ARTIE Collaboration

# Measurement of the total neutron cross section on argon
in the 20 to 70 keV energy range

S. Andringa  Laboratório de Instrumentação e Física Experimental de Partículas
(LIP), Av. Prof. Gama Pinto, 2, 1649-003, Lisboa, Portugal  Y.Bezawada
University of California at Davis, Department of Physics and Astronomy, Davis,
CA 95616, U.S.A.  T. Erjavec  University of California at Davis, Department of
Physics and Astronomy, Davis, CA 95616, U.S.A.  J. He  University of
California at Davis, Department of Physics and Astronomy, Davis, CA 95616,
U.S.A.  J. Huang  University of California at Davis, Department of Physics and
Astronomy, Davis, CA 95616, U.S.A.  P. Koehler  Los Alamos National
Laboratory, Physics Division, Los Alamos, NM 87545, U.S.A.  M. Mocko  Los
Alamos National Laboratory, Physics Division, Los Alamos, NM 87545, U.S.A.  M.
Mulhearn  University of California at Davis, Department of Physics and
Astronomy, Davis, CA 95616, U.S.A.  L. Pagani  University of California at
Davis, Department of Physics and Astronomy, Davis, CA 95616, U.S.A.  E. Pantic
University of California at Davis, Department of Physics and Astronomy, Davis,
CA 95616, U.S.A.  L. Pickard  University of California at Davis, Department of
Physics and Astronomy, Davis, CA 95616, U.S.A.  R. Svoboda  University of
California at Davis, Department of Physics and Astronomy, Davis, CA 95616,
U.S.A.  J. Ullmann  Los Alamos National Laboratory, Physics Division, Los
Alamos, NM 87545, U.S.A.  J. Wang  University of California at Davis,
Department of Physics and Astronomy, Davis, CA 95616, U.S.A.  South Dakota
School of Mines and Technology, Physics Department, Rapid City, SD 57701 USA

###### Abstract

The cross section for neutron interactions on argon is an important design and
operational parameter for a number of neutrino, dark matter, and neutrinoless
double beta decay experiments which use liquid argon as a detection or
shielding medium. There is a discrepancy between the evaluated total cross
section in the $20$ to $70~{}\rm$ keV neutron kinetic energy region given in
the ENDF database and a single measurement conducted by an experiment with a
thin target (0.2 atoms/barn) optimized for higher cross sections. This gives
rise to significant uncertainty in the interaction length of neutrons in
liquid argon. This discrepancy is now resolved by new results presented here
from the Argon Resonance Transport Interaction Experiment (ARTIE), a thick
target experiment (3.3 atoms/barn) optimized for the small cross sections in
this energy region.

###### pacs:

28.20.Cz

##  I Introduction

Liquid argon (LAr) is used in a wide range of particle physics experiments
investigating neutrinos Antonello:2013ypa ; Acciarri:2016smi ; Abi:2017aow ;
Acciarri:2016ooe , dark matter DEAP:2019 ; Aalseth:2017fik , and neutrinoless
double beta decay Ackermann:2012xja ; Abgrall:2017syy . Achieving the
scientific objectives sought by these experiments relies on understanding the
transport of neutrons through LAr at a level of precision which has only
recently emerged as a critical experimental requirement. Recent studies have
shown that understanding the behavior of neutrons presents a special challenge
in liquid-argon-based experiments Ankowski_2015 ; Friedland_2019 ;
Gardiner_2021 .

The neutron-argon total cross section from the ENDF ENDF  evaluation has a
destructive interference feature at $57$ keV, which appears as a dip in the
cross section around this energy, where interaction length in LAr, for a
natural abundance of isotopes, is $30$ m. This is important, as at this energy
scale, neutrons only lose a small fraction of their kinetic energy in single
elastic collisions with relatively massive argon nuclei. Thus, even neutrons
with kinetic energy well above $57$ keV have a significant probability of
reaching the low cross section region with the resulting long interaction
length. The results of the most recent previous measurement Winters:1991 ,
contained in the EXFOR databaseexfor:2014 , are inconsistent with ENDF
evaluation in the region of this feature, with an inferred interaction length
of $4.2$ m. This discrepancy makes it impossible to reliably predict the
performance of LAr in transporting and/or shielding neutrons. This paper
presents the results of the Argon Resonant Transport Interaction Experiment
(ARTIE) which was designed specifically to resolve the neutron cross section
discrepancy.

##  II Experimental Method

The transmission through a target in a neutron beam $T(E)$ is defined as the
fraction of neutrons in a medium which pass through a distance $d$ without
scattering. This is related to the cross section by the equation:

$\sigma(E)=-\frac{m}{\rho_{eff}\,d}\,\ln\,T(E)$ (1)

Where $m$ is the mass of an argon atom, $d$ is the target thickness, and
$\rho_{eff}$ is the effective density. The ARTIE target was designed with a
thickness approximately twenty times larger than used in Ref. Winters:1991 .
Thus, the inconsistent cross section values reported by ENDF and Ref.
Winters:1991  result in a 20% difference in $T(E)$ at the dip energy. While
well-suited to the dip region, the target becomes essentially opaque at
energies where the cross section is higher, restricting our energy Region Of
Interest (ROI) to $20$ to $70$ keV. To make the measurement, Flight Path 13
(FP13) at the Lujan Neutron Scattering Center LANSCEBeam:1990  was used. FP13
has a total flight path of about 64 meters, allowing for excellent Time Of
Flight (TOF) energy resolution up to several hundred keV.

The ARTIE experimental configuration is shown in Fig. 1. The ARTIE target
consisted of a column of LAr of length $168$ cm and diameter $25$ mm, held at
atmospheric pressure, and contained in a vessel constructed from standard
components. The target was inserted into FP13 at a distance of about $31$ m
from the upper-tier liquid hydrogen moderator of the proton-accelerator-driven
pulsed neutron source. The proton beam current (proportional to the resulting
neutron flux) was monitored by a Current Transformer (CT), with the integral
output recorded every minute for relative beam normalization.

Figure 1: Configuration of the beam line elements. 1) Mark III LANSCE target
and modrator. 2) cadmium filter for suppression of thermal neutrons. 3) $\sim
30\;m$ beam pipe where carbon and aluminum filters were mounted. 4) brass
collimators. 5) ARTIE target. 6) $\sim 30\;m$ beam pipe. 7) 6-Li glass
detector at $64\;m$ followed by beam dump.

The ARTIE neutron detector was located at the end of the beam line, about $30$
m downstream of the target and consisted of a $9$ cm diameter by $1$ mm thick
6Li-glass scintillator, viewed edge-on by two RCA 8854 five-inch
photomultiplier tubes (PMTs). A triggered event was defined as a pulse above
threshold in either PMT. The data acquisition system (DAQ) recorded the time
and integrated PMT charge for each trigger.

Two-inch thick brass cylinders with $6$ mm holes in the center were used to
collimate the beam through the target. Two collimators were located upstream
of the target, and two downstream. These constrained the beam to the center of
the $25$ mm diameter target, and produced a beam spot with an $8$ cm diameter
at the $9$ cm diameter neutron detector. Final alignment of the collimators
was performed by first maximizing the DAQ-reported event rate. Collimators
were then fine-tuned in order to produce a symmetric, fully contained image on
storage-phosphor image plates at the detector location.

For insulation, the target was covered with a rigid foam designed for
cryogenic applications. Near each end an upward opening feeds into commercial
nalgene dewars. As argon boiled in the target, Gaseous Argon (GAr) vented
through the openings and was replaced by LAr from the dewars. During
operation, dewars were refilled about once per hour to ensure that the target
remained full. Data was collected during a two-week period, with most runs
being in either target-in (LAr fill) or target-out (GAr fill) mode. The use of
GAr during target-out runs was accounted for by defining the effective target
density as $\rho_{\rm eff}=\rho_{\rm in}-\rho_{\rm out}$. The transmission was
then experimentally determined as:

$T(E)=\frac{N_{\rm in}(E)-B_{\rm in}}{N_{\rm out}(E)-B_{\rm
out}}\cdot\frac{Q_{\rm out}}{Q_{\rm in}}$ (2)

where $N_{\rm in/out}$ is the number of neutrons observed during target-in/out
runs, $Q_{\rm in/out}$ is the time-integrated beam current from the CT
monitor, and $B_{\rm in/out}$ is the experimentally determined background
rate.

The collected data was subjected to both run quality and individual event
selection cuts. Firstly, to minimize the impact of potential non-linearity
between the CT sensor and the neutron beam intensity, the analysis only
included the $95\%$ of collected data taken while the CT was near the maximum
value. Secondly, data taken around the time of target filling were rejected
since during these times, a roughly 30% excursion in the beam-intensity
normalized neutron event rate was observed, which we surmise was caused by the
unavoidable spilling of LAr vapor into the brass collimators from the filling
dewar located above them. The rate returned to the nominal value about 15
minutes after the end of the fill. Thus, these filling times were removed from
the analysis by requiring that the post-fill rate return to at least 95% of
the pre-fill plateau value. This cut removed about 12% of the target-in data.
Thirdly, both detector PMTs were required to have pulses within a $100$ ns
coincidence window, and to pass a cut which removed re-triggered events (i.e.
two triggers from a single event). This last cut removed a negligible number
of actual signal events. Following these cuts, there were 197k events recorded
in the ROI for target-in runs, and 85k for target-out.

##  III Calibration

Energy: The neutron detector recorded a time ($t_{n}$) and the proton beam
pulse provided a start time ($t_{0}$). Together these gave a TOF which was
used to determine the velocity ($v$) and hence the kinetic energy ($E$) of
each event. These times were corrected for the average time a neutron
scattered inside the moderator ($t_{\rm mod}$), usually referred to as the
Moderator Function (MF). The MF had been previously determined for LANSCE via
Monte Carlo simulation moderator . In the ROI this correction is typically
1-2% with a smearing of roughly 10% about the mean. The velocity $v$ is then a
function of the TOF $t\equiv t_{n}-t_{0}$:

$v(t)=\frac{L_{\rm fit}}{t-t_{\rm mod}(t)+t_{\rm fit}}$ (3)

where $L_{\rm fit}$ and $t_{\rm fit}$ were parameters determined by fitting
LAr data to known resonances of aluminum and cadmium (both present in the beam
line) and argon. The best fit value of $L_{\rm fit}$ was $63.82\pm 0.06$ m
which agrees well with physical measurements made along the beam line. The
parameter $t_{\rm fit}$ (best fit $420\pm 29$ ns) accounted for time delays in
the detector, cables, and DAQ, as well as any residual difference between the
actual and simulated moderator response. In addition to the MF time-smearing,
the incident triangular-shaped neutron pulse had a FWHM of $125$ ns which led
to a $53$ ns uncertainty in $t_{0}$. These two factors dominated the energy
resolution.

