# A multiplexed synthesizer for non-Gaussian photonic quantum state generation

M. F. Melalkia  Université Côte d’Azur, CNRS, Institut de Physique de Nice
(INPHYNI), UMR 7010, Parc Valrose, Nice Cedex 2, France  J. Huynh  Université
Côte d’Azur, CNRS, Institut de Physique de Nice (INPHYNI), UMR 7010, Parc
Valrose, Nice Cedex 2, France  S. Tanzilli  Université Côte d’Azur, CNRS,
Institut de Physique de Nice (INPHYNI), UMR 7010, Parc Valrose, Nice Cedex 2,
France  V. D’Auria  Université Côte d’Azur, CNRS, Institut de Physique de Nice
(INPHYNI), UMR 7010, Parc Valrose, Nice Cedex 2, France  Institut
Universitaire de France (IUF), France  J. Etesse  Université Côte d’Azur,
CNRS, Institut de Physique de Nice (INPHYNI), UMR 7010, Parc Valrose, Nice
Cedex 2, France

###### Abstract

Disposing of simple and efficient sources for photonic states with non-
classical photon statistics is of paramount importance for implementing
quantum computation and communication protocols. In this work, we propose an
innovative approach that drastically simplifies the preparation of non-
Gaussian states as compared to previous proposals, by taking advantage from
the multiplexing capabilities offered by modern quantum photonics tools. Our
proposal is inspired by iterative protocols, where multiple resources are
combined one after the other for obtaining high-amplitude complex output
states. Here, conversely, a large part of the protocol is performed in
parallel, by using a single projective measurement along a mode which
partially overlaps with all the input modes. We show that our protocol can be
used to generate high-quality and high-amplitude Schrödinger cat states as
well as more complex states such as error-correcting codes. Remarkably, our
proposal can be implemented with experimentally available resources,
highlighting its straightforward feasibility.

## Introduction

Efficient coding of quantum information relies on the capability to generate
high-quality states of light with non-classical photon statistics, both in the
continuous- [1] and discrete-variable [2] regimes. In particular, states with
negative Wigner functions have shown to be crucial for implementing
sophisticated protocols and information processing, such as error-correction
[3] and one-way quantum computing [4]. Different strategies for generating
such states were reported, relying for instance on high-order nonlinearity [5,
6, 7], qubit interaction [8] and linear optical interaction [9, 10, 11, 12]. A
class of protocols of particular interest is that of iterative schemes [13,
14, 10, 9, 11, 12], where photonic ‘resource’ states are combined together in
an iterative fashion, allowing for growing step-by-step the size of the
states. However, the experimentally implemented protocols are to date
restricted to few operations, due to protocol scalability issues arising from
the simultaneous need for multiple heralding signals and from the increase of
experimental overhead [15, 16]. This ultimately limits the average photon
number in the generated state.
In this article, we propose a novel scheme to overcome the current
experimental limitations, by relying on a single shot operation with a
spectrally multiplexed resource. Our proposal takes advantage of the multimode
aspect of state-of-the-art experimentally investigated quantum photonic
sources in order to greatly enhance the capabilities of iterative generation
protocols. This improvement occurs at two levels: (i) the number of operations
to implement is reduced to one, and, as a consequence, (ii) the success
probability of the overall operation is strongly increased. To illustrate our
proposal, we show that Schrödinger cat state (SCS) and GKP-codes [3] can be
generated in an efficient manner, proving the validity of the approach, and
opening a new path in the domain of non-Gaussian photonic state preparation.
In Sec. I we recall the principle of iterative protocols and propose a fully
parallelized counterpart relying on spectral multimode quantum optics. Section
II is devoted to the investigation of parallelized generation of cat states
and GKP codes. Finally, in Sec. III we show that our protocol can be used with
experimental readily accessible ressources such as single photon Fock states.

##  I Multimode iterative protocol

###  I.1 Framework: iterative protocols

The protocol proposed in this paper takes its essence from iterative
generation protocols, an elementary brick of which is represented in Fig. 1.

Figure 1: Elementary brick of an iterative protocol.

Such operation consists in mixing two input ‘resource’ states,
$\left|\psi_{0}\right\rangle$ and $\left|\psi_{1}\right\rangle$, on a
beamsplitter with transmittivity $\tau_{0}$, to herald the generation of an
ouput state $\left|\psi_{\rm out}\right\rangle$. The heralding process can be
performed in different ways [10, 12], and even if the results derived here can
easily be extended to the photon counting case, we focus in the following on
homodyne conditioning. In this scenario, one of the two output arms of the
beamsplitter is measured by a homodyne detector (along a certain quadrature
$x_{\theta}$). When the measurement outcome is close enough to an assigned
value ($x_{\theta}=x_{0}$ on Fig.1), the target output state is successfully
heralded. The strength of such a protocol is that the average photon number of
the output state can be higher than the one of the individual input states.
Accordingly, it can be used to ‘breed’ complex photonic states by cascading
the protocol using $\left|\psi_{\rm out}\right\rangle$ as one of the input of
a subsequent operation.
Figure 2 shows such a configuration, where states of increasing size are
combined with resource states; tree-like configurations are also commonly
investigated, thanks to the parallelization capabilities they offer [14].

Figure 2: A cascaded architecture for an iterative protocol.

This idea is at the core of the ‘cat breeding’ scheme, proposed in [9], where
balanced beamsplitter ($\tau=1/\sqrt{2}$) and $x_{i}=0$ homodyne conditioning
are used [17, 18]. In this case, mixing two SCSs allows to create a single SCS
with an amplitude increased by a factor $\sqrt{2}$. Iterating this protocol
$N$ times then allows to create a SCS with an amplitude amplified by a factor
$2^{N/2}$. An alternative manner has also been proposed, where the input cat
states were replaced by single photons, allowing the production of squeezed
SCSs, robuster to losses [10, 19]. By modifying the quadrature along which the
homodyne conditioning is performed ($p=0$ homodyne conditionings), GKP codes
can be generated all-optically [20, 14]. More generally, all superpositions of
up to $N$ photons can be generated by properly adjusting the beamsplitter
reflectivities and homodyne conditionings [10].
Proof-of principle experimental validations are currently limited to single
stage implementations due to scalability issues [15, 16]: in bulk optics,
increasing the number of stages by increasing the number of input state
sources indeed leads to intractable complexities. Temporal multiplexing
represents a promising alternative that is also investigated [21], but could
ultimately lead to a decrease in the overall generation rate of the states.
Integrated optics, on the other hand, offers an attractive solution to address
spatial scalability issues, but inevitable losses associated with such
platforms should be carefully taken into consideration [22].
We propose in this article to rely instead on spectral multiplexing, which
offers an increased generation efficiency as well as noticeable experimental
simplifications.

