displaystyle\Pr(S_{1_{n}},S_{2_{n}},\psi^{\mp}_{n})=$ (41)
$\displaystyle\frac{1}{8}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\\!\left[K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})+K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})\right]$
$\displaystyle=\frac{1}{4}\\!\left(\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\,\Phi_{n}(\omega_{S},\omega_{S})\right)^{2}$
$\displaystyle=\frac{1}{4}\\!\left(\int\\!\frac{{\rm
d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})|^{2}\right)^{2}$
$\displaystyle=[\Pr(S_{n},I_{n})]^{2}/4,$

so that

$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}=\frac{\tilde{\rho}_{S_{1_{n}}S_{2_{n}},\psi^{\mp}_{n}}}{[\Pr(S_{n},I_{n})]^{2}/4}.$
(42)

Equation (41) makes perfect sense. Given that the $I_{+}$ and $I_{-}$ photons
pass through their DWDMI filters’ $n$th-channel passbands, the joint state of
the $I_{+n}$ and $I_{-n}$ photons is equally likely to be any of the four
polarization Bell states. Thus, assuming they have passed through those
passbands, the probability of their resulting in a $\psi^{\mp}_{n}$ herald is
1/4. Meanwhile, assuming the ZALM QTX’s Sagnac-configured SPDCs each produce a
signal-idler biphoton, the probability that both signals and both idlers pass
through their DWDM filter’s $n$th-channel passbands is
$[\Pr(S_{n},I_{n})]^{2}$. Combining these two results we get Eq. (41). We now
see that the partial-BSM’s heralding efficiency satisfies

$\Pr(S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n})=\frac{\Pr(S_{1_{n}},S_{2_{n}},\psi^{\mp}_{n})}{\Pr(\psi^{\mp}_{n})}=\left[\frac{\Pr(S_{n},I_{n})}{\Pr(I_{n})}\right]^{2}.$
(43)

Our next goal is to find the purity of the heralded biphoton state sent to
Alice and Bob, ${\rm
Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})$. En route to that
goal it will be fruitful to take a detour, in the next subsection, to express
$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}$ in terms of the single
Sagnac source’s SVD.

###  IV.2 Single-Sagnac SVD representation of
$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}$

Substituting Eq. (24) into Eqs. (36) and (37) gives us

$\displaystyle
K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\lambda^{2}_{\ell}\lambda_{\ell^{\prime}}^{2}$
(44) $\displaystyle\times$
$\displaystyle[\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{1}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{2}})$
$\displaystyle\,\,+\,\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{2}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{1}})],$

and

$\displaystyle
K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\lambda^{2}_{\ell}\lambda_{\ell^{\prime}}^{2}$
(45) $\displaystyle\times$
$\displaystyle[\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{1}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{2}})$
$\displaystyle\,\,-\,\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{2}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{1}})].$

It is immediately evident from these results that both
$K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
and
$K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
are Hermitian kernels, i.e.,

$K^{(p)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}^{\prime}_{S};{\boldsymbol{\omega}}_{S})=K^{(p)*}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S}),\mbox{
for $p=c,e$}.$ (46)

As a result, they have eigenvalue-eigenfunction decompositions,

$K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\sum_{m=1}^{\infty}\mu_{m}\,\xi_{m}({\boldsymbol{\omega}}_{S})\xi^{*}_{m}({\boldsymbol{\omega}}^{\prime}_{S}),$
(47)

and

$K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\sum_{m=1}^{\infty}\nu_{m}\,\zeta_{m}({\boldsymbol{\omega}}_{S})\zeta^{*}_{m}({\boldsymbol{\omega}}^{\prime}_{S}),$
(48)

with real eigenvalues, $\\{\mu_{m}\\}$ and $\\{\nu_{m}\\}$, and orthonormal
eigenfunctions, $\\{\xi_{m}({\boldsymbol{\omega}}_{S})\\}$ and
$\\{\zeta_{m}({\boldsymbol{\omega}}_{S})\\}$, for $\omega_{S_{1}}$ and
$\omega_{S_{2}}$ in the DWDMS filter’s $n$th-channel passband [39]. If
eigenfunctions with zero eigenvalues are included, the
$\\{\xi_{m}({\boldsymbol{\omega}}_{S})\\}$ and
$\\{\zeta_{m}({\boldsymbol{\omega}}_{S})\\}$ can be taken to be CON function
sets on their domains.

To find the preceding eigenvalues and eigenfunctions we need to solve the
following Fredholm integral equations,

$\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})\xi_{m}({\boldsymbol{\omega}}^{\prime}_{S})=\mu_{m}\,\xi_{m}({\boldsymbol{\omega}}_{S}),$
(49)

and

$\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})\zeta_{m}({\boldsymbol{\omega}}^{\prime}_{S})=\nu_{m}\,\zeta_{m}({\boldsymbol{\omega}}_{S}),$
(50)

for $\omega_{S_{1}},\omega_{S_{2}}$ in the DWDMS filter’s $n$th-channel
passband. Ordinarily, solving these equations would be a formidable task.
However, because of Eqs. (44) and (45), we note that numerical solutions to
Eqs. (49) and (50) can be readily obtained via available numerical techniques
for finding the SVD of an arbitrary
$\Psi_{S_{n}I_{n}}(\omega_{S},\omega_{I})$. Let us see how this comes about.

We start by changing the scalar ($m$) indices in Eqs. (49) and (50) to vector
indices ${\bf m}\equiv(m_{1},m_{2})$ and choosing the $\\{\xi_{\bf
m}({\boldsymbol{\omega}}_{S})\\}$ and $\\{\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})\\}$ to be

$\xi_{\bf
m}({\boldsymbol{\omega}}_{S})=\left\\{\begin{array}[]{cl}\phi_{m}(\omega_{S_{1}})\phi_{m}(\omega_{S_{2}}),&\mbox{for
$m_{1}=m_{2}=m=1,2,\ldots,$}\\\\[3.61371pt]
[\phi_{m_{1}}(\omega_{S_{1}})\phi_{m_{2}}(\omega_{S_{2}})+\phi_{m_{1}}(\omega_{S_{2}})\phi_{m_{2}}(\omega_{S_{1}})]/\sqrt{2},&\mbox{for
$m_{1}>m_{2}=1,2,\ldots,$}\end{array}\right.$ (51)

and

$\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})=[\phi_{m_{1}}(\omega_{S_{1}})\phi_{m_{2}}(\omega_{S_{2}})-\phi_{m_{1}}(\omega_{S_{2}})\phi_{m_{2}}(\omega_{S_{1}})]/\sqrt{2},\mbox{
for $m_{1}>m_{2}=1,2,\ldots$}$ (52)

Because
$\\{\phi_{m_{1}}(\omega_{S_{1}})\phi_{m_{2}}(\omega_{S_{2}}):m_{1},m_{2}=1,2,\ldots\\}$
are CON on $\omega_{S_{1}},\omega_{S_{2}}$ in the DWDMS filter’s $n$th-channel
passband, it follows that the $\\{\xi_{\bf m}({\boldsymbol{\omega}}_{S})\\}$
and $\\{\zeta_{\bf m}({\boldsymbol{\omega}}_{S})\\}$ are, collectively, a CON
function set on the DWDMS filter’s $n$th-channel passband. Furthermore, it is
easily verified, see Appendix A, that the $\\{\xi_{\bf
m}({\boldsymbol{\omega}}_{S})\\}$ and $\\{\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})\\}$ are, respectively, solutions to Eqs. (49)
and (50) with eigenvalues

$\mu_{\bf m}=\left\\{\begin{array}[]{cl}2\lambda^{4}_{m},&\mbox{for
$m_{1}=m_{2}=m=1,2,\ldots,$}\\\\[3.61371pt]
2\lambda^{2}_{m_{1}}\lambda^{2}_{m_{2}},&\mbox{for
$m_{1}>m_{2}=1,2,\ldots,$}\end{array}\right.$ (53)

and

$\nu_{\bf m}=2\lambda^{2}_{m_{1}}\lambda^{2}_{m_{2}},\mbox{ for
$m_{1}>m_{2}=1,2,\ldots$}$ (54)

Finally, because the $\\{\mu_{\bf m}\\}$ and $\\{\nu_{\bf m}\\}$ are all non-
negative, checking for consistency with Eq. (41) viz., verifying that [40]

$\displaystyle\frac{1}{8}$ $\displaystyle\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\\!\left[K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})+K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})\right]$
$\displaystyle=\sum_{\bf m}\frac{\mu_{\bf m}+\nu_{\bf
m}}{8}=[\Pr(S_{n},I_{n})]^{2}/4,$ (55)

will ensure that Eqs. (51) and (52) include _all_ the eigenfunctions with non-
zero eigenvalues. Using Eqs. (53) and (54) we have that

$\displaystyle\sum_{\bf m}$ $\displaystyle\frac{\mu_{\bf m}+\nu_{\bf
m}}{8}=\frac{1}{4}\sum_{m=1}^{\infty}\lambda_{m}^{4}+\frac{1}{2}\sum_{m_{1}=2}^{\infty}\sum_{m_{2}=1}^{m_{1}-1}\lambda^{2}_{m_{1}}\lambda^{2}_{m_{2}}$
$\displaystyle=\frac{1}{4}\\!\left(\sum_{m=1}^{\infty}\lambda^{2}_{m}\right)^{2}=[\Pr(S_{n},I_{n})]^{2}/4,$
(56)

and the verification is complete.

We can now rewrite the state of the heralded biphoton sent to Alice and Bob,
$\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}$, as

$\displaystyle\hat{\rho}$
${}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}=\sum_{\bf m}\tilde{\mu}_{\bf
m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\xi_{\bf
m}({\boldsymbol{\omega}}_{S})$
$\displaystyle\,\,\,\,\times|\psi^{\mp}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{\mp}({\boldsymbol{\omega}}^{\prime}_{S})|\xi^{*}_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S})$ $\displaystyle+\sum_{\bf
m}\tilde{\nu}_{\bf m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})$
$\displaystyle\,\,\,\,\times|\psi^{\pm}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{\pm}({\boldsymbol{\omega}}^{\prime}_{S})|\zeta^{*}_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S}),$ (57)

where

$\tilde{\mu}_{\bf m}\equiv\frac{\mu_{\bf m}}{\sum_{\bf m}(\mu_{\bf m}+\nu_{\bf
m})},$ (58)

and

$\tilde{\nu}_{\bf m}\equiv\frac{\nu_{\bf m}}{\sum_{\bf m}(\mu_{\bf m}+\nu_{\bf
m})},$ (59)

are its normalized eigenvalues. We note that Eqs. (51)–(54) allow us to
instantiate Eq. (57) for an arbitrary single-source biphoton wave function
using straightforward numerical evaluation of Eq. (21)’s single-source
temporal modes and singular values.

