# Ancilla quantum measurements on interacting chains: Sensitivity of
entanglement dynamics to the type and concentration of detectors

Elmer V. H. Doggen  Institute for Quantum Materials and Technologies,
Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Institut für
Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76131
Karlsruhe, Germany Igor V. Gornyi  Institute for Quantum Materials and
Technologies, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of
Technology, 76131 Karlsruhe, Germany Alexander D. Mirlin  Institute for
Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76131
Karlsruhe, Germany Institut für Theorie der Kondensierten Materie, Karlsruhe
Institute of Technology, 76131 Karlsruhe, Germany

###### Abstract

We consider a quantum many-body lattice system that is coupled to ancillary
degrees of freedom (“detectors”), which are periodically measured by means of
strong projective measurements. The concentration $\rho_{a}$ of ancillae and
their coupling $M$ to the main system are considered as parameters. We explore
the dynamics of density and of entanglement entropy in the chain, for various
values of $\rho_{a}$ and $M$ for two models of the detector-chain interaction
that couple the local density in the chain to a detector degree of freedom. It
is found that, for the density-density ($S_{z}s_{z}$-type in spin language)
coupling, the critical values $M_{c}$ for the measurement-induced entanglement
transition depends sensitively on $\rho_{a}$. Moreover, our results indicate
that for a sufficiently small $\rho_{a}$ the transition in this model
disappears, i.e., a finite density of detectors is needed to reach a
disentangling phase. The behavior is qualitatively different for the second
model, with density-hopping ($S_{z}s_{x}$-type) coupling. Specifically, the
dynamics is much less sensitive to the concentration $\rho_{a}$ of detectors
than in the first model. Furthermore, the dependence of entanglement on the
coupling strength $M$ is strongly non-monotonic, indicating re-entrance of the
entangling phase at large $M$.

##  I Introduction

The dynamics of monitored open quantum systems [1, 2] has been of interest
since the early days of quantum mechanics. Indeed, the Born rule providing a
probabilistic interpretation of the wave function assumes that the system of
interest is interacting with some external “observer” [3, 4] that can induce
wave function collapse. Recently, a renewed surge of interest in the topic has
emerged because of the relevance of the problem of quantum information
processing [5]. In this context, one aims at detailed understanding of the
effect of external monitoring (measurements) on the otherwise unitary dynamics
of a quantum many-body system. This effect generically depends on the type and
strength of interaction between the system and its environment (measurement
apparatus).

A key prediction in this field is the existence of a dynamical phase
transition in monitored quantum systems. In the absence of measurements, a
generic interacting many-body system would become maximally entangled, with
volume-law scaling of the entanglement entropy. The transition is driven by
the rate and strength of measurements, with sufficiently strong and frequent
measurements driving the system to a disentangled state. Hence, this type of
phase transitions has been dubbed a _measurement-induced entanglement
transition_ , which has been theoretically studied in various settings [6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. Experimental
studies of the measurement-induced entanglement transition have been
undertaken in superconducting qubit [82, 83] and trapped-ion [84, 85]
architectures.

It is understood that the volume-law behavior shows up only in the presence of
interparticle interaction. At the same time, numerical modeling of interacting
many-body systems is very costly from the point of view of computational
resources. For this reason, most of the previous numerical works on
measurement-induced transition in interacting systems were carried out for
special models that are particularly convenient for exact simulations of
quantum dynamics. Only a few works have addressed measurement-induced
transitions in interacting models described by “conventional” time-independent
Hamiltonians. Hence, only little is known about sensitivity of the transition
in such systems to the specifics of the measurement protocol. These include
implementation of measurements (e.g., projective or generalized ancilla-based,
continuous or stroboscopic), type and strength of coupling between the system
and ancillae, concentration of the detectors, etc.

In a previous work [64], we have started to address this problem by
considering an interacting chain, with every site coupled to an ancillary
degree of freedom (the concentration of two-level detectors $\rho_{a}=1$). To
overcome very stringent system-size limitations in exact simulations of
quantum dynamics, we have applied the matrix-product-state (MPS) approach to
the measurement-transition problem. The results exhibited clear signatures of
a transition between area-law and volume-law phases, driven by strength $M$ of
the system-ancilla coupling. This coupling was chosen to be of density-density
type (i.e., of $S_{z}s_{z}$ type in the spin language), and the ancillae were
projectively measured in the $s_{z}$ basis. Interestingly, it was also found
in Ref. [64] that, even if only one or two ancillae of this type are coupled
to the system (close to the bipartition cut), the entanglement increase after
a quench gets strongly suppressed by the measurements at a strength $M$ close
to the critical $M_{c}$ for the transition at $\rho_{a}=1$. This poses a
question about the dependence of $M_{c}$ on the concentration of detectors
$\rho_{a}$.

One can note a certain similarity between $\rho_{a}$ and the probability $p$
of measurement in models with random locations and times of measurements such
as random quantum circuits where the entanglement transition driven by $p$ was
originally studied [7, 6, 8]. However, there is a clear difference between
these two situations: in the first one, the positions of measurements are
fixed, while in the second one, they change randomly in time. In particular,
the importance of this difference manifests in the case of special one-
dimensional (1D) circuit models that can be mapped to percolation [7, 11, 44,
45]: while in the random case, there is a two-dimensional (2D) percolation
transition, fixed positions of detectors “cut” the system, implying an area
law for the entanglement.

Another important question concerns the dependence of the entanglement
dynamics on the measurement operator (including the ancilla Hamiltonian in the
case the measurement is implemented with the help of an ancilla).
Specifically, one can choose different monitored observables in the system and
also different ancilla degrees of freedom involved in the coupling to the
system (for a fixed projection basis of the ancilla). It is a priori unclear
how this choice would influence the measurement-induced transition.

Figure 1: Schematic depiction of the setup (shown here for system size $L=8$
and the ancilla concentration $\rho_{a}=1/2$). The main chain (cyan symbols)
is characterized by Hamiltonian (1) with hopping $J$ and nearest-neighbor
interaction $U$. Detectors (two-level ancillas, each represented by a pair of
red and blue sites) are coupled periodically to the main chain (the system-
ancilla coupling indicated by green dashed lines). Projective measurements are
performed on the red ancillary site at regular intervals $\Delta T$. At the
beginning of the protocol, the chain is initialized in the ground state of its
Hamiltonian, while each ancilla is initialized in the state with its red site
occupied and the blue site empty. We consider the two models of detectors:
Model (i) with density-density ($S^{z}s^{z}$-type) coupling, Eqs. (3),(4) (top
panel); Model (ii) with density-hopping ($S^{z}s^{x}$-type) coupling, Eq. (5)
(bottom panel). The bipartite entropy of entanglement is computed with respect
to the division in the middle of the chain (as indicated by the thick green
line).

In this paper, we investigate the sensitivity of the entanglement dynamics in
an interacting chain subjected to ancilla-based measurements (Fig. 1) with
respect to the concentration $\rho_{a}$ of detectors and to the type of
measurement. We use a numerical approach involving the MPS formalism to study
sufficiently large systems. Starting with the model introduced in Ref. [64]
with the density-density coupling at $\rho_{a}=1$ as a reference point, we
first decrease $\rho_{a}$ in this setup to $\rho_{a}=1/2$ and $\rho_{a}=1/4$.
Next, we modify the model by replacing the density-density coupling with the
density-hopping one, where the occupation of the chain site affects the
hopping in the ancilla. In the latter setup, we also consider ancilla
concentrations $\rho_{a}=1,1/2$, and $1/4$. For both models and for each value
of $\rho_{a}$, we scan over a broad range of the coupling strength $M$. This
allows us to construct qualitative phase diagrams of entanglement entropy in
the parameter planes $\rho_{a},\,M$ for both models.

##  II Model

We consider a system that consists of a main chain coupled to ancillary
qubits. The main chain is defined by the Hamiltonian

$\mathcal{H}_{\mathrm{s}}=\sum_{i=1}^{L-1}\left[-\frac{J}{2}\left(b_{i}^{\dagger}b_{i+1}+\mathrm{H.c.}\right)+U\hat{n}_{i}\hat{n}_{i+1}\right],$
(1)

where $L$ is the chain length, $b^{\dagger}_{i}$ creates a hard-core boson on
lattice site $i$, $\hat{n}_{i}\equiv b_{i}^{\dagger}b_{i}$ is the density, $J$
is a hopping parameter, and $U$ is the nearest-neighbor interaction strength.
Below we set $J=1$ to fix the energy units. The measurement procedure is
implemented by performing projective density measurements on the ancillary
sites coupled to the main system, as schematically depicted in Fig. 1. We
consider two different types of coupling of the ancillae to the main system.