Target Density: Since the LAr in the target was always slightly boiling, there
was always a small fraction of GAr present, which affected the overall target
density. Thus, a separate experiment to directly measure the density of the
liquid-gas mixture in situ was performed. The mass of the target assembly
($M$) as a function of dewar liquid height ($h$) is given by:
$M(h)=M_{0}+\left(\rho_{eff}-\rho_{air}\right)V(h)$ where $\rho_{eff}$ is the
effective density of the argon mixture, $\rho_{air}$ is the density of air,
$M_{0}$ is the mass of the empty target, and $V(h)$ is the volume of the
target as a function of liquid height. $M_{0}$ was measured using a precision
scale ($\pm 1$ g in the range of the roughly $25$ kg target assembly mass),
and $V(h)$ was determined by filling the target with known amounts of water
and noting the level on a steel ruler inside the dewar. The dry, empty target
was then filled with liquid argon while sitting on the precision scale. During
the subsequent boil off, a camera was used to simultaneously record the scale
mass $M$ and the liquid level $h$, which were then analyzed to give $M(h)$.
The observed boil off rate of $1.56$ L/hr during this test was consistent with
that observed during the actual neutron beam runs. A target-in density of
$\rho_{in}=1.318\pm 0.017$ g/cm3 was obtained, which included a correction for
ice-buildup on the target. This is $5.9$% lower than the nominal density NBS74
of pure liquid phase, and implies that this fraction of argon gas was mixed in
the target during the beam runs. This resulted in an effective density
$\rho_{eff}=3.30\pm 0.04$ atoms/barn. Additionally, a 1.3% upward adjustment
in LAr density was made to account for the difference in density at the
altitude of Los Alamos ($2300$ m) as compared to the lab where the density
test was performed ($16$ m). The uncertainty in the density measurement was
taken into account when calculating the overall experimental uncertainty.

##  IV BACKGROUNDS AND UNCERTAINTIES

Backgrounds: Measurement of the transmission from Eq. 2 relied on subtracting
background events, especially for the GAr runs. The two major backgrounds
were: (i) scattered neutrons from times earlier than the ROI that hit the
detector at random later times, and (ii) gammas from neutron capture on water
in the moderator. Other backgrounds such as “wrap-around” neutrons, prompt
moderator gammas, and random backgrounds from other beam lines were determined
to be insignificant.

To measure these major backgrounds, a standard technique Brown:2017accapp
MUCCIOLA2022100  that utilizes resonances that scatter or absorb nearly all
incident neutrons was used. Many beam line components are made of aluminum,
and thus there are aluminum resonance features present in the data at $5.9$,
$35$ and $88$ keV. In addition, dedicated runs were made with an additional
$2.54$ cm aluminum filter inserted into the beam to further enhance these
features. The background was then extracted at the resonance energy by
subtracting the expected counts given the calculated aluminum transmission
from the observed counts, given by

$B_{out}(E)=\frac{N_{in,f}(E)-R_{in,f/out}(E)\,N_{out}(E)\,T(E)}{R_{in,f/out}(E)\left(1-T(E)\right)}$
(4)

where $T(E)$ is the calculated effective transmission of neutrons with the
aluminum filter, $N(E)$ are the counts in the bins at the resonance dips at
$5.9$, $35$, and $88$ keV for the filtered case ($in,f$) and unfiltered case
($out$), and $R_{in,f/out}(E)$ is the ratio of filtered background to the
unfiltered background. This ratio is introduced to correct background
reduction due to filter attenuation.

Both the random-scatter neutron and the moderator capture gammas are seen to
be nearly flat in time within our ROI Stamat:2022  for our neutron beam. As
the filter attenuation effect is non-negligible for the $2.54$ cm aluminum
filter, it is corrected by $R_{in,f/out}(E)=B_{in,f}(E)/B_{out}(E)$ using an
external measurement of the background components from Stamat:2022  and
private communication with the authors privCommKoehler . The ratio is
$R_{in,f/out}(E_{ROI})=0.755_{-0.004}^{+0.021}$, with total uncertainties
determined by applying Eq. 4 for situations when $B_{out}(E)$ consists
entirely of gammas or neutrons.

For LAr fill runs, gammas from the moderator were heavily suppressed due to
target thickness. A small background still remained, and thus an argon
resonance at $77$ keV was used to evaluate this near the ROI. This background
is also expected to be flat in TOF.

Figure 2: The event rates and energy for a LAr and GAr filled target as a
function of TOF. Filled points are the extracted background rates.

Fig. 2 shows the GAr and LAr event rates as a function of TOF and energy. The
background rates are fit to a constant resulting in a background contribution
relative to the signal in the ROI of about 0.14% for LAr and 7.1% for GAr,
where the one-sigma uncertainties (dotted lines) are given by the flat fit to
all background measurements.

Long-term Beam Line Stability: The cross section calculation relies on the
proportional nature between neutrons detected and the total integrated beam
current, yet other than the 6Li detectors at the end of the beam line, there
is no monitor of the neutron flux once the proton beam strikes the tungsten
target. Since gaseous and LAr data runs involve different target setups, the
uncertainties associated with the neutron beam in the target hall, after the
neutrons are created, are not canceled naturally in Eq.2. To assess these
uncertainties, event rates normalized by beam current were analyzed as a
function of time for both air and LAr data, discussed below.

Air: The target was periodically run with air inside instead of liquid or
gaseous argon. These air data, due to similar densities and neutron cross
sections, was used as a surrogate for long-term beam line stability of GAr
data. This assessment includes any systematic effects from changing
atmospheric pressure. Three days of air data showed a daily modulation
correlated closely to outside air temperature, and consistent across energies
below, within, and above our ROI. Due to the close correlation to temperature,
we suspect the effect is due to misalignment from the thermal expansion and
contraction along the entire length of the beam line. The combined uncertainty
on median air event rate is taken as an asymmetric systematic of $+3.14\%$ and
$-3.93\%$ Erjavec2023 .

LAr: LAr data did not show the same modulation as the air data. A measurement
of the event rate as a function of time for LAr includes event rate decrease
from ice build-up on the kapton windows and also the cuts used to remove
periods of refilling. The combined uncertainty on median LAr event rate is
taken as an asymmetric systematic of $+0.69\%$, and $-1.06\%$ Erjavec2023 .

Target Density: As described in the calibration section, this was measured
with a systematic uncertainty of $1.3\%$.

After the selection cuts, several other systematic uncertainties were
determined to be negligible, including those from: non-linearity between the
beam intensity and CT measurement, dead time in the DAQ system Erjavec2023 ,
PMT afterpulsing Erjavec2023 , and contamination of the argon gas.

The absolute energy calibration was limited by the statistical uncertainties
on the fitted parameter $\delta(t_{\rm fit})=29~{}\rm$ ns, and the
contribution from $\delta(L_{\rm fit})=0.064~{}\rm$ m was negligible. No
systematic uncertainty for energy resolution was applied to the cross section
results, which are reported here as a function of measured energy.

The total systematic uncertainties on transmission and cross section are
estimated for each energy point by using a toy Monte Carlo simulation where
all relevant parameters are allowed to vary simultaneously around their
central values. The resulting spreads within the 68th percentile around the
central-value measurements of transmission and cross section are taken as the
total systematic uncertainties.

##  V Conclusions

Using the TOF data and measured backgrounds, the transmission $T$ is
calculated from Eq. 2 and the cross section from Eq. 1. The central value of
each TOF bin is converted to energy using Eq 3. Fig. 3 shows the transmission
as a function of energy for the range $20$-$70$ keV.

Figure 3: The measured transmission of argon compared to EXFORexfor:2014  and
ENDF. The points are the central values and the bars represent the statistical
uncertainty. The shaded grey regions are the total uncertainty, including
systematics. For most energies, statistical error bars are smaller than the
symbols.

The measured neutron-argon total cross section as a function of kinetic energy
is shown in Fig. 4. The dashed red and blue lines represent the EXFOR database
exfor:2014  and ENDF evaluation, and the solid blue line is the ENDF
prediction smeared by the ARTIE energy resolution. The beam energy resolution
does not allow us to see the sharp features near $60$ keV.

Figure 4: Neutron-argon total cross section as a function of energy. Also
shown are the EXFOR evaluation and the ENDF evaluation smeared by the ARTIE
energy resolution.

It can be seen that, for energies below 40 keV the data is in good agreement
with the EXFOR database, Winters:1991 , while the ENDF evaluation shows
slightly higher cross sections. Between 40 and 70 keV, ARTIE data is in better
agreement with ENDF, confirming the existence of the cross section dip. As a
cross-check of our experimental setup and analysis technique, the transmission
for carbon was measured from data collected with two $0.125\pm 0.010$” thick
carbon (99.999% purity) disks CarbonLesker  attached to the target while
filled with GAr. There was good agreement between the measured
($0.73^{+0.03}_{-0.05}$) and predicted ($0.72$) transmission in the ROI with
$\chi^{2}/\text{NDF}=2.7/6$, which confirms the analysis’ methodology.
In conclusion, our results confirm a dip in the total cross section in the
region of $50-60$ keV. The point at $54.9$ keV is found to have the lowest
cross section of
$\sigma=0.0339\pm{0.0087(stat.})_{-0.0112}^{+0.0175}(sys.)~{}b$. These results
can now be used to reliably predict neutron transport to the level required in
current and future experiments using liquid argon.

##  VI ACKNOWLEDGEMENTS

Work at UC Davis was supported by the U.S. Department of Energy (DOE) Office
of Science under award number DE-SC0009999, and by the DOE National Nuclear
Security Administration through the Nuclear Science and Security Consortium
under award number DE-NA0003180. Support for LIP was from the Fundação para a
Ciência e a Tecnologia, I.P., project CERN/FIS-PAR/0012/2019. Major support
was also provided by the U.S. Department of Energy through the Los Alamos
National Laboratory. Los Alamos National Laboratory is operated by Triad
National Security, LLC, for the National Nuclear Security Administration of
U.S. Department of Energy (Contract No. 89233218CNA000001). Finally, we
gratefully acknowledge the logistical and technical support and the access to
laboratory infrastructure provided to us by LANSCE and its personnel.

##  VII Bibliography

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Enriching Diagrams with Algebraic Operations

Alejandro Villoria Henning Basold Alfons Laarman

Villoria et al.