###  IV.3 Purity, fidelity, and error probability

Subsection IV.2’s results allow us to make short work of finding the partial-
BSM’s purity, fidelity, and error probability. For its purity we start from

$\displaystyle\hat{\rho}$
${}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}=\sum_{\bf
m}\tilde{\mu}^{2}_{\bf m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\xi_{\bf
m}({\boldsymbol{\omega}}_{S})$
$\displaystyle\,\,\,\,\times|\psi^{\mp}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{\mp}({\boldsymbol{\omega}}^{\prime}_{S})|\xi^{*}_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S})$ $\displaystyle+\sum_{\bf
m}\tilde{\nu^{2}}_{\bf m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})$
$\displaystyle\,\,\,\,\times|\psi^{\pm}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\\!\langle\psi^{\pm}({\boldsymbol{\omega}}^{\prime}_{S})|\zeta^{*}_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S}),$ (60)

which immediately gives us

$\displaystyle{\rm Tr}(\hat{\rho}$
${}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})=\sum_{\bf
m}(\tilde{\mu}^{2}_{\bf m}+\tilde{\nu}^{2}_{\bf m})$ (61)
$\displaystyle=\sum_{\bf m}\tilde{\lambda}^{8}_{\bf
m}+2\sum_{m_{1}=2}^{\infty}\sum_{m_{2}=1}^{m_{1}-1}\tilde{\lambda}^{4}_{m_{1}}\tilde{\lambda}^{4}_{m_{2}}$
(62)
$\displaystyle=\left(\sum_{m=1}^{\infty}\tilde{\lambda}^{4}_{m}\right)^{2}=\left[{\rm
Tr}(\hat{\rho}^{2}_{S_{n_{P}}\mid I_{n}})\right]^{2},$ (63)

where the last equality shows, as could easily be expected, that the purity of
the ZALM transmission to Alice and Bob is proportional to the product of its
two Sagnac sources’ purities. Note that although the Sagnac source’s SVD was a
valuable tool for deriving Eq. (63), we can evaluate ${\rm
Tr}[(\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})^{2}]$ directly from
the channelized biphoton wave function by squaring the single-source result
from Eq. (20).

Turning now to the partial-BSM’s fidelity, i.e., the probability,
$\Pr(c_{n})$, that a $\psi^{\mp}_{n}$ herald will result in a $\psi^{\mp}_{n}$
biphoton being sent to Alice and Bob, we have that

$\displaystyle\Pr(c_{n})$ $\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,{}_{S_{1_{n}}S_{2_{n}}}\langle\psi^{\mp}({\boldsymbol{\omega}}_{S})|$
$\displaystyle\hat{\rho}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}|\psi^{\mp}_{n}({\boldsymbol{\omega}}_{S})\rangle_{S_{1_{n}}S_{2_{n}}}$
(64) $\displaystyle=\sum_{\bf m}\tilde{\mu}_{\bf
m}=\sum_{m=1}^{\infty}\tilde{\lambda}^{4}_{m}+\sum_{m_{1}=2}^{\infty}\sum_{m_{2}=1}^{m_{1}-1}\tilde{\lambda}^{2}_{m_{1}}\tilde{\lambda}^{2}_{m_{2}}$
(65)
$\displaystyle=\frac{1}{2}\\!\left(1+\sum_{m=1}^{\infty}\tilde{\lambda}^{4}_{m}\right),$
(66)

where the last equality uses $\sum_{m=1}^{\infty}\tilde{\lambda}^{2}_{m}=1$.
Here too the single-source SVD has been a useful tool, but $\Pr(c_{n})$ can be
found directly from the channelized biphoton wave function because

$\Pr(c_{n})=\frac{1+\sqrt{{\rm
Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})}}{2}.$ (67)

To complete this subsection we compute the error probability, $\Pr(e_{n})$.
Because $\hat{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{\mp}_{n}}$ is diagonal in the
$\psi^{\mp}_{n}$ and $\psi^{\pm}_{n}$ Bell states, $\Pr(e_{n})$ is the
probability that a $\psi^{\mp}_{n}$ herald leads to a $\psi^{\pm}_{n}$
biphoton being sent to Alice and Bob. As we must have
$\Pr(c_{n})+\Pr(e_{n})=1$, we get

$\Pr(e_{n})=\frac{1-\sqrt{{\rm
Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})}}{2},$ (68)

which can be found directly from the channelized biphoton wave function
without performing an SVD.

###  IV.4 Partial-BSM performance for the all-Gaussian special case

Figures 9–12 plot, versus channel number, the $n$th-channel partial-BSM’s
normalized heralding probability, $\Pr(\psi^{\mp}_{n})$, its heralding
efficiency, $\Pr(S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n})$, its purity, ${\rm
Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})$, and its error
probability, $\Pr(e_{n})$, which equals $1-\Pr(c_{n})$ where $\Pr(c_{n})$ is
the partial-BSM’s fidelity. These figures assume the all-Gaussian biphoton
wave function from Eq. (9) and brickwall filtering with parameter values from
Table 1.

The behaviors seen in Figs. 9–11 are readily understood from what we saw for
the single Sagnac source. In particular, like their single-source
counterparts, the partial-BSM’s normalized heralding probability and its
heralding efficiency are respectively, insensitive to the pump-pulse duration
and the phase-matching bandwidth. They also exhibit the tradeoff between
heralding probability and heralding efficiency, albeit exacerbated by their
requiring two photons to pass through DWDM filters instead of just one.
Comparing Fig. 11 with its single-source counterpart from Fig. 8, we see that
Case 2 has lost very little purity in going from twice the single source’s
99.2% signal-photon purity to the partial-BSM’s 98.5%-purity heralded
biphoton. In Case 1, however, squaring has a more dramatic effect, with the
partial-BSM’s 7.50% purity being well below the 27.4% value for twice the
single source’s signal-photon purity.

Error probability is a new metric that arises because the partial-BSM’s
$\psi^{\mp}_{n}$ herald may result in a $\psi^{\pm}$ state being sent to Alice
and Bob. Figure 12 shows that Case 2 does well on error probability,
$\Pr(e_{n})=3.87\times 10^{-3}$, because of its high purity, while Case 1
suffers with $\Pr(e_{n})=0.363$ across all 81 channels, owing to its low
purity.

Figure 9: Partial-BSM normalized heralding probability, $\Pr(\psi^{\mp}_{n})$
from Eq. (29), versus channel number assuming Eq. (9)’s all-Gaussian biphoton
wave function with brickwall filtering and parameter values from Table 1.
Figure 10: Partial-BSM heralding efficiency,
$\Pr(S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n})$ from Eq. (43), versus channel
number assuming Eq. (9)’s all-Gaussian biphoton wave function with brickwall
filtering and parameter values from Table 1.  Figure 11: Partial-BSM purity,
${\rm Tr}(\hat{\rho}^{2}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}})$ from Eq.
(63), versus channel number assuming Eq. (9)’s all-Gaussian biphoton wave
function with brickwall filtering and parameter values from Table 1.  Figure
12: Partial-BSM error probability, $\Pr(e_{n})$ from Eq. (68), versus channel
number assuming Eq. (9)’s all-Gaussian biphoton wave function with brickwall
filtering and parameter values from Table 1.

##  V Entanglement Loading of Duan-Kimble Quantum Memories

We now move on to our final major task, deriving the density operators,
$\hat{\rho}^{({\bf s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}$ for ${\bf
s}\equiv(s_{1},s_{2})=(\pm,\pm)$, of Alice and Bob’s intra-cavity color-center
quantum memories, given they were illuminated by an $n$th-channel heralded
biphoton from the ZALM QTX and Fig. 4’s SPD${}_{1_{s_{1}}}$ and
SPD${}_{2_{s_{2}}}$ detected photons after reflection from the cavities and
interference at the 50–50 beam splitters.

In our derivation we shall ignore all nonidealities in free-space propagation
from the ZALM QTX to Alice and Bob’s QRXs except for an identical
transmissivity, $0<\eta_{\rm prop}<1$, associated with each propagation path.
Likewise, we shall ignore all nonidealities in Alice and Bob’s mode
converters, i.e., their frequency converters and bandwidth compressors, except
for symmetrically-distributed losses that we combine into an efficiency,
$0<\eta_{\rm mc}<1$. The modeling situation is slightly different, however,
for Fig. 4’s quantum memories, where we choose the SiV color center as the
prototypical quantum memory and use the push-pull protocol’s state-dependent
reflectivities from [21], which explicitly include leakage from the memory
cavity’s HR mirror and spontaneous emission from the SiV color center. The
rest of the Duan-Kimble memories’ properties will be assumed ideal, except for
symmetrically-distributed exo-cavity losses that we combine into identical
efficiency factors, $0<\eta_{\rm mem}<1$. The composite receiver efficiency,
$\eta_{\rm qrx}\equiv\eta_{\rm mc}\eta_{\rm mem}$, is then assigned to the
SPDs in Alice and Bob’s QRXs. Note that neither $\eta_{\rm prop}$ nor
$\eta_{\rm qrx}$ appear in this section’s density operator, as it applies to a
post-selected event. That said, $\eta_{\rm qtx}$, $\eta_{\rm prop}$, and
$\eta_{\rm qrx}$ will show up in Sec. VI, where we discuss ZALM’s
entanglement-distribution rate and its tradeoff with inter-channel
interference.

###  V.1 Mode-converted density operator,
$\hat{\rho}_{\tilde{S}_{1_{n}},\tilde{S}_{2_{n}}\mid\psi^{\mp}_{n}}$

For a ZALM QTX that uses PPLN SPDCs, the $\psi^{\mp}_{n}$-heralded biphoton
sent to Alice and Bob has signal-photon components with
$\omega_{S_{0}}/2\pi-n\Delta B\sim 200$ THz center frequencies. After they
both successfully pass through Alice and Bob’s DWDMS filters’ $n$th-channel
passbands, they have bandwidths $\delta B\sim 25$ GHz in our Table 1’s Cases 1
and 2. In contrast, Alice and Bob’s SiV quantum memories have a center
frequency $\tilde{\omega}_{S_{0}}/2\pi\sim 400$ THz and bandwidth
$\delta\tilde{B}\sim 600\,$MHz, quantifying the task to be performed by the
QRX’s mode converters. To derive the mode-converted density operator,
$\hat{\rho}_{\tilde{S}_{1_{n}},\tilde{S}_{2_{n}}\mid\psi^{\mp}_{n}}$, we make
use of results from Sec. IV, which indicate that it suffices for us to derive
the mode-converted form of $\Phi_{n}(\omega_{S_{1}},\omega_{S_{2}})$ and then
use Eqs. (35)–(37) to get the mode-converted density operator.

