NGC2146NGC2146variables.txt NGC2976NGC2976variables.txt

11institutetext:  Institute for Research in Fundamental Sciences (IPM), School
of Astronomy, Tehran, Iran 22institutetext: Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
33institutetext: School of Physics and Astronomy, Cardiff University, CF24
3AA, UK 44institutetext: Université Paris-Saclay, UniversitÃ Paris CitÃ, CEA,
CNRS, AIM, 91191, Gif-sur-Yvette, France 55institutetext: Université Grenoble
Alpes, CNRS, Grenoble INP, LPSC-IN2P3, 38000 Grenoble, France 66institutetext:
Max Planck Institute for Extraterrestrial Physics, 85748 Garching, Germany
77institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
88institutetext: Université Grenoble Alpes, CNRS, Institut Néel, France
99institutetext: Institut de RadioAstronomie Millimétrique (IRAM), Grenoble,
France 1010institutetext: Dipartimento di Fisica, Sapienza Università di Roma,
I-00185 Roma, Italy 1111institutetext: Univ. Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 1212institutetext: Centro de Astrobiología (CSIC-INTA),
Torrejón de Ardoz, 28850 Madrid, Spain 1313institutetext: National Observatory
of Athens, IAASARS, GR-15236, Athens, Greece 1414institutetext: Faculty of
Physics, University of Athens, GR-15784 Zografos, Athens, Greece
1515institutetext: High Energy Physics Division, Argonne National Laboratory,
Lemont, IL 60439, USA 1616institutetext: Instituto de Radioastronomía
Milimétrica (IRAM), Granada, Spain 1717institutetext: LERMA, Observatoire de
Paris, PSL Research Univ., CNRS, Sorbonne Univ., UPMC, 75014 Paris, France
1818institutetext: School of Earth & Space and Department of Physics, Arizona
State University, AZ 85287, USA 1919institutetext: School of Physics and
Astronomy, University of Leeds, Leeds LS2 9JT, UK 2020institutetext: INAF-
Osservatorio Astronomico di Cagliari, 09047 Selargius, Italy
2121institutetext: LPENS, ENS, PSL Research Univ., CNRS, Sorbonne Univ.,
Université de Paris, 75005 Paris, France 2222institutetext: Department of
Physics and Astronomy, University of Pennsylvania, PA 19104, USA
2323institutetext: Institut d’Astrophysique de Paris, CNRS (UMR7095), 75014
Paris, France 2424institutetext: University of Lyon, UCB Lyon 1, CNRS/IN2P3,
IP2I, 69622 Villeurbanne, France 2525institutetext: Sterrenkundig
Observatorium Universiteit Gent, Krijgslaan 281 S9, B-9000 Gent, Belgium
2626institutetext: Institut d’Astrophysique Spatiale (IAS), CNRS, Université
Paris Sud, Orsay, France 2727institutetext: IRAP, UniversitÃ de Toulouse,
CNRS, UPS, IRAP, Toulouse Cedex 4, France

# Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2: the
case of NGC2146 and NGC2976

G. Ejlali Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976<EMAIL_ADDRESS>R. Adam
Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case
of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies
with NIKA2: the case of NGC2146 and NGC2976 P. Ade Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 H. Ajeddig Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 P. André Constraining Millimeter Dust Emission
in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976 E. Artis Constraining Millimeter Dust Emission in Nearby Galaxies
with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976 H. Aussel Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976 M. Baes Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Beelen Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Benoît Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 S. Berta Constraining Millimeter Dust Emission
in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976 L. Bing Constraining Millimeter Dust Emission in Nearby Galaxies
with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976 O.
Bourrion Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976 M. Calvo Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976 A. Catalano Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 M. De Petris Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 F.-X. Désert Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 S. Doyle Constraining Millimeter Dust Emission
in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976 E. F. C. Driessen Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976
F. Galliano Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in
Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976 A. Gomez
Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case
of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies
with NIKA2: the case of NGC2146 and NGC2976 J. Goupy Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. P. Jones Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 C. Hanser Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Hughes Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 S. Katsioli Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976
F. Kéruzoré Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in
Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976 C. Kramer
Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case
of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies
with NIKA2: the case of NGC2146 and NGC2976 B. Ladjelate Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976 G. Lagache Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 S. Leclercq Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 J.-F. Lestrade Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 J. F. Macías-Pérez Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 S. C. Madden Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Maury Constraining Millimeter Dust Emission
in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976 P. Mauskopf Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976 F. Mayet Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976 A. Monfardini Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Moyer-Anin Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 M. Muñoz-Echeverría Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Nersesian Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 L. Pantoni Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 D. Paradis Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 L. Perotto Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 G. Pisano Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 N. Ponthieu Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 V. Revéret Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. J. Rigby Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Ritacco Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining Millimeter
Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976
C. Romero Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976Constraining Millimeter Dust Emission in Nearby
Galaxies with NIKA2: the case of NGC2146 and NGC2976 H. Roussel Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with
NIKA2: the case of NGC2146 and NGC2976 F. Ruppin Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 K. Schuster Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 A. Sievers Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 M. W. S. L. Smith Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 F. S. Tabatabaei Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 J. Tedros Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 C. Tucker Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 E. M. Xilouris Constraining Millimeter Dust
Emission in Nearby Galaxies with NIKA2: the case of NGC2146 and
NGC2976Constraining Millimeter Dust Emission in Nearby Galaxies with NIKA2:
the case of NGC2146 and NGC2976 R. Zylka Constraining Millimeter Dust Emission
in Nearby Galaxies with NIKA2: the case of NGC2146 and NGC2976Constraining
Millimeter Dust Emission in Nearby Galaxies with NIKA2: the case of NGC2146
and NGC2976

###### Abstract

This study presents the first millimeter continuum mapping observations of two
nearby galaxies, the starburst spiral galaxy NGC2146 and the dwarf galaxy
NGC2976, at 1.15 mm and 2 mm using the NIKA2 camera on the IRAM 30m telescope,
as part of the Guaranteed Time Large Project IMEGIN. These observations
provide robust resolved information about the physical properties of dust in
nearby galaxies by constraining their FIR-radio SED in the millimeter domain.
After subtracting the contribution from the CO line emission, the SEDs are
modeled spatially using a Bayesian approach. Maps of dust mass surface
density, temperature, emissivity index, and thermal radio component of the
galaxies are presented, allowing for a study of the relations between the dust
properties and star formation activity (using observations at 24$\mu$m as a
tracer). We report that dust temperature is correlated with star formation
rate in both galaxies. The effect of star formation activity on dust
temperature is stronger in NGC2976, an indication of the thinner interstellar
medium of dwarf galaxies. Moreover, an anti-correlation trend is reported
between the dust emissivity index and temperature in both galaxies.

##  1 Introduction

Dust grains in galaxies are important for the formation of protostellar cores
and significantly impact the heating and cooling processes of the InterStellar
Medium (ISM). Modeling the Spectral Energy Distribution (SED) of dust allows
us to infer the physical properties of the grains. Dust emits across a range
of temperatures, with the warmer component emitting in the Mid-InfraRed (MIR)
and colder dust emitting in the Far-InfraRed (FIR). The sub-
millimeter/millimeter waveband is crucial for detecting this cold dust
component and estimating the total dust mass. Telescopes such as IRAS, ISO,
Spitzer, and Herschel have allowed us to study dust emission up to 500 $\mu$m,
but studying longer millimeter wavelengths is crucial for modeling the mass
and temperature of cold dust and constraining the radio component
Galliano+2018 .
The New IRAM KIDs Array (NIKA2) on the IRAM 30m telescope brings the unique
opportunity to map full galaxies for the first time at 1.15 mm and 2 mm.
NIKA2-performance . Within the Guaranteed Time Large Project IMEGIN
(Interpreting Millimeter Emission of Galaxies with IRAM and NIKA2, PI: S.
Madden), a sample of 22 nearby galaxies with varying ranges of mass,
morphological types, star formation rate (SFR), and ISM properties have been
observed. This paper presents observations of two IMEGIN galaxies of widely
different physical properties: NGC2146 a starburst spiral galaxy (D=3.5 Mpc),
and NGC2976 a peculiar dwarf galaxy (D=17.2 Mpc) (Ejlali et al. in prep.).