The first measurement protocol [model (i)] involves a density-density
interaction between the system and the ancillae, with the total Hamiltonian
given by

$\mathcal{H}=\mathcal{H}_{\mathrm{s}}+{\sum_{j}}^{\prime}\left[\mathcal{H}_{\mathrm{a}}^{(j)}+\mathcal{H}_{\mathrm{sa}}^{(j)}\right],$
(2)

where the sum goes over sites coupled to ancillae, where

$\mathcal{H}_{\mathrm{a}}^{(j)}=-\frac{1}{2}\left(a_{j,1}^{\dagger}a_{j,2}+a_{j,2}^{\dagger}a_{j,1}\right)\quad\mathrm{model\,(i)}$
(3)

is the own ancilla Hamiltonian (with the hopping parameter equal to that in
the main chain), and

$\mathcal{H}_{\mathrm{sa}}^{(j)}=-M\hat{n}_{j}a_{j,1}^{\dagger}a_{j,1}\quad\mathrm{model\,(i)}$
(4)

describes the system-ancilla coupling. Here $a_{j,1},a_{j,2}$ are annihilation
operators of a hard-core boson in the ancillary pair.

For the second measurement protocol [model (ii)], the Hamiltonian has the same
form (2), where now $\mathcal{H}_{\mathrm{a}}^{(j)}=0$ and
$\mathcal{H}_{\mathrm{sa}}^{(j)}$ describes a density-hopping coupling:

$\mathcal{H}_{\mathrm{sa}}^{(j)}=-\frac{M}{2}\hat{n}_{j}\left(a_{j,1}^{\dagger}a_{j,2}+a_{j,2}^{\dagger}a_{j,1}\right),\quad\mathrm{model\,(ii)}.$
(5)

In this setup, the hopping between the ancilla sites is mediated by the
coupling to the density on the main chain: the ancilla sites are connected
only when the corresponding system site is occupied. This also means that the
ancillae are effectively frozen in the limit of zero coupling $M\rightarrow
0$.

For both models, a projective Born-rule measurement of the density
$a_{j,1}^{\dagger}a_{j,1}$ on ancilla sites 1 is performed at regular
intervals $\Delta T$. When the model (1) is represented in terms of spin-$1/2$
operators $\mathbf{S}_{i}$, one can view the coupling (4) in setup (i) as
$S^{z}s^{z}$ coupling, and the coupling (5) in setup (ii) as $S^{z}s^{x}$
coupling, where $\mathbf{s}$ refers to the ancilla and $s_{x}$ corresponds to
hopping between the ancilla sites.

We consider the ancilla concentrations $\rho_{a}=1$, 1/2, and 1/4, where
$\rho_{a}$ is equal to the number of ancilla pairs divided by the length $L$
of the main chain. The number of ancillae therefore scales extensively with
$L$, and the total number of lattice sites is
$L_{\mathrm{tot}}=(1+2\rho_{a})L$. We choose $L$ divisible by 4 and attach the
ancillae starting from the first site, see Fig. 1.

The final ingredient of the protocol is absence of resetting of the ancillae
after measurements 111Measurement model (ii) for $L=2$ (single-particle main
chain) was considered in Refs. [55] and [92] in the presence of resetting.
This induces feedback loops, with the measurement of the ancilla affecting the
dynamics in the main system, which in turn affects consecutive measurement
results, and so on. When the measurement frequency and the ancilla Rabi
oscillation frequency are commensurate, this leads to very long-lived
correlations between measurements, that can significantly affect the dynamics
of the system. In Ref. [64], this was observed for setups with one and two
detectors (with density-density coupling) and dubbed the _quantum-Zeno-valve
effect_. For $\rho_{a}=1$, this effect was smeared out. Here we will
investigate whether this effect shows up in the entanglement dynamics for
smaller $\rho_{a}$ and also for an alternative [model (ii)] coupling 222Upon
inspection of Eqs. (4)-(5), one can observe that in the mean-field limit
$\hat{n}\rightarrow\langle\hat{n}\rangle$ there is a well-defined Rabi
oscillation frequency for the ancilla boson in both setups (i) and (ii)..

The dynamics is computed using the time-dependent variational principle [88],
with a procedure similar to the one used in Ref. [64]. The whole setup (main
chain and ancillary pairs) is mapped onto a 1D chain using matrix product
operators, where the ancillae are “folded” into the chain on consecutive
sites, and the main chain sites are then coupled through next-next-nearest
neighbor terms. The numerical approach is based on MPS [89, 90], a type of
tensor network wherein we consider a variational subspace of the whole Hilbert
space of the system. The total Hilbert space of the model scales as $\propto
2^{L_{\mathrm{tot}}}$; in the MPS approximation, we consider only a subspace
with polynomial complexity, controlled by a numerical parameter $\chi$ called
the bond dimension. The lower computational complexity of the cases
$\rho_{a}=1/2$ and 1/4 allows us to consider a larger size of the main chain
($L=40$) than in Ref. [64].

##  III Results

###  III.1 Observables

We focus on the following observables. To quantify entanglement, we compute
the von Neumann bipartite entropy of entanglement $S(t)$, where the
bipartition is taken in the middle of the main system as indicated in Fig. 1,

$S(t)=-\mathrm{Tr}\Big{(}\rho_{A}\ln\rho_{A}\Big{)},\quad\rho_{A}=\mathrm{Tr}_{\mathrm{B}}\rho.$
(6)

Here $\rho$ is the density matrix of the whole system (including the
detectors), and $\rho_{A}$ is the reduced density matrix corresponding to a
part $A$ of a bipartition in parts $A$ and $B$. Furthermore, we track the
dynamics of the density $n_{i}(t)\equiv\langle\hat{n}_{i}\rangle(t)$ at every
site in the system (where $\langle\ldots\rangle$ denotes the averaging over a
quantum state), as a function of time $t$. From this, we can also compute the
probability density function $P(n;t)$, which quantifies the fluctuations of
the density throughout the system.

For each choice of parameters $\rho_{a}$ and M, we numerically compute an
ensemble (typically $\sim 40$, in some cases up to 200) quantum trajectories.
In what follows, we use the MPS bond dimension $\chi=128$, which establishes
an upper cutoff for the entanglement entropy $S=\ln 128\simeq 4.85$.

###  III.2 Setup (i): Density-density coupling

Figure 2: Average entanglement entropy $S(t)$ for model (i) at various values
of the system-ancilla density-density coupling $M$ for system length $L=40$.
Left panel: $\rho_{a}=1/2$; right panel: $\rho_{a}=1/4$.

We first consider model (i) with the density-density interaction, as defined
in Sec. II. We start with the case where there is an ancilla pair at every
site, $\rho_{a}=1$, as in Ref. [64]. (Note that our definition of the system-
ancilla coupling is different by the sign from that in Ref. [64]. This,
however, does not lead to any essential difference in the results.) For
$\rho_{a}=1$, the data provide evidence for a measurement-induced entanglement
transition at $M_{c}\approx 5$, see Ref. [64]. Specifically, $M>M_{c}$ leads
to disentangling behavior, with $S(t)$ saturating at a relatively small value,
smaller than $S(t=0)$ and way smaller than the cutoff set by the MPS bond
dimension $\chi$. On the other hand, for $M<M_{c}$, the entanglement entropy
$S(t)$ grows with time, providing an indication of the volume law. For
sufficiently large systems and moderate times, the volume-law phase manifests
itself in the linear growth of $S(t)$ with time until it approaches the
saturation cutoff set by the bond dimension $\chi$.

We now proceed by analyzing the effect of reduction of the ancilla
concentration $\rho_{a}$. In Fig. 2, we show the entanglement entropy $S(t)$
averaged over quantum trajectories for $\rho_{a}=1/2$ and $\rho_{a}=1/4$.
First of all, the average entropy is higher for smaller $\rho_{a}$, so that
having fewer ancillae disentangles less, as may be expected. From this, we can
infer that the transition, if any, will be at a larger value of $M$.

For $\rho_{a}=1/2$, the behavior of entanglement is qualitatively similar to
that for $\rho_{a}=1$: the entangling behavior for small $M$ and disentangling
behavior at large $M$ are clearly visible. This provides an indication of the
entanglement transition at $M_{c}\approx 8$. Importantly, this value of
$M_{c}$ is significantly larger than the critical value $M_{c}\approx 5$
observed for $\rho_{a}=1$. Thus, $M_{c}$ in model (i) strongly depends on the
ancilla concentration $\rho_{a}$.

To further explore the evolution of entanglement with decreasing $\rho_{a}$,
we show in the right panel of Fig. 2 $S(t)$ for $\rho_{a}=1/4$. We observe a
drastic change in its behavior: all curves show $S(t)$ increasing with time,
even for such large coupling as $M=20$. Although for large values of $M$ the
growth is relatively slow (and we cannot exclude saturation at longer times),
we do not have clear evidence of the transition at all. This further confirms
that $M_{c}$ increases with decreasing $\rho_{a}$ and likely diverges at some
$\rho_{a}$ not far from 1/4.

Figure 3: Color plot of the density $n_{i}(t)$ for model (i) with parameters
$\rho_{a}=1/2,M=3,L=40,\Delta T=2$. Left panel: $n_{i}(t)$ averaged over the
ensemble of quantum trajectories; middle panel: $n_{i}(t)$ for the quantum
trajectory with the smallest $S(t=50)$; right panel: $n_{i}(t)$ for the
quantum trajectory with the largest $S(t=50)$. Figure 4: The same as in Fig. 3
for model (i) with $\rho_{a}=1/2,M=10,L=40,\Delta T=2$. Figure 5: The same as
in Fig. 3 for model (i) with $\rho_{a}=1/4,M=3,L=40,\Delta T=2$. Figure 6: The
same as in Fig. 3 for model (i) with $\rho_{a}=1/4,M=10,L=40,\Delta T=2$.