Ideal frequency conversion from center frequency $\omega_{S_{0}}-2\pi n\Delta
B$ to center frequency $\tilde{\omega}_{S_{0}}$ changes
$\Phi_{n}(\omega_{S_{1}},\omega_{S_{2}})$ to
$\Phi_{n}(\omega_{S_{1}}+\delta\omega_{S},\omega_{S_{2}}+\delta\omega_{S})$,
where $\delta\omega_{S}\equiv\tilde{\omega}_{S_{0}}-\omega_{S_{0}}+2\pi
n\Delta B$. After bandwidth compression by $\beta\equiv\delta
B/\delta\tilde{B}$, we get the following
$\tilde{\Phi}_{n}(\omega_{S_{1}},\omega_{S_{2}})$ to use in lieu of
$\Phi_{n}(\omega_{S_{1}},\omega_{S_{2}})$ in obtaining
$\hat{\rho}_{\tilde{S}_{1_{n}},\tilde{S}_{2_{n}}\mid\psi^{\mp}_{n}}$ from Eqs.
(35)–(37),

$\tilde{\Phi}_{n}(\omega_{S_{1}},\omega_{S_{2}})=\beta^{2}\Phi_{n}[\beta(\omega_{S_{1}}+\delta\omega_{S}),\beta(\omega_{S_{2}}+\delta\omega_{S})].$
(69)

Note that ideal mode conversion transforms
$K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$
and
$K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$’s
eigenfunctions from $\xi_{\bf m}({\boldsymbol{\omega}}_{S})$ to
$\tilde{\xi}_{\bf m}({\boldsymbol{\omega}}_{S})$ and from $\zeta_{\bf
m}({\boldsymbol{\omega}}_{S})$ to $\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})$ as follows:

$\tilde{\xi}_{\bf m}({\boldsymbol{\omega}}_{S})=\beta^{2}\xi_{\bf
m}[\beta({\boldsymbol{\omega}}_{S}+{\boldsymbol{\delta\omega}}_{S})],$ (70)

and

$\tilde{\zeta}_{\bf m}({\boldsymbol{\omega}}_{S})=\beta^{2}\zeta_{\bf
m}[\beta({\boldsymbol{\omega}}_{S}+{\boldsymbol{\delta\omega}}_{S})],$ (71)

where
${\boldsymbol{\delta\omega}}_{S}\equiv(\delta\omega_{S},\delta\omega_{S})$. It
does _not_ , however, change those kernel’s normalized eigenvalues,
$\\{\tilde{\mu}_{\bf m}\\}$ and $\\{\tilde{\nu}_{\bf m}\\}$. Thus, as expected
for ideal frequency conversion and bandwidth compression, the state’s purity,
fidelity, and error probability are unaffected and, as exemplified in Figs. 11
and 12, independent of channel number.

###  V.2 Push-pull Duan-Kimble entanglement loading

Our approach to determining the density operator, $\hat{\rho}^{({\bf
s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}$, for Alice ($M_{1}$) and Bob’s ($M_{2}$)
loaded quantum memories is straightforward. We have that
$\hat{\rho}_{\tilde{S}_{1_{n}},\tilde{S}_{2_{n}}\mid\psi^{\mp}_{n}}$’s
eigenkets are

$\displaystyle|\tilde{\xi}^{\mp}_{\bf
m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}\equiv\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})|\psi^{\mp}({\boldsymbol{\omega}}_{S})\rangle_{\tilde{S}_{1}\tilde{S}_{2}}$
(72)

and

$\displaystyle|\tilde{\zeta}^{\mp}_{\bf
m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}\equiv\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})|\psi^{\pm}({\boldsymbol{\omega}}_{S})\rangle_{\tilde{S}_{1}\tilde{S_{2}}},$
(73)

where dependence on the channel number $n$ is implicit in $\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})$ and $\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})$. From $|\tilde{\xi}^{\mp}_{\bf
m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}$ and $|\tilde{\zeta}^{\mp}_{\bf
m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}$ we will determine the unnormalized
density operators, $\tilde{\rho}^{(\bf
s)}_{M_{1}M_{2}\mid\tilde{\xi}^{\mp}_{\bf m}}$, and $\tilde{\rho}^{(\bf
s)}_{M_{1},M_{2}\mid\tilde{\zeta}^{\mp}_{\bf m}}$ of Alice and Bob’s memories
after these eigenkets have undergone the loading process sketched in Fig 4
with SPD${}_{1_{s_{1}}}$ and SPD${}_{2_{s_{2}}}$ having detected photons. Then
we get the quantum memories’ unnormalized density operators via

$\tilde{\rho}^{({\bf s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}=\sum_{\bf
m}\left[\mu_{\bf m}\,\tilde{\rho}^{(\bf
s)}_{M_{1}M_{2}\mid\tilde{\xi}^{\mp}_{\bf m}}+\nu_{\bf m}\,\tilde{\rho}^{(\bf
s)}_{M_{1},M_{2}\mid\tilde{\zeta}^{\mp}_{\bf m}}\right],$ (74)

whose trace provides the normalization constant needed to get the normalized
density operators, $\hat{\rho}^{({\bf s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}$.

Section II has already provided a high-level description of the loading
process. The rest of this subsection is devoted to the details, specifically
finding general expressions for the $\\{\tilde{\rho}^{(\bf
s)}_{M_{1}M_{2}\mid\tilde{\xi}^{\mp}_{\bf m}}\\}$ and $\\{\tilde{\rho}^{(\bf
s)}_{M_{1},M_{2}\mid\tilde{\zeta}^{\mp}_{\bf m}}\\}$ and then specializing
them to the push-pull memory-loading operation.

Suppose that $|\tilde{\xi}_{\bf m}^{\mp}\rangle_{S_{1_{n}}S_{2_{n}}}$ has
reached Alice and Bob’s quantum memories, which have been prepared in the
equal superpositions of their ground states, so that

$|\tilde{\xi}_{{\bf m}_{\rm in}}^{\mp}\rangle_{\tilde{\bf S}{\bf
M}}\equiv|\tilde{\xi}_{\bf
m}^{\mp}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}\otimes|\psi\rangle_{\bf M},$ (75)

with $|\psi\rangle_{\bf
M}\equiv(|g_{1}\rangle_{M_{1}}+|g_{2}\rangle_{M_{1}})(|g_{1}\rangle_{M_{2}}+|g_{2}\rangle)_{M_{2}})/2$,
is the initial biphoton-memories product state. After the state-dependent
reflections from the memory cavities, and $\tilde{S}^{\prime}_{1_{H}}$ and
$\tilde{S}^{\prime}_{2_{H}}$’s passage through Fig. 4’s HWP,
$|\tilde{\xi}_{{\bf m}_{\rm in}}^{\mp}\rangle_{{\bf S}{\bf M}}$ becomes the
entangled biphoton-memories state

$\displaystyle|\tilde{\xi}_{{\bf m}_{\rm out}}^{\mp}\rangle_{\tilde{\bf S}{\bf
M}}$ $\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})}{\sqrt{2}}\left[|\omega_{S_{1}}\rangle_{\tilde{S}^{\prime}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}_{2_{V}}}|\psi^{\prime}_{1}\rangle_{\bf
M}\right.$
$\displaystyle\,\,\mp\left.|\omega_{S_{1}}\rangle_{\tilde{S}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}^{\prime}_{2_{V}}}|\psi^{\prime}_{2}\rangle_{\bf
M}\right],$ (76)

where

$\displaystyle|\psi^{\prime}_{1}\rangle_{\bf M}$
$\displaystyle=\left[r_{1}(\omega_{S_{1}})|g_{1}\rangle_{M_{1}}+r_{2}(\omega_{S_{1}})|g_{2}\rangle_{M_{1}}\right]$
$\displaystyle\,\,\times\left[|g_{1}\rangle_{M_{2}}+|g_{2}\rangle_{M_{2}}\right]/2,$
(77)

and

$\displaystyle|\psi^{\prime}_{2}\rangle_{\bf M}$
$\displaystyle=\left[|g_{1}\rangle_{M_{1}}+|g_{2}\rangle_{M_{1}}\right]$
$\displaystyle\,\,\times\left[r_{1}(\omega_{S_{2}})|g_{1}\rangle_{M_{2}}+r_{2}(\omega_{S_{2}})|g_{2}\rangle_{M_{2}}\right]/2,$
(78)

with $r_{1}(\omega_{S_{k}})$ and $r_{2}(\omega_{S_{k}})$, for $k=1,2$, being
the memories’ state-dependent field reflectivities at frequency
$\omega_{S_{k}}$.

Similarly, if $|\tilde{\zeta}_{\bf m}^{\mp}\rangle_{S_{1_{n}}S_{2_{n}}}$ has
reached Alice and Bob’s quantum memories, which are again prepared in the
equal superpositions of their ground states, the initial biphoton-memories
product state is

$|\tilde{\zeta}_{{\bf m}_{\rm in}}^{\mp}\rangle_{\tilde{\bf S}{\bf
M}}\equiv|\tilde{\zeta}_{\bf
m}^{\mp}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}\otimes|\psi\rangle_{\bf M}.$ (79)

Thus, after the state-dependent reflections from the memory cavities, and
$\tilde{S}^{\prime}_{1_{H}}$ and $\tilde{S}^{\prime}_{2_{H}}$’s passage
through Fig. 4’s HWP, $|\tilde{\zeta}_{{\bf m}_{\rm in}}^{\mp}\rangle_{{\bf
S}{\bf M}}$ becomes the entangled biphoton-memories state

$\displaystyle|\tilde{\zeta}_{{\bf m}_{\rm out}}^{\mp}\rangle_{\tilde{\bf
S}{\bf M}}$ $\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})}{\sqrt{2}}\left[|\omega_{S_{1}}\rangle_{\tilde{S}^{\prime}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}_{2_{V}}}|\psi^{\prime}_{1}\rangle_{\bf
M}\right.$
$\displaystyle\,\,\pm\left.|\omega_{S_{1}}\rangle_{\tilde{S}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}^{\prime}_{2_{V}}}|\psi^{\prime}_{2}\rangle_{\bf
M}\right].$ (80)

Now, with $|\tilde{\xi}_{{\bf m}_{\rm out}}^{\mp}\rangle_{\tilde{\bf S}{\bf
M}}$ and $|\tilde{\zeta}_{{\bf m}_{\rm out}}^{\mp}\rangle_{\tilde{\bf S}{\bf
M}}$ in hand, we find $\tilde{\rho}^{(\bf
s)}_{M_{1},M_{2}\mid\tilde{\xi}^{\mp}_{\bf m}}$ and $\tilde{\rho}^{(\bf
s)}_{M_{1},M_{2}\mid\tilde{\xi}^{\mp}_{\bf m}}$ from

$\tilde{\rho}^{(\bf s)}_{M_{1}M_{2}\mid\tilde{\upsilon}^{\mp}_{\bf
m}}\equiv{\rm Tr}\\!\left[\hat{\prod}_{\bf s}|\tilde{\upsilon}_{{\bf m}_{\rm
out}}^{\mp}\rangle_{\tilde{\bf S}{\bf M}}\,{}_{\tilde{\bf S}{\bf
M}}\langle\tilde{\upsilon}_{{\bf m}_{\rm out}}^{\mp}|\right],\mbox{ for
$\tilde{\upsilon}=\tilde{\xi},\tilde{\zeta}$},$ (81)

where $\hat{\prod}_{\bf s}$ is the positive operator-valued measurement for
photons being detected by Fig. 4’s SPD${}_{S_{s_{1}}}$ and SPD${}_{S_{s_{2}}}$
[41]. Substituting Eq. (81) into Eq. (74) for ${\bf s}=(\pm,\pm)$ gives the
loaded memories’ unnormalized joint density operator as a function of the
detectors that registered photon counts.