Figure 1: Observed NIKA2 maps of NGC2146 (top) and NGC2976 (bottom) at 1.15 mm
(left, resolution 12′′) and 2 mm (right, resolution 18′′). The rms noise level
for NGC2146 are 0.9 (1.15 mm) and 0.24 (2 mm) mJy/beam and for NGC2976 is 0.8
(1.15 mm) and 0.23 (2 mm) mJy/beam.

##  2 Data

We observed NGC2146 and NGC2976 with the NIKA2 camera for 5 and 5.5 hours,
respectively. The data were reduced using Scanam_NIKA pipeline Roussel+2020
and are shown in Fig. 1.
The complementary data used in this work to model the spatial SEDs include
Spitzer MIPS observations at 24 $\mu m$ Clark+2018 , Herschel PACS and SPIRE
observations at 70, 100, 160, and 250 $\mu$m Kennicutt+2011 , Planck
observations at 1.38 mm Clark+2018 , Radio Continuum (RC) observations with
WSRT at 18 cm and 21 cm Braun+2007 , Effelsberg 100m telescope observations at
6.2 cm Tabatabaei+2017 , CO(2-1) data from the HERACLES survey Leroy+2009 .
All the maps are pre-processed to have the same resolution (18′′) and geometry
(pixel size 6′′). A 6′′ pixel size corresponds to a physical size of about
$100pc$ and $500pc$ in NGC2976 and NGC2146, respectively.
As the NIKA2 bandpass (NIKA2-performance ) encompasses CO(2-1) line, part of
the observed flux at 1.15 mm is due to line emission Drabek+2012 . The mean
contribution of the CO(2-1) line emission to the observed NIKA2 continuum
emission at 1.15 mm (within pixels above 3$\sigma$ limit) is
$(\DTLfetch{NGC2146}{variable}{COcontr_{m}ean}{value}\pm\DTLfetch{NGC2146}{variable}{COcontr_{s}td}{value})\%$
(error is the standard deviation) in NGC2146 and
$(\DTLfetch{NGC2976}{variable}{COcontr_{m}ean}{value}\pm\DTLfetch{NGC2976}{variable}{COcontr_{s}td}{value})\%$
for NGC2976. Before including the NIKA2 1.15 mm data point in the SED modeling
process, we subtract the contamination by CO.

##  3 Model

Continuum emission at 1.15 mm and 2 mm consists of radio continuum emission
$S^{\rm RC}_{\nu}$ and thermal emission from dust $S^{\rm dust}_{\nu}$. To
describe the radio continuum emission, we employ two power-laws defined as
$S^{\rm RC}_{\nu}=A_{1}\nu^{-0.1}+A_{2}\nu^{-\alpha_{syn}}$, in which $A_{1}$
and $A_{2}$ are free parameters quantifying contributions from thermal free-
free and nonthermal synchrotron components. $\alpha_{syn}$ is the synchrotron
spectral index, equal to 0.71 and 1.13 for NGC2146 and NGC2976, respectively
Tabatabaei+2013b ; Tabatabaei+2017 . We model the thermal emission of dust
using a Modified Black-Body (MBB) model defined as $S^{\rm
dust}_{\nu}=\kappa_{0}\left(\frac{\nu}{\nu_{0}}\right)^{\beta}\left(\frac{M}{D^{2}}\right)B_{\nu}(T)$
with dust temperature $T_{\rm dust}$, dust mass $M_{\rm dust}$, and dust
emissivity index $\beta$ as free parameters
[$\kappa_{0}(250\,\rm{GHz})=0.4\,{\rm m}^{2}/{\rm kg}$] Hunt+2015 . We find
the best-fit values for the five free parameters ($M_{\rm{dust}}$,
$T_{\rm{dust}}$, $\beta$, $A_{1}$, $A_{2}$) with the Bayesian approach. We use
the MCMC method via Python package emcee and fit the model to the observations
emcee .

##  4 Results

Fig. 2 shows the best-fit values of four of the free parameters of our model
across the two galaxies, namely $T_{\rm{dust}}$, $M_{\rm{dust}}$, $\beta$, and
thermal free-free fraction at 21 cm defined as
$f_{\rm{th}}\,(21\,cm)=A_{1}\nu_{21\,cm}^{-0.1}/S^{\rm{RC}}_{21\,cm}$. The
mean value and standard deviation for both galaxies are reported in Table 1.
In NGC2146, the dust mass peaks at the central region, and the inner $\sim$1
kpc of the disk contains $\sim$40% of the total dust mass in this galaxy. In
NGC2976, dust mass peaks at two star-forming regions, which contain more than
25% of the total dust mass. The dust mass in NGC2146 varies within a range
that is three orders of magnitude larger than in NGC2976. The dust temperature
varies across each galaxy, and peaks in the outer disk of NGC2146
($\DTLfetch{NGC2146}{variable}{temp_{m}ax}{value}\pm\DTLfetch{NGC2146}{variable}{temp_{m}ax_{e}r}{value}\,{\rm
K}$) and in the northern star-forming region of NGC2976
($\DTLfetch{NGC2976}{variable}{temp_{m}ax}{value}\pm\DTLfetch{NGC2976}{variable}{temp_{m}ax_{e}r}{value}\leavevmode\nobreak\
{\rm K}$). Moreover, we report values of $3.7\times 10^{7}\,{\rm M}_{\odot}$
and $4.9\times 10^{5}\,{\rm M}_{\odot}$ for the total dust mass (within pixels
above 3$\sigma$ limit) in NGC2146 and NGC2976, respectively.
Mapping dust emissivity index provides hints about grain formation and
evolution, but the MBB model does not take into account the mixing of physical
conditions along the line of sight. This can cause systematic underestimation
of dust mass compared to more complex dust models Remy+2015 . In both
galaxies, $\beta$ reaches its maximum in the outskirts of the disk. We find
$\beta$ ranging from
NGC2146variablebeta_minvalue$\pm$NGC2146variablebeta_min_ervalue to
NGC2146variablebeta_maxvalue$\pm$NGC2146variablebeta_max_ervalue in NGC2146.
On the contrary, $\beta$ in NGC2976 has lower values throughout the galaxy,
ranging from NGC2976variablebeta_minvalue$\pm$NGC2976variablebeta_min_ervalue
to NGC2976variablebeta_maxvalue$\pm$NGC2976variablebeta_max_ervalue. This is
in agreement with previous studies (Remy+2015 ) showing lower values of
emissivity index in dwarf or low metallicity spiral galaxies. The maxima of
$T_{\rm dust}$ and $\beta$ are located in different locations in both galaxies
so we expect a negative correlation among these parameters; this is explored
more in Sec. 5.

