# The double quasar Q2138-431: detection of a lensing galaxy

M. R. S. Hawkins 1
1Institute for Astronomy (IfA), University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

This paper reviews the question of whether the wide separation double quasar
Q2138-431 is a gravitational lens. From early work, the two quasar images are
known to have almost identical spectra and redshifts, but no lensing galaxy
has so far been detected. In this paper we used recent deep surveys in
infrared and optical bands to search for the presence of a galaxy with the
expected properties of a gravitational lens. The search revealed a $5\sigma$
detection of a faint galaxy between the two quasar images on a deep $J$-band
frame from the VISTA Science Archive, with apparent magnitude $J=20.68$. Non-
detection in the $I$-band implied a redshift $z>0.6$, and mass modelling of
the quasar system gave a mass of $1.31\times 10^{12}M_{\odot}$ for the lensing
galaxy, with mass-to-light ratio $M_{\odot}/L_{\odot}=9.0$. Archival
photographic data from the UK 1.2m Schmidt telescope covering 25 years were
used to construct light curves for the two quasar images, which were then
cross-correlated to measure any time lag. This showed image B to lead image A
by around a year, consistent with 410 days from the mass model. Although the
similarity of the spectra and the detection of the lensing galaxy are the most
compelling arguments for the classification of Q2138-431 as a gravitational
lens, the time delay and mass-to-light ratio provide a consistent picture to
support this conclusion. The wide separation of the quasar images and the
simplicity of the mass model make Q2138-431 an excellent system for the
measurement of the Hubble constant.

###### keywords:

quasars: individual (Q2138-431) – gravitational lensing: strong

††pubyear: 2021††pagerange: The double quasar Q2138-431: detection of a
lensing galaxy–References

##  1 Introduction

The double quasar Q2138-431 was discovered as part of a photographic survey
for quasars based on optical variability (Hawkins et al., 1997). It was
detected as an elongated blue image which varied by over a magnitude in a few
years, and closer inspection showed it to comprise two star-like images
separated by about 4.5 arcsec. Subsequent analysis, which we briefly review in
Section 2, revealed that the two images were quasars with the same redshift,
$z=1.641$. This raised the possibility that the quasar images were part of a
gravitational lens system, although there was no obvious sign of a lensing
galaxy. In Fig. 1 we have constructed a composite 3-colour image using frames
from the Dark Energy Survey (DES) (Abbott et al., 2018) in the $g$, $r$ and
$i$ passbands. The two quasar images are clearly visible near the centre,
together with a sparse population of mostly red galaxies with a limiting
magnitude of $R\approx 24$.

Wide separation gravitational lenses with image separations greater than two
arsecs are rare, and have proved particularly useful in investigating dark
matter distributions (Fadely et al., 2010), and the measurement of the Hubble
constant (Eigenbrod et al., 2005; Suyu et al., 2013; Wong et al., 2017).
Should Q2138-431 be shown to be a two-component gravitational lens, it would
be among the widest separation systems known, and very well suited to mass
modelling and the measurement of a time delay for calculating the Hubble
constant. In the CASTLES111https://www.cfa.harvard.edu/castles/ database of
gravitational lenses, there are only five systems with separations greater
than 4 arcsec. Two of these are quadruple cluster lenses (Oguri et al., 2008;
Fohlmeister et al., 2008; Hawkins, 2020), which although interesting in their
own right present major difficulties for the measurement of the Hubble
constant. The remaining three are doubles, of which RXJ 0921+4529 has now been
shown to be a binary system (Popović et al., 2010). CLASS B2108+213, which was
discovered as a double lensed radio source with a smooth optical spectrum, is
tentatively identified as a BL Lac object (McKean et al., 2005, 2010). So far
the source redshift has not been measured, and the complexity of the lensing
group or cluster suggests some difficulty in measuring the value of the Hubble
constant. The final large separation lens is the well-known double Q0957+561
(Walsh et al., 1979) which has indeed proved useful for the measurement of the
Hubble constant (Fadely et al., 2010), but even here the asymmetry of the
positions of the quasar images and galaxy lenses has complicated the modelling
of the system.