The foregoing procedure is carried out in Appendix A with the following
results:

$\displaystyle\tilde{\rho}^{(a)}_{M_{1},M_{2}}$
$\displaystyle=\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\psi^{-}_{n}}=\tilde{\rho}^{(-,-)}_{M_{1},M_{2}\mid\psi^{-}_{n}}$
$\displaystyle=\tilde{\rho}^{(+,-)}_{M_{1},M_{2}\mid\psi^{+}_{n}}=\tilde{\rho}^{(-,+)}_{M_{1},M_{2}\mid\psi^{+}_{n}},$
(82)

and

$\displaystyle\tilde{\rho}^{(b)}_{M_{1},M_{2}}$
$\displaystyle=\tilde{\rho}^{(+,-)}_{M_{1},M_{2}\mid\psi^{-}_{n}}=\tilde{\rho}^{(-,+)}_{M_{1},M_{2}\mid\psi^{-}_{n}}$
$\displaystyle=\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\psi^{+}_{n}}=\tilde{\rho}^{(-,-)}_{M_{1},M_{2}\mid\psi^{+}_{n}},$
(83)

where

$\displaystyle\tilde{\rho}^{(p)}_{M_{1},M_{2}}$ $\displaystyle=\sum_{\bf
m}\mu_{\bf m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{|\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})|^{2}}{4}\left[|\phi^{+}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\phi^{-}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\psi^{+}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\psi^{-}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}\right]$
$\displaystyle\,\,\times\,\left[{}_{M_{1}M_{2}}\langle\phi^{+}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\phi^{-}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\psi^{+}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\psi^{-}_{p_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})|\right]$ $\displaystyle+\,\sum_{\bf m}\nu_{\bf
m}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{|\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})|^{2}}{4}\left[|\phi^{+}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\phi^{-}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\psi^{+}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}+|\psi^{-}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}\right]$
$\displaystyle\,\,\times\,\left[{}_{M_{1}M_{2}}\langle\phi^{+}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\phi^{-}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\psi^{+}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})|+{}_{M_{1}M_{2}}\langle\psi^{-}_{p_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})|\right],\mbox{ for $p=a,b$},$ (84)

with

$\displaystyle|\phi^{+}_{a_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\phi^{+}\rangle_{M_{1}M_{2}}$
(85) $\displaystyle|\phi^{-}_{a_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\phi^{-}\rangle_{M_{1}M_{2}}$
(86) $\displaystyle|\psi^{+}_{a_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\psi^{+}\rangle_{M_{1}M_{2}}$
(87) $\displaystyle|\psi^{-}_{a_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\psi^{-}\rangle_{M_{1}M_{2}}$
(88) $\displaystyle|\phi^{+}_{a_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\phi^{+}\rangle_{M_{1}M_{2}}$
(89) $\displaystyle|\phi^{-}_{a_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\phi^{-}\rangle_{M_{1}M_{2}}$
(90) $\displaystyle|\psi^{+}_{a_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\psi^{+}\rangle_{M_{1}M_{2}}$
(91) $\displaystyle|\psi^{-}_{a_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\psi^{-}\rangle_{M_{1}M_{2}},$
(92)

and

$\displaystyle|\phi^{+}_{b_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})\right]|\phi^{+}\rangle_{M_{1}M_{2}}$
(93) $\displaystyle|\phi^{-}_{b_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\phi^{-}\rangle_{M_{1}M_{2}}$
(94) $\displaystyle|\psi^{+}_{b_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\psi^{+}\rangle_{M_{1}M_{2}}$
(95) $\displaystyle|\psi^{-}_{b_{\mu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\psi^{-}\rangle_{M_{1}M_{2}}$
(96) $\displaystyle|\phi^{+}_{b_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\phi^{+}\rangle_{M_{1}M_{2}}$
(97) $\displaystyle|\phi^{-}_{b_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|\phi^{-}\rangle_{M_{1}M_{2}}$
(98) $\displaystyle|\psi^{+}_{b_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})+e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\psi^{+}\rangle_{M_{1}M_{2}}$
(99) $\displaystyle|\psi^{-}_{b_{\nu{\bf
m}}}({\boldsymbol{\omega}}_{S})\rangle_{M_{1}M_{2}}$
$\displaystyle=4^{-1}\\!\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})-e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|\psi^{-}\rangle_{M_{1}M_{2}},$
(100)

for

$|\phi^{\mp}\rangle_{M_{1}M_{2}}\equiv\frac{|g_{1}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}\mp|g_{2}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}}{\sqrt{2}},$
(101)

and

$|\psi^{\mp}\rangle_{M_{1}M_{2}}\equiv\frac{|g_{1}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}\mp|g_{2}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}}{\sqrt{2}},$
(102)

being the memories’ Bell states.

Note that Appendix A obtained the preceding results using an augmented version
of the Duan-Kimble loading protocol, in which Bob applies a $\pi$ pulse to his
memory for a $\psi^{-}_{n}$ herald from the ZALM QTX if Alice and Bob’s
detectors record ${\bf s}=(+,-)$ or $(-,+)$ coincidences, and for a
$\psi^{+}_{n}$ herald from the ZALM QTX if Alice and Bob’s detectors record
${\bf s}=(+,+)$ or $(-,-)$ coincidences. With this conditional $\pi$-pulse
usage, Alice and Bob’s memories are primed to load a
$|\psi^{-}\rangle_{M_{1}M_{2}}$ singlet state, _regardless_ of which state the
ZALM QTX has heralded, and which combination of Alice and Bob’s detectors have
registered photon counts. Indeed, if the ZALM QTX has a separable channelized
biphoton wave function, and Alice and Bob’s memories are ideal (in the sense
defined below), the augmented Duan-Kimble protocol achieves unit-fidelity
loading of that singlet state, as will be seen in the next subsection.
Consequently, we _define_ $|\psi^{-}\rangle_{M_{1}M_{2}}$ to be the loaded-
memories target state, and

$F_{p}\equiv{}_{M_{1}M_{2}}\langle\psi^{-}|\hat{\rho}^{(p)}_{M_{1}M_{2}}|\psi^{-}\rangle_{M_{1}M_{2}},\mbox{
for $p=a,b$}$ (103)

to be their entangled-state fidelities for type-a and type-b coincidences.

The preceding development applies to _any_ quantum memory whose loading relies
on state-dependent reflectivities, including, e.g., the original Duan-Kimble
protocol—as demonstrated in Refs. [27, 28, 29]—which Raymer  _et al._ [21]
refers to as on-off operation. Our interest, however, is in push-pull
operation, for which we have [21]

$r_{1}(\omega_{S})=\frac{(\gamma+i\Delta_{12}/2-i\Delta\tilde{\omega}_{S})(\kappa-\kappa_{J}+i\Delta\tilde{\omega}_{S})-g^{2}}{(\gamma+i\Delta_{12}/2-i\Delta\tilde{\omega}_{S})(\kappa+\kappa_{J}-i\Delta\tilde{\omega}_{S})+g^{2}},$
(104)

and

$r_{2}(\omega_{S})=\frac{(\gamma-i\Delta_{12}/2-i\Delta\tilde{\omega}_{S})(\kappa-\kappa_{J}+i\Delta\tilde{\omega}_{S})-g^{2}}{(\gamma-i\Delta_{12}/2-i\Delta\tilde{\omega}_{S})(\kappa+\kappa_{J}-i\Delta\tilde{\omega}_{S})+g^{2}}.$
(105)

Here: $\gamma$ is the SiV’s spontaneous-emission rate;
$\Delta\tilde{\omega}_{S}\equiv\omega_{S}-\tilde{\omega}_{S_{0}}$ is the
detuning from the photons’ center frequency; $\kappa$ is the memory cavities’
field-coupling rate from its output coupler to the incoming and outgoing
optical fields; $\kappa_{J}$ is the field-coupling rate for loss due to
leakage from the cavities’ HR; and $g$ is the SiV-photon coupling rate, i.e.,
the single-photon Rabi frequency. (The above field damping rates equal one-
half the associated energy-damping rates.)

The next subsection reports the loaded-memories entangled-state fidelity for:
(A) ideal operation, in which $|r_{1}(\omega_{S})|=|r_{2}(\omega_{S})|=1$ and
$r_{2}(\omega_{S})=-r_{1}(\omega_{S})$ over the biphoton’s bandwidth; and (B)
narrowband push-pull operation, in which Eqs. (104) and (105) apply with
$r_{1}(\omega_{S})\approx r_{1}(\tilde{\omega}_{S_{0}}))$ and
$r_{2}(\omega_{S})\approx r_{2}(\tilde{\omega}_{S_{0}})$ over the biphoton’s
bandwidth. Neither of these cases require solving Eqs. (49) and (50). Thus we
postpone consideration of broadband push-pull operation, in which Eqs. (104)
and (105) apply but the biphoton’s bandwidth no longer permits approximating
the reflectivities by their zero-detuning values, to a future paper. There,
because broadband operation’s memory-loading evaluation requires knowledge of
the $\\{\mu_{\bf m},\xi_{\bf m}({\boldsymbol{\omega}}_{S})\\}$ and the
$\\{\nu_{\bf m},\zeta_{\bf m}({\boldsymbol{\omega}}_{S})\\}$, Eqs. (51)–(54)
and existing numerical techniques for finding SVDs will be key to getting
results.

###  V.3 Two examples of loaded-memories entangled-state fidelity

Here, we present our two examples’ general loaded-memories entangled-state
fidelity expressions, which can be evaluated for arbitrary channelized
biphoton wave functions, and their loaded-memories entangled-state fidelities
for the all-Gaussian biphoton wave function with brickwall filtering and
parameter values from Table 1’s Case 2.
_Case A: Ideal operation_ When $|r_{1}(\omega_{S})|=|r_{2}(\omega_{S})|=1$ and
$r_{2}(\omega_{S})=-r_{1}(\omega_{S})$ over the biphoton’s bandwidth, Alice
and Bob’s QRXs can omit Fig. 4’s time delay because, with $T=0$, Eqs.
(84)–(100) collapse to

$\displaystyle\tilde{\rho}^{(a)}_{M_{1},M_{2}}$
$\displaystyle=\tilde{\rho}^{(b)}_{M_{1},M_{2}}$ $\displaystyle=\sum_{\bf
m}\mu_{\bf m}\,|\psi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\langle\psi^{-}|$
$\displaystyle\,\,+\,\sum_{\bf m}\nu_{\bf
m}\,|\phi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\langle\psi^{+}|.$ (106)

Ideal operation is indeed ideal; its normalized loaded-memories density
operators are

$\displaystyle\hat{\rho}$ ${}^{({\bf
s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}=\sum_{\bf m}\tilde{\mu}_{\bf
m}\,|\psi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\langle\psi^{-}|$
$\displaystyle\,\,+\,\sum_{\bf m}\tilde{\nu}_{\bf
m}\,|\phi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\langle\phi^{-}|,\mbox{ for
${\bf s}=(\pm,\pm)$,}$ (107)

which indicates that $\hat{\rho}^{\mp}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}$
from Eq. (57) has been perfectly transferred to Alice and Bob’s quantum
memories. In general, this ideality means that the loaded-memories entangled-
state fidelity obeys

$\displaystyle F_{\bf s}\equiv{}_{M_{1}M_{2}}\langle\psi^{-}|\hat{\rho}^{({\bf
s})}_{M_{1},M_{2}\mid\psi^{\mp}_{n}}|\psi^{-}\rangle_{M_{1}M_{2}}$ (108)
$\displaystyle=$
$\displaystyle{}_{S_{1_{n}}S_{2_{n}}}\\!\langle\psi^{\mp}|\hat{\rho}^{\mp}_{S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n}}|\psi^{\mp}\rangle_{S_{1_{n}}S_{2_{n}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\\!\left(1+\sum_{m=1}^{\infty}\tilde{\lambda}^{4}_{m}\right)$
(109) $\displaystyle=$ $\displaystyle\frac{\displaystyle\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,K_{S_{1_{n}}S_{2_{n}}}^{(c)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})}{\displaystyle\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\sum_{v=c,e}K_{S_{1_{n}}S_{2_{n}}}^{(v)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S})},$
(110)

where the second equality follows from Eq. (66) and the third equality, which
follows from Eqs. (47) and (48), can be evaluated directly from the Sagnac
sources’ channelized biphoton wave function.