Figure 2: Maps of dust temperature ($T_{\rm{dust}}$), mass ($M_{\rm{dust}}$) and emissiviry index ($\beta$) and thermal fraction ($f_{\rm th}$) at 21 cm for NGC2146 (top) and NGC2976 (bottom). They are determined with the Bayesian approach and using the MCMC method, in each pixel with value larger than $3\sigma$ rms of all used maps. Table 1: Mean values of four free parameters of the pixel-by-pixel SED modeling are reported. Their standard deviation is reported as statistical error. SED parameter | NGC2146 | NGC2976
---|---|---
$T_{\rm{dust}}\,({\rm K})$ | $\DTLfetch{NGC2146}{variable}{temp_{m}ean}{value}\pm\DTLfetch{NGC2146}{variable}{temp_{s}td}{value}$ | $\DTLfetch{NGC2976}{variable}{temp_{m}ean}{value}\pm\DTLfetch{NGC2976}{variable}{temp_{s}td}{value}$
$M_{\rm{dust}}\,({\rm M}_{\odot})$ | $(\DTLfetch{NGC2146}{variable}{mass_{m}ean}{value}\pm\DTLfetch{NGC2146}{variable}{mass_{s}td}{value})\times 10^{5}$ | $(\DTLfetch{NGC2976}{variable}{mass_{m}ean}{value}\pm\DTLfetch{NGC2976}{variable}{mass_{s}td}{value})\times 10^{3}$
$\beta$ | $\DTLfetch{NGC2146}{variable}{beta_{m}ean}{value}\pm\DTLfetch{NGC2146}{variable}{beta_{s}td}{value}$ | $\DTLfetch{NGC2976}{variable}{beta_{m}ean}{value}\pm\DTLfetch{NGC2976}{variable}{beta_{s}td}{value}$
$f_{\rm th}$ | $(\DTLfetch{NGC2146}{variable}{thfr_{m}ean}{value}\pm\DTLfetch{NGC2146}{variable}{thfr_{s}td}{value})\times 10^{-2}$ | $(\DTLfetch{NGC2976}{variable}{thfr_{m}ean}{value}\pm\DTLfetch{NGC2976}{variable}{thfr_{s}td}{value})\times 10^{-1}$

##  5 Discussion

We explore the relation between dust emissivity index $\beta$ and temperature
$T_{\rm{dust}}$. An anti-correlation between the two has often been reported
(JuvelaYsard2012 ,Paradis+2010 ), while Shetty+2009 cautions it might be
created by uncertainties in measurements. Fig. 3 shows this relation in both
galaxies, including only pixels above 3$\sigma$ level. While the general trend
between $T_{\rm{dust}}$ and $\beta$ is an anti-correlation, pixels with
relatively higher flux at 1.15 mm do not completely follow the general inverse
trend. In other words, the higher S/N pixels are less affected by the
degeneracy between $\beta$ and $T_{\rm{dust}}$ in the MBB model.

Figure 3: Pixel-by-pixel relation of dust temperature $T_{\rm{}_{dust}}$ with
$\beta$ in (left) NGC2146 and (right) NGC2976. Only pixels above 3$\sigma$ rms
level are included. The color scheme shows the flux density in the observed
NIKA2 map at 1.15 mm.

We next explore the relation of $T_{\rm{dust}}$ with star formation activity
in these galaxies. To do so, we use 24 $\mu$m observation as a standard tracer
of star formation rate density $\Sigma_{\rm{SFR}}$ (Calzetti+2007 ). The
relation between $T_{\rm{dust}}$ and $\Sigma_{\rm{SFR}}$ is shown in Fig. 4.
The high values found for Pearson correlation coefficients $r_{P}=0.7$ and
$0.9$ indicate tight correlations in both galaxies, even with different ranges
of $\Sigma_{\rm{SFR}}$. The positive relation indicates that in regions with
higher star-forming activity, InterStellar Radiation Field (ISRF) is
intensified by larger amounts of energetic UV photons, heating the dust.
Comparison of the slopes in the $T_{\rm{dust}}-\log\Sigma_{\rm{SFR}}$ plane
shows that dust temperature increases faster with $\log\Sigma_{\rm{SFR}}$ in
the dwarf galaxy NGC2976 than in NGC2146 by a factor of two. This can be
linked to their different ISM densities: dwarf galaxies have a less-dense ISM
due to lower dust opacity, resulting in a stronger ISRF and faster heating
rate by the radiation field. On the contrary, a thicker ISM in a starburst
spiral galaxy like NGC2146 means more efficient shielding of dust grains from
an energetic radiation field and a slower heating rate (Remy+2015, ).

Figure 4: Pixel-by-pixel relation of temperature $T_{\rm{dust}}$ with
$\Sigma_{\rm{SFR}}$ in NGC2146 (green dots) and NGC2976 (red dots) for pixels
above 5$\sigma$ rms level. The slope of the best-fit line and Pearson
coefficient $r_{P}$ are included in the legend.

##  6 Conclusions

This research introduces the millimeter observations of two nearby galaxies,
NGC2146 (a starburst spiral) and NGC2976 (a dwarf), at 1.15 mm and 2 mm, in
the framework of the Guaranteed Time Large Project IMEGIN. These observations
offer detailed information about the physical characteristics of dust in
nearby galaxies by constraining their SED in the FIR-radio wavelength domain.
We generate dust mass, temperature, and emissivity index maps of the galaxies
and investigate relationships between dust properties and other ISM components
such as SFR. An anti-correlation is reported between dust emissivity index and
temperature, but the high S/N pixels do not follow the general inverse trend.
Additionally, we report a strong correlation between dust temperature and SFR
in both galaxies. The impact of star formation activity on dust temperature is
more pronounced in the dwarf galaxy NGC2976 than in NGC2146.

## Acknowledgements

We would like to thank the IRAM staff for their support during the observation
campaigns. The NIKA2 dilution cryostat has been designed and built at the
Institut Néel. In particular, we acknowledge the crucial contribution of the
Cryogenics Group, and in particular Gregory Garde, Henri Rodenas, Jean-Paul
Leggeri, Philippe Camus. This work has been partially funded by the Foundation
Nanoscience Grenoble and the LabEx FOCUS ANR-11-LABX-0013. This work is
supported by the French National Research Agency under the contracts "MKIDS",
"NIKA" and ANR-15-CE31-0017 and in the framework of the "Investissements
dâavenirâ program (ANR-15-IDEX-02). This work is supported by the Programme
National Physique et Chimie du Milieu Interstellaire (PCMI) and the Programme
National Cosmology et Galaxies (PNCG) of the CNRS/INSU with INC/INP co-funded
by CEA and CNES. This work has benefited from the support of the European
Research Council Advanced Grant ORISTARS under the European Union’s Seventh
Framework Programme (Grant Agreement no. 291294). A. R. acknowledges financial
support from the Italian Ministry of University and Research - Project
Proposal CIR01$\\_00010$. S. K. acknowledges support provided by the Hellenic
Foundation for Research and Innovation (HFRI) under the 3rd Call for HFRI PhD
Fellowships (Fellowship Number: 5357). M.B., A.N., and S.C.M. acknowledge
support from the Flemish Fund for Scientific Research (FWO-Vlaanderen,
research project G0C4723N.

