# A positive operator-valued measure for two-photon detection via sum-
frequency generation

Sofiane Merkouche  Corresponding author<EMAIL_ADDRESS>Valérian Thiel
Brian J. Smith  Oregon Center for Optical, Molecular, and Quantum Science, and
Department of Physics, University of Oregon, Eugene, OR 97403

###### Abstract

Spontaneous parametric down conversion (PDC), in the perturbative limit, can
be considered as a probabilistic splitting of one input photon into two output
photons. Conversely, sum-frequency generation (SFG) implements the reverse
process of combining two input photons into one. Here we show that a single-
photon projective measurement in the temporal-mode basis of the output photon
of a two-photon SFG process effects a generalized measurement on the input
two-photon state. We describe the positive operator-valued measure (POVM)
associated with such a measurement, and show that its elements are
proportional to the two-photon states produced by the time-reversed PDC
process. Such a detection acts as a joint measurement on two photons, and is
thus an important component of many quantum information processing protocols
relying on photonic entanglement. Using the retrodictive approach, we analyze
the properties of the two-photon POVM that are relevant for quantum protocols
exploiting two-photon states and measurements.

###### pacs:

##  I Introduction

Entangled photon pairs are an extremely useful system for studying both the
fundamentals Aspect (2015) and applications of quantum mechanics, and are the
workhorse of experimental quantum optics. This is mainly due to their ease of
generation in the laboratory through spontaneous parametric downconversion
(PDC), whereby a nonlinear medium such as a crystal is pumped with a bright
laser beam and mediates the probabilistic splitting of one pump photon into a
pair of photons, subject to energy and momentum conservation. Over the past
three decades, much progress has been made in the generation of PDC photon
pairs with well-engineered polarization, spectral-temporal, and spatial
structure, exhibiting varying degrees of correlation in all of these degrees
of freedom. Particular attention has been given recently to encoding quantum
information in the spectral-temporal degree of freedom of light. This is
because time-frequency modes of light, generally referred to as temporal
modes, can encode a large amount of information, are particularly well-suited
to integrated optics technology, and are robust to communication channel noise
Brecht  _et al._ (2015). In addition, time-frequency entangled photons are
useful for applications such as large-alphabet quantum key distribution Nunn
_et al._ (2013), quantum-enhanced spectroscopy Raymer  _et al._ (2013);
Schlawin and Buchleitner (2017); Dayan (2007), and quantum-enhanced sensing
Zhuang  _et al._ (2017).

Complementary to two-photon state generation is two-photon joint detection,
which is an example of the more general concept of a joint quantum measurement
on two systems. It is known that joint quantum measurements on separately
prepared systems can inherently reveal more information than accessible
through separate measurements relying on local operations and classical
communication Bennett  _et al._ (1999). In addition entangled measurements,
joint measurements whose eigenstates are entangled states, are as crucial a
resource as entangled states in quantum protocols such as quantum
teleportation Bennett  _et al._ (1993), remote state preparation Bennett  _et
al._ (2001), entanglement swapping Halder  _et al._ (2007); Sangouard  _et
al._ (2011), superdense coding, and quantum illumination Lloyd (2008). In
fact, the equal footing that entangled states and entangled measurements have
in quantum protocols such as teleportation has only recently been given due
attention Gisin (2019).

One way to implement a two-photon joint measurement is to use the complement
of PDC, sum-frequency generation (SFG). Here two photons interact in a
nonlinear medium and are upconverted to a single photon, conserving energy and
momentum. Two-photon measurement via SFG has been explored theoretically Dayan
(2007) and experimentally Dayan  _et al._ (2005). In addition, it has been
pointed out that the theory of two-photon detection by SFG closely parallels
that of two-photon absorption in a molecule, and a unified framework
describing both of these processes can be found in reference Dayan (2007).

In this work we construct and analyze the positive operator valued measure
(POVM) associated with joint two-photon measurements relying on SFG followed
by mode-selective detection of the upconverted photon in the time-frequency
domain. Our development of the two-photon POVM closely parallels that of the
POVM for a single photon detected after a filter, as described in reference
van Enk (2017). We then give some figures of merit for such measurements that
are relevant to some of the aforementioned protocols, namely the projectivity,
orthogonality, and entanglement of the measurement operators. We illustrate
the role of entanglement in measurements with a model of the spectral quantum
teleportation scenario. We conclude by highlighting some questions and
possible future directions left open by this work.

##  II Framework

###  II.1 The three-wave mixing interaction

We begin by writing down the transformation describing three-wave mixing,
which includes both parametric down-conversion and sum-frequency generation,
in the interaction picture. We assume a given polarization configuration and
assume that all the interacting fields occupy a single transverse spatial
mode, so that only the time-frequency degrees of freedom of the field are
relevant. Under these conditions the transformation may be expressed as

$\begin{gathered}\hat{H}=\hat{H}_{PDC}+\hat{H}_{SFG},\\\
\hat{H}_{PDC}=\chi\int\text{d}\omega_{s}\text{d}\omega_{i}\Phi(\omega_{s},\omega_{i})\
\hat{a}_{p}(\omega_{s}+\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i}),\\\
\hat{H}_{SFG}=(\hat{H}_{PDC})^{\dagger},\end{gathered}$ (1)

where $\hat{a}^{(\dagger)}_{j}(\omega_{j})$ is the annihilation (creation)
operator for a single photon at monochromatic mode $j$ with frequency
$\omega_{j}$, and $j=p,s,i$ label the pump, signal, and idler frequencies;
$\chi\ll 1$ is a parameter characterizing the efficiency of the process,
describing the second-order nonlinearity and containing all the parameters
that are constant or slowly-varying over the integration; and
$\Phi(\omega_{s},\omega_{i})$ is the phase-matching function, which has the
form

$\Phi(\omega_{s},\omega_{i})\propto\mathrm{sinc}\left(\frac{\Delta\mathbf{k}\cdot\mathbf{L}}{2}\right),$
(2)

where $\mathbf{L}$ is the vector quantifying the length of the interaction
medium, and
$\Delta\mathbf{k}=\mathbf{k}_{p}(\omega_{s}+\omega_{i})-\mathbf{k}_{s}(\omega_{s})-\mathbf{k}_{i}(\omega_{i})$
is the wavevector mismatch for the three fields. $\Phi$ takes on its maximum
value when $\Delta\mathbf{k}=\mathbf{0}$, and thus corresponds to momentum
conservation in the process. Finally, we have separated the transformation
explicitly into $\hat{H}_{PDC}$, the term responsible for PDC, and its
Hermitian conjugate, $\hat{H}_{SFG}$, responsible for SFG.