###  I.2 Multiplexed single-stage generation protocol

The main idea of the proposal, sketched in Fig.3, consists in the following
key ingredients. To begin with, a ‘seed’ state $\left|\psi_{0}\right\rangle$
initially populates a spectral mode $\chi_{0}(\omega)$ in which the quantum
engineering process is to be performed. In output of the protocol, we will
focus on the state $\rho_{\rm out}$ that populates this same mode
$\chi_{0}(\omega)$. Subsequently, we select a collection of states
$|\psi^{\prime}_{0}\rangle$, $|\psi^{\prime}_{1}\rangle$, …
$|\psi^{\prime}_{N-2}\rangle$ respectively in modes $\varphi_{0}(\omega)$,
$\varphi_{1}(\omega)$, … $\varphi_{N-2}(\omega)$ that are used to ‘feed’ the
protocol, so as to grow the overall size of the output state $\rho_{\rm out}$.
As will be discussed with a concrete example, the spectral modes
$\varphi_{i}(\omega)$ have to overlap $\chi_{0}(\omega)$ in a clever way, so
that each $\left|\psi^{\prime}_{i}\right\rangle$ contributes in an efficient
manner to the generation protocol. This collection of states is merged in a
single spatial mode and sent to the first input port (port $a$) of a
beamsplitter of transmittivity $\tau$ so as to interfere with the seed state
$\left|\psi_{0}\right\rangle$ sent to the other input port of this
beamsplitter (port $b$). Importantly, a perfect spatial mode matching of the
two states populating modes $a$ and $b$ must be guaranteed. After interaction,
one output port of the beamsplitter is measured by a homodyne detector along
mode $\chi_{0}(\omega)$, to perform the desired conditioning
$x_{\theta}=x_{0}$, for successfully heralding the generation of a state
$\rho_{\rm out}$ on the other output port.

Figure 3: The multiplexed generation protocol (see text for description).

A major advantage of this method is that it allows mixing $N$ states in a
single heralding step, therefore drastically increasing the success
probability of the protocol and reducing the experimental complexity as
compared with the standard spatial multiplexing strategy, as will be
quantitatively shown later.
The multimode description that we use complies with realistic implementations
of gaussian state sources, such as, for instance, multimode squeezed vacuum
sources [23]. It is now indeed well established that spontaneous parametric
down-conversion (SPDC) through three-wave mixing process produces a collection
of squeezed vacuum states in separate orthogonal spectral modes
$\\{\varphi_{k}(\omega)\\}_{k\in\mathbb{N}}$, each with a different squeezing
parameter [24, 25].
More generally, each spectral mode $\varphi_{k}(\omega)$ can be associated
with a bosonic anihilation operator $\hat{A}_{k}$ :

$\displaystyle\hat{A}_{k}=\int\varphi_{k}^{*}(\omega)\hat{a}(\omega)d\omega,$
(1)

$\\{a(\omega)\\}$ being the bosonic operators of individual spectral
components at the SPDC output. When two different sets of modes
$\\{\varphi_{k}(\omega)\\}_{k\in\mathbb{N}}$ and
$\\{\chi_{l}(\omega)\\}_{l\in\mathbb{N}}$ are involved, one can express their
‘likeness’ by using the overlap formula

$\displaystyle c_{k,l}=\int\varphi_{k}^{*}(\omega)\chi_{l}(\omega)d\omega.$
(2)

This formula allows switching from one mode basis to the other thanks to

$\displaystyle\varphi_{k}(\omega)=\sum_{l}c^{*}_{k,l}\chi_{l}(\omega).$ (3)

We show in the next paragraph that the proposed protocol can be used to
generate highly-fidel SCSs and GKP codes with increased success probabilities
compared to standard iterative protocols.

##  II Paralellized generation of complex photonic states

###  II.1 Parallel cat ‘breeding’

(a) $N=2$ (b) $N=3$

(c) $N=6$ (d) $N=8$

Figure 4: Fidelity of the output state with a cat state of amplitude $\gamma$
as a function of the beamsplitter reflectivity $r$, for (a) $N=2$, (b) $N=3$,
(c) $N=6$ and (d) $N=8$ input cat states of amplitude $\alpha=3$. In each
plot, a star spots the highest fidelity point, and the corresponding
parameters are given in the insets. The Wigner functions corresponding to this
maximal fidelity point are represented in (e), (f), (g), (h) respectively.

The previously described scheme can be applied to the generation of large
amplitude SCSs, inspired by cat ‘breeding’ iterative protocols. To perform
such a generation, we simply consider here $N\geqslant 2$ identical SCS input
states of amplitude $\alpha$ in the $N-1$ input modes $\varphi_{k}(\omega)$:

$\displaystyle\forall k,\in[\\![0,N-2]\\!],\
\left|\psi_{k}^{\prime}\right\rangle=\left|\psi_{0}\right\rangle=\frac{1}{\mathcal{N}}\left(\left|\alpha\right\rangle+\left|-\alpha\right\rangle\right),$
(4)

where $\mathcal{N}=\sqrt{2\left(1+e^{-2|\alpha|^{2}}\right)}$ is a
normalization factor. Note that we restrict the discussion to even cat states,
but similar reasoning is also valid with odd cat states, as treated in the
Suppl. Mat.. Then we set that each mode $\varphi_{k}(\omega)$ is of equal
contribution in the generation process, namely $\forall k\in[\\![0,N-2]\\!]$:

$\displaystyle
c_{k,0}=\int\varphi_{k}^{*}(\omega)\chi_{0}(\omega)d\omega=\frac{1}{\sqrt{N-1}}.$
(5)

In this way, each input ‘feeding’ cat $\left|\psi_{k}^{\prime}\right\rangle$
will ‘breed’ the input seed cat $\left|\psi_{0}\right\rangle$ by the same
amount. Subsequently, homodyne conditioning is performed along the $x$
quadrature (in-phase with the SCS) around $x=0$, within a given acceptance
window $[-\Delta x/2,\Delta x/2]$. An analytic formula for the output Wigner
function (see Suppl. Mat.) can then be derived, allowing us to predict the
performances of the protocol as a function of key parameters such as $r$, $N$,
$\alpha$ and the heralding width $\Delta x$. Many different metrics can be
used to quantify the distance between an output state and a target state.
Here, the fidelity of $\rho_{\rm out}$ with the nearest ideal SCS is used
(amplitude $\gamma$).