## References

  * (1) L. Perotto, N. Ponthieu, J.-F. Macías-Pérez, et al., Astron. Astrophys. 637, A71 (2020)
  * (2) H. Roussel, et al. European Physical Journal Web of Conferences 228, 00024 (2020)
  * (3) F. S. Tabatabaei, et al. Astron. Astrophys. 555, A128 (2013b)
  * (4) A. K. Leroy, et al. The Astronomical Journal, 137, 4670-4696 (2009)
  * (5) E. Drabek et al. Mon. Not. R. Astron. Soc. 426, 23-39 (2012)
  * (6) F. S. Tabatabaei, et al. The Astrophysical Journal, 836, 185 (2017)
  * (7) D. Foreman-Mackey et al., Pub. of the Astron. Soc. of the Pacific, 925, 306 (2013)
  * (8) F. Galliano, M. Galametz, & A. P. Jones, ARA&A 56, 673 (2018)
  * (9) C. J. R. Clark, et al. 2018, Astron.& Astrophys. 609, A37 (2018)
  * (10) R. C. Kennicutt, D. Calzetti et al. PASP 123, 1347 (2011)
  * (11) R. Braun, T. A. Oosterloo, et al. Astron. Asrophys. 461, 455 (2007)
  * (12) D. Calzetti, R. C. Kennicutt, et al. ApJ. 666, 870 (2007)
  * (13) A. RÃmy-Ruyer, S. C. Madden, F. Galliano, et al. Astron. Asrophys. 582, A121 (2015)
  * (14) M. Juvela, N. Ysard, A&A 541, A33 (2012)
  * (15) D. Paradis, M. Veneziani, et al. Astron. Asrophys. 520, L8 (2010)
  * (16) R. Shetty, J. Kauffmann, et al., ApJ, 696, 2234 (2009)
  * (17) L. K. Hunt, S. GarcÃa-Burillo, V. Casasola, et al., Astron. Asrophys., 583, A114 (2015)
  * (18)


# Hamiltonians for one-way quantum repeaters

Filippo M. Miatto  Institute for Quantum Computing and Department of Physics
and Astronomy, University of Waterloo, 200 University Ave. W, N2L 3G1
Waterloo, Ontario, Canada  Télécom ParisTech, LTCI, Université Paris Saclay,
46 Rue Barrault, 75013 Paris, France  Michael Epping  Institute for Quantum
Computing and Department of Physics and Astronomy, University of Waterloo, 200
University Ave. W, N2L 3G1 Waterloo, Ontario, Canada  Norbert Lütkenhaus
Institute for Quantum Computing and Department of Physics and Astronomy,
University of Waterloo, 200 University Ave. W, N2L 3G1 Waterloo, Ontario,
Canada

###### Abstract

Quantum information degrades over distance due to the unavoidable
imperfections of the transmission channels, with loss as the leading factor.
This simple fact hinders quantum communication, as it relies on propagating
quantum systems. A solution to this issue is to introduce quantum repeaters at
regular intervals along a lossy channel, to revive the quantum signal. In this
work we study unitary one-way quantum repeaters, which do not need to perform
measurements and do not require quantum memories, and are therefore
considerably simpler than other schemes. We introduce and analyze two methods
to construct Hamiltonians that generate a repeater interaction that can beat
the fundamental repeaterless key rate bound even in the presence of an
additional coupling loss, with signals that contain only a handful of photons.
The natural evolution of this work will be to approximate a repeater
interaction by combining simple optical elements.

##  1 Introduction

Quantum physics plays an increasingly important role in the development of
future technologies. The transmission of information is an example of this
trend, where the benefit of exploiting quantum effects has motivated several
ideas, such as quantum money, quantum key distribution (QKD), quantum
fingerprinting, quantum digital signatures, and several others [3, 2, 7, 12,
11, 19]. Eventually, we will build a quantum internet: a communication
infrastructure that will enable distributed quantum information tasks on a
global scale [13, 17].

It would be very convenient if we could rely as much as possible on the
current infrastructure, such as the optical fiber network, to propagate
quantum states of light as the main carrier of quantum information. Light is
cheap, fast, and it does not require vacuum or low temperatures. However, at
the present moment, we cannot send photons reliably to distant parties:
optical fibers are “leaky” and deteriorate the quantum properties of the
signal. The fact that quantum information cannot travel far enough constitutes
one of the outstanding challenges in the field [30], and it is the main focus
of our work.

Note that as optical modes are like harmonic oscillators, photon loss is
equivalent to the relaxation of a harmonic oscillator towards equilibrium
[20]. This makes our findings applicable to any domain in which one needs to
fight relaxation (e.g. fields in a cavity, nanoscale resonators, etc).

##  2 Preliminaries

We know from the theory of quantum information that we can “stretch” the
distance between two entangled systems by a process that involves entanglement
swapping, which is a core component of the first generation of quantum
repeaters [5]. The newest generation of repeater architectures relies instead
on quantum error correction, which allows for “one-way” functionality by
removing the need to broadcast measurement results backwards and forwards [14,
23, 24, 25]. It is even possible to omit measurements altogether: just as one
can implement quantum error correction by measuring the error syndrome and
then applying the appropriate correction, it is also possible to do so
coherently by using an ancillary system initialized in a fiducial state
$|0\rangle$ to store the index $i$ of each syndrome and of the corresponding
Kraus operator:

$\displaystyle
U_{\mathrm{rep}}=\sum_{i=1}^{\mathrm{syndromes}}R^{(i)}\otimes|i\rangle\langle
0|.$ (1)

In this expression, $R^{(i)}=U_{i}P^{(i)}$ where $P^{(i)}$ is the projector
onto the $i$-th syndrome space, and $U_{i}$ is the unitary that brings the
system from the $i$-th syndrome space back into codespace. Note that
expression (1) is intended as an isometry. The need for an ancilla is a simple
consequence of the Stinespring dilation theorem [28], according to which any
CPTP channel can be described through a unitary interaction with a larger
system.

In this work we present two recipes to build families of Hamiltonians that
generate such unitary one-way quantum repeaters.

In order to evaluate the performance of our repeaters, we will compare the
entanglement between two systems (one of which is travelling in a lossy
channel interspersed with repeaters), to the repeaterless key rate bound [33,
29], see also [36] for results in the non-asymptotic regime. In particular,
[29] found that the tightest upper bound of the QKD rate per optical mode that
can be achieved using only a lossy channel with transmissivity $\eta$ is equal
to:

$\displaystyle R(\eta)=\log_{2}\frac{1}{1-\eta}.$ (2)

What we found is that we can beat this repeaterless bound even if each
repeater introduces a mild coupling loss between the optical fibre and the
repeater device. Furthermore, the optimized separation between the repeater
stations turns out to be rather reasonable, on the order of a few kilometers.

The complexity of our repeater Hamiltonians depends on the code space that
they protect and on the fiducial ancilla states. Consequently, if we pick them
wisely, we have the potential to obtain _simple_ Hamiltonians that can be
implemented by combining a few known optical elements, such as single photon
sources, beamsplitters, squeezers, four wave mixers, etc. Such a quest for
simplification is outside the scope of the present work, but it is part of our
ongoing research.

##  3 Working principles of a one-way quantum repeater

Our task is to protect the quantum information that is encoded in a photonic
system. Although in this particular context we lack the typical qubit
structure of QEC codes, if we treat photon loss as an “error”, we can still
apply QEC successfully to our problem. Note that the set of correctable errors
cannot include all of the possible loss operators (e.g. including the loss of
all photons), but rather a few of them. How many depends on the particular
code in question.

Furthermore, in practice the correction process is not going to be perfect and
it can itself introduce errors and noise. This enforces an optimal separation
between the repeaters, as well as other important considerations regarding the
noise parameters. In this work we describe an ideal repeater, i.e. one that
does not introduce any additional noise. In this we consider the method of
exact error correction [32, 15] to illustrate our approach of Hamiltonian
Quantum Repeaters. It will be interesting to extend this line of consideration
also to approximate error correction [34, 4, 27, 21] though we leave that to
upcoming work. For this reason it is sufficient for us to consider exact and
not approximate quantum error correction.