The interacting fields evolve unitarily under this transformation, and for our
analysis, we will consider only the weak-interaction limit, so that, for an
input state $\ket{\Psi_{\text{in}}}$, the output state is given by

$\ket{\Psi_{\text{out}}}=\exp{[-i\hat{H}]}\ket{\Psi_{\text{in}}}\approx\left(1-i\hat{H}\right)\ket{\Psi_{\text{in}}}.$
(3)

Note that, in a slight abuse of notation, we are using $\hat{H}$ to reflect
the fact that this transformation is derived from the interaction Hamiltonian
for three-wave mixing, although the latter is a time-dependent quantity with a
different dimensionality (see Appendix A).

###  II.2 PDC photon pairs and the joint spectral amplitude

Figure 1: Two-dimensional plot of the magnitude of a typical JSA. The solid
lines contour a Gaussian pump mode $\phi_{p}(\omega_{s}+\omega_{i})$, and the
dashed lines contour the phasematching function $\Phi(\omega_{s},\omega_{i})$.
This shows how spectral correlations arise in the JSA. Frequencies are in
arbitrary units.

It is instructive to briefly review the spectral-temporal structure of photon
pairs generated by PDC, governed by the $\hat{H}_{PDC}$ term. In most
applications PDC is pumped by a strong coherent state occupying a spectral
mode function $\phi_{p}(\omega)$, which can be treated as a classical field
amplitude $E_{p}(\omega)=E_{0}\phi_{p}(\omega)$, where $E_{0}$ quantifies the
field strength, and $\phi_{p}(\omega)$ is normalized as $\int\text{d}\omega\
|\phi_{p}(\omega)|^{2}=1$. However, since we are working in the perturbative
limit, it is equivalent to consider a single-photon pump in the state

$\ket{\Psi_{\text{in}}}=\ket{\phi_{p}}=\int\text{d}\omega\phi_{p}(\omega)\hat{a}_{p}^{\dagger}(\omega)\ket{\text{vac}}.$
(4)

After this state undergoes unitary evolution according to equation (3), we
obtain the output state

$\ket{\Psi_{\text{out}}}=\ket{\phi_{p}}-i\sqrt{w}\ket{\Psi_{\text{PDC}}},$ (5)

where

$\ket{\Psi_{\text{PDC}}}=\frac{\chi}{\sqrt{w}}\int\text{d}\omega_{s}\text{d}\omega_{i}\phi_{p}(\omega_{s}+\omega_{i})\Phi(\omega_{s},\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})\ket{\text{vac}}$
(6)

is a normalized two-photon state, and where

$w=\int\text{d}\omega_{s}\text{d}\omega_{i}|\chi\
\phi_{p}(\omega_{s}+\omega_{i})\Phi(\omega_{s},\omega_{i})|^{2}$ (7)

is a normalization factor.

It is convenient here to define the joint spectral amplitude (JSA)

$f(\omega_{s},\omega_{i})=\frac{\chi}{\sqrt{w}}\phi_{p}(\omega_{s}+\omega_{i})\Phi(\omega_{s},\omega_{i}),$
(8)

so that

$\ket{\Psi_{\text{PDC}}}=\int\text{d}\omega_{s}\text{d}\omega_{i}f(\omega_{s},\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})\ket{\text{vac}}$
(9)

The JSA can be viewed as a two-photon wavefunction, and its modulus squared,
$|f(\omega_{s},\omega_{i})|^{2}$, is the probability density function for the
photon pair in frequency space, normalized as
$\int\text{d}\omega_{s}\text{d}\omega_{i}|f(\omega_{s},\omega_{i})|^{2}=1$.
Considerable progress has been made in engineering the temporal-mode structure
of PDC photon pairs, which is completely characterized by the JSA, and this is
done by shaping of the pump spectral amplitude
$\phi_{p}(\omega_{s}+\omega_{i})$ and engineering of the phasematching
$\Phi(\omega_{s},\omega_{i})$ in the nonlinear medium. We plot schematically
in Fig. 1 a typical JSA configuration showing its dependence on the pump
amplitude and the phasematching function. A thorough review of the state-of-
the-art in two-photon state engineering in the time-frequency domain can be
found in reference Ansari  _et al._ (2018a).

###  II.3 Two-photon SFG and the two-photon POVM

Figure 2: PDC uses a $\chi^{(2)}$ interaction medium to convert a single-
photon state $\ket{1_{\phi}}$ in the mode $p$ to a pair of photons in modes
$s$ and $i$, described by the state $\ket{\psi_{\text{PDC}}}$ given in the
text. In the time-reverse picture, a projective measurement
$\hat{\text{P}}_{n}$ of a single photon produced by SFG implements measurement
with POVM element $\hat{\Pi}_{n}$ on the two input photons.

We now turn our attention to the SFG term in equation (1), explicitly given by

$\hat{H}_{SFG}=\chi^{*}\int\text{d}\omega_{s}\text{d}\omega_{i}\Phi^{*}(\omega_{s},\omega_{i})\hat{a}^{\dagger}_{p}(\omega_{s}+\omega_{i})\hat{a}_{s}(\omega_{s})\hat{a}_{i}(\omega_{i})$
(10)

and consider the upconversion of an arbitrary pure two photon state given by

$\ket{\Psi_{\text{in}}}=\ket{\psi_{g}}=\int\text{d}\omega_{s}\text{d}\omega_{i}g(\omega_{s},\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})\ket{\text{vac}},$
(11)

where $g(\omega_{s},\omega_{i})$ is a two-photon JSA. The output state will
then be

$\ket{\Psi_{\text{out}}}=\ket{\psi_{g}}-i\chi^{*}\ket{\sigma},$ (12)

where

$\ket{\sigma}=\int d\nu\sigma(\nu)\hat{a}^{\dagger}_{p}(\nu)\ket{\text{vac}},$
(13)

with the (unnormalized) spectral amplitude function

$\sigma(\nu)=-\frac{1}{2}\int d\nu^{\prime}\
\tilde{\Phi}^{*}\left(\nu,\nu^{\prime}\right)\tilde{g}\left(\nu,\nu^{\prime}\right).$
(14)

We obtain this last equation by changing variables to the sum and difference
frequencies $\nu=\omega_{s}+\omega_{i}$ and
$\nu^{\prime}=\omega_{s}-\omega_{i}$, and defining
$\tilde{\Phi}^{*}(\nu,\nu^{\prime})=\Phi^{*}\left(\frac{\nu+\nu^{\prime}}{2},\frac{\nu-\nu^{\prime}}{2}\right)$
(and likewise for $\tilde{g}(\nu,\nu^{\prime})$).