In order to shed more light on the physics behind the entanglement dynamics,
we illustrate in Figs. 3-6 the temporary evolution of the density $n_{i}(t)$
for $\rho_{a}=1/2$ and $\rho_{a}=1/4$ at relatively weak coupling $M=3$ and
relatively strong coupling $M=10$. In each of the figures, the left panel
shows $n_{i}(t)$ averaged over the ensemble of quantum trajectories, while the
middle (right) panel shows $n_{i}(t)$ for the quantum trajectory with the
smallest (respectively, largest) $S(t=50)$.

Compared to the corresponding plots for $\rho_{a}=1$ (see Ref. [64]), we see a
striking difference. While for $\rho_{a}=1$ a density clusterization at large
$M$ was found, here we observe a well-pronounced striped structure. This is
related to the fact that ancillae are coupled only to a subset of sites of the
main chain. Long-lived metastable “pajama” structures form, beyond the size of
a single site/ancilla pair. These structures prevent entanglement growth and
can be attributed to the quantum-Zeno-valve effect, which was discovered in
Ref. [64] for setups with one or two ancillae.

For both cases $\rho_{a}=1/2$ and $\rho_{a}=1/4$, the contrast of plots for
individual trajectories is enhanced with increasing $M$, which implies
stabilization of the quantum-Zeno-valve effect. We further note that, for not
too strong coupling $M=3$, the least and most entangled quantum trajectories
produce similar density plots. On the contrary, for $M=10$, we observe much
stronger fluctuations within the ensemble of quantum trajectories, as
visualized by the striking difference between the middle and right panels of
Fig. 4.

For $\rho_{a}=1/4$, the effect of measurements on the density profiles is
considerably weaker: even for $M=10$, the least and most entangled
trajectories are characterized by visually similar densities, see the middle
and right panels in Fig. 6. The large difference in entanglement in these two
trajectories is because the bipartition cut in the right panel (most entangled
trajectory) occurs in the middle of the pajama-free region.

The averaged (over the ensemble of quantum trajectories) densities shown in
the left panels of Figs. 3-6 also exhibit pajama-like patterns but with much
weaker contrast. This reduction of contrast originates from the two
possibilities of the metastable frozen states formed by the ancilla site and
the site of the chain to which it is attached: either an ancilla-particle or
an ancilla-hole quasi-bound state. This is well seen in all panels showing
individual quantum trajectories, where the red vertical stripes correspond to
particles and the blue stripes to holes. Red stripes are somewhat more stable,
leading to residual pajamas in the panels for averaged densities.

The qualitatively different physics of quantum states at small and large $M$
is reflected also in the distribution function $P(n;t)$ of $n(t)$, see plots
in Supplemental Material [91]. For sufficiently small $M$, this distribution
is relatively narrow and peaked at $n=1/2$. On the other hand, at large $M$
the distribution broadens over the whole range $[0,1]$ of densities, with
peaks emerging at $n=0$ and $n=1$ that reflect the high-contrast pajama
structure. The emergence of these peaks in $P(n;t)$ is correlated with the
suppression of the entanglement growth, see the above discussion of the
quantum-Zeno-valve effect.

###  III.3 Setup (ii): Density-hopping coupling

Figure 7: Average entanglement entropy $S(t)$ for model (ii) at various values
of the system-ancilla density-hopping coupling $M$ for $L=40$. Left panel:
$\rho_{a}=1/2$; right panel: $\rho_{a}=1/4$.

We turn now to model (ii), in which the density on a site of the main chain is
coupled to ancilla hopping, see Sec. II. The time dependence of the average
entanglement entropy for this model with $\rho_{a}=1/2$ and 1/4 and various
values of coupling $M$ is shown in Fig. 7. Comparing to Fig. 2, we observe an
essential difference between the models (i) and (ii). While an initial
increase of $M$ in model (ii) leads to a suppression of the entanglement, a
re-entrant behavior is observed when $M$ grows further. This non-monotonic
behavior can be related to the “frustration” between the coupling term and the
measurement in model (ii): in the spin language, the former involves the
$s_{x}$ component and the latter the $s_{z}$ component of ancilla spin. While
the measurements try to freeze the ancilla in the $z$ basis, the large-$M$
coupling leads to rapid oscillations in this basis, precluding the formation
of a quasi-bound state, in contrast to model (i). Another clear difference is
that the results in model (ii) are much less sensitive to the ancilla
concentration.

Results for the density evolution in model (ii) are presented in Figs. 8 and 9
for $\rho_{a}=1/2$. For relatively small coupling, $M=3$, we observe a very
low contrast: the density is typically close to its average value $n_{i}=1/2$,
as also seen in the distribution function $P(n;t)$, see Supplemental Material
[91]. This is in correspondence with a fast increase of the entropy $S(t)$ in
the left panel of Fig. 7. On the other hand, for $M=10$ a pajama-like pattern
is seen as for the model (i). There is, however, a difference as compared to
model (i): the long-lived metastable structures in the case of model (ii) are
statistically particle-hole symmetric (red and blue stripes in the color
density plot have similar appearance). In terms of the $P(n;t)$, this
manifests itself in the approximate symmetry $n\to 1-n$.

Figure 8: The same as in Fig. 3 for model (ii) with
$\rho_{a}=1/2,M=3,L=40,\Delta T=2$ Figure 9: The same as in Fig. 3 for model
(ii) with $\rho_{a}=1/2,M=10,L=40,\Delta T=2$

###  III.4 Phase diagram

Figure 10: Qualitative phase diagrams for models (i) (left) and (ii) (right)
in the parameter plane $(\rho_{a},M)$. The color code shows the value
$S(t=50)$ of the entanglement entropy and the end of time evolution. The
square symbols indicate the points for which MPS calculations were performed;
the remainder is obtained by interpolation. The red lines correspond to
$S(t=50)=1.0$ and serve as a rough estimate of the phase boundaries between
the disentangling (area-law) and entangling (presumably, volume-law) phases).

Based on the dynamics of the entropy, we are now in a position to investigate
the phase diagrams for both models. Figure 10 shows the average entropy $S$ at
the final time $t=50$ of the simulation, for various choices of the ancilla
concentration $\rho_{a}$ and measurement strength $M$. Of course, strictly
speaking, this value does not automatically distinguish between the possible
phases. However, inspecting the entropy curves in Figs. 2 and 7, we clearly
see that large values of $S(t=50)$ correspond to unsaturated growth of $S$
with time, while low values are associated with decreasing entropy that tends
to saturate at long times. With this indicator, in both panels of Fig. 10, we
find a qualitative change between disentangling (blue regions) and entangling
(yellow regions) types of behavior, with the border between them belonging to
the green regions. This border is additionally visualized by a red line
corresponding to $S(t=50)=1.0$, which can be viewed as an estimate of the
phase boundary between the area-law and volume-law phases.

For model (i), the increase of $M_{c}$ with increasing ancilla concentration
$\rho_{a}$ is clearly visible. The phase diagram for model (ii) has a more
complex structure, reflecting the reentrance and weaker dependence on
$\rho_{a}$ discussed above.

##  IV Summary and discussion

###  IV.1 Summary

In this paper, we have numerically modeled a quantum many-body lattice system
(with a conserved particle number) coupled to a finite concentration of
detectors (“ancillae”) that are subject to periodic measurements. The central
questions that we addressed are how the dynamics of the entropy and of the
density depend on the ancilla concentration $\rho_{a}$, the ancilla coupling
$M$ to the main system, and on the type of this coupling. Specifically, we
considered two types of coupling: model (i) with a density-density coupling,
i.e., $S_{z}s_{z}$-type in the spin language, and model (ii) with a density-
hopping coupling, i.e., $S_{z}s_{x}$-type, where $S$ and $s$ refer to the
lattice site and to the detector coupled to it, respectively. In both models,
the detectors are measured in the $s_{z}$ basis, so that the measured operator
commutes with the coupling in the first model but does not commute in the
second model. By using the MPS-based computational approach, we studied chains
of the length $L=40$ for $\rho_{a}=1/2$ and $1/4$ (i.e., with $L/2$ and $L/4$
detectors, respectively) and of the length $L=24$ for $\rho_{a}=1$ (i.e., with
$L$ detectors).

We have found that, for model (i), the critical value $M_{c}$ of the
measurement-induced entanglement transition is strongly dependent on the
ancilla concentration $\rho_{a}$, see Fig. 10. Furthermore, our results
indicate that $M_{c}$ diverges at some critical concentration $\rho_{a}^{(c)}$
(which is close to 1/4), so that for $\rho_{a}<\rho_{a}^{(c)}$ the system is
in the entangling phase for any value of $M$.

For model (ii), the behavior is different in two key aspects. First, the
dependence on $M$ is strongly non-monotonic, suggesting a re-entrance of the
entangling phase at large $M$. Secondly, the system is much less sensitive to
the concentration of detectors $\rho_{a}$ than in the case of model (i).