In all that follows, we will seek the protection of a two-dimensional subspace
within each signal from the loss of a single photon. If this is done
frequently enough, and if we separate the repeaters such that the single-
photon loss is the dominant error, we can witness the benefits of a repeater
action. All that we need from QEC is encapsulated in the QEC condition in the
form $PE^{\dagger}_{i}E_{j}P=\delta_{ij}c_{i}P$ where $c_{i}$ is the
probability of the $i$-th error, $P$ is the projector onto the protected
subspace (called the “code space”) and {$E_{i}$} are the correctable effect
operators of the undesired interaction, which map $P$ to the orthogonal spaces
$P^{(i)}=E_{i}PE_{i}^{\dagger}/\mathrm{\frac{1}{2}Tr}(E_{i}PE_{i}^{\dagger})$
[28]. The fact that such spaces are orthogonal to each other and to $P$ means
that the quantum information has not left the system, it is just in a
different subspace and it can be brought back.

To understand what the error operators are, we need a description of the lossy
channel. There are three main ways to describe a single-mode lossy channel of
transmissivity $\eta$:

(i) As a quantum channel in Kraus form (also e.g. in [10]):

$\displaystyle\mathcal{L}_{\eta}[\rho]=\sum_{k=0}^{\infty}\frac{(1-\eta)^{k}}{k!}\sqrt{\eta}^{N}{a}^{k}\rho{a^{\dagger}}^{k}\sqrt{\eta}^{N},$
(3)

where $N=a^{\dagger}a$ is the number operator and $a$ is the annihilation
operator. The Kraus operators are therefore
$A_{k}=\frac{(1-\eta)^{k/2}}{\sqrt{k!}}\sqrt{\eta}^{N}a^{k}$.

(ii) As the action of a beamsplitter in each mode (effectively the Stinespring
representation of the channel):

$\displaystyle\mathcal{L}_{\eta}[\rho]=\mathrm{Tr}_{B}\left[U_{AB}(\rho_{A}\otimes|0\rangle\langle
0|_{B})U_{AB}^{\dagger}\right],$ (4)

where $U_{AB}=\exp[i\phi(a^{\dagger}b+ab^{\dagger})]$ is a beamsplitter
between the signal and a local environment in the vacuum state.

(iii) Through the master equation [20]:

$\displaystyle\dot{\rho}=\frac{\gamma}{2}\left(2a\rho
a^{\dagger}-\\{N,\rho\\}\right).$ (5)

The connection between the three representations is the relation
$\eta=\cos(\phi)^{2}=\exp(-\gamma t)$. This channel generalizes to an
$M$-modes lossy channel as a simple tensor product of $M$ channels. We will
indicate the Kraus operator corresponding to losing $k$ photons from mode $i$
and none from the other modes as
$A_{k}^{(i)}=\frac{(1-\eta)^{k/2}}{\sqrt{k!}}\sqrt{\eta}^{N}a_{i}^{k}$ (where
$N=\sum_{i}a_{i}^{\dagger}a_{i}$). As we concentrate on correcting single
photon losses, we need the single-photon loss operator in the $i$-th mode:
$E_{i}:=A_{1}^{(i)}=\sqrt{1-\eta}\sqrt{\eta}^{N}{a_{i}}$. If we replace these
operators into the error-correction conditions and take the trace of both
sides, we obtain that the probability of the $i$-th error ($i\geq 1$) is:

$\displaystyle
c_{i}=\frac{1-\eta}{d}\mathrm{Tr}\left(Pa_{i}^{\dagger}a_{i}\eta^{N-1}\right)$
(6)

where we used the operator identity $f(N)a=af(N-1)$ and $d$ is the dimension
of the protected subspace, which comes from $\mathrm{Tr}(P)=d$. In our case,
as we are protecting a two-dimensional subspace, $d=2$. Eq. (6) means that,
understandably, the more photons a code uses the easier it is to lose one,
which gives rise to an interesting trade-off between the probability to lose a
photon and the ability to protect from photon loss. From inspection, it
appears that a good choice is to make sure we can correct for a single loss in
each mode using as few photons as possible in the code.

Note that in case of no loss, we have a non-trivial evolution: the no-loss
operator is $E_{0}=A_{0}^{(i)}=\sqrt{\eta}^{N}$, which is not the identity
operator. This means that if the code space $P$ is not an eigenspace of the
total photon number operator, it will evolve non-trivially even if no losses
occur.

As anticipated, a unitary operation is sufficient to rotate $P^{(i)}$ back to
$P$, but there is a catch: the operation that we should apply to the system
depends on the error syndrome. As we have more than one subspace that should
rotate back to $P$, this is not a unitary operation unless we introduce an
ancillary system, as in Eq. (1). So the quantum repeater must produce a fresh
local ancilla, it must have the incoming system interact with it, and then it
can discard the used ancilla. The output of the repeater is the recovered
system (unless more losses occurred than the code can handle). Note that in
principle we could measure the used ancilla to detect the syndrome, but the
repeater would work equally well if we did not.

To summarize: we choose to put our quantum information in protected subspaces
that do not become entangled with the environment upon photon loss, but rather
they rotate to an orthogonal subspace. This enables us to rotate them back
without involving measurements if we supply this operation with a ‘catalytic’
ancilla which we can discard at the end of the operation.

##  4 Repeater Hamiltonians

In this section we present two families of Hamiltonians that generate a
repeater unitary. Each Hamiltonian will depend on the code space $P$ as well
as the initial ancilla state, so one can view them as a _resource_ , which can
be chosen wisely to yield simpler Hamiltonians in the sense mentioned in the
introduction. Note that a well-designed repeater action will not let the
uncorrectable error spaces to eventually couple to code space (for instance
after a series of several losses and repeaters). So the action of
$U_{\mathrm{rep}}$ on spaces that will never interfere with code space can be
thought of as another resource, as we can use the freedom to choose this
otherwise irrelevant action to simplify the Hamiltonian. We present both
architectures because although they are related by a system-ancilla swap after
the repeater, the Hamiltonians are rather different.

###  4.1 Direct unitary

Figure 1: In the direct architecture, the system interacts with a local
ancilla, it “transfers the error” and then proceeds onwards. Measuring the
used ancilla only gives us information about the loss, which does not
influence the performance of the repeater.

The first architecture (Fig. 1) that we introduce is that of a repeater which
lets the input system “transfer” the loss errors to the ancilla. Once this
operation is complete, we have essentially recovered the initial state. In
this construction the initial state of the ancilla is free to live in a
$K$-dimensional space, and one is free to pick any value for $K\geq 1$. The
Hamiltonian is

$\displaystyle
H_{\mathrm{direct}}=\sum_{k=1}^{K}\sum_{i=1}^{M}\left(\Psi_{0k}^{(i)}+\Psi_{1k}^{(i)}\right),$
(7)

where $M$ is the number of modes, $\Psi_{jk}^{(i)}$ is a projector onto
$|\Psi_{jk}^{(i)}\rangle=\frac{1}{\sqrt{2}}(|j\rangle_{s}|k^{(i)}\rangle_{a}-|j^{(i)}\rangle_{s}|k\rangle_{a}$,
and the subscripts $s$ and $a$ indicate the system and the ancilla. Here we
must comply with the error spaces that are imposed by the loss operators, i.e.
for the system we have $|j^{(i)}\rangle=E_{i}|j\rangle/\sqrt{\langle
j|E_{i}^{\dagger}E_{i}|j\rangle}$, however we have complete freedom (even in
terms of number of modes) to pick the ancilla states $|k\rangle_{a}$ and
$|k^{(i)}\rangle_{a}$ as long as all together they form an orthonormal set. As
this Hamiltonian acts as the identity operator on the space of non-correctable
errors (two or more photon lost), it automatically avoids these error spaces
to couple back to the code space. It is possible to simplify the direct
Hamiltonian by minimizing the dimension of the initial ancilla space (i.e.
$K=1$):