Leiden Institute of Advanced Computer Science, The Netherlands

{a.d.villoria.gonzalez, h.basold<EMAIL_ADDRESS>
In this paper, we extend diagrammatic reasoning in monoidal categories with algebraic
operations and equations.
We achieve this by considering monoidal categories that are enriched in the category
of Eilenberg-Moore algebras for a monad.
Under the condition that this monad is monoidal and affine, we construct an adjunction between
symmetric monoidal categories and symmetric monoidal categories enriched over
algebras for the monad.
This allows us to devise an extension, and its semantics, of the ZX-calculus with probabilistic
choices by freely enriching over convex algebras, which are the algebras of the
finite distribution monad.
We show how this construction can be used for diagrammatic reasoning of noise in quantum systems.

§ INTRODUCTION

Monoidal categories are one way of generalising algebraic reasoning and they can be used to draw
intuitive diagrams that encapsulate this reasoning graphically.
That monoidal categories are a powerful abstraction has been demonstrated in countless areas,
such as linear logic [18] or
quantum mechanics [1], just to name a few, and are amenable to
graphical reasoning [46].
Another abstraction of algebraic reasoning are monads <cit.>
and their algebras, or
representations thereof [20, 34],
which are distinct from monoidal
categories in that identities (like associativity) always hold strictly and they allow rather
arbitrary algebraic operations.
In this paper, we set out to combine these two approaches into one framework, in which
monoidal category diagrams can be composed not only sequentially and with a tensor product but
also with additional algebraic operations.

[bend angle=20, x=0.5cm, y=0.5cm]
[probabilistic split] (s) ;
[gate, below left = 0.9 and 0.3 of s] (g) $G$;
[gate, below right = 0.9 and 0.3 of s] (e) $E$;
[probabilistic join, below right = 0.9 and 0.3 of g] (j) ;

[above = 0.3 of s] (i);
[below = 0.3 of j] (o);

[right = 0.1 of i] $A$;
[right = 0.1 of o] $B$;

(s) to[bend right] node[left] $0.9$ (g);
(g) to[bend right] (j);
(s) to[bend left] node[right] $0.1$ (e);
(e) to[bend left] (j);
(i) – (s);
(j) – (o);

[bend angle=20, x=0.5cm, y=0.5cm]

[probabilistic split] (s) ;
[gate, below left = 0.9 and 0.3 of s] (g1) $G_{1}$;
[gate, below right = 0.9 and 0.3 of s] (e) $E$;
[probabilistic join, below right = 0.9 and 0.3 of g] (j) ;

[above = 0.3 of s] (i1);
[below = 0.3 of j] (o1);

[right = 0.1 of i1] $A$;
[right = 0.1 of o1] $B$;

(s) to[bend right] node[left] $0.9$ (g1);
(g1) to[bend right] (j);
(s) to[bend left] node[right] $0.1$ (e);
(e) to[bend left] (j);
(i1) – (s);
(j) – (o1);

[right = 0.1 of e] (tens) $\otimes$;

[gate, right = 0.1 of tens] (g2) $G_{2}$;
let 1 = (i1), 2 = (o1), 3 = (g2)
coordinate (i2) at (3,1)
coordinate (o2) at (3,2)
(i2) – (g2);
(g2) – (o2);

[right = 0.1 of i2] $C$;
[right = 0.1 of o2] $D$;

[right = 0.1 of g2] (eq) $=$;

[gate, right = 0.1 of eq] (g12) $G_{1}$;
[gate, right = 0.1 of g12] (g21) $G_{2}$;
[gate, right = 0.1 of g21] (e2) $E$;
[gate, right = 0.1 of e2] (g22) $G_{2}$;
[probabilistic split, above right = 0.9 and 0 of g21 ] (s2) ;
[probabilistic join, below right = 0.9 and 0 of g21] (j2) ;

[above = 0.3 of s2] (i3);
[below = 0.3 of j2] (o3);

[right = 0.1 of i3] $A \otimes C$;
[right = 0.1 of o3] $B \otimes D$;

[above right = 0.5 and 0 of g12] (ms1);
[above left = 0.5 and 0 of g22] (ms2);
[double] (s2) – node[left, inner sep = 10pt, pos = 0.1] $0.9$ (ms1);
(ms1) to[bend right] node[left, inner sep = 4pt] $A$ (g12);
(ms1) to[bend left] node[right, inner sep = 4pt] $C$ (g21);
[double] (s2) – node[right, inner sep = 10pt, pos = 0.1] $0.1$ (ms2);
(ms2) to[bend right] node[left, inner sep = 4pt] $A$ (e2);
(ms2) to[bend left] node[right, inner sep = 4pt] $C$ (g22);

[below right = 0.5 and 0 of g12] (mj1);
[below left = 0.5 and 0 of g22] (mj2);
[double] (mj1) – (j2);
[double] (mj2) – (j2);
(g12) to[bend right] node[left, inner sep = 4pt] $B$ (mj1);
(g21) to[bend left] node[right, inner sep = 4pt] $D$ (mj1);
(e2) to[bend right] node[left, inner sep = 4pt] $B$ (mj2);
(g22) to[bend left] node[right, inner sep = 4pt] $D$ (mj2);

[double] (s2) – (i3);
[double] (j2) – (o3);

Left: Probabilistic mix of a gate $G$ with an error $E$.
Right: Interaction of tensor and convex sum, where double wires visually indicate a
tensor product

One such operation is the formation of convex combinations, which can be used to create
a probabilistic mix of two or more diagrams.
This occurs, for instance, when reasoning about the behaviour of noise in quantum circuits.
<Ref> shows on the left two gates, one called $G$ and one called $E$ that,
respectively, model the wanted behaviour and a possible error.
These two gates are mixed, where $G$ gets a probability of $0.9$ and $E$ of $0.1$.
The trapezoids delimit the combination of the gates, and $A$ and $B$ are the input and output types
of the gates[We read diagrams from top to bottom.].
In monoidal categories, the gates in the picture represent morphisms $G, E \from A \to B$
and our aim is then to interpret the trapezoid block as a convex sum $G +_{0.9} E$ of these
morphisms, where we define $G +_{p} E = p G + (1-p) E$.
Such sums should also nicely interact with the tensor product.
For instance, if $G_{1} \from A \to B$ and $G_{2} \from C \to D$ are gates, then an identity such
as $(G_{1} +_{0.9} E) \otimes G_{2} = (G_{1}  \otimes G_{2}) +_{0.9} (E \otimes G_{2})$
should hold for these morphisms of type $A \otimes C \to B \otimes D$, see
<Ref> on the right.
Having an operation to form convex combinations together with intuitive identities enables
reasoning about e.g. probabilistic combination and noise in quantum circuits.

The difficulty lies then in combining monoidal diagrams with algebraic operations such that this
interaction is coherent with both the monoidal identities and the algebraic identities.
We will handle this difficulty by using enriched monoidal categories, where the enrichment
yields the algebraic operations and the monoidal structure the parallel composition.
More precisely, we will assume that algebraic theory is given form of a monad $T$ and that the
monoidal categories are enriched over the Eilenberg-Moore category $\emt$ of algebras for this
Our aim in this paper is to construct for an arbitrary monoidal category $\CC$ an $\emt$-enriched
monoidal category $F\CC$ that is free in the sense that there is an inclusion
$\iota_{\CC} \from \CC \to (F\CC)_0$ into the underlying category of $F\CC$ and for every
$\emt$-enriched monoidal category $\eD$ and monoidal functor $G \from \CC \to \eD_{0}$, there is a
unique $\emt$-enriched monoidal functor $\bar{G}_0 \from F\CC \to \eD$ that makes the following
diagram commute.

\begin{equation*}
% https://q.uiver.app/#q=WzAsMyxbMCwwLCIoRlxcQ0MpXzAiXSxbMSwwLCJcXGVEXzAiXSxbMCwxLCJcXENDIl0sWzIsMSwiRyIsMl0sWzIsMCwiXFxpb3RhX3tcXENDfSJdLFswLDEsIlxcYmFye0d9XzAiXV0=&macro_url=https%3A%2F%2Fgist.githubusercontent.com%2Fhbasold%2F5b205ad0b469224b4b006c401b7872bd%2Fraw%2F349e29a93d61269cf30340b0fd265dd92be38e27%2Fgistfile1.txt
\begin{tikzcd}
{(F\CC)_0} & {\eD_0} \\
\CC
\arrow["G"', from=2-1, to=1-2]
\arrow["{\iota_{\CC}}", from=2-1, to=1-1]
\arrow["{\bar{G}_0}", from=1-1, to=1-2]
\end{tikzcd}
\end{equation*}

This free construction does not work for all monads, but we show that the free enrichment
always exists for monoidal affine monads.

Specifically, we contribute a definition of monoidal diagrams that are enriched with algebraic
operations and a construction of free enrichment over algebras for monoidal affine monads.
This enables composing diagrams in arbitrary monoidal categories with algebraic operations.
We show how the theory can be applied to obtain convex combinations for ZX-diagrams and what the
resulting identities of diagrams are.
By exploiting the mapping property of the free enrichment, we obtain automatically sound
interpretations of these operations and identities.

Related Work

ZX-diagrams are universal in the sense that they can technically represent any linear map
between Hilbert spaces of dimension $\bC^{2^n}$ [14].
Indeed, sums and linear/convex combinations [23, 48]
of ZX-diagrams can be encoded within the language, but in practice these representations
oftentimes lead to either very large diagrams or to diagrams that do not reveal upon visual
inspection the (linear/convex) structure that the diagram is technically representing,
which in return lessens the advantages gained by reasoning in terms of abstract graphical
Our perspective of using enrichment keeps the abstraction barrier and thus makes reasoning about
convex combinations of diagrams tractable.
In general, our theory covers also linear combinations such as
[49, 38]
and other, so far unexplored, algebraic operations such as those of join-semilattices.
Moreover, the identities that have to be crafted carefully by hand and proven to be sound
fall automatically out of our theory.
Other related work is that of Sheet diagrams [15]
and Tape diagrams [5],
recently developed graphical languages for rig categories,
which are categories with two monoidal structures – one for addition and one for multiplication.

The paper is organised as follows.
We start by introducing notation and recalling some background of enriched and
monoidal categories in <Ref>.
In <Ref>, we establish the necessary theory to define categories enriched
over Eilenberg-Moore algebras and we construct a free enrichment over those algebras.
Our next step in <Ref> is to extend these definitions and the free construction to
also include monoidal structures on categories, which ensures that these enrichment and monoidal
structure coherently interact.
<Ref> is devoted to applying our theory to enrich ZX-diagrams with convex sums to reason
about probabilistic processes such as quantum noise.
We conclude the paper with directions for future work in <Ref>.

§ BACKGROUND

In this section we recall some needed definitions concerning category theory, for which we
assume some (basic) familiriaty, which otherwise we refer the reader to [33].
For brevity, we sometimes omit coherence conditions that some constructions have to satisfy.