We are now equipped to develop the two-photon POVM corresponding to a
detection of the upconverted single-photon state $\ket{\sigma}$, which closely
mirrors the one-photon, pre-filter POVM described in reference van Enk (2017).
Consider performing an ideal, projective measurement of the upconverted photon
onto an orthonormal set of temporal mode single photon states
$\\{(\hat{\text{P}}_{n}=\ket{\phi_{n}}\bra{\phi_{n}})_{n=1}^{\infty}\\}$ with

$\ket{\phi_{n}}=\int\text{d}\omega\phi_{n}(\omega)\hat{a}^{\dagger}_{p}(\omega)\ket{\text{vac}},$
(15)

satisfying

$\braket{\phi_{n}}{\phi_{m}}=\int\text{d}\omega\
\phi^{*}_{n}(\omega)\phi_{m}(\omega)=\delta_{nm}.$ (16)

Such a measurement can in principle be realized using a quantum pulse gate,
recently described and demonstrated in references Ansari  _et al._ (2018b);
Reddy and Raymer (2018), whereby a strong pump field in a particular temporal
mode selects out that same mode from an input signal field and upconverts it
through SFG to a register mode which can be easily detected with a
spectrometer. The probability for a successful detection for this measurement
will be given by

$\begin{gathered}p_{n}=|\chi^{*}\braket{\phi_{n}}{\sigma}|^{2}\\\
=\left|-\frac{\chi^{*}}{2}\int\ \text{d}\nu\text{d}\nu^{\prime}\
\phi_{n}^{*}(\nu)\tilde{\Phi}^{*}\left(\nu,\nu^{\prime}\right)\tilde{g}\left(\nu,\nu^{\prime}\right)\right|^{2}\\\
=\left|\chi^{*}\int\text{d}\omega_{s}\text{d}\omega_{i}\phi_{n}^{*}(\omega_{s}+\omega_{i})\Phi^{*}(\omega_{s},\omega_{i})g(\omega_{s},\omega_{i})\right|^{2}\end{gathered}$
(17)

However, this same probability can be obtained by applying the Born rule to
the input state
$\hat{\rho}_{\text{in}}=\ket{\Psi_{\text{in}}}\bra{\Psi_{\text{in}}}$ in the
two-photon space:

$p_{n}=\text{Tr}(\hat{\rho}_{\text{in}}\hat{\Pi}_{n}),$ (18)

if we define a POVM element

$\hat{\Pi}_{n}=w_{n}\ket{\Psi_{n}}\bra{\Psi_{n}},$ (19)

where

$\ket{\Psi_{n}}=\frac{\chi}{\sqrt{w_{n}}}\int\text{d}\omega\text{d}\omega^{\prime}\phi_{n}(\omega+\omega^{\prime})\Phi(\omega,\omega^{\prime})\hat{a}^{\dagger}_{s}(\omega)\hat{a}^{\dagger}_{i}(\omega^{\prime})\ket{\text{vac}},$
(20)

and

$w_{n}=\int\text{d}\omega\text{d}\omega^{\prime}|\chi\
\phi_{n}(\omega+\omega^{\prime})\Phi(\omega,\omega^{\prime})|^{2}.$ (21)

We immediately recognize $\ket{\Psi_{n}}$ as the normalized two-photon state
that would result from PDC with a pump photon in the state $\ket{\phi_{n}}$.
That is, a projective measurement of an upconverted photon with projector
$\hat{\text{P}}_{n}=\ket{\phi_{n}}\bra{\phi_{n}}$ implements a generalized
measurement of the two input photons with POVM element $\hat{\Pi}_{n}$. This
is schematically shown in Fig. 2. Furthermore, the properties of
$\hat{\Pi}_{n}$ follow immediately from the properties of the PDC state
$\ket{\Psi_{n}}$, as we will see in the following section. It is convenient to
associate with the POVM element $\hat{\Pi}_{n}$ a measurement JSA

$f_{n}(\omega+\omega^{\prime})=\frac{\chi}{\sqrt{w_{n}}}\phi_{n}(\omega+\omega^{\prime})\Phi(\omega,\omega^{\prime}).$
(22)

To complete the POVM, we note that we are considering an ideal detector in the
SFG mode, such that any upconverted photon is detected with certainty. We are
thus justified in defining an element corresponding to no detection as

$\hat{\Pi}_{\text{null}}=\mathds{1}-\sum_{n=1}^{\infty}\hat{\Pi}_{n},$ (23)

where $\mathds{1}$ denotes the identity operator in the relevant two-photon
subspace. Using the fact that the $\phi_{n}$ mode functions form a complete
orthonormal set, we can evaluate

$\sum_{n=1}^{\infty}\hat{\Pi}_{n}=|\chi|^{2}\int\text{d}\omega\text{d}\omega^{\prime}|\Phi(\omega,\omega^{\prime})|^{2}\ket{\omega,\omega^{\prime}}\bra{\omega,\omega^{\prime}},$
(24)

where
$\ket{\omega,\omega^{\prime}}=\hat{a}^{\dagger}_{s}(\omega)\hat{a}^{\dagger}_{i}(\omega_{i})\ket{\text{vac}}$.
Noting that the identity in the two-photon subspace can be resolved as

$\mathds{1}=\int\text{d}\omega\text{d}\omega^{\prime}\ket{\omega,\omega^{\prime}}\bra{\omega,\omega^{\prime}},$
(25)

we can express $\hat{\Pi}_{\text{null}}$ explicitly as

$\hat{\Pi}_{\text{null}}=\int\text{d}\omega\text{d}\omega^{\prime}\left(1-|\chi|^{2}|\Phi(\omega,\omega^{\prime})|^{2}\right)\ket{\omega,\omega^{\prime}}\bra{\omega,\omega^{\prime}}.$
(26)

Finally we may write down the complete two-photon POVM as

$\left\\{(\hat{\Pi}_{n})_{n=1}^{\infty},\hat{\Pi}_{\text{null}}\right\\},$
(27)

satisfying

$\sum_{n=1}^{\infty}\hat{\Pi}_{n}+\hat{\Pi}_{\text{null}}=\mathds{1}.$ (28)

##  III Properties of the measurement operator

###  III.1 Projectivity

We will now take advantage of the well-studied properties of the two-photon
PDC state $\ket{\Psi_{n}}$ to analyze some of the useful properties of the
POVM element $\hat{\Pi}_{n}$. We begin by defining the retrodicted two-photon
state Amri  _et al._ (2011), corresponding to an outcome $n$, as

$\hat{\rho}_{n}=\frac{\hat{\Pi}_{n}}{\text{Tr}(\hat{\Pi}_{n})}=\ket{\Psi_{n}}\bra{\Psi_{n}}.$
(29)

We consider the measurement projective, if $\hat{\rho}_{n}$ is a pure state,
satisfying $\text{Tr}(\hat{\rho}_{n}^{2})=1$, which is indeed the case for
equation (29).