#### Beamsplitter reflectivity

The aforementioned fidelity of the output state is plotted in Figs. 4(a) to
(d) for $N=$2, 3, 6 and 8 input SCSs of amplitude $\alpha=3$, as a function of
the beamsplitter reflectivity $\tau$ and of the output state amplitude
$\gamma$. These plots clearly reveal that the output amplitude of the cat
state in mode $\chi_{0}(\omega)$ that corresponds to optimal fidelities with
$\rho_{\rm out}$ grows with the number of input states. This successfully
proves the breeding of the input SCSs. Figure 4 also shows that the
reflectivity should be optimized in a non-trivial way, according to the number
of input breeding states, in order to maximize the output state fidelity with
high-amplitude SCSs.

To gain a qualitative intuition on the quality of the multimode breeding
operation, the Wigner functions of the corresponding output states are plotted
in Figs.4 (e) to (h), showing their strong non-classical signature, as the
Wigner functions contain oscillations with negative values. To be more
quantitative on this result, the numerical values of fidelity, optimal
reflectivity and output SCS amplitudes are also written on Figs. 4 (a) to (d),
revealing the very high fidelities of the output states with high-amplitude
SCSs. In addition, it confirms that the amplitude of the output SCS increases
with $N$, with a scaling of the order of $\gamma~{}\sim~{}\sqrt{N}\alpha$.

#### Amplitude of the input states

The protocol is also sensitive to the amplitude of the input cat states. This
is illustrated in Fig. 5, where the output state fidelity with the nearest cat
state is represented as a function of the input SCS amplitude $\alpha$, for
different numbers of input states $N$. Two important trends shall be noticed:
(i) for a given $\alpha$, increasing $N$ decreases the overall output state
fidelity. However, (ii) increasing $\alpha$ for a given $N$ leads to an
increased global fidelity. For instance, to create states with 99% fidelity
with $N=2$, an input amplitude $\alpha=1.13$ is sufficient, whereas
$\alpha=2.49$ is required to reach the same fidelity with $N=4$ input states.
In the case of even cat states that are considered here, there is also an
increase of the fidelity for vanishing $\alpha$ simply because vacuum is a
perfect even cat state with zero amplitude, a case of no interest for us.

Figure 5: Fidelity of the output state with the nearest SCS as a function of
$\alpha$, for different $N$. The cases represented in fig.4 correspond to the
vertical blue line ($\alpha=3$).

#### Comparison with the iterative cat breeding protocol

Interestingly, we can draw a direct parallel with the iterative cat breeding
protocol, where $N-1$ heralding steps are performed on $N$ input resources.
The main difference, however, is that the multimode protocol that we propose
here is inevitably accompanied with information loss due to partial overlap of
the spectral modes $\varphi_{k}(\omega)$ with the output mode of interest
$\chi_{0}(\omega)$. Equation (5) directly shows that each mode
$\varphi_{i}(\omega)$ overlaps only partially with $\chi_{0}(\omega)$,
therefore all of the components of the input states
$\left|\psi^{\prime}_{i}\right\rangle$ along $\chi_{l>0}(\omega)$ are purely
lost during the process, where $\\{\chi_{l}(\omega)\\}_{l\in\mathbb{N}}$
constitutes a complete spectral basis. Therefore, the output state $\rho_{\rm
out}$ cannot be a pure state as long as more than two modes are populated in
total.
To give an intuitive model, we propose a simple parallel between spatial and
spectral modes: the operation depticted in Fig. 3 by the merging arrows from
modes $\varphi_{k}(\omega)$ to $\chi_{0}(\omega)$ can actually be simply
modeled as a $N-1$ input / $N-1$ output beamsplitter transformation, where
$N-2$ outputs are lost in the environment. This transformation is exactly the
one depicted in Fig. 2 for input states $\left|\psi_{0}\right\rangle$ to
$\left|\psi_{N-2}\right\rangle$, with $\tau_{i}=1/\sqrt{i+2}$ so as to obtain
$c_{0,k}=1/\sqrt{N-1}$. In other words, our protocol is equivalent to the
iterative breeding scheme of Fig. 2, where only the last heralding step
$x=x_{N-1}$ is performed and all the other ones are simply ignored. The direct
consequence for this is that the overall success probability of the protocol
is increased by several orders of magnitude as compared to the iterative
scheme. This is illustrated in Fig. 6, where the success probability of our
scheme for input SCS amplitudes of $\alpha=3$ and $N=4$ is plotted, in
comparison with the iterative scheme of Fig.2 in the case where all the
heralding widths are identical. The cost obviously is the overall fidelity,
which is degraded in the multimode case, but nevertheless remains very high,
as we have seen in Fig. 4. The multimode protocol therefore takes advantage of
the robustness of the breeding protocol for imperfect heralding at
sufficiently high SCS amplitudes.

Figure 6: Comparison between the success probability of the multimode scheme
and the iterative scheme for $N=4$, as a function of the heralding window
$\Delta x$ normalized to the shot noise standard deviation $\sigma_{0}$.

###  II.2 GKP states preparation

In addition to the generation of SCSs, our protocol can also advantageously be
applied to for the generation of more complex states such as the GKP codes
[3]. These states were first introduced as a means to encode information in an
oscillator, with an error-correcting capability. They were recently generated
and used in trapped ions [26] and superconducting circuits [27], but still
remain out of reach of all-optical protocols. Here, we can use the same
protocol as the one depicted in Sec. II.1 ($N-1$ input cat states in modes
$\varphi_{k}$ and one cat state in mode $\chi_{0}$, all of amplitude
$\alpha$), with the only difference that homodyne conditioning is performed
along $x_{\pi/2}:=p=0$. Such conditioning has already been investigated in the
context of all-optical generation of GKP with iterative protocol [20, 14]. We
show here that it can be transposed to our multiplexed approach. As a
reminder, the wavefunctions of approximate GKP codes
$\left|\tilde{0}\right\rangle$ and $\left|\tilde{1}\right\rangle$ take the
form

$\displaystyle\langle x|\tilde{0}\rangle$
$\displaystyle=\dfrac{1}{\mathcal{N}_{0}}\sum_{k\in\mathbb{Z}}e^{-\frac{s_{1}^{2}}{2}(ka)^{2}}e^{-\frac{1}{2s_{2}^{2}}(x-ka)^{2}}$
(6a) $\displaystyle\langle x|\tilde{1}\rangle$
$\displaystyle=\dfrac{1}{\mathcal{N}_{0}}\sum_{k\in\mathbb{Z}}e^{-\frac{s_{1}^{2}}{2}(k+1/2)^{2}a^{2}}e^{-\frac{1}{2s_{2}^{2}}\left(x-(k+1/2)a\right)^{2}},$
(6b)

where $1/s_{1}$ gives the overall envelope size of the state, $s_{2}$ is the
standard deviation of each tooth in the distribution, and $a$ is the distance
between consecutive teeth along the $x$ quadrature.