We denote the collection of objects of a category C as $|\mathbf{C}|$,
and the morphisms from object $A$ to $B$ as $\mathbf{C}(A,B)$.
A monoidal category $(\mathbf{C},\otimes,\mathit{I})$ is a category C together with
a functor $\otimes:\mathbf{C} \times \mathbf{C} \to \mathbf{C}$ called the tensor product
and an object $\mathit{I} \in |\mathbf{C}|$ called the tensor unit subject
to some conditions [26].
We will often refer to a monoidal category $(\mathbf{C},\otimes,\mathit{I})$ as just C.
A monoidal category is a symmetric monoidal category (SMC) when it also has a
braiding $\sigma_{A,B}: A\otimes B \to B\otimes A$ such that
$\sigma_{B,A}\circ \sigma_{A,B} = \text{Id}_{A\otimes B}$.

Given a monoidal category $(\VC, \times, \timesUnit)$, a $\VC$-(enriched) category $\eC$ consists

* a class $\obj{\eC}$ of objects,

* for each pair $A,B \in \obj{\eC}$, an object $\eC(A,B) \in \obj{\VC}$ that we refer to as the

* for all objects $A,B,C \in \obj{\eC}$, a composition morphism
$\enr{\circ} \from \eC(B,C) \times \eC(A,B) \to \eC(A,C)$
in $\mathbf{V}$, and

* for all $A \in \obj{\eC}$, an identity element $j_A \from  \timesUnit \to \eC(A,A)$

subject to associativity and unit axioms [26].
We say that $\VC$ is the base of enrichment for $\eC$.
A way to look at the above definition is that we construct a $\VC$-enriched category
$\eC$ by identifying collections of morphisms of some category $\mathbf{C}$ as objects from
$\VC$, which we are able to compose by using the tensor product of $\VC$.
The most well-known example is that of locally small categories, in which the morphisms
between two objects form a set, and thus we can see them as objects in the monoidal category
($\mathbf{Set},\times,\timesUnit$) for $\times$ the Cartesian product and $\timesUnit$ the singleton

With a suitable definition of $\VC$-functors and $\VC$-natural transformation, $\VC$-categories
organise themselves into a 2-category [26], that we denote
by $\VCat$.
This category is also a symmetric monoidal category as follows.
We define for $\VC$-categories $\eC$ and $\eD$ a $\VC$-category $\eC \otimes \eD$ with objects
$\obj{\eC \otimes \eD} = \obj{\eC} \times \obj{\eD}$ and hom-objects
$(\eC \otimes \eD)((A, B), (C, D)) = \eC(A, C) \times \eD(B, D)$.
The unit is given by
$j_{(A, B)}
= \timesUnit
\cong \timesUnit \times \timesUnit
\xrightarrow{u_{A} \times v_{B}} \eC(A, A) \times \eD(B,B)$
in terms of the units $u$ of $\eC$ and $v$ of $\eD$.
Similarly, one also defines the composition for $\eC \otimes \eD$ in terms of the composition
morphisms of $\eC$ and $\eD$, appealing to the symmetry in $\VC$ <cit.>.
The tensor product also extends to $\VC$-functors and $\VC$-natural transformations,
which makes it a 2-functor.
Finally, one defines $\otimesUnit$ to be the unit $\VC$-category with one object $0$ and
$I(0,0) = \timesUnit$ and we thus obtain, with suitable definitions of associators etc., a
symmetric monoidal 2-category $(\VCat, \otimes, \otimesUnit)$.

Most of the categories we are interested in are also dagger-compact
categories ($\dagger\text{-}CC$).
These are SMC with some additional structure. First, they are equipped with an endofunctor
$\dagger: \mathbf{C}^{op} \to \mathbf{C}$, that satisfies $(\text{Id}_A)^\dagger = \text{Id}_A$,
$(g \circ f)^\dagger = f^\dagger \circ g^\dagger$, $(f^\dagger)^\dagger=f$, and
$(f \otimes g)^\dagger = f^\dagger \otimes g^\dagger$.
And secondly, for every object $A$ there exists a dual $A^*$ such that there exists
unit $\eta_A: I\to A \otimes A^*$ and counit $\epsilon_A: A^*\otimes A \to I$
morphisms subject to some conditions [19].

We are interested in categories that let us reason about quantum mechanics.
One of them is FdHilb, the category of finite dimensional Hilbert spaces of the form
$\bC^n$ and linear maps as morphisms.
The category $\qbit$ is the (full) subcategory of FdHilb with objects Hilbert spaces of
the form $\bC^{2^n}$ and linear maps.
Similarly, the category $\cpm$ [47] has objects $\bC^{2^n}$ and
morphisms completely positive linear maps between them [12].
We usually work in $\qbit$ when reasoning about pure quantum evolutions and in $\cpm$ when
impure quantum evolutions (such as noise) can take place.
All of these categories are $\dagger\text{-}CC$, with the monoidal structure $\otimes$ given by
the usual Kronecker product of vector spaces and the dagger $\dagger$ being the conjugate transpose.

§ ALGEBRAIC ENRICHMENT

In this section, we are going to recall the concept of monoidal and affine monads,
and discuss some properties of the Eilenberg-Moore category of a monad.
We also start applying the Distribution monad and the Multiset monad to
running examples that will be of interest in later sections.

The Distribution monad $\md$ is the functor in the category of sets
$\md:\SetC\to\SetC$ that maps a set $A$ to the set $\md(A)$ of (finitely supported)
probability distributions over elements of $A$.
We write probability distributions as formal convex sums: $\sum_{a} p_a [a]\in \md(A)$ such that
$a\in A, p_a \in [0,1]$, and $\sum_a p_a = 1$.
$\md$ acts on a morphism $f$ by simply applying $f$ to the underlying set:
$(\md f)(\sum_{a} p_a [a]) = \sum_{a} p_a [f(a)]$.
The unit of $\md$ is the map $\eta: A\to\md(A): a \mapsto 1[a]$
(the Dirac distribution) and the multiplication
$\mu$ of $\md$ “flattens” a distribution of distributions by multiplying the probabilities together:
$\mu: \md(\md(A)) \to \md(A): \sum_q p_q [\sum_a q_a [a]]\mapsto \sum_{a}r_a[a]$ where
$r_a = \sum_q p_q q_a$ [22].
Moreover, the monad $\md$ is monoidal, meaning that there is a map:

$$\nabla: \md(A)\times\md(B)\to\md(A\times B): \left( \sum_{a} p_a [a], \sum_{b} p_b [b] \right) \mapsto \sum_{a,b} p_ap_b [(a,b)]$$

for every $A,B\in |\mathbf{Set}|$.
It is also affine, meaning that $\md(\timesUnit) \cong \timesUnit$.

If $\md$ is a monad for expressing convex combinations of elements of a set,
the Multiset monad $\mm$ is its analogue for linear combinations with coefficients
over some semiring $S$.

We recall that given any monad $\mt$ in $\CC$ we can construct its Eilenberg-Moore category,
with objects $\mt$-algebras of the form $(A,\alpha_A)$ for $A\in\obj{\CC}$ and
$\mt$-action $\alpha_A:\mt(A)\to A$ such that
$\alpha_A \circ T(\alpha_{T(A)}) = \alpha_A \circ \mu_A$ and $\alpha_A \circ \eta_A = \text{Id}_A$.
Algebra homomorphisms $f:(A,\alpha_A)\to (B,\alpha_B)$ are morphisms of the underlying objects
$f: A\to B$ that commute with the action: $f \circ \alpha_A = \alpha_B \circ T(f)$.
The identity and composition follow from the ones for the underlying objects [33].

For a monad $\mt$ on $\CC$, we have that $\emt$ is complete whenever $\CC$ is complete.
Cocompleteness is not as immediate, but if $\CC = \SetC$ then $\emt$ is also cocomplete
[3, 21].
This makes $\emt$ over monads on $\SetC$ a complete and cocomplete category and
in particular, $\emt$ has reflexive coequalizers, which we use to define the tensor
product of algebras.

When $\mt$ is a monoidal monad on a monoidal category $(\CC,\otimes,I)$,
the tensor product of $\mt$ algebras $(A,a), (B,b)$, denoted $(A,a)\otimes_T(B,b)$,
is (if it exists) defined as the following coequalizer diagram [45, 9].

\begin{equation}
\label{eq:coeq}
% https://q.uiver.app/#q=WzAsMyxbMCwwLCJGKFQoQSlcXG90aW1lcyBUKEIpKSJdLFsyLDAsIkYoQVxcb3RpbWVzIEIpIl0sWzMsMCwiKEEsYSlcXG90aW1lc19UIChCLGIpIl0sWzAsMSwiXFxtdVxcY2RvdCBGKFxcbmFibGEpIiwwLHsib2Zmc2V0IjotMn1dLFswLDEsIkYoYVxcb3RpbWVzIGIpIiwyLHsib2Zmc2V0IjoyfV0sWzEsMiwicSJdXQ==
\begin{tikzcd}
{F(T(A)\otimes T(B))} && {F(A\otimes B)} & {(A,a)\otimes_T (B,b)}
\arrow["{\mu\cdot F(\nabla)}", shift left=2, from=1-1, to=1-3]
\arrow["{F(a\otimes b)}"', shift right=2, from=1-1, to=1-3]
\arrow["q", from=1-3, to=1-4].
\end{tikzcd}
\end{equation}

Where $F:\CC\to \emt: A\mapsto (\mt(A),\mu)$ is the left adjoint to the
forgetful functor $U:\emt\to\CC: (A,\alpha_A)\mapsto A$ that maps objects to
their free algebras over $\mt$.
Given that we need $\emt$ to be monoidal in order to use it as a base of enrichment,
diagram (<ref>) above is a convenient representation of the tensor product of algebras.
The rest of the structure to make $\emt$ a (symmetric) monoidal category follows under certain
conditions, in particular when $(\CC,\otimes,I)$ is a closed (S)MC and the coequalizer
(<ref>) exists for all algebras $(A,a), (B,b)$ [45].