In general, however, single-photon detectors are not perfectly resolving. In
the case of the quantum pulse gate, a detector click may not correspond to
single pulse mode, but rather an incoherent mixture of a few modes. In the
case of a non-ideal spectrally resolving detection, one either uses a filter
of finite bandwidth, or a spectrometer with finite resolution. In all of these
cases, it is more accurate to describe a non-ideally resolving, that is, non-
projective, single-photon measurement by

$\hat{\text{P}}_{q}=\sum_{n}q_{n}\hat{\text{P}}_{n}$ (30)

where $0\leq q_{n}\leq 1$ are weighting coefficients. This leads to a two-
photon POVM element

$\hat{\Pi}_{q}=\sum_{n}q_{n}\hat{\Pi}_{n},$ (31)

and a retrodicted state

$\hat{\rho}_{q}=\frac{\hat{\Pi}_{q}}{\text{Tr}(\hat{\Pi}_{q})},$ (32)

which has $\text{Tr}(\hat{\rho}_{q}^{2})\leq 1$ and is not in general a pure
state. Evidently, the two-photon POVM elements are projective if and only if
the single-photon measurement operators are projective.

Projective two-photon measurements are of particular importance in quantum
teleportation and remote-state preparation, and entanglement swapping, because
in these schemes the measurement acts as a herald to a single photon state or
a two-photon entangled state, respectively. Ideally the heralded states should
be pure to be useful for quantum information processing. And the purity of the
heralded state is limited by both the purity of the input states and the
purity (projectivity) of the heralding measurement Amri  _et al._ (2011).

###  III.2 Orthogonality

Orthogonal measurements are measurements which project onto orthogonal states,
and thus satisfy

$\hat{\Pi}_{n}\hat{\Pi}_{m}\propto\delta_{nm}\hat{\Pi}_{n}.$ (33)

We note here that orthogonal measurements of the SFG photon do not correspond
to orthogonal two-photon POVM elements in general. This is analogous to the
fact that PDC pumped with orthogonal pulse modes does not produce orthogonal
PDC states in general. The non-orthogonality of the two-photon states can be
seen by taking

$\begin{gathered}\braket{\Psi_{n}}{\Psi_{m}}=\\\
\frac{|\chi|^{2}}{\sqrt{w_{n}w_{m}}}\int\text{d}\omega\text{d}\omega^{\prime}\phi^{*}_{n}(\omega+\omega^{\prime})\phi_{m}(\omega+\omega^{\prime})|\Phi(\omega,\omega^{\prime})|^{2}\neq\delta_{nm}.\end{gathered}$
(34)

This is due to the filtering induced by the phasematching function. This is
indeed analogous to what happens when two orthogonal modes are subjected to
linear filtering (see reference van Enk (2017) on this point): in general the
transmitted modes considered alone are not orthogonal, even though filtering
is a unitary process. The orthogonality is preserved only when considering all
of the modes involved in the transformation, whereas here we are only
considering the signal and idler modes and not the pump.

Figure 3: JSA’s for the configuration described in the text where the
phasematching function is engineered through group-velocity matching makes an
angle $\theta=45^{\text{o}}$ with respect to the $\omega_{s}$-axis. Then it
becomes independent of the sum frequency $\nu=\omega_{s}+\omega_{i}$, and thus
orthogonal measurements of the SFG photon correspond to orthogonal two-photon
POVM elements. Blue (red) indicates positive (negative) amplitudes. In the
case of PDC, the amount of correlations in the JSA can be controlled by
shaping of the pump pulse, as described in reference Ansari  _et al._ (2018a).
Here we plot the JSA’s obtained by shaping the pump into the (a) zeroth-, (b)
first-, and (c) second-order Hermite-Gauss modes, resulting into mutually-
orthogonal two-photon states. Frequencies are in arbitrary units.

An obvious question that arises then is, in what cases do the POVM elements,
in fact, correspond to orthogonal measurements? The answer to this question
becomes obvious when we rewrite equation (34) in terms of the sum and
difference frequencies $\nu$ and $\nu^{\prime}$,

$\begin{gathered}\braket{\Psi_{n}}{\Psi_{m}}=\\\
\frac{|\chi|^{2}}{4\sqrt{w_{n}w_{m}}}\int\text{d}\nu\text{d}\nu^{\prime}\phi^{*}_{n}(\nu)\phi_{m}(\nu)\left|\tilde{\Phi}\left(\nu,\nu^{\prime}\right)\right|^{2}.\end{gathered}$
(35)

Clearly, only when the phasematching function does not depend on the sum-
frequency $\nu$, that is, $\Phi=\Phi(\nu^{\prime})$, then do we obtain

$\braket{\Psi_{n}}{\Psi_{m}}=\delta_{nm},$ (36)

and the $\hat{\Pi}_{n}$ then satisfy

$\hat{\Pi}_{n}\hat{\Pi}_{m}=\delta_{nm}w_{n}\hat{\Pi}_{n}.$ (37)

Orthogonality of the two-photon POVM elements is of interest, for example, in
the quantum illumination scheme as originally described by Lloyd Lloyd (2008).
Here an entangled two-photon state $\ket{\Psi_{n}}$ is prepared and one of the
photons sent to reflect off a possibly present target, while the other photon
is kept in the lab. The two photons are then to be jointly measured, whereupon
a successful projection onto the initial state $\ket{\Psi_{n}}$ indicates the
presence of the target. If one is to implement this scheme using SFG as the
two-photon measurement, non-orthogonal measurements would suffer from the
possibility that the desired state $\ket{\Psi_{n}}$ could give a positive
outcome corresponding to the “wrong” measurement associated with a non-
orthogonal state $\ket{\Psi_{m}}$.