$\displaystyle
H_{\mathrm{direct}}^{\prime}=\sum_{i=1}^{M}\left(\Psi_{0}^{(i)}+\Psi_{1}^{(i)}\right),$
(8)

where $\Psi_{j}^{(i)}$ projects onto
$|\Psi_{j}^{(i)}\rangle=\frac{1}{\sqrt{2}}(|j\rangle_{s}|0^{(i)}\rangle_{a}-|j^{(i)}\rangle_{s}|0\rangle_{a})$.
Note that in the construction of $H_{\mathrm{direct}}$ (as opposed to the
construction of $H_{\mathrm{swap}}$ in the next subsection) we excluded the
no-loss ($i=0$) element, as the repeater should act as the identity if no
error occurs. We remark that in case a code had a non-trivial evolution under
zero loss, one might need to include a term to act in this eventuality.

The fact that the Hamiltonian is a projector simplifies enormously the
computation of the unitary, which implements the correction at time $t=\pi$:

$\displaystyle U_{\textrm{direct}}=\exp\left(i\pi
H_{\textrm{direct}}\right)=\mathbbm{1}-2H_{\mathrm{direct}}.$ (9)

It is a simple computation to verify that $U_{\mathrm{direct}}$ indeed
corrects an error by “transferring” it to the ancilla:

$\displaystyle
U_{\mathrm{direct}}|\psi^{(i)}\rangle_{s}\otimes|\varphi\rangle_{a}=|\psi\rangle_{s}\otimes|\varphi^{(i)}\rangle_{a},$
(10)

where
$|\psi^{(i)}\rangle_{s}=\alpha|0^{(i)}\rangle_{s}+\beta|1^{(i)}\rangle_{s}$ is
the state of the system after the error $i$, and
$|\varphi\rangle_{a}=\sum_{k}\gamma_{k}|k\rangle_{a}$ is the initial ancilla
state ($|\varphi\rangle_{a}$ is fixed, for example the vacuum).

###  4.2 SWAP unitary

Figure 2: In the “swap” repeater configuration, the system and the ancilla
remain in their respective spaces (the error space $P^{(i)}$ and code space),
but they swap states: the system acquires the state of the ancilla and the
ancilla acquires the state of the system and then proceeds onwards.

The second architecture (Fig. 2) is a repeater that “swaps” the state of the
input system with the state of an ancilla prepared in code space, while
leaving the errors behind. More specifically, the SWAP operation occurs
between $P$ (where the ancilla needs to be initialized) and the space where
the system currently lives, which is $P$ if no error occurred and it is
$P^{(i)}$ if a single photon loss from the $i$-th mode occurred. The
Hamiltonian of this repeater is:

$\displaystyle H_{\mathrm{swap}}$ $\displaystyle=\sum_{i=0}^{M}\Phi^{(i)}$
(11)

where $\Phi^{(i)}$ projects onto
$|\Phi^{(i)}\rangle=\frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{s}|1\rangle_{a}-|1^{(i)}\rangle_{s}|0\rangle_{a})$.
Similar to $H_{direct}$, this Hamiltonian also automatically avoids the
uncorrectable error spaces to couple back to the code space. Note that now in
case of no error the repeater still needs to swap, so we have to include the
$i=0$ case. The correction unitary is again given by

$\displaystyle U_{\textrm{swap}}=\exp\left(i\pi H_{\textrm{swap}}\right).$
(12)

And it is again a simple computation to verify that the unitary corrects by
swapping (compare with Eq. (10)):

$\displaystyle
U_{\mathrm{swap}}|\psi^{(i)}\rangle_{s}\otimes|\varphi\rangle_{a}=|\varphi^{(i)}\rangle_{s}\otimes|\psi\rangle_{a},$
(13)

where
$|\psi^{(i)}\rangle_{s}=\alpha|0^{(i)}\rangle_{s}+\beta|1^{(i)}\rangle_{s}$ is
the state of the system after the error $i$, and
$|\varphi\rangle_{a}=\gamma_{0}|0\rangle_{a}+\gamma_{1}|1\rangle_{a}$ is the
initial ancilla state in code space.

##  5 Beating the repeaterless bound

In this section we evaluate how well our repeaters protect the quantum
information along a lossy channel. Our goal is to beat the repeaterless bound
asymptotically, which in terms of distance $x$ with $\alpha$ dB/Km of loss is
$R(x)=-\log_{2}(1-10^{-\alpha x/10})$. For a telecom fibre, $\alpha\approx
0.2$ dB/Km.

We present a single-mode code to show the effect of the uncorrected no-loss
operator $\sqrt{\eta}^{N}$ and we will compare a four-photon two-mode code [8]
with a three-photon three-mode code [35] to show the advantage of using fewer
photons. We supply the logical codewords that span $P$ explicitly in terms of
the number states in Fock space:

  1. 1.

_single-mode code_ : Logical states $|0_{L}\rangle=|1\rangle$ and
$|1_{L}\rangle=|3\rangle$.

  2. 2.

_two-mode code_ : Logical states
$|0_{L}\rangle=\frac{|4,0\rangle+|0,4\rangle}{\sqrt{2}}$ and
$|1_{L}\rangle=|2,2\rangle$.

  3. 3.

_three-mode code_ : Logical states $|0_{L}\rangle=|1,1,1\rangle$ and
$|1_{L}\rangle=\frac{|0,0,3\rangle+|0,3,0\rangle+|3,0,0\rangle}{\sqrt{3}}$.

These are just simple didactical examples, for an excellent in-depth
presentation of bosonic codes see [22, 16, 1].

We initialize an entangled state
$|\psi\rangle_{AB}=\frac{1}{\sqrt{2}}(|0,0_{L}\rangle+|1,1_{L}\rangle)$ using
the logical states in one of the example codes presented above, and we
propagate the $B$ system through a lossy channel with repeaters separated by a
distance $L$. For codes that are eigenstates of the total photon number, after
one segment comprising loss and repeater action, we find the original state in
code space with probability $p_{s}$ (defined below). Since the uncorrectable
error spaces do not couple back to the code space, we can extract $p_{s}^{n}$
secret bits of key after $n$ segments, which means that the condition to beat
the repeaterless bound asymptotically is that the logarithmic slope of the key
rate be larger than the logarithmic slope of the repeaterless bound:
$p_{s}>\eta_{s}$, where $\eta_{s}$ is the transmissivity of a segment. The
same condition also appears from different arguments, see [26, 9].

We also investigate the stability if we have coupling losses between an
optical fibre and a repeater station. To do this, we consider the total loss
of a segment to be composed of two parts: $\eta=\eta_{c}\eta_{s}$, where
$\eta_{c}$ is the coupling efficiency of the fibre into the repeater and
$\eta_{s}$ is the regular loss of the channel. As the success probability
$p_{s}$ depends on the code space $P$, on the repeater separation $L$ and on
the coupling efficiency $\eta_{c}$, we can find for which values of these
quantities we beat the repeaterless bound (see Fig. 3). For the two-mode code,
we have $p_{s}=\eta^{4}+4(1-\eta)\eta^{3}$, for the three-mode code we have
$p_{s}=\eta^{3}+3(1-\eta)\eta^{2}$, where $\eta=\eta_{c}\eta_{s}$. Here we
assume that a repeater does not couple the error spaces with the code space in
undesired ways (e.g. by adding more photons than necessary, which may lead to
the system entering back into code space upon further loss).