In the paper reporting the discovery of the double quasar Q2138-431, Hawkins
et al. (1997) showed that the spectra of the two quasar images are almost
identical, and that the redshifts are the same to within very small errors.
This made a strong case for the two images to be gravitationally lensed, but
long CCD integrations in the $R$-band failed to reveal a lensing galaxy. For
this paper we have taken advantage of modern deep photometric surveys,
especially in the infrared, to look again for a lensing galaxy between the two
quasar images. Image subtraction of the quasar images has resulted in the
positive $5\sigma$ detection of a galaxy in the $J$-band, but no significant
detections in the $r$-, $i$\- or $z$-bands. We used the photometry in these
passbands to estimate a lower limit to the redshift of the lensing galaxy, and
hence its absolute magnitude. We also used astrometric measurements of the
positions of the quasar images and lensing galaxy to fit a mass model to the
system, and obtain a value for the mass of the deflector. The resulting
parameters imply that the lens is a massive luminous red galaxy (LRG). The
survey data on which this analysis is based is described in Section 2, and the
image analysis techniques and photometry in Section 3.

For a gravitational lens system to be useful for the measurement of the Hubble
constant, it is necessary to measure time delays between the light paths to
the images. The photographic survey which resulted in the discovery of
Q2138-431 is described in more detail in Section 2, and comprised long runs of
yearly observations in several passbands between the years 1977 and 2001.
These plates yield light curves for the two quasar images which can be cross-
correlated to provide a preliminary estimate of the time lag between
variations in the two images, which we describe in Section 4. We then compare
this result with the predicted time delay from the mass model. Throughout the
paper we assume $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$ and $H_{0}=71$ km
s-1 Mpc-1 for cosmological calculations.

Figure 1: Composite 3-colour image from DES frames in $g$, $r$ and $i$
passbands, showing the galaxy environment of the two quasar images near the
centre of the frame, with the brighter image A to the right and image B to the
left. The field also includes some of the local standard stars used for the
measurement of the light curve. The frame is approximately 20 arcsec on a
side, and north is up the page, east to the left.

##  2 Observations

The double quasar Q2138-431 was discovered as part of a survey for variable
objects with an ultra-violet excess and an elongated image on photographic
plates from the 1.2m UK Schmidt telescope at Siding Springs Observatory,
Australia (Hawkins et al., 1997). The purpose of the survey was to find
gravitationally lensed quasars, and Q2138-431 was the most promising candidate
which on closer examination was seen to comprise two images separated by 4.5
arcsec. Spectroscopy of the two images was obtained with the ESO 3.6m
telescope at La Silla, Chile, which showed them to be quasars with no
detectable difference between the two high signal-to-noise spectra, as
illustrated in Figure 2 of Hawkins et al. (1997). The redshifts of the two
quasar images were the same at $z=1.641$, and cross-correlation between the
two spectra showed a difference in velocity of $0\pm 114$ km s-1. Such a small
velocity difference is hard to account for in a random alignment or binary
pair, and strongly suggested that the quasar images were part of a
gravitational lens system. There was however no obvious evidence for a lensing
galaxy. In order to clarify the situation, deep CCD observations of the quasar
images were obtained in the $R$-band, and the area between the two quasar
images investigated using image subtraction techniques to reduce contamination
from the quasar light. No lensing galaxy was detected, with a lower limit of
$R>23.8$.