In general, the orthogonality condition (36) can be approximately satisfied as
long as the phase-matching function varies slowly enough in the $\nu$
direction, in comparison to the support of the detection mode function. This
happens, for example, in a sufficiently short interaction medium. However,
there are two limiting cases that are of note. The first is the spectrally
resolved detection limit, which corresponds to simply measuring the output
with an ideal spectrometer. In this limit, the detection mode can be
approximated by a delta function,

$\phi_{n}(\omega)\rightarrow\delta(\omega-\omega_{n}),$ (38)

and

$f_{n}(\omega,\omega^{\prime})\propto\delta(\omega+\omega^{\prime}-\omega_{n}),$
(39)

where $\omega_{n}$ is the measured frequency at the spectrometer. This is the
analogue of pumping a PDC source with monochromatic, or continuous-wave (cw),
light. In both of these cases, orthogonal pump (or measurement modes) with
frequencies $\omega_{n}$ and $\omega_{m}$ correspond to orthogonal two-photon
states (or measurements) with sum frequencies $\omega_{n}$ and $\omega_{m}$.

The second case of interest is achieved by extended phase-matching techniques,
as described in reference Ansari  _et al._ (2018a). For certain nonlinear
materials and field configurations, it is possible, using group-velocity
matching, to make the phase-matching function approximately constant in the
$\nu$ direction over some range of interest. More precisely, the phase-
matching function can be engineered to make an angle $\theta=45^{\text{o}}$ in
the $\omega_{s}$-$\omega_{i}$ plane, perpendicular to the angle that the pump
function makes. This configuration has been used by Ansari et al to generate
PDC states with a controllable temporal-mode structure and degree of
entanglement through pump pulse-shaping Ansari  _et al._ (2018b). This concept
is illustrated schematically in Fig. 3. More recently, similarly exotic two-
photon states have been obtained through phasematching shaped by the periodic
poling of the nonlinear crystal, rather than pulse-shaping of the pump
Graffitti  _et al._ (2020).

An interesting result that follows from the limit where $\Phi$ is independent
of $\nu$ is the possibility of downconverting an arbitrary pulse shape in a
nonlinear medium into an entangled photon pair, and recovering the pump pulse
shape by upconverting the photon pair in an identical medium. This can be seen
by taking $\tilde{g}(\nu,\nu^{\prime})=\phi(\nu)\tilde{\Phi}(\nu^{\prime})$ in
equation (14), and obtaining

$\sigma(\nu)=\phi(\nu)\int d\nu^{\prime}|\tilde{\Phi}(\nu^{\prime})|^{2},$
(40)

which is evidently proportional to the input $\phi(\nu)$. The spatial analogue
of this result, whereby a pump beam shaped in a specific transverse spatial
mode is downconverted, and the photon resulting from the upconversion of the
PDC pair is shown to recover the transverse spatial mode, has recently been
experimentally demonstrated by Jimenez et al Jimenez  _et al._ (2019).

###  III.3 Entanglement

We now turn to perhaps a more interesting question regarding the two-photon
measurement operator: when is the POVM element $\hat{\Pi}_{n}$ a projector
onto an entangled two-photon state, and thus can be said to enact an entangled
measurement on the input photons? Vértesi and Navascués (2011); Renou  _et
al._ (2018) We can answer this question readily: $\hat{\Pi}_{n}$ is an
entangled measurement, if the retrodicted state $\rho_{n}$ is an entangled
state. Entangled measurements play a central role in quantum teleportation,
superdense coding, and quantum illumination, among many other protocols, and
recently the role of entanglement in joint measurements has been recognized to
be equally important to the role of entanglement of states as a shared
resource Gisin (2019).

To illustrate the role of entangled measurements in a quantum protocol, we
will investigate briefly the spectral quantum teleportation scenario,
described by Molotkov Molotkov (1998) and by Humble Humble (2010) (and whose
spatial analogue was described by Walborn et al Walborn  _et al._ (2007)). In
this protocol, Alice and Bob share a two-photon entangled state described by a
JSA $f_{s}(\omega_{a},\omega_{b})$, and Alice is to teleport a single photon
state with spectral amplitude $\psi_{c}(\omega_{c})$ by performing an SFG
measurement on this photon and her half of the entangled state, and
communicating the measurement result to Bob.

Figure 4: Spectral teleportation scenario considered in the text. Alice and
Bob share entangled photons $a$ and $b$ in the state $\ket{\Psi_{s}}$. Alice
performs a two-photon SFG measurement $\hat{\Pi}_{m}$ on her photon $a$ and
photon $c$, in the state $\ket{\psi_{c}}$, and communicates the result of her
measurement to Bob, whereupon Bob reconstructs the state $\ket{\psi_{b|m}}$.

Reference Molotkov (1998) considers only the case of a maximally-correlated
pair of entangled photons shared between Alice and Bob, while reference Humble
(2010) generalizes this result to the case of a Gaussian JSA, which is a good
approximation to what can be produced using pulsed lasers as a pump. In both
references however, Alice’s joint measurement is a spectrally-resolved
measurement of the SFG photon. Here we use our formalism to generalize further
to a pulse-mode resolved measurement of the SFG photon, as can be realized
with a quantum pulse gate, by considering a generalized measurement JSA
$f_{m}(\omega_{a},\omega_{c})$. It was first pointed out in the original
proposal of quantum teleportation Bennett  _et al._ (1993) that in addition to
the maximally-entangled state (generalized Bell-state) shared by Alice and
Bob, quantum teleportation with unit fidelity is achieved when Alice’s joint
measurement projects onto a maximally-entangled state. Here we show behavior
that is consistent with this result by quantifying the teleportation fidelity
as a function of the entanglement of both the shared state and the joint
measurement. It is worth clarifying that our current goal is not to
demonstrate that the POVM element is entangled, but rather, it is to show that
our POVM formalism is sufficient to describe quantum teleportation in the
time-frequency domain, provided we stipulate entanglement as a property of the
measurement. This is in keeping with the more familiar case of the Bell-state
measurement’s role in qubit teleportation.