For the all-Gaussian biphoton wave function with brickwall filtering and the
Case 2 parameter values from Table 1, Fig. 12 shows that ideal operation of
our Duan-Kimble quantum memories has error probability $\Pr(e_{n})=0.386\%$,
and fidelity $\Pr(c_{n})=1-\Pr(e_{n})=99.6\%$ across all 81 channels.
_Case B: Narrowband push-pull operation_ In narrowband push-pull operation we
assume that $r_{1}(\omega_{S})\approx r_{1}(\tilde{\omega}_{S_{0}})$ and
$r_{2}(\omega_{S})\approx r_{2}(\tilde{\omega}_{S_{0}})$ over the biphoton’s
bandwidth. Equations (104) and (105) show the push-pull reflectivities have
Hermitian symmetry about $\tilde{\omega}_{S_{0}}$, viz.,
$r_{2}(\tilde{\omega}_{S_{0}}+\Delta\tilde{\omega}_{S})=r_{1}^{*}(\tilde{\omega}_{S_{0}}-\Delta\tilde{\omega}_{S})$,
implying that
$r_{2}(\tilde{\omega}_{S_{0}})=r_{1}^{*}(\tilde{\omega}_{S_{0}})$. Thus Alice
and Bob’s QRX’s can again omit Fig. 4’s time delay, because narrowband
operation only attenuates and phase shifts incoming wave packets hence it does
not significantly delay or reshape them. Under these conditions,
$\tilde{\rho}_{M_{1}M_{2}}^{(a)}$ and $\tilde{\rho}_{M_{1}M_{2}}^{(b)}$ from
Eqs. (84)–(100) reduce to

$\displaystyle\tilde{\rho}_{M_{1}M_{2}}^{(a)}$ $\displaystyle=\sum_{\bf
m}\frac{\mu_{\bf m}}{4}\,{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]|\psi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\\!\langle\psi^{-}|$
$\displaystyle\,\,+\,\sum_{\bf m}\frac{\nu_{\bf m}}{4}\left\\{{\rm
Re}[r_{1}(\tilde{\omega}_{S_{0}})](|\phi^{+}\rangle_{M_{1}M_{2}}+|\psi^{+}\rangle_{M_{1}M_{2}})+i\,{\rm
Im}[r_{1}(\tilde{\omega}_{S_{0}})]|\phi^{-}\rangle_{M_{1}M_{2}}\right\\}$
$\displaystyle\,\,\times\,\left\\{{\rm
Re}[r_{1}(\tilde{\omega}_{S_{0}})]({}_{M_{1}M_{2}}\\!\langle\phi^{+}|+{}_{M_{1}M_{2}}\\!\langle\psi^{+}|)-i\,{\rm
Im}[r_{1}(\tilde{\omega}_{S_{0}})]{}_{M_{1}M_{2}}\\!\langle\phi^{-}|\right\\},$
(111)

and

$\displaystyle\tilde{\rho}_{M_{1}M_{2}}^{(b)}$ $\displaystyle=\sum_{\bf
m}\frac{\mu_{\bf m}}{4}\left\\{{\rm
Re}[r_{1}(\tilde{\omega}_{S_{0}})](|\phi^{+}\rangle_{M_{1}M_{2}}+|\psi^{+}\rangle_{M_{1}M_{2}})+i\,{\rm
Im}[r_{1}(\tilde{\omega}_{S_{0}})]|\psi^{-}\rangle_{M_{1}M_{2}}\right\\}$
$\displaystyle\,\,\times\,\left\\{{\rm
Re}[r_{1}(\tilde{\omega}_{S_{0}})]({}_{M_{1}M_{2}}\\!\langle\phi^{+}|+{}_{M_{1}M_{2}}\\!\langle\psi^{+}|)-i\,{\rm
Im}[r_{1}(\tilde{\omega}_{S_{0}})]{}_{M_{1}M_{2}}\\!\langle\psi^{-}|\right\\}$
$\displaystyle\,\,+\,\sum_{\bf m}\frac{\nu_{\bf m}}{4}\,{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]|\phi^{-}\rangle_{M_{1}M_{2}}\,{}_{M_{1}M_{2}}\\!\langle\phi^{-}|,$
(112)

leading to fidelities to the target singlet-state
$|\psi^{-}\rangle_{M_{1}M_{2}}$ given by

$F_{a}=\frac{\displaystyle\sum_{\bf m}\mu_{\bf m}\,{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]}{\displaystyle\sum_{\bf m}\,\mu_{\bf
m}\,{\rm Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]+\sum_{\bf m}\nu_{\bf
m}\left(2\,{\rm Re}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]+{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]\right)},$ (113)

and

$F_{b}=\frac{\displaystyle\sum_{\bf m}\mu_{\bf m}\,{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]}{\displaystyle\sum_{\bf m}\mu_{\bf
m}\left(2\,{\rm Re}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]+{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]\right)+\sum_{\bf m}\nu_{\bf m}\,{\rm
Im}^{2}[r_{1}(\tilde{\omega}_{S_{0}})]}.$ (114)

These fidelities are easily evaluated, given a set of memory parameters,
because

$\sum_{\bf m}\mu_{\bf m}=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}),$
(115)

and

$\sum_{\bf m}\nu_{\bf m}=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}),$
(116)

are easily computed for the all-Gaussian case. We choose, however, to postpone
consideration of memory parameters until the future paper, where performance
evaluation for narrowband push-pull operation will serve as a prelude to the
broadband case. Nevertheless, there is something else to be said here about
narrowband push-pull memory loading: with the appropriate, potentially
accessible, parameter values for the quantum memories, our results further
reduce to an attenuated version of ideal operation, as we now show.

Raymer  _et al_. [21] rewrite the memories’ zero-detuning, state-dependent
reflectivities for push-pull operation as

$\displaystyle
r_{1}(\tilde{\omega}_{S_{0}})=r_{2}^{*}(\tilde{\omega}_{S_{0}})=\frac{(1+i\Delta_{12}/2\gamma)(1-\kappa_{J}/\kappa)-C}{(1+i\Delta_{12}/2\gamma)(1+\kappa_{J}/\kappa)+C},$
(117)

where $C\equiv g^{2}/\kappa\gamma$ is the cavities’ cooperativity, with $C>1$
being the strong-coupling regime in which the memories are operated. With this
rewriting, and assuming $\kappa>\kappa_{J}$, they then prove that

$C=C_{\pi}\equiv\sqrt{1+(1-\kappa_{J}^{2}/\kappa^{2})\Delta^{2}_{12}/4\gamma^{2}}-\kappa_{J}/\kappa$
(118)

makes the narrowband reflectivities pure-imaginary

$r_{1}(\tilde{\omega}_{S_{0}})=i\frac{\sqrt{\gamma^{2}\kappa^{2}+\Delta^{2}_{12}(\kappa^{2}-\kappa^{2}_{J})/4}-\gamma\kappa}{\Delta_{12}(\kappa+\kappa_{J})/2},$
(119)

giving the desired $\pi$-rad phase shift between the actions of the two color-
center states. It immediately follows that the loaded memories’ normalized
density operators and entangled-state fidelities match those of ideal
operation, as given by Eqs. (107) and (110). So, for the QTX parameters from
Table 1’s Case 2, the loaded memories have 99.6% entangled-state fidelity
across all 81 channels.

There is, of course, a difference between ideal operation and narrowband push-
pull operation with $C=C_{\pi}$. It comes as a reduction in the entanglement-
distribution rate. Ideal operation’s memories have no intra-cavity losses, so
its probability that a particular pump pulse succeeds in loading both Alice
and Bob’s memories is $\eta_{\rm qtx}\,\eta^{2}_{\rm prop}\,\eta^{2}_{\rm
qrx}$, where $\eta_{\rm qrx}$ only accounts for _exo-cavity_ memory losses.
For narrowband push-pull operation with $C=C_{\pi}$, however, that success
probability gets an additional sub-unit factor of $\eta_{\rm
cavity}\equiv|r_{1}(\tilde{\omega}_{S_{0}})|^{2}$, to account for _intra-
cavity_ memory losses.

Broaching the issue of ZALM’s entanglement-distribution rate is a perfect
segue into the next section, where the tradeoff between ZALM’s entanglement-
distribution rate and its inter-channel interference will be the first topic
on the agenda.

##  VI Discussion

In this culminating section we will address a variety of additional issues,
but before doing so it is worthwhile to appraise what we did accomplish, as
opposed to what remains for the future. We set out to do a deep dive into
ZALM’s heralded source of polarization-entangled biphotons and their coupling
to a pair of intra-cavity color-center quantum memories, and we accomplished
those objectives. In particular, we derived the density operator for the
heralded biphoton, which quantifies its heralding performance and fidelity. We
also derived the loaded memories’ density operator and its fidelity with the
memories’ entangled-state target. Although we have not undertaken a
comprehensive exploration of ZALM performance as its source and memory
parameters are varied, our theory is set up for just such an endeavor.
Nevertheless, the examples we have evaluated already demonstrate two critical
features of Sec. II’s ZALM architecture: (1) the necessity of achieving a
near-separable channelized biphoton wave function to ensure the biphoton sent
to Alice and Bob is of high purity; and (2) the premium placed on Alice and
Bob’s bandwidth compressors’ enabling narrowband push-pull memory loading to
ensure the arriving biphoton’s state is faithfully transferred to the intra-
cavity color centers.

We turn now to our first issue for discussion: ZALM’s entanglement-
distribution rate, i.e., its per-pump-pulse rate (probability) of Alice and
Bob’s QRXs sharing the $|\psi^{-}\rangle_{M_{1}M_{2}}$ target state, and that
rate’s tradeoff with inter-channel interference, which occurs when the signal
photon from an $n$th-channel herald does not pass through its DWDMS filter’s
$n$th-channel passband but a signal photon whose idler companion lies in a
different channel does. As detailed below, ZALM’s operating with near-
deterministic creation of one polarization-entangled pair per pump pulse
across its dual-Sagnac source’s full phase-matching bandwidth implies a non-
trivial probability that there will be a two-pair event over that bandwidth.
But, before quantifying that interference, we address the entanglement-
distribution rate.