For codes that are not so well-behaved (such as the single-mode code in the
examples), things are more complicated as the code spaces become distorted. In
the case of our single-mode code, we need to compute explicitly the action of
the lossy channel $\mathcal{L}_{\eta}$ from Eq. (3) (where the total loss
$\eta=\eta_{s}\eta_{c}$ is composed of the segment loss and the coupling loss)
followed by the channel $\mathcal{R}$ that maps the signal from the input to
the output of the repeater, which we can describe with $M+1$ Kraus operators:
$R^{(0)}=(\mathbbm{1}-\sum_{i}P^{(i)})$ and $R^{(i)}=|0\rangle\langle
0^{(i)}|+|1\rangle\langle 1^{(i)}|$ for $1\leq i\leq M$ (see Eq. (1)). Note
that we do not correct the distortion on the zero-photon loss (for some codes
it can be possible, especially for small $\eta$ [22]). We find:

$\displaystyle\mathcal{R}\circ\mathcal{L}_{\eta}|1\rangle\langle 1|$
$\displaystyle=|1\rangle\langle 1|$
$\displaystyle\mathcal{R}\circ\mathcal{L}_{\eta}|1\rangle\langle 3|$
$\displaystyle=(\eta^{2}+(1-\eta)\sqrt{3}\eta)|1\rangle\langle 3|$
$\displaystyle\mathcal{R}\circ\mathcal{L}_{\eta}|3\rangle\langle 1|$
$\displaystyle=(\eta^{2}+(1-\eta)\sqrt{3}\eta)|3\rangle\langle 1|$
$\displaystyle\mathcal{R}\circ\mathcal{L}_{\eta}|3\rangle\langle 3|$
$\displaystyle=(3\eta^{2}-2\eta^{3})|3\rangle\langle
3|+(1-3\eta^{2}+2\eta^{3})|1\rangle\langle 1|$

This mapping allows us to compute the $n$-fold application of
$\mathcal{R}\circ\mathcal{L}_{\eta}$. After $n$ segments, the state is

$\displaystyle\psi_{AB}^{(n)}$ $\displaystyle=\frac{1}{2}|0,1\rangle\langle
0,1|+\frac{1}{2}(1-(3\eta^{2}+2\eta^{3})^{n})|1,1\rangle\langle 1,1|$
$\displaystyle\frac{1}{2}(\eta^{2}+(1-\eta)\sqrt{3}\eta)^{n}(|0,1\rangle\langle
1,3|+|1,3\rangle\langle 0,1|)$
$\displaystyle+\frac{1}{2}(3\eta^{2}-2\eta^{3})^{n}|1,3\rangle\langle 1,3|$
(14)

If we perform the six-states protocol [6, 18] with this state, we can compute
the key rate by following the results in appendix A of [31]. We resort to the
six-state protocol because the distortion causes errors, and thus the simple
consideration of success probabilities will not be sufficient to guarantee the
cross-over. What we find is that the region of parameter space that allows us
to beat the repeaterless bound _at some distance_ (note: not asymptotically)
is very small, but it exists (see Fig. 3). We remark that it might be possible
to increase the single-mode key rate by other means (correcting the zero-loss
effect, noisy pre-processing, two-way codes, etc…).

We can finally plot for each code the parameter region that allows us to beat
the repeaterless bound (non-asymptotically for the single-mode code and
asymptotically for the two- and three-mode codes) in a telecom fiber (Fig. 3).

Figure 3: The shaded regions indicate for which coupling efficiency $\eta_{c}$
and repeater separation $L$ we can extract more key than the repeaterless
bound allows (non-asymptotically for the single-mode code and asymptotically
for the two- and the three-mode code). Notably, the three-mode code can
withstand up to about 11% of additional coupling loss between a fibre and a
repeater station.

##  6 Considerations

We can see that unitary one-way quantum repeaters can beat the repeaterless
bound asymptotically, which means that the combination of loss and repeater
stations effectively turns the lossy bosonic channel into a different channel
with lower effective loss, to the point that in a one-way exchange of quantum
states we can extract more secret key than one could obtain from a pure-loss
channel. In Fig. 4 we include an example of the key rate over distance
obtained with the three-mode code over a range of 600 Km including coupling
loss, and compare it to the repeaterless bound.

Figure 4: Example key rates per mode for the three-mode code. For each curve
we have optimized the repeater separation (in parenthesis). Note that even
with 10% of coupling loss between fibres and repeaters, the three-mode code
can still beat the repeaterless bound starting around the 400 km mark.

In this work we have assumed that the repeaters are not introducing any
additional noise, which is an unrealistic assumption. We address the
limitations due to added noise in our upcoming work [9].

This result shows that it is possible to build unsupervised, one-way unitary
stations that act as quantum repeaters, which could be placed conveniently
every few kilometers along the optical fibre network, allowing us to perform
QKD and other distributed quantum information tasks at much increased
distances than what we can currently achieve.

Furthermore, some codes show a remarkable resilience to the _additional_
coupling loss between fibres and repeater stations, which is encouraging from
a technological point of view.

The actual physical realization of a repeater station is a very challenging
problem. There are many possible families of bosonic codes and within each
family there are many possible choices of specific codes. For each code there
is in turn a continuum of Hamiltonians that depend on the initial ancilla
state. So there are innumerable Hamiltonians that one could use, but we do not
yet understand how to determine if a given Hamiltonian can be implemented (or
approximated to a satisfactory degree) by arranging a reasonable number of
optical components. Understanding thed connection between a Hamiltonian and
its implementation is part of our ongoing work.

##  7 Conclusions

We introduced two families of Hamiltonians that generate a quantum repeater
interaction which protects one qubit of information propagating in a lossy
bosonic channel. We presented a few examples to show that it is possible to
overcome the repeaterless bound even in the presence of a mild coupling loss
between the lossy channel and the repeater stations. Our construction is
general and it gives the freedom to choose a code space and an initial ancilla
state, which translates into the potential to discover particularly convenient
Hamiltonians that can be decomposed into a feasible arrangement of a few
simple optical elements.

##  8 Acknowledgements

The authors thank Marco Piani, Ryo Namiki and John Jeffers for helpful
discussions. This work was supported by NSERC Discovery Grant, Industry Canada
and the Army Research Lab.