The teleportation scenario we consider is shown schematically in Fig. 4. Alice
and Bob share entangled photons a and b, respectively, described by a Gaussian
JSA similar to the one in reference Humble (2010):

$\begin{gathered}\ket{\Psi_{s}}=\int\text{d}\omega_{a}\ \text{d}\omega_{b}\
f_{s}(\omega_{a},\omega_{b})\hat{a}^{\dagger}_{a}(\omega_{a})\hat{a}^{\dagger}_{b}(\omega_{b})\ket{\text{vac}}\\\
f_{s}(\omega_{a},\omega_{b})=N_{s}\text{Exp}\left[-\frac{1}{\gamma_{s}^{2}(1-\alpha^{2})}\left(\frac{\omega_{a}^{2}}{2}+\frac{\omega_{b}^{2}}{2}+\alpha\omega_{a}\omega_{b}\right)\right]\end{gathered}$
(41)

where $\alpha\in[-1,1]$ is the correlation between the the photon frequencies,
with $\alpha=1$ corresponding to maximal frequency anticorrelation, such as
would be obtained from a cw pump; $\gamma_{s}$ is the characteristic bandwidth
of the PDC photons, and $N_{s}$ is the normalization constant. Alice provides
a single photon c to be teleported, described by the state

$\ket{\psi_{c}}=\int\text{d}\omega_{c}\psi_{c}(\omega_{c})\hat{a}^{\dagger}_{c}(\omega_{c})\ket{\text{vac}}$
(42)

where $\psi_{c}(\omega_{c})$ is an arbitrary spectral amplitude function.
Alice initiates the teleportation by performing an SFG measurement on photons
a and c, represented by an operator
$\hat{\Pi}_{m}=w_{m}\ket{\Psi_{m}}\bra{\Psi_{m}}$, with

$\begin{gathered}\ket{\Psi_{m}}=\int\text{d}\omega_{a}\ \text{d}\omega_{c}\
f_{m}(\omega_{a},\omega_{c})\hat{a}^{\dagger}_{a}(\omega_{a})\hat{a}^{\dagger}_{c}(\omega_{c})\ket{\text{vac}}\\\
f_{m}(\omega_{a},\omega_{c})=N_{m}\text{Exp}\left[-\frac{1}{\gamma_{m}^{2}(1-\beta^{2})}\left(\frac{\omega_{a}^{2}}{2}+\frac{\omega_{c}^{2}}{2}+\beta\omega_{a}\omega_{c}\right)\right]\end{gathered}$
(43)

with parameters defined similarly to $\ket{\Psi_{s}}$.

Figure 5: Behavior of the teleportation fidelity for the different cases
described in the text. Plot (a) shows the behavior with $\alpha$, the state
entanglement, and $\sigma=\gamma_{c}/\gamma_{s}$, for the ideal SFG
measurement, with $\beta=1$ and $\gamma_{m}\rightarrow\infty$, as considered
in Ref Humble (2010). The same plot describes the fidelity as a function of
$\beta$ and $\sigma=\gamma_{c}/\gamma_{m}$ for the case of a maximally
entangled state with $\alpha=1$ and $\gamma_{s}\rightarrow\infty$. Plots (b)
and (c) illustrate the behavior of the fidelity when the entangled state and
the entangled measurement have comparable bandwidths (here
$\gamma_{s}=\gamma_{m}=1$). Here the fidelity behaves differently with
$\alpha$ and with $\beta$, because $f_{s}$ and $f_{m}$ are not in general
interchangeable in the expression for $\psi_{b|m}$. All quantities are
dimensionless.

We point out here that we have centered both $f_{s}$ and $f_{m}$ at 0 in
frequency space, without loss of generality. This is because, in the protocol
described in reference Humble (2010), Alice communicates her obtained
frequency $\omega_{a}+\omega_{c}$ to Bob, whereupon he performs the
appropriate frequency translation to his photon $b$ to recover the state that
would have resulted, had Alice obtained $\omega_{a}+\omega_{b}$ in her
measurement. Further note that we are using the parameters $\alpha$ and
$\beta$ to quantify the entanglement of the shared state and the joint
measurement, respectively, rather than a more familiar measure of entanglement
for pure states, such as the Schmidt number Parker  _et al._ (2000). We have
made this choice because, although the Schmidt number $K$ bears a simple
relationship with our parameter $\alpha$ (or $\beta$), satisfying
$K=\frac{1}{\sqrt{1-\alpha^{2}}}$ (see Appendix B), the latter has the
convenient feature of being bounded by the interval $[-1,1]$, whereas the
Schmidt number diverges for maximal entanglement.

With all of this in consideration, Alice’s joint measurement on photons a and
c heralds Bob’s photon b in the teleported state

$\begin{gathered}\ket{\psi_{b|m}}=\int\text{d}\omega_{b}\psi_{b|m}(\omega_{b})\hat{a}^{\dagger}_{b}(\omega_{b})\ket{\text{vac}},\\\
\psi_{b|m}(\omega_{b})=N_{b|m}\int\text{d}\omega_{a}\text{d}\omega_{c}f^{*}_{m}(\omega_{a},\omega_{c})f_{s}(\omega_{a},\omega_{b})\psi_{c}(\omega_{c}).\end{gathered}$
(44)

where $N_{b|m}$ is the appropriate normalization constant. The teleportation
fidelity is then given by the modulus squared of the overlap,

$F=\left||^{2}=\left|\int\text{d}\omega\psi_{c}^{*}(\omega)\psi_{b|m}(\omega)\right|^{2}$
(45)

For this analysis, we let $\psi_{c}$ be a Gaussian function with
characteristic width $\gamma_{c}$,

$\psi_{c}(\omega)=\frac{1}{\sqrt{\gamma_{c}\sqrt{\pi}}}e^{-\omega^{2}/2\gamma_{c}^{2}}.$
(46)

Using this form for the states and measurements, we obtain an algebraic
expression for the fidelity which depends on five parameters,
$F=F(\alpha,\beta,\gamma_{s},\gamma_{m},\gamma_{c})$. The full expression is
unwieldy and not very instructive to display here. We shall verify that our
formalism reproduces the result of reference Humble (2010) in the appropriate
limits. That reference studies the behavior of the fidelity as a function of
$\alpha$ and $\sigma=\gamma_{c}/\gamma_{s}$ for a uniformly phasematched SFG
process followed by an ideally-resolved frequency detection. This corresponds
to taking the limit $\gamma_{m}\rightarrow\infty$ and $\beta=1$. In these
limits, our formalism exactly recovers the fidelity

$\begin{gathered}F_{\gamma_{m}\rightarrow\infty}=\sqrt{\frac{4\sigma^{2}(\sigma^{2}+1)(\sigma^{2}+1-\alpha^{2})}{((\sigma^{2}+1)^{2}-\alpha^{2})^{2}}},\end{gathered}$
(47)

which is displayed in Fig. 5 (a). In that reference, an interesting feature of
this behavior of the fidelity was noted. That is, although the fidelity
increases monotonically with the source entanglement $\alpha$ for $\sigma\ll
1$, this is no longer true for when $\gamma_{c}$ is comparable to
$\gamma_{s}$. In particular, the fidelity is equal to one along the curve
$\alpha^{2}=1-\sigma^{4}$, and is equal to $\sqrt{8/9}$ at the upper-right
hand corner of the plot, where $\alpha=1$ and $\sigma=1$. In the language of
our formalism, given the ideal entangled measurement, with infinite SFG
bandwidth and ideal spectral resolution, there is a trade-off between spectral
bandwidth and spectral entanglement of the sources.