The failure to find a lensing galaxy suggested the possibility that the lens
might be a dark galaxy, an idea which was receiving some attention at that
time (Hawkins, 1997). However, the limit obtained in the $R$-band is only
useful for relatively low redshift lenses. The 4000 Å feature starts to leave
the $R$-band at around $z\approx 0.4$ and from the $I$-band at $z\approx 0.8$,
making the detection of a lensing galaxy in this regime very challenging. On
this basis we moved the search into the infrared, and made use of the $J$-band
survey from the VISTA Science Archive222http://horus.roe.ac.uk/vsa/ (VSA). The
archive contains image tiles which we used for the detection of the lensing
galaxy, and catalogue data which we used for calibration. The details of the
analysis are described in Section 3. In addition to the $J$-band data, we also
made use of $r$-, $i$\- and $z$-band data from the Dark Energy
Survey333https://des.ncsa.illinois.edu/releases/dr1 (DES) (Abbott et al.,
2018) to look for detections or upper limits in the reddest available optical
bands. Data from this survey were also used for the construction of a local
sequence to calibrate the light curves described in Section 4. The light
curves themselves were measured from an extension of the series of
photographic plates used in the discovery of Q2138-431, and the SuperCOSMOS
scans form part of the SuperCOSMOS Science Archive444http://ssa.roe.ac.uk
(SSA). The measurement and analysis of the light curves of the two quasar
images is described in detail in Section 4.

##  3 The Lensing galaxy

###  3.1 Infrared observations

The Q2138-431 system has celestial coordinates $21^{\rm h}$ $41^{\rm m}$
$16.27^{\rm s}$, -42∘ $57^{\prime}$ $10.0^{\prime\prime}$ (2000) and in the
VISTA Science Archive is covered by frames in the $J$ and $K_{s}$ passbands.
The frames have a pixel size of 0.341 arcsec, and median seeing FWHM = 1.18
arcsec. In Fig. 2 we show cutouts from these frames, with the double quasar at
the top, and below it a star which has been helpful in the photometric
calibration and for image subtraction. Cursory examination of the $J$-band
image in the left hand panel shows a suggestive bulge along the line of
centres of the two quasar images, which has the potential to be the lensing
galaxy. The $K_{s}$-band image in the right hand panel of Fig. 2 does not show
this feature, but the frame does not go very deep, and indeed image B of the
double quasar is barely visible. We discuss detection limits in detail below.

In order to investigate the possibilty that the feature between the quasar
images is the lensing galaxy, we undertook a PSF subtraction procedure to
remove the contaminating quasar light. Attempts to use stars in the field as
models for the PSF, including the one visible in Fig. 2, were not very
successful due to difficulties in registration and poor signal-to-noise. The
best approach proved to be to use a Moffat profile (Trujillo et al., 2001) of
the form:

$P(r)=h\biggl{[}1+\Bigl{(}\frac{r}{\alpha}\Bigr{)}^{2}\biggr{]}^{-\beta}$ (1)

fitted to a star of similar magnitude to the quasar images. We first rebinned
the array to improve registration giving the images shown in the left hand
panel of Fig. 3, and then measured the image centroids with the Starlink GAIA
package. The next step was to fit the Moffat profile to the star in Fig. 2. We
followed Trujillo et al. (2001) and set $\beta=4.765$, and then varied
$\alpha$ and $h$ to give the best fit. For the PSF subtraction we kept these
values of $\alpha$ and $\beta$ and for each of the two quasar images varied
$h$ to get the best fit. The resulting profiles $P(r)$ were then subtracted
from each image using the measured centroids. The results of this procedure
are shown in the right hand panel of Fig. 3. Contours of the subtracted quasar
images are included to show the location and shape of the galaxy image.

In order to assess the likelihood that we had a detection of the lensing
galaxy, the first step was to measure its brightness. To do this, we used
local photometric standards from the VSA catalogue, and then with photometry
routines from the GAIA package we measured the apparent magnitude of the
lensing galaxy to be $J=20.68\pm 0.09$. Based on the rms variation of the sky
background, this is a $5\sigma$ detection. The detection limit of the frame
was determined from the faintest $5\sigma$ detections listed in the VSA
catalogue, and verified by our photometry of the faintest images observed on
the frame covering the quasar position. The $5\sigma$ detection limit we
derived was $J\lesssim 20.7$.

Figure 2: Frames from the VISTA Science Archive in the $J$-band (left hand
panel) and $K_{s}$-band (right hand panel). The frames are approximately 30
arcsec on a side, and north is up the page, east to the left. The frames show
the double image of Q2138-431 at the top, with image A to the right and image
B to the left, and a nearby star used to fit the PSF below.