Let $N_{p}$ be the random number of photon pairs produced by the ZALM QTX’s
dual-Sagnac source from a single pump pulse, and let $\mathbb{E}(N_{p})\sim 1$
be its expected value [42]. We then have that the ZALM system’s entanglement-
distribution rate is $R=\sum_{n=-(N-1)/2}^{(N-1)/2}R_{n}$, where $R_{n}$, the
entanglement-distribution rate for an $n$th-channel herald, is [44]

$\displaystyle R_{n}$ $\displaystyle=\mathbb{E}(N_{p})[\eta_{\rm
qtx}\Pr(\psi^{\mp}_{n})][\eta^{2}_{\rm
prop}\Pr(S_{1_{n}},S_{2_{n}}\mid\psi^{\mp}_{n})]$
$\displaystyle\,\,\times\,[\eta^{2}_{\rm qrx}\eta_{\rm
cavity}][(F_{a}+F_{b})/2],$ (120)

where $F_{a}$ and $F_{b}$ are the fidelities for type-a and type-b coincidence
detections at Alice and Bob’s QRXs, as defined in Eqs. (82) and (83), viz.,
$(+,+),(-,-),(+,-),$ or $(-,+)$, which are equally likely to occur. The first
bracketed term in Eq. (120) is the probability of a $\psi^{\mp}_{n}$ herald,
given that the dual-Sagnac source has emitted a biphoton. The second bracketed
term there is the probability that the $n$th-channel signal photons will reach
Alice and Bob’s QRXs _and_ pass through their DWDMS filters’ $n$th-channel
passbands, given that there was a $\psi^{\mp}$ herald. The third bracketed
term in Eq. (120) is the probability that Alice and Bob’s QRXs register a
photon-detection coincidence, given that a pair of signal photons passed
through their DWDMS filters $n$th-channel passbands. The final bracketed term
in that equation is the average $|\psi^{-}\rangle_{M_{1}M_{2}}$ fidelity,
given that Alice and Bob’s QRXs register a photon-detection coincidence.

The inter-channel interference problem—alluded to above—now becomes clear. To
realize near-deterministic generation of heralded polarization-entangled
biphotons for transmission to Alice and Bob, the ZALM QTX will operate with
$\mathbb{E}(N_{p})\sim 1$, $\Omega_{\rm PM}\gg\Delta B$, and $N\sim\Omega_{\rm
PM}/\Delta B$, so that each DWDM channel’s heralding probability will be low
enough, e.g., $\Pr(\psi^{\mp}_{n})\sim 10^{-2}$, that multi-pair events can be
ignored on a per-channel basis. That said, $\mathbb{E}(N_{p})\sim 1$ implies
that a 2-pair event, over the dual-Sagnac’s full phase-matching bandwidth,
will occur with $\sim$20% probability. Then, because the ZALM QTX must operate
at low heralding efficiency to maintain high fidelity, cf. Figs. 10 and 11, a
2-pair event that heralds on channels $n$ and $m$ can have its $n$th-channel’s
signal photon appear in the $m$th channel and vice versa causing memory-load
errors. This effect is most severe for $|n-m|=1$, as illustrated, for the
simple case of a single Sagnac-configured source, in Fig. 13, assuming the
all-Gaussian special case with parameter values from Table 1. There we have
plotted the normalized $m=n+1$ inter-channel interference,
$\chi_{1}(n)\equiv\Pr(S_{n+1},I_{n})/\Pr(S_{n},I_{n})$ for
$n=-40,-39,\ldots,40,$ with

$\Pr(S_{m},I_{n})\equiv\int\\!\frac{{\rm d}\omega_{S}}{2\pi}\int\\!\frac{{\rm
d}\omega_{I}}{2\pi}\,|\Psi_{SI_{n}}(\omega_{S},\omega_{I})H_{S_{m}}(\omega_{S})|^{2}$
(121)

being the probability of the Sagnac SPDC’s biphoton has a signal component
that passes through the DWDMS filter’s $m$th channel and an idler component
that passes through the DWDMI filter’s $n$th channel. As expected, Fig. 13
shows that Case 1, with its high heralding efficiency, has very low inter-
channel interference, but Case 2, with its high fidelity coming at the cost of
low heralding efficiency, has very significant inter-channel interference.

We can dramatically reduce the inter-channel interference, at the expense of
entanglement-distribution rate, by having ZALM’s QTX impose artificially
enhanced guard bands, e.g., by only allowing heralds from channels spaced two
or three apart. Figure 14 shows the reductions in normalized inter-channel
interference offered by $\chi_{2}(n)\equiv\Pr(S_{n+2},I_{n})/\Pr(S_{n},I_{n})$
for $n=-40,-38,\ldots,40,$ and
$\chi_{3}(n)\equiv\Pr(S_{n+3},I_{n})/\Pr(S_{n},I_{n})$ for
$n=-39,-36,\ldots,39,$ assuming the all-Gaussian special case with Table 1’s
Case 2 parameter values. These benefits are accompanied by a reduction in the
number of usable channels to 41 for $\chi_{2}(n)$, and to 27 for
$\chi_{3}(n)$. A full exploration of the tradeoff between entanglement-
distribution rate and inter-channel interference is postponed to a future
paper.

Figure 13: Normalized inter-channel interference, $\chi_{1}(n)$ for
$n=-40,-39,\ldots 40,$ using Eq. (9)’s all-Gaussian biphoton wave function
with brickwall filtering and parameter values from Table 1.  Figure 14:
Normalized inter-channel interferences, $\chi_{2}(n)$ for
$n=-40,-38,\ldots,40,$ and $\chi_{3}(n)$ for $n=39,-36,\ldots,39,$ using Eq.
(9)’s all-Gaussian biphoton wave function with brickwall filtering and the
Case 2 parameter values from Table 1.

In truth, inter-channel interference will not be much of a problem when
$\mathbb{E}(N_{p})\sim 1$ and $N\gg 1$, because it is unlikely that a two-pair
event will herald on adjacent DWDM channels. Future availability of high-
speed, high-quantum-efficiency, number-resolving photodetectors, however, will
lead to ZALM’s using $\mathbb{E}(N_{p})\gg 1$ so as to approach
$\Pr(\psi^{\mp}_{n})\sim 0.2$. In that case, inter-channel interference will
be a significant concern.

Our final comments have to do with extending our analysis to account for
equipment nonidealities, and to consider alternative SPDC sources as well as
alternative quantum-memory protocols. With respect to equipment nonidealities,
culprits missing from our treatment so far include: (1) slight asymmetry in
the 50–50 beam splitters in the partial BSMs and the Duan-Kimble quantum
memories; (2) dark counts in the partial-BSMs and the Duan-Kimble quantum
memories’ heralding detectors; (3) background-light counts in the Duan-Kimble
quantum memories’ heralding detectors when ZALM relies on free-space
propagation; and (4) imperfect mode conversion in Alice and Bob’s QRXs.
Evaluating their impact on ZALM’s entanglement-distribution rate and its
entangled-state fidelity should be part of an accurate assessment of its
utility, and the techniques we have employed here can be extended to account
for them.

Turning to the possibility of alternative SPDC sources for use in ZALM, the
key issue is to find better ways to approach a separable channelized wave
function for the Sagnac sources’ polarization-entangled biphotons _without_
sacrificing heralding efficiency, which is the bane of our DWDM-filtered
sources. Two possibilities here are intra-cavity dual-SPDC sources and SPDCs
whose nonlinear crystals have many phase-matching islands. Shapiro and Wong
[45] proposed an intra-cavity dual-SPDC source that could directly generate
narrowband signal and idler for interfacing to the Rb-atom quantum memories in
an early proposal for long-distance qubit teleportation [46]. Reference [45]
only considered the output from a single cavity resonance. For ZALM, however,
multiple cavity resonances would be employed to provide the protocol’s
frequency multiplexing. Cavity losses may be a significant issue for this
approach, but a detailed design study for multi-channel, cavity-based sources
is certainly warranted and our analysis could be adapted for this purpose and
our techniques are capable of addressing the resulting entangled-state
fidelity.

SPDC sources with phase-matching-islands eschew intra-cavity operation.
Instead, they engineer a nonlinear crystal’s phase-matching function to
produce a joint spectral intensity comprised of near-separable, non-
overlapping islands well-suited to frequency-multiplexed entanglement
distribution. Reference [47] realized an 8-island demonstration of this
technique. Extending their work to the desired $\sim$100 islands for ZALM may
be challenging, but here too further study is needed.

The final area we will mention for extending our work is consideration of
alternative memory protocols, specifically those associated with distributing
time-bin entanglement. The dual-Sagnac source we have assumed affords a large
number of frequency-multiplexed channels for high-rate transmission of
polarization-entangled biphotons. Polarization entanglement is the preferred
choice for satellite-to-ground entanglement distribution, because atmospheric
turbulence does not disturb polarization [48]. For terrestrial entanglement
distribution over optical fiber, however, time-bin entanglement is the way to
go, because standard fiber does not preserve polarization and polarization-
maintaining fiber is currently too lossy for long-distance operation. Our
dual-Sagnac source’s heralded polarization-entangled biphotons can be
converted to time-bin entanglement using linear optics, as shown in Fig. 15,
and then loaded into Duan-Kimble quantum memories set up for time-bin
entanglement as in Nguyen _et al_. [27]. Raymer  _et al._ ’s reflectivity
analysis needs only a minor modification, to account for the $\pi$ pulse Ref.
[27] applies to the memories between the arrival of $S_{k_{e}}$ and
$S_{k_{\ell}}$, after which the same state-dependent reflectivities as those
for Duan-Kimble loading of polarization entanglement are obtained.

Figure 15: Schematic for converting the polarization-entangled biphoton sent
to Alice ($S_{1}$) and Bob ($S_{2}$) to time-bin entanglement. PBS:
polarization beam splitter. HWP: half-wave plate. M: mirror. SW: fast switch
to first pass $S_{k_{H}}$ and later pass $S^{\prime}_{k_{H}}$ into the output.
$S_{k_{\ell}}\cdots S_{k_{e}}$ for $k=1$ and $2$: the time-bin entangled ($e$
= early, $\ell$ = late) state.

###### Acknowledgements.

This work was supported by the Engineering Research Centers Program of the
National Science Foundation under Grant #1941583 to the NSF-ERC Center for
Quantum Networks. The authors thank Saikat Guha and Dirk Englund for advice
and encouragement and Prajit Dhara for valuable discussions.

##  Appendix A Derivation Details for Sec. IV

In this appendix we supply derivation details for results that were stated,
but not proven, in Sec. IV.