We have complemented the analysis of the entanglement entropy with the
particle density $n_{j}(t)$ in the chain. Importantly, we studied $n_{j}(t)$
for individual quantum trajectories, as the average of $n_{j}(t)$ over
trajectories misses the physics related to the entanglement transition. For
relatively weak couplings $M$, i.e., in the entangling phase, the density
$n_{i}(t)$ for a given quantum trajectory fluctuates weakly around $n=1/2$. On
the other hand, for large $M$, we observe clear long-living striped (“pajama”)
patterns. This freezing of density suppresses the entanglement growth and is
attributed to the quantum-Zeno-valve effect.

###  IV.2 Outlook

Let us conclude by briefly discussing prospects for future research. An
important open question in the physics of measurement-induced transitions is
the effect of particle-number conservation. For monitored non-interacting
fermions (with explicit particle-number conservation), the behavior of the
entanglement entropy can be captured by the analysis of the particle-number
cumulants [75], with the entanglement transition in 2D systems coinciding with
the transition for the density correlations [76, 79]. At the same time, in a
certain special class of random quantum circuits representing interacting
systems (involving Haar-random gates and qudits with $d\to\infty$ states), a
related “charge sharpening” transition was predicted to be distinct from the
entanglement transition and to take place within the volume-law phase [45,
44]. It is thus a key open question whether the entanglement transition and
the particle-number-fluctuation (or “charge sharpening”) transitions coincide
or are distinct for a realistic problem of interacting fermions (or hard-core
bosons), like the one considered in the present work. In both cases, it is
also important to understand how the entanglement and the density correlations
influence each other (and, in particular, the corresponding scaling behavior).

Importantly, the above density correlations should be evaluated in a given
quantum state (and only after this can be averaged over quantum trajectories),
in similarity to density patterns for individual trajectories and to the
distribution function $P(n;t)$ discussed in this paper (and also to the
density clusterization that was observed for $\rho_{a}=1$ in Refs. [55] and
[64]). We expect that the MPS-based approach developed in this work may be
extended to study quantitatively density fluctuations in models of monitored
interacting 1D fermions or bosons and, in this way, to provide responses to
the above questions from the computational perspective. It remains to be seen
how universal the resolution of the above dichotomy (one vs. two transitions)
is and, in particular, whether it may depend on the interaction strength, the
type of the measurement protocol (like models (i) and (ii) in this paper) and
on resetting (as in this paper) or non-resetting of detectors. Finally, it is
interesting to study whether a commensurability of the ancilla periodicity
with that of the main lattice is important in this context and whether random
placement of ancillas would lead to any essential modifications.

##  V Acknowledgments

We acknowledge collaboration with Y. Gefen and D.G. Polyakov on earlier
related projects. We also thank D.G. Polyakov for useful discussions and
comments on the manuscript. IVG acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) via the grant GO 1405/6-1.

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Supplemental Material to
“Ancilla quantum measurements on interacting chains: Sensitivity of
entanglement dynamics to the type and concentration of detectors”

In this Supplemental Material, we provide additional information on the
results of numerical modeling for both models (i) and (ii), for the chain
length $L=40$ and various values of the ancilla density $\rho_{a}$. Each
figure corresponds to a point $(\rho_{a},M)$ in the parameter space of the
corresponding model, cf. Fig. 10 of the main text. In each figure, the top row
shows the time evolution of average density in the chain (left), average
entanglement entropy (middle), and average occupation of the “red” ancilla
site (right), cf. Fig. 1 of the main text. The second (third) row shows
analogous quantities for the quantum trajectory that is least (respectively,
most) entangled at $t=50$. The bottom panel displays the distribution function
$P(n)$ of the local density in the main chain at times $t=0.25$, 25, and 50.
For each model and $\rho_{a}$, the values of $M$ are chosen in such a way that
the data illustrate the disentangling and entangling regions of the phase
diagram, Fig. 10 of the main text.

Figure S11: Model (i), $\rho_{a}=1/2$, $M=3$. Top row: time evolution of
average density in the chain (left), average entanglement entropy (middle),
and average occupation of the “red” ancilla site (right), cf. Fig. 1 of the
main text. Second (third) row: analogous quantities for the quantum trajectory
that is least (respectively, most) entangled at $t=50$. Bottom panel:
distribution function $P(n)$ of the local density in the main chain at times
$t=0.25$, 25, and 50.  Figure S12: Same as Fig. S11 for model (i),
$\rho_{a}=1/2$, $M=20$. Figure S13: Same as Fig. S11 for model (i),
$\rho_{a}=1/4$, $M=3$. Figure S14: Same as Fig. S11 for model (i),
$\rho_{a}=1/4$, $M=20$. Figure S15: Same as Fig. S11 for model (ii),
$\rho_{a}=1$, $M=3$. Figure S16: Same as Fig. S11 for model (ii),
$\rho_{a}=1$, $M=15$. Figure S17: Same as Fig. S11 for model (ii),
$\rho_{a}=1$, $M=25$. Figure S18: Same as Fig. S11 for model (ii),
$\rho_{a}=1/2$, $M=3$. Figure S19: Same as Fig. S11 for model (ii),
$\rho_{a}=1/2$, $M=15$. Figure S20: Same as Fig. S11 for model (ii),
$\rho_{a}=1/4$, $M=3$. Figure S21: Same as Fig. S11 for model (ii),
$\rho_{a}=1/4$, $M=15$.


# Sound Field Reconstruction Using a Compact Acoustics-informed Neural Network

Fei Ma  Sipei Zhao  Electronic mail<EMAIL_ADDRESS>Center for Audio,
Acoustics and Vibration, Faculty of Engineering and IT, University of Sydney
Technology, Ultimo, NSW 2007, Australia.  Ian S. Burnett  DVC and VP STEM,
RMIT University, Melbourne, VIC 3000, Australia

###### Abstract

Sound field reconstruction (SFR) augments the information of a sound field
captured by a microphone array. Conventional SFR methods using basis function
decomposition are straightforward and computationally efficient, but may
require more microphones than needed to measure the sound field. Recent
studies show that pure data-driven and learning-based methods are promising in
some SFR tasks, but they are usually computationally heavy and may fail to
reconstruct a physically valid sound field. This paper proposes a compact
acoustics-informed neural network (AINN) method for SFR, whereby the Helmholtz
equation is exploited to regularize the neural network. As opposed to pure
data-driven approaches that solely rely on measured sound pressures, the
integration of the Helmholtz equation improves robustness of the neural
network against variations during the measurement processes and prompts the
generation of physically valid reconstructions. The AINN is designed to be
compact, and is able to predict not only the sound pressures but also sound
pressure gradients within a spatial region of interest based on measured sound
pressures along the boundary. Numerical experiments with acoustic transfer
functions measured in different environments demonstrate the superiority of
the AINN method over the traditional cylinder harmonic decomposition and the
singular value decomposition methods.

##  I Introduction

Microphone arrays are commonly used for measuring a sound field [2] and to
maximize the information about the source, a large-aperture array with
densely-spaced microphones, is preferred for sound field measurements.
However, that is not always possible due to practical considerations such as
cost and microphone arrangement [20]. This necessitates sound field
reconstruction (SFR) [28, 29, 6], a task that aims to reconstruct a sound
field apart from the limited (sparse) measurements.

Existing SFR methods can be broadly classified into two categories:
conventional methods based on basis function decomposition and recent
learning-based methods. The conventional methods decompose sound field
measurements into some basis functions, such as cylinder harmonics (CHs) [28],
spherical harmonics [28, 27, 3, 26, 25], prolate spheroidal wave functions
[30], plane waves [28, 6, 1, 22], and their corresponding weights. The basis
functions are solutions of the Helmholtz equation [24, 28], the governing
partial differential equation (PDE) of acoustic wave propagation, and are
continuous spatial functions which can be evaluated at arbitrary positions.
These two factors make the conventional methods easy to compute and generate a
physically valid reconstruction of the sound field away from the measurement
positions. However, the basis functions are designed with respect to some
coordinate systems [28] without considering the statistical characteristics of
sound fields. Thus, conventional methods may require more than the absolutely
necessary number of measurements (spatial sampling points) to determine the
basis function weights to reconstruct a sound field. The reality is that the
statistical characteristics of a sound field can be used to allow the sampling
requirement to be relaxed based on singular value decomposition (SVD) [32,
33], compressive sensing [27, 26], statistical learning [11], or Bayesian
inference [22].

In contrast to conventional methods, recent learning-based methods do not rely
on pre-designed basis functions. Instead, they exploit the learned statistical
characteristics of sound fields for SFR. Lluıs et al. [16] and Kristoersen et
al. [14] developed U-net-like neural networks, which were trained with
simulated or measured room impulse responses. The U-net-like neural networks
achieved superior SFR performance than some of the conventional methods in the
low frequency range ($<300$ Hz). Hahmann et al. [11] proposed to learn basis
functions in local subdomains. The learned basis functions generalized across
different rooms and frequencies, and showed potential for modeling complex
sound fields according to their local (spatial) or statistical
characteristics. By further enforcing self-similarity between adjacent local
subdomains [10], the method attained better SFR performance when few
measurements were available. Most recently, Fernandez-Grande et al. examined
the use of generative adversarial networks for SFR [7]. The generative
adversarial networks recovered some of the sound field energy at high
frequencies that would otherwise be lost due to under-sampling [7],
demonstrating the promise of using statistical learning methods to overcome
the sampling limitations. Although the learning-based methods outperformed the
conventional methods in some SFR tasks, their computations are time-consuming.
Furthermore, they are purely data-driven and thus do not necessarily
reconstruct physically valid sound fields [7].