Figure 7: Performance of the multiplexed protocol for the generation of
approximate GKP codes. (a) Wigner function of the output state with $N=3$
input SCSs of amplitude $\alpha=3$. (b) GKP state amplitude when the amount of
input cat states $N$ is augmented.

Figure 7(a) shows the shape of the Wigner function of the output state for
$N=3$, revealing the grid structure specific to GKP codes. Such likeness is,
here again, quantified by the fidelity with the nearest GKP code, that we plot
in Fig. 7(b). This figure shows that augmenting the amount of input resources
indeed increases the size of the output state along the opposite quadrature
($1/a$), while requiring an optimization of $r$. Similarly to the SCS case, it
also shows that the fidelity decreases with $N$.

##  III Proposal with experimentally accessible resources

Figure 8: Wigner functions of the output state of the protocol fed by single
photon Fock states, with (a) $N=2$, (b) $N=3$, (c) $N=4$ and (d) $N=5$.

An important aspect of our strategy is that it can directly be implemented
with already accessible experimental resources. Even if high-amplitude SCSs
are not trivial to generate experimentally, good approximation of them at low
amplitudes (also called Schrödinger ‘kitten’ states) are commonly produced
states. Moreover, the breeding principle has been experimentally demonstrated
with kittens, showing the validity of the approach [16]. In addition, single
photon Fock states also constitute a valid alternative to them, as
theoretically and experimentally demonstrated [14, 15], to generate squeezed
SCS-like output states with high fidelity. It is worth noting that squeezing
the output states constitutes a high advantage, as it can increase their
robustness to losses [28, 19]. Accordingly, the results presented here will
focus on input single photon Fock states. Very similar results are expected
with kitten states.
We recall that our protocol requires the use of a multiplexed input state
source. This can simply be achieved by taking advantage of the spectral
tailoring possibility that processes like spontaneous parametric down-
conversion offer [24]. Indeed, depending on the pumping and phase-matching
conditions, two-mode squeezing can be generated along a large number of
spectral modes, offering the ideal platform for multiplexed generation [25].
Then, single photon Fock states are simply heralded by performing a projective
measurement over one of the two mode squeezing subsystems. This method,
although very commonly used for the generation of high-quality single photon
Fock states for a couple of decades now, has recently been extended
experimentally to high-dimension mode-selective quantum state tailoring [29].
This allows to tweak the supermode in which the single photon detection
occurs, therefore heralding a single photon Fock state in a given supermode.
Here we can take advantage of such capabilities, by using multiplexed photon
sources to feed the protocol and generate complex photonic states in a single
shot.
The output Wigner functions of the protocol fed by an increasing number $N$ of
single photon Fock states are shown in Fig. 8. They clearly reveal that the
output state contains more and more oscillations at its center with increasing
$N$. To be more quantitative, Tab. 1 gathers the parameters of the closest
SCS, where $\alpha$ is the amplitude, $s$ its compression parameter and
$\mathcal{F}$ the fidelity between the two states. This table reveals that the
amplitude of the state increases with $N$ for low values, but degrades after
$N=4$. Further simulations show that this is true for larger $N$ as well. Such
an evolution can qualitatively be explained by the phase invariance of the two
states impinging the beamsplitter: as the only rotation symmetry breaking
operation is the homodyne measurement, the fidelity of the created state with
a highly non-phase invariant state decreases with its size.

|  $N=2$ |  $N=3$ |  $N=4$ |  $N=5$
---|---|---|---|---
$\alpha$ |  1.63 |  2.03 |  2.30 |  2.12
$s$ |  0.66 |  0.65 |  0.66 |  0.74
$\mathcal{F}$ |  0.99 |  0.90 |  0.66 |  0.49
Table 1: Fidelity of the output state with a squeezed SCS of amplitude
$\alpha$ and squeezing $s$.

It should also be pointed that the optimal generation condition (highest
amplitude and fidelity) is fulfilled with the use of a low-reflectivity
beamsplitter (see Suppl. Mat. for numerical simulations).
This analysis reveals that few-mode single photon states are relevant
resources for our protocol in order to produce high-quality SCSs, essential
brick in continuous-variable quantum information processing.

## conclusion

We have proposed and discussed an innovative protocol for the generation of
non-Gaussian states of light. Our scheme is inspired by iterative protocols
but takes advantage of multiplexing capabilities of photonics resources. It
relies on the use of multiple states in a single spatial mode but multiplexed
in an other degree of freedom, mixed with a resource state whose mode
partially overlaps all input states. By performing a homodyne conditioning
after mixing the states at a beamsplitter, complex output states can be
produced. The major strength of our proposal is that no cascade manipulation
steps are required for the state breeding, therefore strongly simplifying the
experimental implementation in a one-shot scheme, and allowing to access high-
success rates. We have shown that frequency multiplexing can be used for the
generation of high-quality and high-amplitude Schrödinger cat and GKP states,
by using as input states small-amplitude Schrödinger cat states, or
alternatively, readily accessible experimental resources such as multiplexed
single photon Fock states. Note that current state-of-the-art non-linear
sources offer the capability for generating such states by providing the
native generation of frequency multiplexed single- and two-mode squeezing. Due
to its high efficiency and simplified implementation as well as to the
possibility of implementing it with already available experimental resources,
we believe that our approach to non-Gaussian state preparation represents a
major breakthrough in the domain of non-classical photonic state generation,
and will open a wide range of applications.

## acknowledgments

This work has been conducted within the framework of the project HyLight (No.
ANR-17-CE30-0006-01) granted by the Agence Nationale de la Recherche (ANR).