We can define (symmetric) monoidal structure in the category of free algebras as follows.
Using (<ref>) for the category of free algebras over a monoidal monad, we have that
the following diagram forms a coequalizer.

\begin{equation}
\label{eq:freecoeq}
% https://q.uiver.app/#q=WzAsMyxbMCwwLCJGKFRUKEEpXFxvdGltZXMgVFQoQikpIl0sWzIsMCwiRihUQVxcb3RpbWVzIFRCKSJdLFs0LDAsIkYoQVxcb3RpbWVzIEIpIl0sWzAsMSwiXFxtdVxcY2RvdCBGKFxcbmFibGEpIiwwLHsib2Zmc2V0IjotMn1dLFswLDEsIkYoXFxtdVxcb3RpbWVzIFxcbXUpIiwyLHsib2Zmc2V0IjoyfV0sWzEsMiwiXFxtdVxcY2RvdCBGKFxcbmFibGEpIl1d
\begin{tikzcd}
{F(TT(A)\otimes TT(B))} && {F(TA\otimes TB)} && {F(A\otimes B)}
\arrow["{\mu\cdot F(\nabla)}", shift left=2, from=1-1, to=1-3]
\arrow["{F(\mu\otimes \mu)}"', shift right=2, from=1-1, to=1-3]
\arrow["{\mu\cdot F(\nabla)}", from=1-3, to=1-5],
\end{tikzcd}
\end{equation}

making $(\mt A,\mu)\otimes_\mt (\mt B,\mu) := F(A \otimes B)$ <cit.>.
The monoidal unit is $T(I)$, while the associator, unitor, and symmetry (if present) are
the images of the ones in $(\CC,\otimes,I)$ under $F$.

A functor $F: (\VC_1,\otimes_{\VC_1},I_{\VC_1}) \to (\VC_2,\otimes_{\VC_2},I_{\VC_2})$
between two monoidal categories can be lifted to a $2$-functor
$F_*: \ECatC{\VC_1}\to \ECatC{\VC_2}$ [7].
This is called a change of enriching, where we turn a $\VC_1$-category into a
Indeed, given a $\VC_1$-category $\eC$, we can construct the $\VC_2$-category
$F_*\eC$ by defining $\obj{F_*\eC}:= \obj{\eC}$ and, for every $A, B \in \obj{F_*\eC}$,
the hom-objects are $F_*\eC(A,B) := F(\eC(A,B))$ with composition and identity element
following from the ones in $\eC$ under $F$.

The following lemma states explictly the case when one of the enriching categories is $\SetC$.

Let $(V,\otimes,I)$ be a closed symmetric monoidal category with coproducts.
Then the forgetful functor $\VC (I,-):\VC\to\SetC$ has a left adjoint $F$ that is
a strong morphism of symmetric monoidal categories with natural isomorphisms $\mu, \epsilon$.
Both functors induce a forgetful $2$-functor $\VC (I,-)_*:\VCat\to\CatC$ and
a left adjoint $F_*$.

A monoidal monad $\mt$ on $(\SetC,\times,\timesUnit)$ endows the category
of $\mt$-algebras $\emt$ of bicomplete (complete and cocomplete), closed symmetric monoidal
This allows to lift the free-forgetful adjunction of <Ref> as a
change of enriching between $\emt$-categories and $\SetC$-categories for a monoidal $\mt$.

The proof follows from <Ref> and previous arguments.
Given that $\emt$ for $T$ a monad on $\SetC$ is bicomplete, then coequalizer (<ref>)
exists and we can define tensor products of algebras.
We can then make $\emt$ a (symmetric) monoidal category given that $\SetC$ is closed symmetric
Finally, $\emt$ can be made into a closed category following [30] given
that $\SetC$ has equalizers.
We can then use <Ref> to create a change of enriching between
$\emt$-categories and $\SetC$-categories.

Let us construct an example for <Ref> and relate it to graphical languages.
If we have a locally small monoidal category $\CC$ with morphisms $f,g:A\to B, h:B\to C$,
represented graphically as
\scalebox{0.8}{\tikzfig{figures/g}}, \scalebox{0.8}{\tikzfig{figures/h}}$,
we can freely enrich $\CC$ over $\emd$ following the change of enriching category method above.
Then, we can realize graphically a probabilistic process involving $f$ and $g$ with probability
$0.9$ and $0.1$ respectively, followed by applying $h$ deterministically
(that is, it occurs with probability $1$) afterwards as follows

\begin{equation}\label{eq:example1}
\scalebox{1.10}{\tikzfig{figures/example1}}
\end{equation}

Intuitively, we distinguish between probabilistic and deterministic processes by having the former
enclosed within distribution brackets
(in the same way as we would represent them as a formal sum $0.9[f] + 0.1[g]$),
that we choose to depict as trapezoids in this paper.
Deterministic processes are depicted without the bracket enclosing mostly as syntactic sugar,
otherwise they would simply have a single choice with probability $1$.
We can see how wires can have weights inside this environment, and how each wire
represents a probabilistic choice.
Intuition also tells us that we could for example rewrite the diagram above by distributing
$h$ over the two probabilistic branches.
We will discuss in later sections which graphical rules capture the interactions present
in these enriched categories.
It is natural to then ask if our enriched category $\CC$ maintained its monoidal structure,
and if other desired properties (such as braiding and symmetry, if present) would still hold too.
We will address this in the next section.

§ ENRICHED MONOIDAL CATEGORIES

Recall from <Ref> that a symmetric monoidal category $\VC$ gives rise to
a symmetric monoidal 2-category $(\VCat, \otimes, \otimesUnit)$ of $\VC$-categories.
This structure allows us to define an enriched (symmetric) monoidal category to be a (symmetric)
pseudo-monoid in
$\VCat$ [17], which amounts to the following explicit
definition [31, 37].
Let us denote by $S$ the symmetry isomorphism of $\VCat$.
A symmetric monoidal $\VC$-category is a tuple
$(\eC, \odot, \odotUnit, \alpha, \lambda, \rho, \sigma)$
consisting of:

* a $\VC$-functor $\odotUnit \from \otimesUnit \to \eC$

* a $\VC$-functor $\odot \from \eC \otimes \eC \to \eC$

* $\VC$-natural isomorphisms
$\alpha \from \odot \comp (\odot \otimes \id_{\eC}) \to \odot \comp (\id_{\eC} \otimes \odot)$
$\lambda \from \odot \comp (U \otimes \id_{\eC}) \to \id_{\eC}$ (left unitor),
$\rho \from \odot \comp (\id_{\eC} \otimes U) \to \id_{\eC}$ (right unitor), and
$\sigma \from \odot \to \odot \comp S$ (symmetry)

subject to the expected coherence axioms [17].
A symmetric $\VC$-monoidal functor $(\eC, \odot_{1}, U_{1}) \to (\eD, \odot_{2}, U_{2})$
is a lax pseudo-monoid homomorphism, which means that it
consists of a $\VC$-functor $F \from \eC \to \eC$ and two $\VC$-natural transformations
$F^{0} \from U_{2} \to F(U_{1})$ and
$F^{2} \from \odot_{2} \comp (F \otimes F) \to F \comp \odot_{1}$
that are coherent with the associators, unitors and
symmetries [31].
Together, symmetric $\VC$-monoidal categories and functors form a category $\SMVCat$.

Our goal is now to lift the adjunction between enriched categories to also include
enriched monoidal structure.
To this end, we will need to talk about lax monoidal strict $2$-functors, which are
tuples $(G, G^{0}, G^{2})$ where $G \from \VC \to \WC$ is a strict functor of $2$-categories
and  $(G, G^{0}, G^{2}) \from (\VC, \otimes, \otimesUnit) \to (\WC, \times, \timesUnit)$
is a lax monoidal functor on the underlying $1$-categories.
If $G$ has a strict left adjoint $F$, which is also strong monoidal,
we say that $(F, F^{0}, F^{2}) \dashv (G, G^{0}, G^{2})$ is a monoidal $2$-adjunction if the
mates [25] of $G^{0}$ and $G^{2}$ are the inverses of
$F^{0}$ and $F^{2}$, respectively, as in the following equations, where $\beta_{A,B}$ is the
natural isomorphism $\WC(A, GB) \xrightarrow{\cong} \VC(FA, B)$ and $\eta$ the unit of the

\begin{equation*}
(F^{0})^{-1} = \beta_{\otimesUnit, \timesUnit}(G^{0})
\quad \text{and} \quad
(F^{2}_{A,B})^{-1} = \beta(G^{2}_{FA,FB}) \comp F(\eta_{A} \otimes \eta_{B})
\end{equation*}

The following theorem is proven by chasing the appropriate diagrams.
Only constructing the adjunction is somewhat involved and makes use of the above equations
in an essential way.

A lax monoidal strict $2$-functor
$(G, G^{0}, G^{2}) \from (\VC, \otimes, \otimesUnit) \to (\WC, \times, \timesUnit)$
induces a $2$-functor
$\PMon(G) \from \PMon(\VC, \otimes, \otimesUnit) \to \PMon(\WC, \times, \timesUnit)$
between $2$-categories of pseudo-monoids, lax homomorphisms and $2$-cells that are compatible
with the homomorphism structures.
If $G$ has a monoidal left adjoint $F$, then $\PMon(F)$ is left adjoint to $\PMon(G)$.
Finally, if the monoidal categories and functors are symmetric, then the adjunction
can be improved to one between symmetric pseudo-monoids.

The following theorem, which shows that the change of enrichment extends to monoidal enriched
categories, follows from <Ref> using that
$\SMVCat = \PMon(\ECatC{\VC}, \otimes, \otimesUnit)$
and that the change of enrichment gives a lax monoidal $2$-adjunction [7, 16].

If $(G, G^{0}, G^{2}) \from (\VC, \otimes, \otimesUnit) \to (\WC, \times, \timesUnit)$
is a symmetric monoidal functor between symmetric monoidal categories
with a monoidal left adjoint $(F, F^{0}, F^{2})$, then there are adjunctions that commute
with the forgetful functors as in the following diagram.

\begin{equation*}
% https://q.uiver.app/#q=WzAsNCxbMCwxLCJcXFZDYXQiXSxbMSwxLCJcXENhdEMiXSxbMSwwLCJcXFNNQ2F0QyJdLFswLDAsIlxcU01WQ2F0Il0sWzAsMSwiKC0pXzAiLDIseyJjdXJ2ZSI6MX1dLFszLDIsIigtKV8wIiwyLHsiY3VydmUiOjF9XSxbMiwzLCJGIiwyLHsiY3VydmUiOjF9XSxbMSwwLCJGIiwyLHsiY3VydmUiOjF9XSxbMywwXSxbMiwxXSxbNCw3LCIiLDIseyJsZXZlbCI6MSwic3R5bGUiOnsibmFtZSI6ImFkanVuY3Rpb24ifX1dLFs1LDYsIiIsMix7ImxldmVsIjoxLCJzdHlsZSI6eyJuYW1lIjoiYWRqdW5jdGlvbiJ9fV1d&macro_url=https%3A%2F%2Fgist.githubusercontent.com%2Fhbasold%2F5b205ad0b469224b4b006c401b7872bd%2Fraw%2F349e29a93d61269cf30340b0fd265dd92be38e27%2Fgistfile1.txt
\begin{tikzcd}[column sep=large]
\SMVCat & \SMECatC{\WC} \\
\VCat & \ECatC{\WC}
\arrow[""{name=0, anchor=center, inner sep=0}, "{\chEnr{G}}"', curve={height=6pt}, from=2-1, to=2-2]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\chEnr{G}}"', curve={height=6pt}, from=1-1, to=1-2]
\arrow[""{name=2, anchor=center, inner sep=0}, "\chEnr{F}"', curve={height=6pt}, from=1-2, to=1-1]
\arrow[""{name=3, anchor=center, inner sep=0}, "\chEnr{F}"', curve={height=6pt}, from=2-2, to=2-1]
\arrow[from=1-1, to=2-1]
\arrow[from=1-2, to=2-2]
\arrow["\dashv"{anchor=center, rotate=270}, draw=none, from=0, to=3]
\arrow["\dashv"{anchor=center, rotate=270}, draw=none, from=1, to=2]
\end{tikzcd}
\end{equation*}

From the this theorem we can derive the following corollary, which is our main tool to build monoidal
diagrams that are enriched with algebraic operations.