Recently, physical laws have been integrated into neural networks for various
acoustic studies, such as the Kirchhoff–Helmholtz-based convolutional neural
network (CNN) for nearfield acoustic holography [17], the physics-informed CNN
(PI-CNN) for sound field estimation [23], and the PINN for room impulse
response reconstruction [12, 19]. These studies [17, 23, 12, 19] attempted to
reconstruct the sound field within a region of interest (ROI) with few
measurements inside the ROI.

In an alternative approach, this paper proposes an acoustic-informed neural
network (AINN) to reconstruct the sound field within the ROI based on the
sound pressures measured on its boundary alone [21, 4]. The AINN is designed
to approximate the sound field at the measurement positions and is guided by
the Helmholtz equation to generate physically valid reconstructions away from
the measurement positions. The AINN is compact and lightweight, making it
easier to train than large neural networks. In addition, owing to the
automatic-differentiation of deep-learning libraries [19], the AINN is able to
reconstruct the pressure gradient within the ROI. Numerical experiments with
transfer functions measured with two microphone arrays in three different
rooms [31] are conducted to compare the proposed method with the CH [28] and
the SVD [33] methods. The experimental results demonstrate the superiority of
the proposed AINN method over the existing methods.

The remainder of this paper is organized as follows. The problem is formulated
in Sec. II. The CH method and the SVD method are reviewed in Sec. III,
followed the proposed AINN method. Numerical experiments are presented in Sec.
IV to validate the performance of the proposed AINN method, in comparison to
the CH and the SVD methods. Sec. V concludes this work.

##  II Problem formulation

Figure 1: (color online) Problem setup: A number of sources $\star$ generate a
sound field within a ROI, which is measured by microphones $\bullet$ on the
boundary of the ROI. The objective is to reconstruct the sound pressure and
its gradient within the ROI based on the microphone measurement.

The problem of interest is illustrated in Fig. 1, where $(x,y)$ and $(r,\phi)$
denote the Cartesian and polar coordinates with respect to the origin $O$,
respectively. The stars denote sound sources that generate the sound field in
the Region of Interest (ROI) depicted as the gray area. An array of
microphones on the boundary of the ROI, shown as the dots in Fig. 1, measure
the sound pressures at $\\{x_{q},y_{q}\\}_{q=1}^{Q}$ (or
$\\{r_{q},\phi_{q}\\}_{q=1}^{Q}$) as $P(\omega,x_{q},y_{q})_{q=1}^{Q}$ (or
$P(\omega,r_{q},\phi_{q})_{q=1}^{Q}$), where $\omega=2\pi{f}$ is the angular
frequency with $f$ being the frequency. The objective is to estimate the sound
pressures and their gradients inside the ROI based on the measured sound
pressures on the boundary. Hereafter, the symbol $\omega$ is omitted in some
quantities for notational simplicity.

There is no sound source inside the ROI, and hence the sound field within the
ROI is governed by the homogeneous Helmholtz equation [28], i.e.,

$P+\frac{1}{(\omega/c)^{2}}\nabla^{2}P=0,$ (1)

where $c=340$ m/s is the speed of sound, and $\nabla^{2}$ denotes the
Laplacian operator. In Cartesian coordinates, the Laplacian operator is given
by [28]

$\nabla^{2}=\frac{\partial}{\partial{x^{2}}}+\frac{\partial}{\partial{y^{2}}},$
(2)

and in polar coordinates it is given by

$\nabla^{2}=\frac{\partial^{2}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}.$ (3)

In this paper, we build a compact neural network informed by the Helmholtz
equation, Eq. (1), to reconstruct the sound field within the ROI based on the
microphone measurements.

##  III Methodology

This section first reviews the CH method [28] and the SVD method [33] for SFR,
and subsequently proposes the AINN method.

###  III.1 The cylinder harmonics method

Acoustic quantities are expressed in polar coordinates for the ease of
computing the CHs. Sound pressures can be decomposed into CHs as [28]

$\mathbf{P}_{\mathrm{M}}=\mathbf{J}\mathbf{A},$ (4)

where
$\mathbf{P}_{\mathrm{M}}=[P(r_{1},\phi_{1}),P(r_{2},\phi_{2}),...,P(r_{Q},\phi_{Q})]^{\intercal}$
denote the measured sound pressures at $\\{r_{q},\phi_{q}\\}_{q=1}^{Q}$,
$(\cdot)^{\intercal}$ is the transpose operation,
$\mathbf{A}=[A_{-N},A_{-N+1},...,A_{N}]^{\intercal}$ denote the CH weights,
and

$\mathbf{J}=\begin{bmatrix}J_{-N}(\frac{\omega}{c}r_{1})e^{iN\phi_{1}}&...&J_{N}(\frac{\omega}{c}r_{1})e^{-iN\phi_{1}}\\\
J_{-N}(\frac{\omega}{c}r_{2})e^{iN\phi_{2}}&...&J_{N}(\frac{\omega}{c}r_{2})e^{-iN\phi_{2}}\\\
...&...&...\\\
J_{-N}(\frac{\omega}{c}r_{Q})e^{iN\phi_{Q}}&...&J_{N}(\frac{\omega}{c}r_{Q})e^{-iN\phi_{Q}}\\\
\end{bmatrix},$ (5)

is a $Q\times{(2N+1)}$ matrix whose entry
$J_{n}(\frac{\omega}{c}r_{q})e^{-in\phi_{q}}$ is the $n$-order cylinder
harmonic [28] evaluated at $(r_{q},\phi_{q})$, $i$ is the imaginary number
unit, and $J_{n}(\cdot)$ is the Bessel function of order $n$ [28]. In Eqs. (4)
and (5), $N$ is the dimensionality of the sound field under CH decomposition
and is normally chosen as [13]

$N=\lceil{2\pi{f}r/c}\rceil,$ (6)

where $\lceil{\cdot}\rceil$ is the ceiling operation,

The CH method estimates the weights
$\hat{\mathbf{A}}=[\hat{A}_{-N},\hat{A}_{-N+1},...,\hat{A}_{N}]^{\intercal}$
through

$\hat{\mathbf{A}}=\mathbf{J}^{\dagger}\mathbf{P}_{\mathrm{M}},$ (7)

where $(\cdot)^{\dagger}$ denotes the pseudo-inverse operation. The sound
pressure and the radial gradient for an arbitrary position $(r_{e},\phi_{e})$
can be reconstructed as [28]

$\hat{P}_{\mathrm{CH}}(r_{e},\phi_{e})\approx\sum_{n=-N}^{N}\hat{A}_{n}J_{n}(\frac{\omega}{c}r_{e})e^{-in\phi_{e}},$
(8)

and

$\frac{\partial\hat{P}_{\mathrm{CH}}(r_{e},\phi_{e})}{\partial{r_{e}}}\approx\frac{\omega}{c}\sum_{n=-N}^{N}\hat{A}_{n}J_{n}^{\prime}(\frac{\omega}{c}r_{e})e^{-in\phi_{e}},$
(9)

where $J_{n}^{\prime}(\cdot)$ denotes the derivative of the Bessel function
with respect to the argument. The pressure gradient along the $x$-axis and the
$y$-axis at $(x_{e},y_{e})$ can be reconstructed as

$\frac{\partial\hat{P}_{\mathrm{CH}}(x_{e},y_{e})}{\partial{x_{e}}}=\frac{\hat{P}_{\mathrm{CH}}(r_{e},\phi_{e})}{\partial{r_{e}}}\cos(\phi_{e}),$
(10)
$\frac{\partial\hat{P}_{\mathrm{CH}}(x_{e},y_{e})}{\partial{y_{e}}}=\frac{\hat{P}_{\mathrm{CH}}(r_{e},\phi_{e})}{\partial{r_{e}}}\sin(\phi_{e}).$
(11)

###  III.2 The SVD method

The SVD method [32, 33] regards a source at $(x_{s},y_{s})$ as a cluster of
virtual point sources whose positions are $\\{x_{s,j},y_{s,j}\\}_{j=1}^{J}$
and constructs two matrices with respect to the virtual point sources. The
first one is a matrix of transfer functions between the virtual point sources
and the microphones

$\mathbf{H}_{\mathrm{SM}}=\begin{bmatrix}H(x_{1},y_{1},x_{s,1},y_{s,1})&...&H(x_{1},y_{1},x_{s,J},y_{s,J})\\\
...&...&...\\\
H(x_{Q},y_{Q},x_{s,1},y_{s,1})&...&H(x_{Q},y_{Q},x_{s,J},y_{s,J})\end{bmatrix},\quad\;\;$
(12)

where the $q$-th row and $j$-th column entry,
$H(x_{q},y_{q},x_{s,j},y_{s,j})$, is the free-field transfer function [28]
between the virtual point source located at $(x_{s,j},y_{s,j})$ and the
microphone located at $(x_{q},y_{q})$ [28]. The second one is a matrix of
transfer functions between the virtual point sources and the pressure
estimation points