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11institutetext: Department of Physics and Astronomy, University of British
Columbia, 6224 Agricultural Road, Vancouver, B.C. V6T 1Z1,
Canada.22institutetext: Institute for Theoretical Physics and Center for
Extreme Matter and Emergent Phenomena, Utrecht University, 3508 TD Utrecht,
The Netherlands.33institutetext: Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2L 2Y5, Canada.

# Baby Universes born from the Void

Panos Betzios1 Nava Gaddam2, and  Olga Papadoulaki3<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

We propose a novel construction of a third quantised baby universe Hilbert
space $\mathcal{H}_{BU}$ for the quantum gravity path integral. In contrast to
the original description of $\alpha$-parameters, both the bulk and boundary
microscopic parameters are fixed in our proposal. Wormholes and baby universes
appear due to refined observables, of the boundary dual quantum field
theories, that crucially involve the space of representations of the boundary
gauge group. Irreducible representations, on which the path integral
factorises, give rise to field theoretic superselection sectors and replace
the $\alpha$ states.

##  1 Introduction

Among the most important open problems in theoretical physics, the role of
topology change in quantum gravity is particularly tantalising. The intrigue
has only grown with recent interest Bousso:2022ntt  in spacetime wormholes
connecting multiple asymptotic boundaries. In the late eighties, it was argued
that the effects of wormholes are accessible in the effective low energy
theory via averaged couplings to local operators Coleman:1988cy ;
Lavrelashvili:1987jg ; Hawking:1988ae ; Giddings:1988cx ; these couplings were
called the $\alpha$-parameters. The unrealised initial hope was that the
integral over the $\alpha$-parameters would be sharply peaked in favour of an
infinitesimal value for the observed positive cosmological constant.
Nevertheless, these ideas led to a novel _third quantised picture_ of “baby
universes (BU)” that are created and destroyed in superspace Giddings:1988wv ,
resulting in dynamical topology changing processes that affect the physics of
a large semi-classical “parent manifold”—the one we observe at large scales.

On the other hand, it has long been expected that there are no free parameters
in a quantum theory of gravity and that all symmetries are gauged
Krauss:1988zc ; Kallosh:1995hi ; Banks:2010zn . This expectation was further
strengthened by the advent of holography Maldacena:1997re ; Witten:1998qj ,
where in all controllable higher dimensional111In this essay, by “higher
dimensions” we mean $d>2$. examples of string/M-theory, all the parameters on
which the boundary field theory depends are known and fixed. Should the
holographic duality hold, the bulk dual quantum gravity (QG) theory must also
depend on the same fixed quantities, leaving no room for $\alpha$-parameters.
Several authors have since emphasised the apparent conflict between holography
and the existence of $\alpha$-parameters Rey:1998yx ; ArkaniHamed:2007js ;
Hebecker:2018ofv , and between holography and fixed multiboundary saddles
Maldacena:2004rf ; Hertog:2018kbz . On the contrary, others have suggested a
harmony in simple topological Marolf:2020xie  or $2d$ models Blommaert:2022ucs
; Johnson:2022wsr . However, known UV complete variants of these models imply
that their non-trivial topological features arise on the string worldsheet
rather than the target space Betzios:2020nry ; McNamara:2020uza . Whereas in
higher dimensions, Euclidean wormholes also arise in bulk duals of
appropriately _interacting_ QFT’s worm ; VanRaamsdonk:2020tlr ;
Betzios:2021fnm , but these cases leave no room for conventional
$\alpha$-parameters.

In this essay, we provide a bridge between these two apparently conflicting
descriptions of the QG path integral in higher dimensions. Before we do so, we
will first describe why the conventional picture of $\alpha$-parameters fails,
necessitating an alternative approach.

##  2 The failure of the bilocal description and $\alpha$-parameters

Coleman’s $\alpha$-states emerge from a bi-local expansion of the contribution
of Euclidean wormholes to the gravitational path integral. This expansion is
presumed to emerge in the limit of soft metric perturbations (whose
wavelengths far exceed the size of the ends of the wormholes (instantons)). In
such a limit, the two instanton ends of a wormhole were argued to couple to
local bulk operators while preserving their connection to each other,
resulting in a bilocal action. This bilocal action can then be traded for
averaged $\alpha$-parameters. This perspective has been reviewed in
Hebecker:2018ofv ; Kundu:2021nwp , for instance. Let us now analyse the
assumptions going into this emergence of $\alpha$-parameters. To this end, let
us define the following length scales222We assume Planck length $L_{P}$ to be
the smallest of all scales in the theory, for validity of effective field
theory.:

  * •

$L_{inst}$ defines the size of the instanton (or the ends of the wormhole).

  * •

$L_{th}$ is the shortest scale associated with the wormhole (it dictates the
size of the wormhole throat).

  * •

$L_{w}$ is the length of the wormhole (which defines the distance between the
two ends of the wormhole).

  * •

And finally, $L_{probe}$ is the scale at which we probe the theory.

In all known solutions, there is a natural hierarchy among these scales
$L_{P}\ll L_{th}<L_{inst}<L_{w}$. One may further argue more generally that
even certain off shell configurations (such as the double cone of Saad:2019lba
) satisfy $L_{th}<L_{inst}$, since the throat is always the minimal
characteristic scale of the wormhole. The existence of a bilocal action, and
consequently the emergence of averaged $\alpha$ parameters, relies on the
following assumptions:

  1. 1.

A local operator expansion that effectively replaces each wormhole-end or
instanton with an infinite sum of local bulk operators $\sum_{i}c_{i}\int
d^{d+1}x\sqrt{g(x)}\mathcal{O}_{i}(x)$. This is achieved in the long
wavelength limit when the theory is probed at scales much larger than the size
of the instanton $L_{probe}\gg L_{inst}$. This implies that the coefficients
$c_{i}$ must be defined in powers of ${L_{inst}}/L_{probe}$.

  2. 2.

Validity of the dilute gas approximation. This means that $L_{inst}\ll L_{w}$,
so that the two endpoint instantons never overlap.

  3. 3.

The throat must retain a connection between the two instantons or wormhole
endpoints. Moreover, this connection must not depend on the location of the
endpoints. Then, the effective action takes the bi-local form

$S_{EFT}~{}=~{}\int d^{d+1}x\sqrt{g(x)}\,\int
d^{d+1}y\sqrt{g(y)}\,\sum_{i,j}\,\Delta_{ij}\,\mathcal{O}_{i}(x)\,\mathcal{O}_{j}(y)\,.$
(1)

The path integral containing this bilocal action can be recast in terms of
auxiliary $\alpha$-parameters.