Our result allows us to generalize further, however, and also consider the
case of the Gaussian SFG measurement with finite bandwidth. First we consider
the reverse scenario to the one above, where the source is perfectly
entangled, with $\gamma_{s}\rightarrow\infty$ and $\alpha=1$, and look at the
dependence of the fidelity on $\beta$ and $\sigma$. In this case we find that
the fidelity exhibits the same dependence, that is,

$F_{\gamma_{s}\rightarrow\infty}=\sqrt{\frac{4\sigma^{2}(\sigma^{2}+1)(\sigma^{2}+1-\beta^{2})}{((\sigma^{2}+1)^{2}-\beta^{2})^{2}}},$
(48)

and we can conclude that, given an ideal entangled state between Alice and
Bob, there is a trade off between spectral bandwidth and spectral entanglement
of the measurement.

Finally, we arrive at the most realistic case, where both the entangled source
and the measurement have finite bandwidths, corresponding to finite
phasematching in the PDC and SHG processes. Here we set them equal, taking
$\gamma_{s}=\gamma_{m}=1$, and obtain

$\begin{gathered}F_{\gamma_{m}=\gamma_{s}}=\sqrt{\frac{4\sigma^{2}(\beta^{2}-2(1+\sigma^{2}))(\beta^{2}-(2-\alpha^{2})(1+\sigma^{2}))}{(1+\sigma^{2})^{2}(\alpha^{2}+\beta^{2}-2(1+\sigma^{2}))^{2}}}.\end{gathered}$
(49)

In this case we find the interesting and counterintuitive result that the
behaviors of the fidelity with the source entanglement $\alpha$ and with the
measurement entanglement $\beta$ are no longer equivalent. We show this by
plotting the behavior of the limiting cases of $F_{\gamma}(\alpha,1,\sigma)$
(spectral resolution of the SFG) and $F_{\gamma}(1,\beta,\sigma)$
(monochromatic pumping of the PDC) in Fig. 5 (b) and (c), respectively. In the
case of $\beta=1$, the fidelity is maximized along the curve
$\alpha^{2}=\frac{1+\sigma^{2}-2\sigma^{4}}{1+\sigma^{2}}$ and has similar
limiting behaviors to the ideal case considered in reference Humble (2010).
The case of $\alpha=1$ exhibits a starker contrast, taking its maximum value
along the curve $\beta^{2}=\frac{-1+\sigma^{2}+2\sigma^{4}}{-1+\sigma^{2}}$.
Unlike any of the previous cases, the fidelity is no longer equal to unity in
the bottom right-hand corner, for $\sigma=1$, $\beta=0$, but instead it is
equal to $\sqrt{8/9}$.

We emphasize that $\beta<1$ does not represent a non-ideal spectral resolution
of the upconverted photon, since we are only considering projective
measurements, but instead corresponds to a coherent broadband measurement, as
could be obtained using a quantum pulse gate. What this last result suggests
is that, for finite bandwidths of the entangled source and the entangled
measurement, it is not generally the case that spectral resolution maximizes
the teleportation fidelity. Further, the asymmetry between the behaviors of
entangled state and the entangled measurement can be understood from the fact
that the state JSA $f_{s}$ and the measurement JSA $f_{m}$ are not
interchangeable in the expression for $\psi_{b|m}$, with $f_{m}$ having both
of its arguments integrated over. Most notably, we have shown that, by
treating two-photon measurements more generally and on equal footing with the
two-photon states, it is possible not only to recover previously-obtained
results in the limit of ideal measurements, but also to uncover which states
and measurements are optimal for a given task (in this case spectral
teleportation), under more realistic constraints (in this case, finite PDC and
SFG bandwidths).

This brief analysis leaves open the question of how to generalize to a more
realistic, non-ideally resolved SFG measurement. For a mixed bipartite state
$\hat{\rho}$, a convenient measure of entanglement is the negativity Vidal and
Werner (2002). The negativity essentially counts the negative eigenvalues of
$\hat{\rho}$ partially transposed with respect to one of its subsystems, and
it sets an upper bound on the teleportation capacity of the state. This
suggests that we may define a negativity associated with a non-projective POVM
element $\hat{\Pi}_{q}$ as the negativity of its mixed retrodicted state
$\hat{\rho}_{q}$. The role of finite spectral resolution in SFG detection has
been investigated numerically for entanglement swapping in reference Vitullo
_et al._ (2018). However, it could be more elegant to frame this relationship
in terms of the negativities both of the input states and the measurements in
scenarios such as quantum teleportation and entanglement swapping, and this
remains to be explored in future work.

##  IV Conclusion

We have demonstrated how to construct the POVM associated with two-photon
detection by SFG followed by temporal-mode-selective single-photon detection.
We have shown that this POVM is proportional to the two-photon state created
in the time-reverse PDC process pumped with a field in the detected mode. This
allowed us to characterize several aspects of the POVM relevant to its
adequacy for quantum information protocols. In particular, we have shown that
a projective measurement of the SFG photon corresponds to a projective two-
photon POVM element. We have pointed out the special case where orthogonal SFG
single-photon measurements correspond to orthogonal two-photon measurements.
And finally, we have shown the correspondence between the two-photon
entanglement retrodicted by the SFG measurement and the two-photon
entanglement produced by the time-reversed PDC process. These results could
have implications for quantum information experiments relying on PDC and SFG
in terms of exploring the interplay between entangled states and entangled
measurements. Additionally, it remains an open question how best to certify
the entanglement of the SFG measurement Bennet  _et al._ (2014), or even to
perform quantum tomography of the process. Finally, given recent interest in
using quantum light for two-photon absorption Schlawin and Buchleitner (2017)
Landes  _et al._ (2020), our results open the question of whether it’s
possible to have a combined framework of two-photon processes in terms of
quantum measurement theory.

###### Acknowledgements.

We would like to acknowledge M. G. Raymer and S. J. van Enk for valuable
discussions. This work is funded by NSF grant No. 1839216.