$\mathbf{H}_{\mathrm{SV}}=\begin{bmatrix}H(x_{1},y_{1},x_{s,1},y_{s,1})&...&H(x_{1},y_{1},x_{s,J},y_{s,J})\\\
...&...&...\\\
H(x_{V},y_{V},x_{s,1},y_{s,1})&...&H(x_{V},y_{V},x_{s,J},y_{s,J})\end{bmatrix},\quad$
(13)

where the $v$-th row and $j$-th column entry,
$H(x_{v},y_{v},x_{s,j},y_{s,j})$, is the free-space transfer function [28]
between the virtual point source located at $(x_{s,j},y_{s,j})$ and the
pressure estimation point located at $(x_{v},y_{v})$. The two matrices are
decomposed as [33]

$\mathbf{H}_{\mathrm{SM}}=\mathbf{U}_{\mathrm{SM}}\mathbf{\Sigma}_{\mathrm{SM}}\mathbf{V}_{\mathrm{SM}}^{*},$
(14)

and

$\mathbf{H}_{\mathrm{SV}}=\mathbf{U}_{\mathrm{SV}}\mathbf{\Sigma}_{\mathrm{SV}}\mathbf{V}_{\mathrm{SV}}^{*},$
(15)

where $(\cdot)^{*}$ denotes the complex conjugate operation;
$\mathbf{U}_{\mathrm{SM}}$ and $\mathbf{U}_{\mathrm{SV}}$ are unitary matrices
whose columns represent the basis functions of the receiver space;
$\mathbf{V}_{\mathrm{SM}}$ and $\mathbf{V}_{\mathrm{SV}}$ are unitary matrices
whose columns represent the basis functions of the source space; and
$\mathbf{\Sigma}_{\mathrm{SM}}$ and $\mathbf{\Sigma}_{\mathrm{SV}}$ are
diagonal matrices whose elements represent the capability of a source space
basis function to excite a receiver space basis function.

The SVD method [33] reconstructs the sound pressure
$\hat{\mathbf{P}}_{\mathrm{V}}=[\hat{P}(x_{1},y_{1}),...,\\\
\hat{P}(x_{v},y_{v}),...,\hat{P}(x_{V},y_{V})]^{\intercal}$ as

$\hat{\mathbf{P}}_{\mathrm{V}}=\mathbf{U}_{\mathrm{SV}}\mathbf{\Sigma}_{\mathrm{SV}}\mathbf{V}_{\mathrm{SV}}^{*}\mathbf{V}_{\mathrm{SM}}\mathbf{\Sigma}_{\mathrm{SM}}^{-1}\mathbf{U}_{\mathrm{SV}}^{*}\mathbf{P}_{\mathrm{M}},$
(16)

where $(\cdot)^{-1}$ denotes the matrix inversion operation. To reconstruct
the pressure gradient, the SVD method first reconstructs the pressure at
closely spaced positions $(x_{v}\pm\delta_{x},y_{v}\pm\delta_{y})$, and then
approximates the pressure gradient by

$\frac{\partial\hat{P}_{\mathrm{SVD}}(x_{v},y_{v})}{\partial{x_{v}}}\approx\frac{\hat{P}(x_{v}+\delta_{x},y_{v})-\hat{P}(x_{v}-\delta_{x},y_{v})}{2\delta_{x}},$
(17)
$\frac{\partial\hat{P}_{\mathrm{SVD}}(x_{v},y_{v})}{\partial{y_{v}}}\approx\frac{\hat{P}(x_{v},y_{v}+\delta_{y})-\hat{P}(x_{v},y_{v}-\delta_{y})}{2\delta_{y}}.$
(18)

The radial gradient can be reconstructed as

$\frac{\partial{\hat{P}_{\mathrm{SVD}(r_{e},\phi_{e})}}}{\partial{r_{e}}}=\frac{\partial{\hat{P}_{\mathrm{SVD}}(x_{e},y_{e})}}{\partial{x_{e}}}\cos(\phi_{e})\\\
+\frac{\partial{\hat{P}_{\mathrm{SVD}}(x_{e},y_{e})}}{\partial{y_{e}}}\sin(\phi_{e}).$
(19)

###  III.3 The acoustic-informed neural network method

(a) cAINN: One coupled network

(b) dAINN: Two decoupled networks

Figure 2: (color online) Architecture of the AINN: The inputs are Cartesian
coordinates, and the outputs are the real and imaginary parts of the pressure
reconstruction, denoted as $\mathcal{N}^{\Re}(x,y)$ and
$\mathcal{N}^{\Im}(x,y)$, respectively. The data loss and the PDE loss are
calculated with respect the pressure reconstructions and its Laplacian,
respectively.

This section proposes an AINN method for SFR. Instead of using complex-valued
networks [15], whose training can be complicated, to model the frequency-
domain complex-valued acoustic quantities, we design the AINN method using
real-valued networks.

Two designs of the AINN, depending on whether the real and imaginary parts of
the sound pressure are modeled separately or collaboratively, are investigated
in this paper. In the first design, the real and imaginary parts of the sound
pressure are modeled with a single network and with a single loss function, as
shown in Fig. 2 (a). In the second design, by contrast, the real and imaginary
parts of the sound pressure are modeled separately with two small networks
with individual loss functions, as illustrated in Fig. 2 (b). Hereinafter, the
two designs are referred to as the coupled AINN (cAINN) and decoupled AINN
(dAINN), respectively.

To model both the real and imaginary parts of the sound pressure, the one
network design, cAINN, should have more expressive power and hence more hidden
layers or more neurons in each hidden layer [9]. As shown in Fig. 2 (a) and
(b), this will makes the inter-connection between neurons of the one network
design more complicated than that of the two network design, dAINN. Although
in theory cAINN is capable of exploiting both the real and imaginary parts of
the sound pressure for the training process and hence may achieve better SFR
performance, its training is complicated and may not achieve the desired
performance in practice.

As shown in Fig. 2, for both cAINN and dAINN, the inputs are the Cartesian
coordinates $(x,y)$ and the outputs are the real and imaginary parts of the
reconstructed sound pressure, denoted as $\mathcal{N}^{\Re}(x,y)$ and
$\mathcal{N}^{\Im}(x,y)$, respectively. Similarly to the conventional data-
driven methods, a data loss is utilized to minimize the differences between
the reconstructed and the measured sound pressures at the measurement
locations. The data loss for the real parts of the sound pressures
$\mathfrak{L}_{\mathrm{d}}^{\Re}$ is given by

$\mathcal{L}_{\mathrm{d}}^{\Re}=\frac{1}{Q}\sum_{q=1}^{Q}(P^{\Re}(x_{q},y_{q})-\mathcal{N}^{\Re}(x_{q},y_{q}))^{2},$
(20)

where $P^{\Re}(x_{q},y_{q})$ and $\mathcal{N}^{\Re}(x_{q},y_{q})$ denote the
real parts of the measured and reconstructed sound pressure at
$\\{x_{q},y_{q}\\}_{q=1}^{Q}$, respectively.

To incorporate the acoustic information into the design of the neural network,
the Helmholtz Eq. (1) is utilized to calculate an extra partial differential
equation (PDE) loss. It is noted that, different from the data loss that is
calculated only for the measured locations, the PDE loss is calculated for
both the measured locations on the boundary and the SFR locations within the
ROI. By uniformly sampling ${D}$ positions within the ROI at
$\\{x_{d},y_{d}\\}_{d=1}^{D}$ and referring to Eqs. (1) and (2), the PDE loss
for the real part of the sound pressure $\mathfrak{L}_{\mathrm{p}}^{\Re}$ is
given by

$\begin{split}\mathcal{L}_{\mathrm{p}}^{\Re}=&\frac{1}{D}\sum_{d=1}^{D}\Big{(}\mathcal{N}^{\Re}(x_{d},y_{d})\\\
&+\frac{1}{(w/c)^{2}}\Big{[}\frac{\partial^{2}{}\mathcal{N}^{\Re}(x_{d},y_{d})}{\partial{x_{d}^{2}}}+\frac{\partial^{2}{}\mathcal{N}^{\Re}(x_{d},y_{d})}{\partial{y_{d}^{2}}}\Big{]}\Big{)}^{2},\quad\end{split}$
(21)

where $\mathcal{N}^{\Re}(x_{d},y_{d})$ denotes the real-part of the pressure
reconstruction at $\\{x_{d},y_{d}\\}$. The data loss and the PDE loss are
combined and results in the total loss. The definitions of the imaginary-part
data loss $\mathcal{L}_{\mathrm{d}}^{\Im}$ and PDE loss
$\mathcal{L}_{\mathrm{p}}^{\Im}$ are similar to Eqs. (20) and (21),
respectively, and are not shown for brevity.