We now immediately observe that these conditions cannot all be simultaneously
satisfied. Indeed, since the size of the throat is always necessarily smaller
than the size of the instanton, $L_{th}<L_{inst}$, we see that the throat
pinches off before the instantons can be replaced by local operators.
Therefore, in this limit, the wormhole path integral factorises, resulting in
two _disconnected_ local operator expansions at each endpoint. This does not
mean that wormholes cannot exist, it merely implies that they will not be
visible in the infrared where the endpoints are treated as local operator
insertions.

Therefore, we conclude that the bilocal expansion has zero radius of validity
in parameter space, leaving no room for conventional $\alpha$-parameters even
from the point of an IR expansion. Of course, all notions of geometry break
down in the UV.

##  3 Our proposal and motivations

Given the failure of the bilocal expansion described in the previous section,
we now make the following proposal for the Hilbert space of baby universes.

###  3.1 Proposal

In addition to the familiar Hilbert space of asymptotically AdS states in
holography, the QG path integral also describes observables that belong in a
_non trivial_ third quantised baby universe Hilbert space
($\mathcal{H}_{BU}$). The latter gives rise to effective superselection
sectors through the decomposition
$\mathcal{H}_{BU}=\oplus_{R}\,\mathcal{H}_{R}$ in terms of unitary
representations, $R$, of the compact (gauge) group symmetry $\mathcal{G}$.
This refinement is incorporated into the dual field theory partition function
via the insertion of appropriate non-local operators that transform in the
said representations $R$. We thereby arrive at an augmentation of the space of
field theoretic observables which is important for the complete description of
the bulk gravitational physics.

###  3.2 Motivations for our proposal

We will make use of several known examples to both motivate and elucidate
various aspects of our proposal.

#### Liouville theory

The state-operator correspondence is subtle in Liouville quantum gravity owing
to the presence of both microscopic and macroscopic bulk states Seiberg:1990eb
. In analogy333 That the QG path integral should be viewed as a higher
dimensional version of Liouville theory with a dynamical conformal factor was
proposed in tHooft:2014swy ., we define macroscopic operators444In matrix
models, these are the familiar loop operators Saad:2019lba ; Betzios:2020nry ;
McNamara:2020uza . $\hat{Z}[J]$ as functionals of sources, which create large
$AdS$ conformal boundaries. These are associated with partition functions
(Schwinger functionals) of the holographic dual field theory. In contrast, we
introduce microscopic “holes” or “vacuoles” in the geometry555In the
$L_{Pl}\rightarrow 0$ limit, these can be thought of as “defects” or
“punctures” of the geometry. A similar idea was explored in Betzios:2017krj .
that correspond to microscopic states. A basis for describing these is
provided by the representation eigenstates $|R\rangle$ of the gauge group; we
propose that this replaces the $\alpha$-state basis. That is to say that the
bulk state-operator correspondence in any UV complete theory of quantum
gravity bears a resemblance to Liouville theory and not to non-gravitating
CFT’s or topological theories analysed in Marolf:2020xie .

#### Holography

The cogent success of gauge/gravity duality leaves little room for averaged
$\alpha$-parameters that are left unfixed. This motivates our proposal to
abandon the $\alpha$ parameters in favour of microscopic states labelled by
the representations $|R\rangle$, which do _not_ couple to local operators in
the bulk effective action. In contrast to the proposal of Marolf:2020xie ;
McNamara:2020uza , where the Hartle-Hawking state was expanded in a basis of
$\alpha$-eigenstates, our proposal is to expand the dynamical QG vacuum in a
basis of $|R\rangle$ with the Hartle-Hawking state being the trivial or
singlet representation. Therefore, there is a natural selection rule for our
proposed macroscopic $\hat{Z}[J]$ operators that separates different
representation eigenstates. This implies a superselection rule for all
observables in the Hilbert space $\mathcal{H}_{R}$ associated with each
macroscopic boundary arising from the expectation value of $\hat{Z}[J]$ in the
state $|R\rangle$. We describe this further in Section 4.

#### Non-singlet sectors

Our proposal is a significant departure from the original description of
$\alpha$-parameters as random multipliers of local bulk operators that result
in randomness in the parameters of the dual field theory. In our proposal, all
the microscopic parameters are fixed. Moreover, since the boundary field
theory group is a gauge group, only gauge invariant states (“singlets”) must
appear in the partition function. Therefore, the non-trivial representations
are accompanied by a insertions of appropriate operators (characters) in the
field theory path integral666Analyses of non-singlet states in holography have
been reviewed by Callan in Mtheory .. We describe further technical details in
Section 4.

#### Energy scale of topology change

Topology changing processes in quantum gravity are often expected to be
relevant near Planck scale. However, it was shown in Tong:2014era  that they
can in fact emerge at exponentially suppressed scales allowing them to be
accessed in the effective bulk theory. Moreover, the non-trivial topologies
were shown to renormalise the bare coupling constant of concern. This lends
further credibility to the proposal that coupling constants are fixed, and not
averaged. Furthermore, it motivates our proposal for the existence of non-
trivial macroscopic saddles in the sum over representations. Potential
perturbative stability of Euclidean wormholes Marolf:2021kjc ; Loges:2022nuw
provides further encouragement for the contribution of such geometries to the
quantum gravity path integral.

#### Topological strings and LLM geometries

Observables such as Wilson loops backreact on the geometry when they are in
large representations (when the corresponding Young-Tableaux contain
$O(N^{2})$ boxes) Lunin:2006xr . For every such representation, the bulk dual
has a different topology. A sum over all representations in the partition
function would correspond to a sum over the various topologies, giving rise to
a picture of “spacetime foam”. This can be made more precise in the context of
topological strings Iqbal:2003ds ; Dijkgraaf:2005bp  and LLM geometries
Lin:2004nb ; Lunin:2006xr .