## Appendix

##  Appendix A Deriving the three-wave mixing transformation

Strictly speaking, the Hamiltonian describing the nonlinear interactions we
consider is a time-dependent quantity, $\hat{H}(t)$, whereby a state
$\ket{\Psi_{\text{out}}}$ evolves from an initial state
$\ket{\Psi_{\text{in}}}$ according to

$\begin{gathered}\ket{\Psi_{\text{out}}}=\exp{\Big{[}-\frac{i}{\hbar}\int_{0}^{t}\text{d}t^{\prime}\hat{H}(t^{\prime})\Big{]}}\ket{\Psi_{\text{in}}}\\\
\approx\left(1-\frac{i}{\hbar}\int_{0}^{t}\text{d}t^{\prime}\hat{H}(t^{\prime})\right)\ket{\Psi_{\text{in}}}\end{gathered}$
(50)

The relevant Hamiltonian for three-wave mixing has the form

$\hat{H}(t)=\chi\int_{V}\text{d}V\hat{E}^{+}_{p}(\mathbf{r},t)\hat{E}^{-}_{s}(\mathbf{r},t)\hat{E}^{-}_{i}(\mathbf{r},t)+\text{H.c.}$
(51)

where $\hat{E}^{+(-)}_{j}$ denotes the positive (negative) frequency component
of the $j$ field operator, with $j=p,s,i$. $V$ denotes the interaction volume,
which we take to be infinite in the transverse direction (by assuming the
field modes are well-confined within the crystal area), and of length $L$ in
the longitudinal direction. Finally, $\mathbf{r}$ and $t$ denote the space and
time coordinates, and $\tilde{\chi}$ describes the interaction strength. We
expand the field operators into their plane-wave components,

$\begin{gathered}\hat{E}^{+}_{j}(\mathbf{r},t)=\int\text{d}\omega_{j}A_{j}(\omega_{j})\exp{\Big{[}i(\mathbf{k}_{j}(\omega_{j})\cdot\mathbf{r}-\omega_{j}t)\Big{]}}\hat{a}_{j}(\omega_{j}),\\\
\hat{E}^{-}_{j}=(\hat{E}^{+}_{j})^{\dagger},\end{gathered}$ (52)

where $A_{j}(\omega_{j})$ is a slowly-varying function of $\omega$.
Substituting these into the Hamiltonian and absorbing all the slowly-varying
functions into $\chi$, we obtain

$\displaystyle\hat{H}(t)=$
$\displaystyle\chi\int_{V}\text{d}V\int\text{d}\omega_{p}\text{d}\omega_{s}\text{d}\omega_{i}\hat{a}_{p}(\omega_{p})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})$
(53)
$\displaystyle\times\exp\Big{[}i(\mathbf{k}_{p}(\omega_{p})-\mathbf{k}_{s}(\omega_{s})-\mathbf{k}_{i}(\omega_{i}))\cdot\mathbf{r}\Big{]}$
$\displaystyle\times\exp\Big{[}-i(\omega_{p}-\omega_{s}-\omega_{i})t\Big{]}+\text{H.c.}.$

Now we use this form of the Hamiltonian to compute output state (50) to first
order in the expansion, whereupon we carry the integration over the transverse
spatial directions to infinity. Additionally, we carry out the time integral
from negative to positive infinity because the input and output states are
observed long before and after the interaction time $t$, resulting in a delta-
function in $(\omega_{p}-\omega_{s}-\omega_{i})$ (energy conservation). All of
this obtains

$\begin{gathered}\ket{\Psi_{\text{out}}}\approx\Bigg{[}1-i\chi\int_{0}^{L}\text{d}z\int\text{d}\omega_{s}\text{d}\omega_{i}\exp\Big{[}i(\Delta\mathbf{k})_{z}z\Big{]}\\\
\times\hat{a}_{p}(\omega_{s}+\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})+\text{H.c.}\Bigg{]}\ket{\Psi_{\text{in}}},\end{gathered}$
(54)

where we have also absorbed the $\hbar$ into $\chi$. Carrying out the
integration over $z$ provides the phase-matching function
$\Phi(\omega_{s},\omega_{i})$, and we define the transformation

$\hat{H}=\chi\int\text{d}\omega_{s}\text{d}\omega_{i}\Phi(\omega_{s},\omega_{i})\hat{a}_{p}(\omega_{s}+\omega_{i})\hat{a}^{\dagger}_{s}(\omega_{s})\hat{a}^{\dagger}_{i}(\omega_{i})+\text{H.c},$
(55)

such that

$\ket{\Psi_{\text{out}}}\approx\left(1-i\hat{H}\right)\ket{\Psi_{\text{in}}}.$
(56)

##  Appendix B Relating the entanglement parameter $\alpha$ to the Schmidt
number $K$

In section III.3 we used the scenario of spectral teleportation to illustrate
the role of entanglement in the measurement, on par with entanglement in the
state, in a quantum protocol. To that end, we quantified the teleportation
fidelity in terms of the correlation parameters $\alpha$ ($\beta$) of the
bivariate Gaussian state $f_{s}(\omega,\omega^{\prime})$ (measurement
$f_{m}(\omega,\omega^{\prime})$). This parameter has the advantage of being
bounded by the interval $[-1,1]$, with maximal entanglement at the boundaries,
whereas more common measures of entanglement for pure states, such as the
entropy and the Schmidt number, diverge for maximal entanglement. Here we show
for completeness how the Schmidt number $K$ depends functionally on $\alpha$,
while the same analysis holds for $\beta$.

The Gaussian JSA $f_{s}(\omega,\omega^{\prime})$ from (41) has a Schmidt
decomposition of the form

$f_{s}(\omega,\omega^{\prime})=\sum^{\infty}_{j=0}\ \sqrt{\lambda_{j}}\
u_{j}(\omega)v_{j}(\omega^{\prime}),$ (57)

where $\\{u_{j}(\omega)\\}$ is the orthonormal set of Hermite-Gauss functions
spanning the spectral Hilbert space over $\omega$, and the same is true of
$\\{v_{j}(\omega^{\prime})\\}$ Humble (2010). The Schmidt coefficients
$\lambda_{j}$ are given by

$\lambda_{j}=\text{sech}^{2}\ \zeta\ \text{tanh}^{2j}\ \zeta,$ (58)

satisfying $\sum_{j=0}^{\infty}\lambda_{j}=1$, and where $\zeta$ is given by

$\alpha=\tanh\ 2\zeta.$ (59)

The Schmidt number $K$ is then given by

$K=\frac{1}{\sum_{j=0}^{\infty}\lambda_{j}^{2}}=\text{cosh}\ 2\zeta.$ (60)

Combining Eq. (59) and (60), we arrive at the simple relationship

$K=\frac{1}{\sqrt{1-\alpha^{2}}},$ (61)

where, as expected, $K$ is equal to unity for the case of no correlation,
$\alpha=0$, and diverges for maximal correlation, $\alpha=\pm 1$.

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