###  A.1 Derivation of Eq. (35)

We start this derivation by expanding $|\tilde{\psi}\rangle_{{\bf S}_{n}{\bf
I}_{n}}\,{}_{{\bf S}_{n}{\bf I}_{n}}\\!\langle\tilde{\psi}|$ in Eq. (32)’s
first integral using its definition and Eq. (34)’s beam-splitter relations.
The result so obtained is the enormous expression

$\displaystyle\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral 1}}=$ (122) $\displaystyle\int\\!\frac{{\rm
d}\omega_{I_{+}}}{2\pi}\int\\!\frac{{\rm d}\omega_{I-}}{2\pi}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{I}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}^{\prime}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{I}^{\prime}}{4\pi^{2}}\,({}_{I_{+H}}\\!\langle\omega_{I_{+}}|\,{}_{I_{-V}}\\!\langle\omega_{I_{-}}|)\Psi_{S_{n}I_{n}}(\omega_{S_{1}},\omega_{I_{1}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{1}}^{\prime},\omega_{I_{1}}^{\prime})$
$\displaystyle\,\,\times\,\Psi_{S_{n}I_{n}}(\omega_{S_{2}},\omega_{I_{2}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{2}}^{\prime},\omega_{I_{2}}^{\prime})\left[\frac{|\omega_{S_{1}}\rangle_{S_{1H}}(|\omega_{I_{1}}\rangle_{I_{+V}}+|\omega_{I_{1}}\rangle_{I_{-V}})-|\omega_{S_{1}}\rangle_{S_{1V}}(|\omega_{I_{1}}\rangle_{I_{+H}}+|\omega_{I_{1}}\rangle_{I_{-H}})}{2}\right]$
$\displaystyle\,\,\times\,\left[\frac{|\omega_{S_{2}}\rangle_{S_{2H}}(|\omega_{I_{2}}\rangle_{I_{+V}}-|\omega_{I_{2}}\rangle_{I_{-V}})-|\omega_{S_{2}}\rangle_{S_{2V}}(|\omega_{I_{2}}\rangle_{I_{+H}}-|\omega_{I_{2}}\rangle_{I_{-H}})}{2}\right]$
$\displaystyle\,\,\times\,\left[\frac{{}_{S_{2H}}\langle\omega_{S_{2}}^{\prime}|({}_{I_{+V}}\langle\omega_{I_{2}}^{\prime}|-{}_{I_{-V}}\langle\omega_{I_{2}}^{\prime}|)-{}_{S_{2V}}\langle\omega_{S_{2}}^{\prime}|({}_{I_{+H}}\langle\omega_{I_{2}}^{\prime}|-{}_{I_{-H}}\langle\omega_{I_{2}}^{\prime}|)}{2}\right]$
$\displaystyle\,\,\times\,\left[\frac{{}_{S_{1H}}\langle\omega_{S_{1}}^{\prime}|({}_{I_{+V}}\langle\omega_{I_{1}}^{\prime}|-{}_{I_{-V}}\langle\omega_{I_{1}}^{\prime}|)-{}_{S_{1V}}\langle\omega_{S_{1}}^{\prime}|({}_{I_{+H}}\langle\omega_{I_{1}}^{\prime}|-{}_{I_{-H}}\langle\omega_{I_{1}}^{\prime}|)}{2}\right](|\omega_{I_{-}}\rangle_{I_{-V}}\,|\omega_{I_{+}}\rangle_{I_{+H}}).$

Evaluating the resulting bra-ket inner products gives four impulses that
collapse the ${\rm d}^{2}{\boldsymbol{\omega}}_{I}$ and ${\rm
d}^{2}{\boldsymbol{\omega}}_{I}^{\prime}$ integrals in Eq. (122), reducing it
to the slightly less formidable expression

$\displaystyle\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral 1}}=\frac{1}{16}\int\\!\frac{{\rm
d}\omega_{I_{+}}}{2\pi}\int\\!\frac{{\rm d}\omega_{I-}}{2\pi}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}^{\prime}}{4\pi^{2}}$ (123)
$\displaystyle\left\\{\Psi_{S_{n}I_{n}}(\omega_{S_{1}},\omega_{I_{-}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{1}}^{\prime},\omega_{I_{-}})\Psi_{S_{n}I_{n}}(\omega_{S_{2}},\omega_{I_{+}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{2}}^{\prime},\omega_{I_{+}})|\omega_{S_{1}}\rangle_{S_{1V}}|\omega_{S_{2}}\rangle_{S_{2H}}\,{}_{S_{2H}}\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1V}}\langle\omega_{S_{1}}^{\prime}|\right.$
$\displaystyle\,\,+\,\Psi_{S_{n}I_{n}}(\omega_{S_{1}},\omega_{I_{+}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{1}}^{\prime},\omega_{I_{+}})\Psi_{S_{n}I_{n}}(\omega_{S_{2}},\omega_{I_{-}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{2}}^{\prime},\omega_{I_{-}})|\omega_{S_{1}}\rangle_{S_{1H}}|\omega_{S_{2}}\rangle_{S_{2V}}\,{}_{S_{2V}}\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1H}}\langle\omega_{S_{1}}^{\prime}|$
$\displaystyle\,\,-\,\Psi_{S_{n}I_{n}}(\omega_{S_{1}},\omega_{I_{-}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{1}}^{\prime},\omega_{I_{+}})\Psi_{S_{n}I_{n}}(\omega_{S_{2}},\omega_{I_{+}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{2}}^{\prime},\omega_{I_{-}})|\omega_{S_{1}}\rangle_{S_{1V}}|\omega_{S_{2}}\rangle_{S_{2H}}\,{}_{S_{2V}}\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1H}}\langle\omega_{S_{1}}^{\prime}|$
$\displaystyle\,\,-\,\left.\Psi_{S_{n}I_{n}}(\omega_{S_{1}},\omega_{I_{+}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{1}}^{\prime},\omega_{I_{-}})\Psi_{S_{n}I_{n}}(\omega_{S_{2}},\omega_{I_{-}})\Psi^{*}_{S_{n}I_{n}}(\omega_{S_{2}}^{\prime},\omega_{I_{+}})|\omega_{S_{1}}\rangle_{S_{1H}}|\omega_{S_{2}}\rangle_{S_{2V}}\,{}_{S_{2H}}\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1V}}\langle\omega_{S_{1}}^{\prime}|\right\\}\\!.$

Equation (123) compacts further, by using Eq. (19) and defining

$K_{S_{1n}S_{2n}}^{(1)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}^{\prime})\equiv\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{1}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{2}})\mbox{
and
}K_{S_{1n}S_{2n}}^{(2)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}^{\prime})\equiv\Phi_{n}(\omega_{S_{1}},\omega^{\prime}_{S_{2}})\Phi_{n}(\omega_{S_{2}},\omega^{\prime}_{S_{1}}),$
(124)

which gives us

$\displaystyle\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral 1}}=\frac{1}{16}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}^{\prime}}{4\pi^{2}}$ (125)
$\displaystyle\left\\{K_{S_{1n}S_{2n}}^{(1)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}^{\prime})(|\omega_{S_{1}}\rangle_{S_{1V}}|\omega_{S_{2}}\rangle_{S_{2H}}\,{}_{S_{2H}}\\!\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1V}}\\!\langle\omega_{S_{1}}^{\prime}|+|\omega_{S_{1}}\rangle_{S_{1H}}|\omega_{S_{2}}\rangle_{S_{2V}}\,{}_{S_{2V}}\\!\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1H}}\langle\omega_{S_{1}}^{\prime}|)\right.$
$\displaystyle\,\,-\,\left.K_{S_{1n}S_{2n}}^{(2)}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}_{S}^{\prime})(|\omega_{S_{1}}\rangle_{S_{1V}}|\omega_{S_{2}}\rangle_{S_{2H}}\,{}_{S_{2V}}\\!\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1H}}\\!\langle\omega_{S_{1}}^{\prime}|+|\omega_{S_{1}}\rangle_{S_{1H}}|\omega_{S_{2}}\rangle_{S_{2V}}\,{}_{S_{2H}}\langle\omega_{S_{2}}^{\prime}|\,{}_{S_{1V}}\langle\omega_{S_{1}}^{\prime}|)\right\\}.$

Now, recognizing that

$K^{(1)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\frac{K^{(c)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})+K^{(e)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})}{2},$
(126)

and

$K^{(2)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})=\frac{K^{(c)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})-K^{(e)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})}{2},$
(127)

we arrive at

$\displaystyle\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral 1}}$ $\displaystyle=\frac{1}{16}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})|\psi^{-}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{-}({\boldsymbol{\omega}}^{\prime}_{S})|$
$\displaystyle\,\,+\,\frac{1}{16}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(e)}_{S_{1n}S_{2n}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})|\psi^{+}({\boldsymbol{\omega}}_{S})\rangle_{S_{1}S_{2}}\,{}_{S_{1}S_{2}}\langle\psi^{+}({\boldsymbol{\omega}}^{\prime}_{S})|.$
(128)

The same steps can be followed to show that

$\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral
2}}=\left.\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}\right|_{\mbox{\scriptsize
integral 1}},$ (129)

completing the derivation of the
$\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{-}_{n}}$ part of Eq. (35). That
equation’s $\tilde{\rho}_{S_{1_{n}}S_{2_{n}}\mid\psi^{+}_{n}}$ part is found
by repeating for Eq. (33) what we have just done for Eq. (32).

###  A.2 Verification of Eqs. (51)–(54)

Substituting Eqs. (44) and (51) into the left-hand side of Eq. (49), we have
that

$\displaystyle\int\\!$ $\displaystyle\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})\xi_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S})$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\lambda^{2}_{\ell}\lambda_{\ell^{\prime}}^{2}[\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{1}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{2}})+\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{2}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{1}})]\phi_{m}(\omega^{\prime}_{S_{1}})\phi_{m}(\omega^{\prime}_{S_{2}})$
$\displaystyle=\sum_{m=1}^{\infty}2\lambda_{m}^{4}\,\phi_{m}(\omega_{S_{1}})\phi_{m}(\omega_{S_{2}}),\mbox{
for $m=1,2,\ldots,$}$ (130)

and

$\displaystyle\int\\!$ $\displaystyle\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})\xi_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S})$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{\ell^{\prime}=1}^{\infty}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\lambda^{2}_{\ell}\lambda_{\ell^{\prime}}^{2}[\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{1}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{2}})+\phi_{\ell}(\omega_{S_{1}})\phi^{*}_{\ell}(\omega^{\prime}_{S_{2}})\phi_{\ell^{\prime}}(\omega_{S_{2}})\phi^{*}_{\ell^{\prime}}(\omega^{\prime}_{S_{1}})]$
$\displaystyle\,\,\times\,[\phi_{m_{1}}(\omega^{\prime}_{S_{1}})\phi_{m_{2}}(\omega^{\prime}_{S_{2}})+\phi_{m_{1}}(\omega^{\prime}_{S_{2}})\phi_{m_{2}}(\omega^{\prime}_{S_{1}})]/\sqrt{2},\mbox{
for $m_{1}>m_{2}=1,2,\ldots,$}$
$\displaystyle=\sum_{m_{1}=2}^{\infty}\sum_{m_{2}=1}^{m_{1}-1}2\lambda_{m_{1}}^{2}\lambda_{m_{2}}^{2}\,[\phi_{m_{1}}(\omega_{S_{1}})\phi_{m_{2}}(\omega_{S_{2}})+\phi_{m_{1}}(\omega_{S_{2}})\phi_{m_{2}}(\omega_{S_{1}})]/\sqrt{2},$
(131)

and our verification is complete for Eqs. (51) and (53)’s being the
eigenfunctions and eigenvalues of
$K^{(c)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$.