For the cAINN in Fig. 2 (a), a single network is used to model the real and
imaginary parts of the sound pressure, and there are $2N$ neurons in each
hidden layer. The trainable parameters of the network are updated to minimize
a single total loss function

$\mathcal{L}=\mathcal{L}_{\mathrm{d}}^{\Re}+\mathcal{L}_{\mathrm{p}}^{\Re}+\mathcal{L}_{\mathrm{d}}^{\Im}+\mathcal{L}_{\mathrm{p}}^{\Im}.$
(22)

For the dAINN in Fig. 2 (b), two independent networks are used to model the
real and imaginary parts of the sound pressure, respectively, with $N$ neurons
in each hidden layer. The trainable parameters of two networks are updated to
minimize the real-part loss

$\mathcal{L}^{\Re}=\mathcal{L}_{\mathrm{d}}^{\Re}+\mathcal{L}_{\mathrm{p}}^{\Re},$
(23)

and the imaginary-part loss

$\mathcal{L}^{\Im}=\mathcal{L}_{\mathrm{d}}^{\Im}+\mathcal{L}_{\mathrm{p}}^{\Im},$
(24)

respectively.

Once trained, the AINN method can reconstruct the sound pressure at an
arbitrary position $(x_{e},y_{e})$ as
$\mathcal{N}(x_{e},y_{e})=\mathcal{N}^{\Re}(x_{e},y_{e})+i\mathcal{N}^{\Im}(x_{e},y_{e})$,
or equivalent $\mathcal{N}(r_{e},\phi_{e})$. The sound pressure gradient at
that position can be reconstructed as
${\partial{\mathcal{N}(x_{e},y_{e})}}/{\partial{x_{e}}}$ along the $x$
direction and as ${\partial{\mathcal{N}(x_{e},y_{e})}}/{\partial{y_{e}}}$
along the $y$ direction through differentiation on the network output. The
pressure gradient along the radial direction can be reconstructed as

$\frac{\partial{\mathcal{N}(r_{e},\phi_{e})}}{\partial{r_{e}}}=\frac{\partial{\mathcal{N}(x_{e},y_{e})}}{\partial{x_{e}}}\cos(\phi_{e})+\frac{\partial{\mathcal{N}(x_{e},y_{e})}}{\partial{y_{e}}}\sin(\phi_{e}).$
(25)

Here are comments on the AINN method and recommended configurations:

  1. 1.

Using $\tanh$ as the activation function:
We use $\tanh$ as the activation function for two reasons. First, the $\tanh$
function is a smooth function, whose second-order gradient can be computed.
This is necessary for the Laplacian operator, Eqs. (2) and (3). Second, the
$\tanh$ function outputs positive or negative values according to the input.
This makes it easier to model sound pressure, whose value can be either
positive or negative.

  2. 2.

Cartesian coordinates vs polar coordinates:
For the AINN method, we express sound pressures in Cartesian coordinates
instead of polar coordinates for two reasons. First, the presence of the $1/r$
term can make the Laplacian operator in polar coordinates, Eq. (3), to be
numerically unstable. Second, for the sound field on a circle, there is no
pressure variation along the radial direction. In the case, the AINN method is
unable to estimate the first- and second-order radial gradient needed for
calculating the Laplacian operator in polar coordinates, Eq. (3).

  3. 3.

Loss function:
The loss functions, Eqs. (22), (23), and (24), consist of both the data loss
and the PDE loss. The data loss prompts the network output to approximate the
measured sound pressure at positions $\\{x_{q},y_{q}\\}_{q=1}^{Q}$, which are
on the boundary of the ROI as shown in Fig. 1. The PDE loss, on the other
hand, regularizes the network output to conform with the Helmholtz equation at
positions $\\{x_{d},y_{d}\\}_{d=1}^{D}$ both on the boundary of and within the
ROI. $D$ should be chosen to be sufficiently large and thus the distance
between adjacent positions is at most half (ideally one tenth) of the wave-
length for the frequency of interest.

  4. 4.

Neuron number:
As shown in Eq. (8), the sound pressure can be expressed as a linear
combination of $2N+1$ CHs $\\{J_{n}(kr)e^{-in\phi}\\}_{n=-N}^{N}$, which are
solutions of the Helmholtz equation, Eq. (1). As shown in Fig. 2 (a) and (b),
the very same sound pressure can also be expressed as a linear combination of
the output of a number of neurons. This fact inspires us to set to the number
of neurons in hidden layers according to the CH decomposition of the sound
pressure. Specifically, for the cAINN, the one network design, the neuron
number is set to be $2N$, and for the dAINN, the two network design, the
neuron number is set to be $N$, where
$N=\lceil{2\pi{}f{}r/c}\rceil\approx{}\lceil{fr/50}\rceil$. For a sound field
whose size approximates the size of human heads, i.e., $r\approx 0.1$ m,
$N\leq 20$ for $f\leq 10$ kHz.

As shown in Sec. IV, an AINN with two hidden layers and less than 10 neurons
on each hidden layers is found to be sufficient for modeling the sound fields
around two arrays. Therefore, the AINN is compact and lightweight in
comparison to other learning-based methods [16, 14, 11, 10, 7, 17, 23, 12,
19].

(a) A 60-loudspeaker array with

a microphone array at the origin.

(b) A 64-microphone planar array: 28

exterior microphones ($\star$) and 36

interior microphones ($\bullet$).

(c) A dual-circular array:

30 exterior microphones ($\star$)

and 30 interior microphones ($\bullet$).

Figure 3: (color online) Experiment setup [31]: the loudspeaker array and the
microphone arrays.

(a) An anechoic chamber

(b) A medium room

(c) A small room

Figure 4: (color online) Experiment setup: three room environments.

##  IV Experiments

Numerical experiments were conducted to validate the performance of the
proposed AINN method, and to compare it with the CH and SVD methods.

###  IV.1 Data processing

The SFR methods were evaluated using the UTS multi-zone sound field
reproduction dataset [31]. The measurement set up is shown in Figs. 3 and 4.
The room impulse responses (RIRs) between a loudspeaker array (Fig. 3 (a)) and
two microphone arrays (Fig. 3 (b) and (c)) were measured in an anechoic
chamber (Fig. 4 (a)), a medium room (Fig. 4 (b)), and a small room (Fig. 4
(c)). The loudspeaker array and the microphone arrays were arranged
concentrically.

The loudspeaker array consisted of 60 Genelec 8010A Studio Monitors [31] as
shown in Fig. 3 (a), and was placed approximately at the center of each room
as shown in Fig. 4. The loudspeaker positions are

$x_{l}=-R\sin((2l-1)\pi/60)\;\mathrm{m},$ (26)
$y_{l}=R\cos((2l-1)\pi/60)\;\mathrm{m},$ (27)

where $l=1,2,...,60$.

Two microphone arrays were constructed using the DPA 4060 Series Miniature
Omni-directional microphones [31], which were calibrated at 1 kHz. The first
array is a 64 microphone planar array with side length of 0.28 m as shown in
Fig. 3 (b). The microphone positions are

$x_{m}=-0.14+0.04\times\mathrm{mod}(m-1,8)\;\mathrm{m},$ (28)
$y_{m}=0.14-0.04\times\lfloor{(m-1)}/{8}\rfloor\;\mathrm{m},$ (29)

where $\lfloor\cdot\rfloor$ is the floor operation, and $m=1,2,...,64$. The
second array is a dual-circular array with 30 microphones uniformly placed on
each circle as shown in Fig. 3 (c). The microphone positions are

$\theta_{m}=2\pi(m-1)/30,$ (30) $x_{m}=-r\times\sin(\theta_{m})\;\mathrm{m},$
(31) $y_{m}=r\times\cos(\theta_{m})\;\mathrm{m},$ (32)

where $m=1,2,...,30$, and $r=r_{1}=0.12$ m for the exterior circle and
$r=r_{2}=0.1$ m for the interior circle.

The RIRs were transformed into frequency-domain transfer functions through the
discrete Fourier transform [18], resulting in the sound pressure used for the
experiments. For more details about the measurement, please refer to [31].

###  IV.2 Implementation

CH method: The CH method was implemented based on Eqs. (4) - (11). Based on
Eq. (6), the dimensionalities of the sound field within the planar array are
$N=3,6,8$ for $f=1,2,3$ kHz, respectively; the dimensionalities of the sound
field within the dual-circular array are $N=3,5,7$ for $f=1,2,3$ kHz,
respectively.

SVD method: The SVD method was implemented based on Eqs. (12) - (19). For a
source (loudspeaker) located at $(x_{s},y_{s})$, the virtual point sources are
uniformly arranged around the source in a 0.1 m $\times$ 0.1 m square, and the
distance between two virtual point sources is 0.01 m. This amounts to 120
virtual point sources in total.

AINN method: We used the TensorFlow library, and initialized the trainable
parameters according to the Xavier initialization [8]. The ADAM algorithm with
a learning rate of 0.001 was used as the optimizer. The AINN method was
trained for $10^{5}$ epochs. The neuron number was set based on the
dimensionality of the sound field under CH decomposition. The hidden layer
number was set as $1$ for $f=1$ kHz and as $2$ for $f=2,3$ kHz based on a
trial-and-error process.

The measured sound pressures at the RIO boundary and their corresponding AINN
reconstructions were used for calculating the data loss. On the boundary of
and within the ROI, we uniformly selected positions with a $0.01$ m adjacent
distance, reconstructed the sound pressures at these positions, and calculated
the PDE loss with respect to them.