For any monoidal affine monad $\mt$ on $\SetC$, the free-underlying adjunction
$(-)_0:\EmtCat \rightleftarrows \CatC:F_*$ lifts to an adjunction between the $2$-categories
of symmetric monoidal $\emt$-enriched and $\SetC$-enriched categories
$(-)_0: \SMemtCat \rightleftarrows \SMCatC: F_*$.
More explicitly, given a monoidal affine monad $T$ on $\SetC$ and a SMC
$(\CC, \otimes, I, \alpha, \lambda, \rho, \sigma)$ we can construct the freely enriched
symmetric monoidal $\emt$-category
$(\eC, \etens, \enr{I}, \enr{\alpha}, \enr{\lambda}, \enr{\rho}, \enr{\sigma})$.

Knowing that we keep the symmetric monoidal structure after doing the free enrichment,
we can justify drawing parallel composition of probabilistic operations in diagram form.
Continuing example (<ref>), let us have another probabilistic process
in which $f'$ and $g'$ occur with probabilities $0.7$ and $0.3$ (respectively) parallelly
composed. Then we can draw the following picture.

\begin{equation}\label{eq:example2}
\scalebox{1.10}{\tikzfig{figures/example2}}
\end{equation}

§ APPLICATIONS: ZX-CALCULUS

In this section, we show an example application of the categorical constructions
of the previous sections.
In particular, we are interested in demonstrating how we can take the Distribution monad and
enrich the quantum categories of interest for reasoning about probabilistic processes
in quantum systems.
Most importantly, we show how the ZX-calculus, a graphical calculus for reasoning
about quantum processes, can be appropiately extended to accomodate for the extra
structure on said categories and how additional graphical rewrite rules capture the
interaction of probabilistic and deterministic quantum operations.
We begin with a general introduction to quantum computing and ZX-calculus, and then follow
with the enrichment of our categories of interest, together with the introduction of the
extended notation, and we finish by giving an example on how we can use this for diagrammatic
reasoning of noise in quantum systems.

§.§ Quantum computing

When referring to quantum systems and operations, we have to make a distinction whenever we take
impure operations into account.
In the pure states formalisms, quantum states are normalized vectors in a Hilbert space of
dimension $\bC^{2^n}$, with $n$ the number of qubits (quantum bits) of the system.
It is common to use Dirac bra-ket notation to represent states, for example some important
single-qubit states are $\ket 0 = \begin{bsmallmatrix} 1\\0 \end{bsmallmatrix},
\ket1 = \begin{bsmallmatrix} 0\\1 \end{bsmallmatrix},
\ket+ = \frac{1}{\sqrt{2}} \cdot (\ket0 + \ket1),
\ket- = \frac{1}{\sqrt{2}}\cdot (\ket0 - \ket1)$.
We operate on qubits by means of performing unitary transformations $U$ on the quantum states.
A multi-qubit quantum system with states $\ket\psi$ and $\ket\phi$ corresponds
to the tensor (Kronecker) product of the quantum states: $\ket\psi \otimes \ket\phi$.
We will represent the $n$-fold tensor product of a state $\ket\psi$ by $\ket{\psi^{n}}$.
Simultaneous (but independent) operations also follow from tensoring unitaries.

When we take into consideration the possibility of applying non-unitary operations we require
a more general framework, which is the density matrix and completely positive maps
In this case, quantum states are positive semi-definite Hermitian matrices $\rho$ of trace one.
We write them as $\rho = \sum_i p_i \ketbra{\psi_i}{\psi_i}$
(where $\bra{\psi_i} = \ket{\psi_i}^\dagger$, for $\dagger$ the conjugate transpose), that is,
a statistical ensemble of quantum states $\ket{\psi_i}$ (as density matrices) with probability $p_i$.
Operations on density matrices are completely positive (CP) maps of the form
$\Phi: \rho \to \sum_i K_i\rho K_i^\dagger$ with the condition $\sum_i K_iK_i^\dagger \le 1$
(notice how unitary maps fall inside this description too).
When we want to reason about quantum systems in presence of noise, we then have to use the density matrix
and CP map formalism. For more information on quantum computing we refer the reader to [39],
and for a more categorical introduction to [19].

§.§ The ZX-calculus

The ZX-calculus [14] is a graphical language for reasoning about quantum
states and processes as diagrams.
The language consists of a set of generators, which are the green and
red[Light and dark in grayscale, respectively.] spiders
(also called $Z$ and $X$ spiders), the
Hadamard box, the identity wire, the swap, the cup,
the cap, and the empty diagram.
In <ref> we can see the generators of the ZX-calculus and their signature,
with input wire(s) coming from the top and outputs going to the bottom.
Spiders have a phase $\alpha\in [0,2\pi)$,
which as we will see later is omitted when $\alpha=0$.
We can also see how to sequentially compose ($\circ$) arbitrary diagrams by connecting inputs
with outputs, and how to parallely compose diagrams
(as a tensor product $\otimes$) by placing them side by side.

ZX-diagrams generators and how to compose them.

Each of the generators has a standard interpretation $\llbracket\cdot \rrbracket$
as a linear map in $\bC^{2^n}$ that we can find in <ref>.

Standard interpretation of ZX-diagrams.

Categorically, ZX-diagrams form the category ZX with $|\mathbf{ZX}|=\mathbb{N}$
(where some $n\in\mathbb{N}$ is the number of wires,
which we can think of as an $n$-qubit quantum system) and morphisms being the generators.
The standard interpreation is a (monoidal) functor
$\llbracket \cdot \rrbracket: \mathbf{ZX}\to \qbit$ that acts on objects as
$\llbracket n \rrbracket=n$ and on morphisms as defined in
<ref> [51].

ZX-diagrams come with a set of rewrite rules that form the ZX-calculus.
These rewrite rules let us transform a diagram into a different one while preserving the semantics
(i.e. the interpretation).
We have collected the rules in <ref>.
There is also an important additional rule that can be summarized as the
only connectivity matters rule, which states that we can deform diagrams at will without
changing their meaning, as long as we maintain the connectivity between the generators unchanged.
For a thorough explanation of each rule we refer the reader to [14, 50].

ZX-calculus ruleset.
All rules also hold when swapping the colors of the spiders.
In $(eu)$ we omit the calculation of the angles, which can be found
in [50].

The ZX-calculus satisfies important properties. ZX-diagrams are universal,
meaning that any linear map $f$ of the form $f: \bC^{2^n}\to\bC^{2^m}$
can be represented as a ZX-diagram.
The rewrite rules are sound, meaning that they do not change the interpretation
of the diagram as a linear map.
They are also complete, which ensures that if two diagrams have the same interpretation,
the ruleset is powerful enough to always let us transform one diagram into the other.
These properties ensure that the ZX-calculus can be used as a tool for reasoning about
quantum computing, as it has been already demonstrated in tasks such as quantum circuit
optimisation [28], verification of quantum circuits [40],
simulation [29], and as a reasoning tool [4, 27].

In (<ref>) we have example one- and two-qubit gates as ZX-diagrams.
We also see the computational basis $\{|0\rangle,|1\rangle\}$ and Hadamard basis
$\{|0\rangle,|1\rangle\}$ states.

\begin{equation}\label{eq:zxExample}
\scalebox{0.8}{\tikzfig{figures/zxExample}}
\end{equation}

§.§ Enriching the categories $\qbit$ and $\cpm$

Our motivation is to highlight certain type of relevant physical phenomena
(probabilistic processes) that are present in quantum systems within our categories.
It is then natural to use the Distribution monad $\md$ together with the construction
explained in the previous sections to enrich our categories for quantum reasoning.

Indeed, we take $\cpm$ and perform a free enrichment over $\md$.
What we get is the category $(F_*\cpm, \etens, I)$ consisting of the same objects as $\cpm$
and morphisms (incl. identity) for objects $A,B$ the free algebras over $\md$ of the
hom-set $\cpm(A,B)$.
Composites of morphisms are the free algebra over the composite in $\cpm$,
and the SMC structure is preserved thanks to <Ref>.

We also define the non-freely enriched category $\ecpm$ so we can
interpret probability distributions as CP maps.
For this, we define SMC-structure in the non-free $(\emd,\otimes_\md,\md(\timesUnit))$,
with a tensor product of algebras
defined by the coequalizer (<ref>), with a more detailed description
in <Ref>.
The category $(\ecpm,\odot,\odotUnit)$ has the same objects as $\cpm$ and for every pair of
objects $A,B$ the hom-object is an algebra $(\cpm(A,B),\alpha)$.
Composition of hom-objects follows from composition in $\cpm$, and for an object $A$
the identity element is $j_A:(\timesUnit,\alpha)\to(\operatorname{Id}_A,\alpha)$.
We define now its symmetric monoidal structure following the definition of enriched SMC from
the beginning of <Ref>.
The tensor product $\odot$ on objects is the same as in $\cpm$, and on hom-objects applies
the tensor product in $\cpm$ to the underlying sets:
$\odot: (\cpm(A,A'),\alpha)\otimes_\md (\cpm(B,B'),\alpha)$ $\to$
$ (\cpm(A\otimes B, A'\otimes B'),\alpha)$.
The unit $U$ is the same as in $\cpm$.
The associator, unitors, and the symmetry all follow from applying the ones in $\cpm$.

In the following sections, we will interpret ZX-diagrams into $F_*\cpm$
as probability distributions of CP maps. From there, to interpret
probability distributions as CP maps themselves, we define the functor
$\llangle\cdot\rrangle: F_*\cpm\to\ecpm$ that sends objects to themselves and applies
the monad algebra to hom-objects i.e. we “evaluate” a probability distribution
over CP maps by multiplying the probabilities with the corresponding map and then
adding all maps together.