## References

  * [1] Victor V. Albert, Kyungjoo Noh, Kasper Duivenvoorden, Dylan J. Young, R. T. Brierley, Philip Reinhold, Christophe Vuillot, Linshu Li, Chao Shen, S. M. Girvin, Barbara M. Terhal, and Liang Jiang.  Performance and structure of single-mode bosonic codes.  Phys. Rev. A, 97:032346, Mar 2018.  doi:10.1103/PhysRevA.97.032346.
  * [2] C. H. Bennett and G. Brassard.  Quantum Cryptography: Public Key Distribution and Coin Tossing.  In Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, pages 175–179, New York, 1984. IEEE Press.  doi:10.1016/j.tcs.2014.05.025.
  * [3] Charles H Bennett, Gilles Brassard, Seth Breidbart, and Stephen Wiesner.  Quantum cryptography, or unforgeable subway tokens.  In Advances in Cryptology, pages 267–275. Springer, 1983.  doi:10.1007/978-1-4757-0602-4_26.
  * [4] Cédric Bény and Ognyan Oreshkov.  General conditions for approximate quantum error correction and near-optimal recovery channels.  Physical review letters, 104(12):120501, 2010.  doi:10.1103/PhysRevLett.104.120501.
  * [5] H-J Briegel, Wolfgang Dür, Juan I Cirac, and Peter Zoller.  Quantum repeaters: the role of imperfect local operations in quantum communication.  Physical Review Letters, 81(26):5932, 1998.  doi:10.1103/PhysRevLett.81.5932.
  * [6] Dagmar Bruß.  Optimal eavesdropping in quantum cryptography with six states.  Phys. Rev. Lett., 81:3018–3021, Oct 1998.  doi:10.1103/PhysRevLett.81.3018.
  * [7] Harry Buhrman, Richard Cleve, John Watrous, and Ronald De Wolf.  Quantum fingerprinting.  Physical Review Letters, 87(16):167902, 2001.  doi:10.1103/PhysRevLett.87.167902.
  * [8] Isaac L Chuang, Debbie W Leung, and Yoshihisa Yamamoto.  Bosonic quantum codes for amplitude damping.  Physical Review A, 56(2):1114, 1997.  doi:10.1103/PhysRevA.56.1114.
  * [9] Michael Epping, Filippo Miatto, and Norbert Lütkenhaus.  Noise bounds for quantum repeaters.  to be submitted, 2018.
  * [10] Hong-yi Fan and Li-yun Hu.  New approach for analyzing time evolution of density operator in a dissipative channel by the entangled state representation.  Optics Communications, 281(22):5571–5573, 2008.  doi:10.1016/j.optcom.2008.08.002.
  * [11] Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden.  Quantum cryptography.  Reviews of modern physics, 74(1):145, 2002.  doi:10.1103/RevModPhys.74.145.
  * [12] Daniel Gottesman and Isaac Chuang.  Quantum digital signatures.  arXiv preprint quant-ph/0105032, 2001.
  * [13] H Jeff Kimble.  The quantum internet.  Nature, 453:1023–1030, 2008.  doi:10.1038/nature07127.
  * [14] Emanuel Knill and Raymond Laflamme.  Concatenated quantum codes.  arXiv preprint quant-ph/9608012, 1996.
  * [15] Emanuel Knill and Raymond Laflamme.  Theory of quantum error-correcting codes.  Physical Review A, 55(2):900, 1997.  doi:10.1103/PhysRevA.55.900.
  * [16] Linshu Li, Chang-Ling Zou, Victor V Albert, Sreraman Muralidharan, SM Girvin, and Liang Jiang.  Cat codes with optimal decoherence suppression for a lossy bosonic channel.  Physical Review Letters, 119(3):030502, 2017.  doi:10.1103/PhysRevLett.119.030502.
  * [17] Seth Lloyd, Jeffrey H Shapiro, Franco NC Wong, Prem Kumar, Selim M Shahriar, and Horace P Yuen.  Infrastructure for the quantum internet.  ACM SIGCOMM Computer Communication Review, 34(5):9–20, 2004.  doi:10.1145/1039111.1039118.
  * [18] Hoi-Kwong Lo.  Proof of unconditional security of six-state quantum key distribution scheme.  arXiv preprint quant-ph/0102138, 2001.
  * [19] Hoi-Kwong Lo, Marcos Curty, and Kiyoshi Tamaki.  Secure quantum key distribution.  Nature Photonics, 8(8):595–604, 2014.  doi:10.1038/nphoton.2014.149.
  * [20] Rodney Loudon.  The quantum theory of light.  OUP Oxford, 2000.  doi:10.1119/1.1987930.
  * [21] Prabha Mandayam and Hui Khoon Ng.  Towards a unified framework for approximate quantum error correction.  Phys. Rev. A, 86:012335, Jul 2012.  doi:10.1103/PhysRevA.86.012335.
  * [22] Marios H Michael, Matti Silveri, RT Brierley, Victor V Albert, Juha Salmilehto, Liang Jiang, and Steven M Girvin.  New class of quantum error-correcting codes for a bosonic mode.  Physical Review X, 6(3):031006, 2016.  doi:10.1103/PhysRevX.6.031006.
  * [23] WJ Munro, AM Stephens, SJ Devitt, KA Harrison, and Kae Nemoto.  Quantum communication without the necessity of quantum memories.  Nature Photonics, 6(11):777–781, 2012.  doi:10.1038/nphoton.2012.243.
  * [24] Sreraman Muralidharan, Jungsang Kim, Norbert Lütkenhaus, Mikhail D Lukin, and Liang Jiang.  Ultrafast and fault-tolerant quantum communication across long distances.  Physical review letters, 112(25):250501, 2014.  doi:10.1103/PhysRevLett.112.250501.
  * [25] Sreraman Muralidharan, Linshu Li, Jungsang Kim, Norbert Lütkenhaus, Mikhail D Lukin, and Liang Jiang.  Optimal architectures for long distance quantum communication.  Scientific reports, 6:20463, 2016.  doi:10.1038/srep20463.
  * [26] Ryo Namiki, Liang Jiang, Jungsang Kim, and Norbert Lütkenhaus.  Role of syndrome information on a one-way quantum repeater using teleportation-based error correction.  Physical Review A, 94(5):052304, 2016.  doi:10.1103/PhysRevA.94.052304.
  * [27] Hui Khoon Ng and Prabha Mandayam.  Simple approach to approximate quantum error correction based on the transpose channel.  Physical Review A, 81(6):062342, 2010.  doi:10.1103/PhysRevA.81.062342.
  * [28] Michael A Nielsen and Isaac Chuang.  Quantum computation and quantum information.  AAPT, 2002.  doi:10.1119/1.1463744.
  * [29] Stefano Pirandola, Riccardo Laurenza, Carlo Ottaviani, and Leonardo Banchi.  Fundamental limits of repeaterless quantum communications.  Nature Communications, 8:15043, 2017.  doi:10.1038/ncomms15043.
  * [30] Nicolas Sangouard, Christoph Simon, Hugues De Riedmatten, and Nicolas Gisin.  Quantum repeaters based on atomic ensembles and linear optics.  Reviews of Modern Physics, 83(1):33, 2011.  doi:10.1103/RevModPhys.83.33.
  * [31] Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J Cerf, Miloslav Dušek, Norbert Lütkenhaus, and Momtchil Peev.  The security of practical quantum key distribution.  Reviews of modern physics, 81(3):1301, 2009.  doi:10.1103/RevModPhys.81.1301.
  * [32] Peter W Shor.  Scheme for reducing decoherence in quantum computer memory.  Physical review A, 52(4):R2493, 1995.  doi:10.1103/PhysRevA.52.R2493.
  * [33] Masahiro Takeoka, Saikat Guha, and Mark M Wilde.  Fundamental rate-loss tradeoff for optical quantum key distribution.  Nature Communications, 5:6235, 2015.  doi:10.1038/ncomms6235.
  * [34] Jon Tyson.  Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates.  Journal of mathematical physics, 51(9):092204, 2010.  doi:10.1063/1.3463451.
  * [35] Wojciech Wasilewski and Konrad Banaszek.  Protecting an optical qubit against photon loss.  Physical Review A, 75(4):042316, 2007.  doi:10.1103/PhysRevA.75.042316.
  * [36] Mark M Wilde, Marco Tomamichel, and Mario Berta.  Converse bounds for private communication over quantum channels.  IEEE Transactions on Information Theory, 63(3):1792–1817, 2017\.  doi:10.1109/TIT.2017.2648825.