We conclude this section with a pictorial illustration of our proposal in Fig.
1. The conformal compactification of Euclidean $AdS_{d+1}$ is a $d+1$
dimensional ball with one macroscopic conformal boundary. Consider excising a
small ball in the center, creating a “vacuole” which is a microscopic (and not
conformal) boundary. It is unnatural to relate this vacuole to a state that
belongs in the usual field theoretic Hilbert space on the macroscopic
conformal boundary, as the bulk QG theory is neither topological nor
conformal777It may, however, be “conformal” in a generalised Liouville sense.
See Footnote 3.. Instead, we propose that such microscopic boundaries are
associated with states in $\mathcal{H}_{BU}$, which is spanned by group
representation eigenstates888The representation basis is _not_ a basis with a
fixed number of microscopic boundaries. Instead, it replaces the
$\alpha$-state basis in spirit. It would be interesting to understand if there
exists a basis that describes a fixed number of vacuoles.. From this
perspective, $\hat{Z}[J]$ are operators that create macroscopic boundaries on
the “parent” manifold and do not belong to the microscopic third quantised BU
Hilbert space $\mathcal{H}_{BU}$.

Figure 1: In the far left, we depict the excision of a microscopic “vacuole”
at the centre of global EAdS (whose topology is that of a ball). On the far
right, two large macroscopic universes are connected by a microscopic wormhole
whereas such a connection between distant locations in the same macroscopic
universe is shown in the middle. Factorisation is restored in the
_representation_ basis $|R\rangle$ (which we propose to replace the
$\alpha$-basis prescription) for the BU Hilbert space $\mathcal{H}_{BU}$.

##  4 Some technical details

We define the “void” state in $\mathcal{H}_{BU}$ \- the analogue of the vacuum
Giddings:1988wv  \- , and denote it by $|\Omega\rangle$. This state can be
expanded in the basis of representation eigenstates as999The overlaps $\langle
R|\Omega\rangle$ could also involve other objects such as Casimirs. Here we
made the minimal choice that simply involves the degeneracy of the
representations.

$|\Omega\rangle=\sum_{R}\,D_{R}\,|R\rangle\quad\text{with}\quad\langle
R|\Omega\rangle=D_{R}\,,$ (2)

where $D_{R}$ counts the degenerate states in each irreducible representation
(the dimension). Given a concrete model, the “void” may be constrained to all
allowed representations with a finite number of boxes in the Young
Tableaux101010This is a natural way to restrict the space of representations
for $U(N)$ gauge groups which come with a maximum number of rows in the Young
diagrams. Discrete gauge groups have a naturally finite dimensional space of
irreducible representations.. In this case, the dimension of
$\mathcal{H}_{BU}$ would be finite dimensional. In either case, for compact
gauge groups, the Peter-Weyl theorem guarantees that this basis spans the
total Hilbert space. In what follows, we divide all amplitudes by the norm of
the void state111111By our definition this counts the entropy of the
collection of microscopic closed baby universes. The Hartle-Hawking state
$|HH\rangle$ for a single microscopic boundary carries no entropy as in
McNamara:2020uza  and corresponds to the trivial or singlet representation..

As a first non-trivial example, a single macroscopic boundary can now be
expressed as an expectation value of a macroscopic operator

$Z[J]\coloneqq\frac{\langle\Omega|\hat{Z}[J]|\Omega\rangle}{\langle\Omega|\Omega\rangle}=\sum_{R}p_{R}Z_{R}[J]\quad\text{where}\quad
p_{R}\coloneqq\frac{D_{R}^{2}}{\sum_{R^{\prime}}D_{R^{\prime}}^{2}}\,,$ (3)

and $Z_{R}[J]\coloneqq\langle R|\hat{Z}[J]|R\rangle$. The spirit of the
description of the $\alpha$-state path integral is now evident, whilst
retaining fixed parameters for the boundary theory; $Z[J]$ implicitly depends
on these fixed parameters. The familiar holographic source functional is
merely the singlet term in (3). Since the boundary dual must necessarily only
describe gauge invariant states, the astute reader might object to a
definition of a quantity such as $Z_{R}[J]$. A purely field theoretic
definition of such a quantity requires a modification of the boundary field
theory functional, with the insertion of an appropriate (non-local) operator
that depends on both the boundary gauge field (say $A^{a}_{\mu}$) and the
representation $R$. For instance, such an operator would be the Wilson loop,
or the character121212Of course, the character depends on the holonomy of the
gauge field around the circle (zero mode) and this can be integrated over in
the path integral i.e. it is not a source $J$. $\chi_{R}(U)\coloneqq{\rm
Tr}_{R}U={\rm Tr}_{R}e^{i\oint A}$, in gauged matrix quantum mechanics on a
circle Betzios:2017yms ; Betzios:2021fnm . The integral over the gauge field
zero mode projects the path integral onto the representation $R$. A two-
dimensional generalisation for theories on a torus involves the Weyl-Kac
characters Betzios:2021fnm . Therefore, according to our proposal, the
existence of bulk Euclidean wormholes is intimately tied to the existence and
construction of such non-local operators. Provided a construction, it is then
possible to define the source functional $Z_{R}[J]$, transforming in the
representation $R$. Consequently, the information contained in
$\mathcal{H}_{BU}$ is encapsulated in general (non-local) observables of the
dual field theory; they cannot be accessed by the insertion of sources coupled
to local gauge invariant operators in the Schwinger functional.

It is now straightforward to extend our construction to describe geometries
with multiple conformal boundaries by inserting additional $\hat{Z}[J]$
operators. The two boundary case of Fig. 1, for instance, corresponds to a
state contributing to the expectation value

$\langle\Omega|\hat{Z}^{(1)}[J_{1}]\,\hat{Z}^{(2)}[J_{2}]|\Omega\rangle~{}=~{}\sum_{R}\,p_{R}\,Z^{(1)}_{R}[J_{1}]\,Z^{(2)}_{R}[J_{2}]\,.$
(4)

The natural question now is to ask whether this equation contains geometric
Euclidean wormhole(s) connecting the two large macroscopic universes. This
question is equivalent to asking if the sum over group representations in (4)
localises on a saddle with the effective characteristics of a macroscopic
geometry connecting the two asymptotic AdS regions. The answer to this
dynamical question is model dependent131313See Betzios:2021fnm  for an
analysis in concrete lower dimensional models.. Nevertheless, some general
comments are in order. Since geometric characteristics of such saddles are
only expected to emerge at large $N$, small representations (with only a few
number of Young-Tableau boxes) do not contribute; they may at best give rise
to non-geometric microscopic quantum wormholes. A saddle point analysis
therefore requires an appropriate limit in the space of representations
(continuous Young-Tabeau diagrams) and study of the resulting equations
Douglas:1993iia .