###  IV.3 Performance metrics

The performance of all methods was evaluated by the reconstruction error

$\xi=10\log
10\frac{\sum_{e=1}^{E}||P(x_{e},y_{e})-\hat{P}(x_{e},y_{e})||_{2}^{2}}{\sum_{e=1}^{E}||P(x_{e},y_{e})||_{2}^{2}},$
(33)

where $||\cdot||_{2}$ denotes the 2-norm, $P(x_{e},y_{e})$ and
$\hat{P}(x_{e},y_{e})$ are the ground-truth sound pressure and its
reconstruction, respectively, and $\\{x_{e},y_{e}\\}_{e=1}^{E}$ are
coordinates of the sound pressure reconstruction positions. The pressure
gradient reconstruction error was defined similarly to Eq. (33).

###  IV.4 Sound pressure reconstruction: loudspeaker 7

Based on the sound pressure measured by the 28 exterior microphones of the
planar array (Fig. 3 (b)), we reconstructed the sound pressure at the 36
interior microphones within the array. Figs. 5, 6, and 7 show the real part of
the sound pressure due to loudspeaker 7 (the black loudspeaker in Fig. 3 (a))
at 1, 2, and 3 kHz, respectively. The figures also show the reconstructions by
the CH method, the SVD method, the AINN method, and corresponding (both real
and imaginary parts) reconstruction errors. The results for the imaginary
parts are similar and are not shown for brevity.

Figure 5: (color online) Pressure reconstruction: reconstruct the pressure
within the planar array based on the measured pressure ($\star$) \- Anechoic
chamber.

Figure 6: (color online) Pressure reconstruction: reconstruct the pressure
within the planar array based on the measured pressure ($\star$) - medium
room.

Figure 7: (color online) Pressure reconstruction: reconstruct the pressure
within the planar array based on the measured pressure ($\star$) - small room.

cAINN vs dAINN: As shown in Figs. 5, 6, and 7, the dAINN, two network design,
outperforms the cAINN, one network design, at all frequencies and in all room
environments. Experiments for the cAINN with more or less neurons on hidden
layers, i.e., $\lceil{3N/2}\rceil$ or $3N$, were also conducted, and the
results were also inferior to the dAINN, thus are not shown for brevity. The
results demonstrate that two independent small neural networks are better than
a single large neural network for modeling the real and imaginary parts of the
sound pressure. Hereinafter, we focus only on the dAINN.

dAINN vs CH and SVD at 2 and 3 kHz: As shown in the second and third columns
of Figs. 5, 6, and 7, the SVD method performs better than the CH method and in
most of the cases for $f=2,3$ kHz, with a decrease in the overall
reconstruction error by 1.0 to 2.6 dB at 2 kHz and 1.2 to 4.1 dB 3 kHz across
the three rooms. This is because the prior information of the sound source
location was used in the SVD method [32, 33]. In contrast, the dAINN
outperforms the SVD method in all the tested rooms at $f=2,3$ kHz, though no
prior information about the sound source location was required by the dAINN
method. This could be attributed to the fact that the CH and SVD methods
relied on only the sound pressures measured on the edge of the planar array
for SFR. The measured pressures did not necessarily contain sufficient
information to fully determine the sound field within the planar array. The
dAINN, on the other hand, exploited the Helmholtz equation for regularizing
the SFR within the array through the PDE loss and reconstructed the sound
field within the planar array more accurately.

Pressure variation at 1 kHz: As shown in the first column of Figs. 5, 6, and
7, at 1 kHz all methods achieve lower than -10 dB reconstruction errors in the
anechoic chamber, but not in the medium room and the small room. This could be
attributed the fact that at 1 kHz in the medium (Fig. 6 (a0)) and the small
rooms (Fig. 7 (a0)), the pressure variations on the edge of the planar array
were so small that the microphones were unable to capture enough information
of the sound field, and thus the methods were unable to accurately reconstruct
the sound field.

###  IV.5 Sound pressure reconstruction: all loudspeakers

Figure 8: (color online) Pressure reconstruction error for all loudspeakers at
3 kHz in three rooms. Small, CH denote the pressure reconstruction error of
the CH method in the small room. Other labels can be interpreted similarly.

The SFR experiment was repeated over all the 60 loudspeakers. We set each
loudspeaker to the planar array’s transfer function as the ground truth, and
used the sound pressure measured by the exterior 28 microphones to reconstruct
the sound pressure at the interior 36 microphones within the planar array.
Fig. 8 compares the sound pressure reconstruction errors of the CH, SVD, and
dAINN methods in the three rooms for all loudspeakers at 3 kHz.

Fig. 8 shows that the reconstruction errors in the anechoic chamber are
smaller than those in other two rooms, as expected, due to the simpler pattern
of the sound field. The dAINN achieves the smallest average reconstruction
errors of -12.5, -8, and -7.2 dB, in the anechoic chamber, the medium room,
and the small room, respectively. Similar experiments were also conducted at 1
kHz and 2 kHz. Except that the reconstruction errors were larger for the
medium room and the small room at 1 kHz the results were similar to Fig. 8,
thus are not shown here for brevity.

###  IV.6 Pressure gradient reconstruction: loudspeaker 7

Figure 9: (color online) Pressure gradient reconstruction: reconstruct the
radial pressure gradient based the measurement of 30 microphones in the
exterior circle of the dual-circular array - Loudspeaker 7. The numbers in
each sub-figures denote the reconstruction errors by the SH, SVD, and dAINN
methods.

The sound pressures measured by the 30 microphones on the exterior circle of
the dual-circular array (Fig. 3 (c)) were used to reconstruct the radial
pressure gradients between the two circles. The ground truth radial pressure
gradients could not be directly measured by the set up shown in Figs. 3 and 4,
thus were approximated as

$\frac{\partial{P}(r,\theta_{m})}{\partial{r}}\approx\frac{{P}(r_{1},\theta_{m})-{P}(r_{2},\theta_{m})}{r_{1}-r_{2}},$
(34)

where $m=1,2,...,30$. The radial pressure gradients were reconstructed by the
CH method, the SVD method and the dAINN method through Eq. (9),Eq. (19), and
Eq. (25), respectively.

Except using the measured sound pressures to reconstruct the pressure
gradient, we conducted one more numerical experiment using simulated data. The
transfer functions between loudspeaker 7 and the dual-circular array were
simulated using the free-field Green’s function [28] with added white Gaussian
noise at a signal-to-noise (SNR) ratio of 20 dB to model disturbances. The
reconstruction errors are shown in Fig. 9. As shown in Fig. 9 (a1), (b1), and
(c1), the reconstruction errors of three methods are relatively small, i.e.,
$<-10$ dB, when using the simulated sound pressures for reconstruction.

However, as shown in Fig. 9 (a2) - (c4), when the measured sound pressures,
which are expected to have more measurement noises, are used, the
reconstruction errors of all the three methods show signs of degradation. The
data-dependent CH and SVD methods exhibit significant deviations from the
ground truth in the medium room and the small room. In contrast, by exploiting
the data-independent Helmholtz equation for regularization, the dAINN method
is less susceptible to the inherent disturbances of the measurement processes.
Thus, the dAINN method achieves the least pressure gradient reconstruction
errors in most of the cases.

Figure 10: (color online) Pressure gradient reconstruction error for all
loudspeakers at 3 kHz in three rooms. Small, CH denote the pressure gradient
reconstruction error of the CH method in the small room. Other labels can be
interpreted similarly.

###  IV.7 Pressure gradient reconstruction: all loudspeakers

The pressure gradient reconstruction experiment was repeated over all 60
loudspeakers. We set each loudspeaker to the dual-circular array’s transfer
function as the sound pressure, approximated the radial pressure gradient
according to Eq. (34), and reconstructed the pressure gradient in the same way
as in Sec. IV.6.

Fig. 10 shows reconstruction errors for all loudspeakers in the three rooms by
the CH , SVD , and dAINN methods at 3 kHz. The experimental results for 1 and
2 kHz showed similar results as Fig. 10, thus are not shown for brevity.
Comparing Fig. 10 with Fig. 8, we can see that the pressure gradient is more
challenging to reconstruct than the sound pressure. The dAINN achieves the
smallest average reconstruction errors of -11.5, -7.0, and -5.5 dB, in the
anechoic chamber, the medium room, and the small room, respectively. The
average reconstruction errors of the SVD method are about 2 dB higher than
those of the dAINN. The reconstruction errors of the CH method are the worst,
which exceed 0 dB in the small room.

It is noted that, in this paper, the pressure gradient was approximated by the
finite difference, Eq. (34). This may not be accurate in some circumstances
and may contribute to the relatively high reconstruction errors shown in Figs.
9 and 10. The velocity sensor [5] may be used for measuring the pressure
gradient and testing the performance of the pressure gradient reconstruction
in the future.

##  V Conclusion

This paper proposed a compact AINN method for SFR. A neural network was
designed to approximate the measured sound pressure and obey the Helmholtz
equation, which regularized the network to generate physically valid output at
and beyond the measurement positions. The performance of the AINN method was
validated by sound pressure and pressure gradient reconstruction experiments,
and outperformed both the CH and SVD methods. An extension of this work is to
exploit knowledge of the sound source(s) to further improve the performance of
the AINN method; this will be our topic for future work.

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