Technically, we can also enrich $\qbit$ in the same way as we did with $\cpm$,
but density matrices and CP maps are the more sensible choices to talk about
probabilistic mixtures of operations.
On the other hand, enriching $\qbit$ (or $\cpm$) over algebras of the multiset monad $\mm$
leads to an enrichment over commutative monoids that exposes
addition of linear maps [19].
This was recently formulated in [38, 49]
as a way to “split” parameterised Pauli rotation
gates in ZX-calculus in such a way that the parameter relocates from its place inside the spider as a phase
to a scalar on a wire using the identity
$e^{i\alpha P} = \cos\alpha I + i \sin\alpha P$ for $P$ a Pauli matrix (or any matrix satisfying

§.§ Enriched ZX-diagrams and their interpretation

In the same manner as the ZX-calculus is a language for reasoning in $\qbit$,
we can create a graphical language with extra structure to reason in our enriched categories.
Since we are going to be enriching $\cpm$, we first need to see how to turn the ZX-calculus
into a graphical language for CP maps.
This is done straightforwardly by adding a discard operation $\gnd$
to the list of generators of <ref> plus additional rewrite rules
(that we choose to omit here) stating that isometries are causal maps [12].
The interpretation  of a ZX-diagram $D$ as a CP map is then a superoperator
$\rho \to \llbracket D \rrbracket\ \rho\ \llbracket D \rrbracket^\dagger$,
for $\llbracket D \rrbracket$ the standard interpretation of $D$
as in <ref> [8].

We then construct an enriched graphical language for $\ecpm$ by building on top of
the ZX-calculus for CP maps. The notation will be similar to the running examples
we have given throughout the text
(cf. <ref>,<ref>,<ref>).
The main idea is as follows.
We take the generators of the ZX-calculus and allow them to be freely wrapped between
opening $\scalebox{0.8}{\tikzfig{figures/openingbracket}}$
and closing $\scalebox{0.8}{\tikzfig{figures/closingbracket}}$
distribution brackets.

Intuitively, we interpret diagrams that are within distribution brackets as a probabilistic
mixture of operations: diagrams placed side by side correspond to different probabilistic
choices with some weight attached to the corresponding wires.
Within each choice, sequential (and as we will see later, parallel) composition is allowed.
The main difference to the usual graphical languages for monoidal categories
is that the parallel composition of each choice does not correspond
to the tensor product.
In a way, we also subsume ZX-diagrams by drawing diagrams that are not enclosed by distribution
brackets, which are then interpreted as an operation that occurs with probability $1$.

For example, we can represent the single-qubit depolarizing channel [39]
$\Phi:\rho\mapsto (1-p)\rho + \frac{p}{3}(X\rho X + Y\rho Y + Z\rho Z)$
that leaves a quantum state $\rho$ unchanged with probability $(1-p)$ or applies
an $X,Y$ or $Z$ error with probability $\frac{p}{3}$ each with the diagram on the left in

Left: Diagrammatic representation of the depolarizing channel.

Right: Diagrammatic representation of a mixture of two-qubit gates.

We need to take extra care when handling scalars inside the brackets.
Indeed, as opposed to what is usually allowed in graphical languages for SMCs, freely moving
scalars around would change the meaning of the diagram.
A simple solution is to consider the probabilities (and any scalar factor if present, such as the
imaginary unit in the example above) to be bound to the wires themselves.
An alternative done in [49, 38] is to encapsulate each
operation in “bubbles”.

A probabilistic mixture of operations with multiple inputs or outputs looks similarly to
the $1$-to-$1$ case, with the caveat that we need to be more careful in the positioning of the
wires as to distinguish between tensor product and probabilistic choice.
[One could use scalable notation [11]
to allow wires to be multi-qubit quantum registers.
This would help with the distinction when diagrams are larger in practice. ]
For example, if we want to represent applying the CNOT gate and the $CZ$ gate with probabilities
$p$ and $1-p$ respectively we would get the diagram on the right in <Ref>.

We can consider this extra notation as the result of a free enrichment of $\zx$ over $\emd$
giving us the category of enriched ZX-diagrams $F_*\zx$.
We can then define the interpretation $\llbracket\cdot\rrbracket_\md$ of an $\emd$-enriched
diagram as a monoidal functor from the category of enriched ZX-diagrams to $\ecpm$.
This functor factors through $F_*\cpm$ as follows:

\[\begin{tikzcd}
{F_*\zx} & {F_*\cpm} \\
& \ecpm
\arrow["{\llbracket \cdot\rrbracket_\md}"', from=1-1, to=2-2]
\arrow["{\llangle \cdot\rrangle}", from=1-2, to=2-2]
\arrow["{\llangle \cdot\rrangle_*}", from=1-1, to=1-2]
\end{tikzcd}\]

Where $\llangle \cdot\rrangle_*$ interprets an enriched ZX-diagram as a probabilistic
mixture of operations which is then evaluated by $\llangle \cdot\rrangle$ as explained
in <Ref>.
An example of the interpretation of an arbitrary distribution of ZX-diagrams of arbitrary
size can be seen in (<ref>).
When using the multiset monad $\mm$ instead the interpretation $\llbracket \cdot\rrbracket_\mm$
is similar.

\begin{equation}\label{eq:interpretation}
\tikzfig{figures/interpretation}
\end{equation}

Enriched ZX-diagrams are universal, that is, any morphism in $\ecpm$ can be represented
by an enriched ZX-diagram.
Indeed, since $\ecpm$ is still made of CP maps between Hilbert spaces,
we can use universality of the ZX-calculus alone to represent any morphism in $\ecpm$.

§.§ Additional rules for enriched ZX-diagrams

With the new notation we can have new rewrite rules too, some of which were already
introduced in [49, 38]
((es), (ep), (ec), and (e$\delta$)) for the case of
linear combinations.
We will display them here, including additional rules.
The ruleset of the enriched ZX-calculus for the distribution and multiset monads is the same as
the one for ZX-calculus plus additional rules that capture the interaction between sums,
products, tensor products, and scalars.
We can see the additional rules arising from the enrichment in <Ref>.
Intuitively, each rule states the following:

Additional rules for the enriched ZX-calculus, alongside the ones of <Ref>.
Diagrams $D, D'$ are arbitrary ZX-diagrams and weights $p,q$ are probabilities.

* (es): The enriched sequential composition rule shows how to sequentially
compose distributions. Intuitively this rule follows from products distributing over addition.

* (ep): The enriched parallel composition rule is the same as (es), but
for parallel composition instead of sequential.

* (ec): The enriched commutativity rule shows that bracketed diagrams are invariant
under permutation of the branches.

* (e$\delta$): The enriched Dirac delta distribution rule provides a shorthand for the trivial Dirac delta distribution.

* (e$+$): The enriched addition rule shows that we can remove a branch if it is identical
to some other by adding the probabilities.

* (e$0$): The enriched $0$-probability rule allows us to remove branches with $0$ probability .

Rules (es),(ep),(ec) and (e$\delta$) were proven to be sound
in [38] but in the context of linear combinations of diagrams interpreted in $\qbit$.
We show that these rules still hold as an enrichment in $\emd$ and interpreted in $\ecpm$
in <Ref>.
Finding a complete ruleset (i.e. one that can show $D_1 = D_2$ whenever
$\llbracket D_1 \rrbracket_\md = \llbracket D_2 \rrbracket_\md$) for enriched diagrams remains
to be done.
A possible direction to tackle this problem would be to translate enriched diagrams into
ZXW [48] diagrams, which is a complete diagrammatic language with a
W-spider that can encode addition of phases.
Another alternative would be to translate into the controlled form of [23].

We conclude with a demonstration on how we can use this extension of the ZX-calculus
to study the effectiveness of Quantum Error Mitigation (QEM) techniques for different noise models.
Quantum Error Mitigation [10] are the series of techniques that are used to reduce the
effects of noise in near-term quantum systems.
One such technique is Symmetry Verification [6],
which states that given a Hamiltonian
(Hermitian operator that determines the evolution of a system) $\hat{H}$,
and a symmetry $S$ (an operator that commutes with $\hat{H}$ i.e. $[\hat{H},S] = \hat{H}S - S\hat{H} = 0$),
one can perform measurements of $S$ to verify if the state that (ideally) evolves under $\hat{H}$
was affected by errors.
Indeed, under the assumption that the initial state is a $(+1)$ eigenvector of $S$, then it will
stay that way under ideal evolution under $\hat{H}$.
This implies that if there is an error $E$ that anti-commutes with $S$ (i.e. $\{E,S\} = ES + SE = 0$)
at some point in the computation, we can measure $S$ to detect a change in the eigenvalue.
Symmetry verification then proposes to perform a postselection on the result $(+1)$,
meaning that we discard computations that give a $(-1)$ outcome when measuring $S$.

Given a noisy state $\rho_{\text{noisy}}$ and a symmetry $S$, the probability of outcome $(+1)$
when measuring $S$ is given by $p(+1)=\tr(P_{+1}\rho_{noisy})$, for $P_{+1} = \frac{I+S}{2}$ the projector
onto the $(+1)$ eigenspace of $S$ and $\tr$ the trace operator.
This value tells us then with which probability the measurement “accepts” a noisy state, and can
be used to compare the effectivenesss of differente choices of $S$ given a certain noise
model [24].
Let us consider $\rho_{\text{noisy}} = \Phi(U\ket{0})$ for $\Phi$ the depolarizing noise channel
and $U$ some single-qubit unitary – in other words, we have a single layer of depolarizing
noise at the end of our computation.
For simplicity, let us further assume that our state before the depolarizing channel
is the $(+1)$ eigenvector of some Pauli operator e.g. the Pauli $X$, then we have $U=H$
(the Hadamard gate) and we can draw $p(+1)=\tr(\frac{I+S}{2}\rho_{noisy})$ diagrammatically
(up to scalar factor, see <Ref>) as the following diagram:

\begin{equation*}\label{eq:exampleSV}
\tikzfig{figures/exampleSV}
\end{equation*}

From top to bottom, the diagram represents applying $H$ to the $\ket{0}$ state, followed by
a depolarizing noise channel and the verification of $X$ in the form of a CNOT gate controlled
on an auxiliary qubit.
The auxiliary qubit on the right starts in the $\ket{0}$ state and has a Hadamard gate applied
to it before and after the CNOT.
It is then postselected into $\bra{0}$, which is the corresponding state for the $(+1)$
The last operation in the form of $\gnd$ corresponds to the trace.
The full diagrammatic calculation is in <Ref>.
With similar diagrams, we can study diagrammatically how well different QEM techniques mitigate
certain noise models, and apply them to representations of quantum algorithms that, for example,
have one layer of errors for every time step.