##  5 Consequences of our proposal

###  5.1 Comparisons with McNamara-Vafa and Marolf-Maxfield

Motivated by swampland constraints, McNamara and Vafa formulated a “baby
universe hypothesis” McNamara:2019rup ; McNamara:2020uza  which says that
$\text{dim}\mathcal{H}_{BU}=1$ in $d>2$ dimensions. This is further
strengthened by the observation that a microscopic closed universe must have
no entropy141414 In our proposal the state of a single microscopic universe
$|HH\rangle$ is described by the unique singlet representation and does not
correspond to the BU “void” $|\Omega\rangle$ .. Then, there can be no
superselection sectors at finite volume leaving no room for randomised
$\alpha$-parameters which result in global symmetries in QG. Indeed, as we
argued in Section 2, if one demands a bilocal expansion in the effective field
theory, the wormhole throat pinches off, leaving only the no-boundary state
behind. In contrast, it was later shown in Betzios:2021fnm  that certain
superselection sectors could be obtained in concrete unitary field theoretic
holographic models at finite volume without averaging over couplings. These
superselection sectors are in correspondence with the representations of the
gauge group of the dual holographic field theory and replace the usual
$\alpha$-states, as we propose in this essay. This inspires us to formulate
the following extension:

#### Extended baby universe hypothesis

“The constraints of gravity are so strong that the baby universe Hilbert space
$\mathcal{H}_{BU}$ is finite dimensional (but non-trivial). The
$\alpha$-states dictating the superselection sectors are replaced by
representation eigenstates $|R\rangle$ of the compact gauge group of the dual
field theory. The number of (degenerate) states in any such superselection
sector is counted by the dimension of the representation.”

In Marolf:2020xie , Marolf and Maxfield proposed that the baby universe
Hilbert space is spanned by boundary creating operators. In contrast, our
proposal distinguishes between microscopic and macroscopic operators. The
microscopic operators span the baby universe Hilbert space. Another basis for
this Hilbert space is the representation eigenstate basis (which replaces the
$\alpha$-state basis in our proposal), in which expectation values of
macroscopic operators factorise. The macroscopic operators create macroscopic
$AdS$ boundaries. Therefore, our proposal implies that the state-operator
correspondence is more subtle in gravitating theories than in the topological
models considered in their work.

###  5.2 Global symmetries and the Weak gravity conjecture

Coleman observed Coleman:1988cy  that in the presence of random
$\alpha$-parameters, baby universes carry charges associated with global
symmetries resulting in superselection sectors with different values for the
said charges. As we described above, our proposal replaces the familiar
$\alpha$-states with representation eigenstates. Superselection sectors arise
in this representation eigenbasis, with fixed representations. Therefore,
there is no room for randomised undetermined parameters in our proposal. This
ensures consistency with the expectation that there are no global symmetries
in the bulk QG theory151515Boundary global symmetries are of course allowed,
which are accompanied by propagating gauge fields in the bulk.. Moreover, it
allows us to formulate a _Generalised Completeness Hypothesis_ , which states
that in addition to the familiar Completeness Hypothesis Banks:2010zn ;
Casini:2021zgr , every allowed representation is present in the spectrum of
the Baby Universe Hilbert Space (see also McNamara:2021cuo ). Evidently, this
is a corollary of the Peter-Weyl theorem. It is therefore tempting to argue
that models with $\alpha$-parameters of Coleman belong in the swampland.

###  5.3 A top-down perspective from $D$-branes

Our proposal is reminiscent of constructions in string theory, arising from
stacks of $D$-branes. Consider well-separated stacks of many backreacting
$D$-branes. Each stack results in an $AdS$ geometry of its own. The singlet
sector of the theory is described by factorised boundary theories in the
strict decoupling limit. However, massive string excitations extending between
the stacks are captured by the non-singlet sectors which in the low energy are
described by $T$-branes Cecotti:2010bp . It would be interesting to use
constructions of intersecting branes Gaiotto:2008ak ; VanRaamsdonk:2020tlr ;
Heckman:2021vzx , together with the use of appropriate operators such as
monopole operators, to explore our proposed representation eigenbasis in
detail.

###  5.4 Evading the Fischler-Susskind-Kaplunovsky catastrophe

In Fischler:1988ia , an IR catastrophe was anticipated owing to the existence
of an overdensity of giant wormholes of macroscopic sizes. In our
construction, there is no room for free moduli governing the size of the
wormhole “instantons” or throats (their sizes are fixed in terms of the
microscopic parameters defining the model). Moreover, the density of wormholes
is never expected to become large due to a version of the stringy exclusion
principle—the size of the large parent manifold, and consequently that of the
representations, depends on the rank of the gauge group. While a detailed
analysis is necessary to demonstrate how the catastrophe is evaded explicitly,
the premises leading to it are evidently not satisfied.

###  5.5 Geometry and entanglement (of representations)

While it is well established that a geometric connection in Lorentzian
signature can be associated to the presence of non-trivial entanglement, such
as in the case of the thermofield double state Maldacena:2001kr ;
VanRaamsdonk:2010pw  dual to the eternal two-sided black hole, it does not
suffice to explain the existence of Euclidean wormholes. A simple
demonstration of this fact involves the analytic continuation of the Kruskal
manifold which turns the Einstein-Rosen bridge into two disconnected Euclidean
cigar geometries worm . Our proposal suggests a form of _representation
theoretic entanglement_ in the purely Euclidean context Betzios:2021fnm . A
simple demonstration of this is provided by the two boundary case of eqn. (4),
where the allowed field theoretic states belong in the “entangled sum”
$\sum_{R}\mathcal{H}_{R}^{1}\otimes\mathcal{H}_{R}^{2}$.

## Acknowledgements

We thank Elias Kiritsis, Mark van Raamsdonk, and Gerard ’t Hooft for various
discussions related to the concerns of this essay over the years. We would
also like to acknowledge helpful conversations with Thomas Grimm, Ji Hoon Lee,
Stefano Lanza, Javi Magan, Juan Maldacena, Jake McNamara, Miguel Montero, and
the string theory groups at Perimeter Institute, the University of British
Columbia, and Utrecht University.

The research of P.B. is supported in part by the Natural Sciences and
Engineering Research Council of Canada. P.B. and O.P. acknowledge support by
the Simons foundation. Research at Perimeter Institute is supported in part by
the Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Colleges and Universities. The research of NG is supported by the Delta-
Institute for Theoretical Physics (D-ITP) that is funded by the Dutch Ministry
of Education, Culture and Science (OCW).

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