Following the same approach, starting with substitution of Eqs. (45) and (52)
into Eq. (50) will verify Eqs. (52) and (54)’s being the eigenfunctions and
eigenvalues of
$K^{(e)}_{S_{1_{n}}S_{2_{n}}}({\boldsymbol{\omega}}_{S};{\boldsymbol{\omega}}^{\prime}_{S})$.

##  Appendix B Derivation Details for Sec. V

Here we provide derivation details for the $\tilde{\rho}^{(p)}_{M_{1},M_{2}}$
results presented in Sec. V.

Suppose Alice and Bob’s QRXs are illuminated by the $|\tilde{\xi}^{-}_{\bf
m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}$ state from Eq. (72), and that they
record photon counts from their $\tilde{S}_{1+}$ and $\tilde{S}_{2+}$
detectors. Their quantum memories are then left in the state whose
unnormalized density operator is

$\displaystyle\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\tilde{\xi}^{-}_{\bf m}}$
$\displaystyle=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{+}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}^{\prime}_{S}}{4\pi^{2}}\,\frac{\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})\tilde{\xi}^{*}_{\bf
m}({\boldsymbol{\omega}}^{\prime}_{S})}{8}\,{}_{\tilde{S}^{\prime}_{1+}}\\!\langle\omega_{1+}|\,\,{}_{\tilde{S}^{\prime}_{2+}}\\!\langle\omega_{2+}|$
$\displaystyle\,\,\times\left\\{|\omega_{S_{1}}\rangle_{\tilde{S}^{\prime}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}_{2_{V}}}\left[r_{1}(\omega_{S_{1}})|g_{1}\rangle_{M_{1}}+r_{2}(\omega_{S_{1}})|g_{2}\rangle_{M_{1}}\right]\left[|g_{1}\rangle_{M_{2}}+|g_{2}\rangle_{M_{2}}\right]\right.$
$\displaystyle\,\,-\left.|\omega_{S_{1}}\rangle_{\tilde{S}_{1_{V}}}|\omega_{S_{2}}\rangle_{\tilde{S}^{\prime}_{2_{V}}}\left[|g_{1}\rangle_{M_{1}}+|g_{2}\rangle_{M_{1}}\right]\left[r_{1}(\omega_{S_{2}})|g_{1}\rangle_{M_{2}}+r_{2}(\omega_{S_{2}})|g_{2}\rangle_{M_{2}}\right]\right\\}$
$\displaystyle\,\,\times\left\\{\left[r^{*}_{1}(\omega^{\prime}_{S_{1}})\,{}_{M_{1}}\\!\langle
g_{1}|+r^{*}_{2}(\omega^{\prime}_{S_{1}})\,{}_{M_{1}}\\!\langle
g_{2}|\right]\left[{}_{M_{2}}\\!\langle g_{1}|+{}_{M_{2}}\\!\langle
g_{2}|\right]{}_{\tilde{S}_{2_{V}}}\\!\\!\langle\omega^{\prime}_{S_{2}}|\,{}_{\tilde{S}^{\prime}_{S_{1_{V}}}}\\!\\!\langle\omega^{\prime}_{S_{1}}|\right.$
$\displaystyle\,\,-\left.\left[{}_{M_{1}}\\!\langle
g_{1}|+{}_{M_{2}}\\!\langle
g_{2}|\right]\left[r^{*}_{1}(\omega^{\prime}_{S_{2}})\,{}_{M_{2}}\langle
g_{1}|+r^{*}_{2}(\omega^{\prime}_{S_{2}})\,{}_{M_{2}}\\!\langle
g_{2}|\right]\,{}_{\tilde{S}^{\prime}_{S_{2_{V}}}}\\!\\!\langle\omega^{\prime}_{S_{2}}|\,{}_{\tilde{S}_{1_{V}}}\\!\\!\langle\omega^{\prime}_{S_{1}}|\right\\}|\omega_{2+}\rangle_{\tilde{S}^{\prime}_{2+}}\,|\omega_{1+}\rangle_{\tilde{S}^{\prime}_{1+}}.$
(132)

Using the beam-splitter relations,

$\displaystyle|\omega_{S_{k}}\rangle_{\tilde{S}^{\prime}_{k_{V}}}$
$\displaystyle=\frac{|\omega_{S_{k}}\rangle_{\tilde{S}^{\prime}_{k+}}+|\omega_{S_{k}}\rangle_{\tilde{S}^{\prime}_{k-}}}{\sqrt{2}},$
(133) $\displaystyle|\omega_{S_{k}}\rangle_{\tilde{S}_{k_{V}}}$
$\displaystyle=\frac{e^{i\omega_{S_{k}}T}(|\omega_{S_{k}}\rangle_{\tilde{S}^{\prime}_{k+}}-|\omega_{S_{k}}\rangle_{\tilde{S}^{\prime}_{k-}})}{\sqrt{2}},\mbox{
for $k=1,2$},$ (134)

in Eq. (132), evaluating the ensuing bra-ket inner products, and integrating
over ${\boldsymbol{\omega}}_{+}$, we get

$\displaystyle\tilde{\rho}$ ${}^{(+,+)}_{M_{1},M_{2}\mid\tilde{\xi}^{-}_{\bf
m}}=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{|\tilde{\xi}_{\bf
m}({\boldsymbol{\omega}}_{S})|^{2}}{32}\,$
$\displaystyle\,\,\times\left\\{\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|g_{1}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}+\left[e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|g_{2}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}\right.$
$\displaystyle\,\,+\,\left.\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|g_{1}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}+\left[e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})-e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|g_{2}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}\right\\}$
$\displaystyle\,\,\times\,\left\\{{}_{M_{2}}\langle g_{1}|\,{}_{M_{1}}\langle
g_{1}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{1}})-e^{-i\omega_{S_{1}}T}r^{*}_{1}(\omega_{S_{2}})\right]+{}_{M_{2}}\langle
g_{2}|\,{}_{M_{1}}\langle
g_{2}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{2}(\omega_{S_{1}})-e^{-i\omega_{S_{1}}T}r^{*}_{2}(\omega_{S_{2}})\right]\right.$
$\displaystyle\,\,+\,\left.{}_{M_{2}}\langle g_{2}|\,{}_{M_{1}}\langle
g_{1}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{1}})-e^{-i\omega_{S_{1}}T}r^{*}_{2}(\omega_{S_{2}})\right]+{}_{M_{2}}\langle
g_{1}|\,{}_{M_{1}}\langle
g_{2}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{2}(\omega_{S_{1}})-e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{2}})\right]|\right\\}.$
(135)

Rewriting this density operator in terms of the quantum memories’ Bell states
from Eq. (101) and (102), and then inserting the result in Eq. (74), we arrive
at the $\sum_{\bf m}\mu_{\bf m}$, $p=a$ term in Eq. (84) that applies to
$\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\psi^{-}_{n}}$ from Eq. (82) with the
memories’ frequency-dependent Bell states being given by Eqs. (85)–(92).

To get the $\sum_{\bf m}\nu_{\bf m}$, $p=a$ term in Eq. (84), we parallel what
we have just done, starting from Alice and Bob’s memories being illuminated by
the $|\tilde{\zeta}^{-}_{\bf m}\rangle_{\tilde{S}_{1}\tilde{S}_{2}}$ state
from Eq. (73) and assuming Alice and Bob’s QRXs record photon counts from
their $\tilde{S}_{1+}$ and $\tilde{S}_{2+}$ detectors. After using the beam-
splitter relations from Eqs. (133) and (134), evaluating the ensuing bra-ket
inner products, and then integrating over ${\boldsymbol{\omega}}_{+}$, we find
that

$\displaystyle\tilde{\rho}$ ${}^{(+,+)}_{M_{1},M_{2}\mid\tilde{\zeta}^{-}_{\bf
m}}=\int\\!\frac{{\rm
d}^{2}{\boldsymbol{\omega}}_{S}}{4\pi^{2}}\,\frac{|\tilde{\zeta}_{\bf
m}({\boldsymbol{\omega}}_{S})|^{2}}{32}\,$
$\displaystyle\,\,\times\left\\{\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|g_{1}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}+\left[e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|g_{2}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}\right.$
$\displaystyle\,\,+\,\left.\left[e^{i\omega_{S_{2}}T}r_{1}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{2}(\omega_{S_{2}})\right]|g_{1}\rangle_{M_{1}}|g_{2}\rangle_{M_{2}}+\left[e^{i\omega_{S_{2}}T}r_{2}(\omega_{S_{1}})+e^{i\omega_{S_{1}}T}r_{1}(\omega_{S_{2}})\right]|g_{2}\rangle_{M_{1}}|g_{1}\rangle_{M_{2}}\right\\}$
$\displaystyle\,\,\times\,\left\\{{}_{M_{2}}\langle g_{1}|\,{}_{M_{1}}\langle
g_{1}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{1}})+e^{-i\omega_{S_{1}}T}r^{*}_{1}(\omega_{S_{2}})\right]+{}_{M_{2}}\langle
g_{2}|\,{}_{M_{1}}\langle
g_{2}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{2}(\omega_{S_{1}})+e^{-i\omega_{S_{1}}T}r^{*}_{2}(\omega_{S_{2}})\right]\right.$
$\displaystyle\,\,+\,\left.{}_{M_{2}}\langle g_{2}|\,{}_{M_{1}}\langle
g_{1}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{1}})+e^{-i\omega_{S_{1}}T}r^{*}_{2}(\omega_{S_{2}})\right]+{}_{M_{2}}\langle
g_{1}|\,{}_{M_{1}}\langle
g_{2}|\left[e^{-i\omega_{S_{2}}T}r^{*}_{2}(\omega_{S_{1}})+e^{-i\omega_{S_{2}}T}r^{*}_{1}(\omega_{S_{2}})\right]|\right\\}.$
(136)

Rewriting this expression in terms of the quantum memories’ Bell states, and
inserting the result in Eq. (74), we recover the $\sum_{\bf m}\nu_{\bf m}$,
$p=a$ term in Eq. (84) that applies to
$\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\psi^{-}_{n}}$ from Eq. (82) with the
memories’ frequency-dependent Bell states being given by Eqs. (85)–(92). This
completes the derivation of
$\tilde{\rho}^{(+,+)}_{M_{1},M_{2}\mid\psi^{-}_{n}}$. Repeating this procedure
three more times will show that the $\tilde{\rho}^{(a)}_{M_{1},M_{2}}$ we have
just verified as applying to $\psi^{-}_{n}$ heralds with ${\bf s}=(+,+)$
detections also applies to $\psi^{-}_{n}$ heralds with ${\bf s}=(-,-)$
detections, as well as to $\psi^{+}_{n}$ heralds with ${\bf s}=(+,-)$ or ${\bf
s}=(-,+)$ detections. We omit the details.

At this point, it only remains for us to demonstrate that
$\tilde{\rho}^{(b)}_{M_{1},M_{2}}$ from Eq. (84) characterizes the density