§ DISCUSSION AND FUTURE WORK

In this work, we have shown how to construct freely enriched symmetric monoidal categories
over the algebras of a commutative affine monad.
We have then taken this construction and developed a graphical language that captures the
additional algebraic structure of the morphisms for the case of the Distribution monad.
We then show how we can use this to study classical probabilistic processes in quantum systems,
a highly relevant type of operation for near-term quantum applications.
In particular, we extend the ZX-calculus to make it a language for reasoning in
an enriched version of $\cpm$.

We believe that this work opens several directions for future research.
The most evident one is to prove completeness of the enriched diagrams,
which in turn would facilitate automated implementations for tasks such as simulation
of noisy quantum systems, fine-tuned quantum circuit optimization techniques for
specific quantum devices, or comparison of the effectiveness of different Quantum
Error Mitigation techniques.
An interesting venue would be to use enrichment over $\mm$ to reason about
quantum circuit pre- and post-processing techniques,
such as circuit cutting [44, 41],
in which quantum circuits are “split” into linear combinations of smaller
ones that are executed separately.
We also believe that it could be possible to integrate monads that capture purely quantum behaviours
into our construction to intuitively represent within
(enriched) ZX-diagrams phenomena such as superposition of execution orders,
in a manner similar to what is done in the Many-Worlds calculus [13].

Strongly related to completeness is to have presentations of the diagrams in terms of generators
and equations.
We have achieved this by hand in <ref> by using that the algebras for the distribution monad
can be presented as convex algebras with a family of operations $+_{p}$.
The question is then what the analogue of convex monads are when we use algebras presented by
Lawvere theories or sketches [20, 34].

We are also interested in finding other monads that could capture interesting processes
outside of the quantum realm.
For example, the non-empty powerset monad could be used to encode
non-deterministic operations and be used for reasoning about a third party operating on a shared
quantum system.

§.§.§ Acknowledgements

This work was funded by the European Union under Grant Agreement 101080142, EQUALITY project.
AV was partly supported by project PRG 946 funded by the Estonian Research Council.

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§ MONOIDAL STRUCTURE IN $\EMD$

For algebras $a \from \Distr(A) \to A$, $b \from \Distr(B) \to B$ and $c \from \Distr(C) \to C$
we denote by $\BiConv{a}{b}{c}$ the set

\begin{equation*}
\setDef*{f \from A \times B \to C}{
\all{\sigma}\all{\gamma} f(a(\sigma), b(\gamma)) = c\parens*{\sum_{x,y} \sigma(x)\gamma(y)[f(x,y)]}}
\tag{$\ast$}
\end{equation*}

of bi-convex maps.
One can show that the coequalizer (<ref>) for $T = \md$ is given universally by the property

\begin{equation*}
\emd(a \otimes b, c) \cong \BiConv{a}{b}{c}
\end{equation*}

as follows.
First of all, by definition of the equaliser and by $\Distr$ being a monad, we have the following
natural isomorphisms.

\begin{align*}
\emd(a \otimes b, c)
& \cong
\setDef{g \in \emd(F(A \times B),c)}{
g \comp \Distr(a \times b) = g \comp \mu \comp \Distr\nabla} \\
& \cong
\setDef{f \from A \times B \to C}{
c \comp \Distr f \comp \Distr(a \times b) = c \comp \Distr f \comp \mu \comp \Distr\nabla}
\tag{$\dagger$}
\end{align*}

So it remains to show that the two conditions ($\ast$) and ($\dagger$) on maps
$f \from A \times B \to C$ are equivalent.
A calculation shows for $\rho \in \Distr(\Distr(A) \times \Distr(B))$ that

\begin{equation*}
(\mu \comp \Distr\nabla)(\rho) = \sum_{x,y}\sum_{\sigma, \gamma}\rho(\sigma, \gamma) \sigma(x) \gamma(y) [(x,y)]
\end{equation*}

\begin{equation*}
\Distr(a \times b)(\rho) = \sum_{\sigma, \gamma} \rho(\sigma, \gamma) [(a(\sigma), b(\gamma))] \, .
\end{equation*}

Since, for $\rho = \eta(\sigma, \gamma)$ we have

\begin{equation*}
(\mu \comp \Distr\nabla)(\rho) = \sum_{x,y}\sigma(x) \gamma(y) [(x,y)]
\end{equation*}

\begin{equation*}
\Distr(a \times b)(\rho) = \eta(a(\sigma), b(\gamma)) \, ,
\end{equation*}

it is immediately clear that condition ($\dagger$) implies ($\ast$).
The other way around, since $c$ is an algebra and $\mu$ a natural transformation, we have that

\begin{equation*}
c \comp \Distr f \comp \mu \comp \Distr\nabla = c \comp \Distr c \comp \Distr^{2} f \comp \Distr\nabla
\end{equation*}

By functoriality of $\Distr$, the condition ($\dagger$) thus holds if
$f \comp (a \times b) = c \comp \Distr f \comp \nabla$, which is exactly condition ($\ast$).
Hence, the two conditions imply each other and algebra morphisms out of $a \otimes b$ are the same
as bi-convex map out of $A \times B$.

§ SOUNDNESS OF NEW RULESET

We show that the interpretation $\llbracket \cdot \rrbracket_\md$ of LHS and RHS are the same:

* (es):

\begin{equation*}
\begin{split}
\text{LHS: } \llangle \left( \sum_{j=1}^\ell q_j [D_j']\right) \circ \left(\sum_{i=1}^k p_i [D_i]\right)  \rrangle & =
\llangle \sum_{j=1}^\ell q_j [D_j']\rrangle \circ \llangle\sum_{i=1}^k p_i [D_i]  \rrangle\\
& = \left(\sum_{j=1}^\ell q_j \llbracket D_j'\rrbracket\right) \circ \left(\sum_{i=1}^k p_i \llbracket D_i\rrbracket\right)\\
& =  \sum_{j=1}^\ell\sum_{i=1}^k q_jp_i \llbracket D_j'\rrbracket \circ \llbracket D_i\rrbracket\\
& =  \sum_{j=1}^\ell\sum_{i=1}^k q_jp_i \llbracket D_j' \circ D_i\rrbracket
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
\text{RHS: } \llangle \sum_{j=1}^\ell\sum_{i=1}^k q_jp_i [ D_j' \circ D_i]\rrangle & =
\sum_{j=1}^\ell\sum_{i=1}^k q_jp_i \llbracket D_j' \circ D_i\rrbracket
\end{split}
\end{equation*}

* (ep): Same as (es), but substituting composition with tensor product.

* (ec):

\begin{equation*}
\text{LHS: } \llangle p_1 [D_1] + p_2[D_2]  \rrangle  = p_1 \llbracket D_1\rrbracket + p_2 \llbracket D_2\rrbracket
\end{equation*}

\begin{equation*}
\text{RHS: } \llangle p_2 [D_2] + p_1[D_1]  \rrangle  = p_2 \llbracket D_2\rrbracket + p_1 \llbracket D_1\rrbracket
\end{equation*}

* (e$\delta$):

\begin{equation*}
\text{LHS: } \llangle 1 [D] \rrangle  = \llbracket D\rrbracket
\end{equation*}

\begin{equation*}
\text{RHS: } \llbracket D\rrbracket
\end{equation*}

* (e$+$):

\begin{equation*}
\text{LHS: } \llangle p_1 [D] + p_2[D]  \rrangle  = p_1 \llbracket D\rrbracket + p_2 \llbracket D\rrbracket = (p_1 + p_2) \llbracket D\rrbracket
\end{equation*}

\begin{equation*}
\text{RHS: } \llangle (p_1+p_2)[D]  \rrangle  = (p_1 + p_2) \llbracket D\rrbracket
\end{equation*}

* (e$0$):

\begin{equation*}
\text{LHS: } \llangle p [D_1] + 0[D_2]  \rrangle  = p \llbracket D_1\rrbracket + 0 \llbracket D_2\rrbracket = p \llbracket D_1\rrbracket
\end{equation*}

\begin{equation*}
\text{RHS: } \llangle p [D_1]  \rrangle  = p \llbracket D_1\rrbracket
\end{equation*}

§ DIAGRAMMATIC STUDY OF SYMMETRY VERIFICATION

To have the correct scalar factors in the diagram, we have to take a look
at Example (<ref>).
This tells us the scalar factors we need to apply to the diagram in order to represent
exactly the diagrams for $\ket{0},\bra{0},$ and CNOT.
We then have the following scalar-corrected diagram:

\begin{equation*}\label{eq:scaledSV}
\tikzfig{figures/scaledSV}
\end{equation*}

Then, applying diagrammatic rules to the diagram results in the following:

\begin{equation*}\label{eq:derivedSV}
\tikzfig{figures/derivedSV}
\end{equation*}

We then use one rule regarding the discard operator from [12], stating
that we can remove the dangling diagram at the bottom if we multiply the enitre diagram by $\sqrt{2}$:

\begin{equation*}\label{eq:derivedSV2}
\tikzfig{figures/derivedSV2}
\end{equation*}

Calculating the interpretation of the last diagram amounts to calculating the interpretation of
each scalar diagram in each branch, multiply them by the probabilities and scalar factor, and
add them together.
More interestingly, we see that there are two possibilities, either a scalar $Z$ spider with phase
$0$ or with phase $\pi$, which respectively have interpretation $2$ and $0$ following <Ref>.
The interpretation of the last diagram then gives us that $\tr(P_{+1}\rho_{noisy}) = 1 - \frac{2p}{3}$.

What this is expressing is that our symmetry commutes with the first two branches
(the identity and the $X$ error) and anti-commutes with the other two, which in practice
tells us that measuring this symmetry only mitigates some part of the depolarizing channel.
Then, by representing different symmetries we can use this form of diagrammatic reasoning
to find and compare symmetries for different error channels.

§ NOTATION

Notation     Meaning
$\SetC$     Category of sets

$\emt$     Eilenberg-Moore category of a monad

$\CatC$     Category of (small) categories

$\obj{\CC}$     Objects of the category $\CC$

$\eC$, $\enr{\Cat{M}}$     enriched categories

$\timesUnit$     Monoidal unit of $\times$ (singleton set in $\SetC$)

$\bullet$     canonical element in $\timesUnit$ in $\SetC$

$\otimesUnit$     Monoidal unit of $\otimes$

$\ecirc$     Composition in an enriched category

$\odot$     Tensor product in a monoidal enriched category

$\otimes_\mt$     Tensor product in $\emt$

$\VCat$, $\ECatC{\Cat{M}}$     2-category of enriched category