The right hand panel of Fig. 2 shows the quasar system on the $K_{s}$ frame
from the VSA survey. Visual inspection of the area between the two quasar
images shows no evidence for the presence of a lensing galaxy. The $K_{s}$
frame is clearly less deep than the one for the $J$-band, and the B image of
the quasar is very faint, but we found it useful to measure the detection
limit to put bounds on the colour of the lensing galaxy and its redshift.
Using the same approach as for the $J$-band we found the detection limit to be
$K_{s}\lesssim 18.7$. To estimate the $K_{s}$ magnitude of the lensing galaxy
we used the intrinsic colours for luminous red galaxies (LRGs) from Mannucci
et al. (2001). For the $J-K$ passband they give $J-K=0.87$, which implies
$K_{s}\gtrsim 19.8$, well below the detection limit of the $K_{s}$ frame and
consistent with non-detection in the $K_{s}$ passband. Applying $K$ and
evolutionary corrections from Poggianti (1997) up to a redshift $z\sim 1$ does
not significantly alter this.

###  3.2 Optical observations

In addition to the $J$\- and $K_{s}$-band images from the VSA survey, the
Q2138-431 system was also observed as part of the Dark Energy Survey in the
$g$, $r$, $i$ and $z$ passbands (Abbott et al., 2018). These frames have a
pixel size of 0.263 arcsec with a median seeing FWHM = 0.94 arcsec, and proved
very useful in putting limits on the redshift of the lensing galaxy. To
estimate the expected magnitudes of the lensing galaxy in these passbands, we
again made use of the intrinsic colours of Mannucci et al. (2001), based on
template spectra of a large sample of galaxies. We converted the Johnson
colours of Mannucci et al. (2001) to the SDSS system using the colour
transformations of Jordi et al. (2006). These were then applied to the
observed $J$ magnitude of the lensing galaxy to obtain estimated magnitudes of
$g=23.56$, $r=22.73$, $i=22.36$ and $z=22.18$ in the galaxy rest frame.
Examination of the DES frames showed no indication of the presence of a
lensing galaxy, in agreement with the results of Hawkins et al. (1997) from
CCD frames of similar depth to the DES observations. The 95% completeness
limits in these passbands (Abbott et al., 2018) are $g<23.72$, $r<23.35$,
$i<22.88$ and $z<22.25$. In all cases the estimated rest wavelength magnitudes
are within these bounds, and so the non-detections can be used to set minimum
values for the $K$ corrections, and hence the minimum redshift of the lensing
galaxy. As might be expected, the non-detection in the $i$-band provides the
strongest constraint on the redshift of the lensing galaxy. Using $K$
corrections from Poggianti (1997), the minimum redshift for the lensing galaxy
to move below the detection threshhold is $z\gtrsim 0.5$.

So far, we have neglected the possible effect of an evolutionary correction in
our calculations. Evolutionary corrections can be unreliable and model
dependent (D’Souza et al., 2015), but this can be allowed for by comparing the
observed lower limit to the $i-J$ colour with the intrinsic $(i-J)_{0}$
colour. In this case, the large evolutionary corrections largely cancel out,
and the redshift can be estimated by determining the point at which the
addition of the combined $K+e$ correction to the intrinsic colour matches the
observation. We find that $(i-J)_{obs}>2.2$, and $(i-J)_{0}=1.7$ implying a
$K+e$ correction of 0.5, corresponding to a redshift of $z>0.6$. The inclusion
of the evolutionary correction thus slightly tightens the redshift limit.

A lower limit to the redshift of the lensing galaxy of around $z\gtrsim 0.6$
is not surprising, given the faintness of the $J$ magnitude, and provides an
explanation as to why the lens galaxy is not detected in optical passbands.
The lower limit on $z$ also implies a lower limit on the luminosity of the
lensing galaxy. Using the $K$ and evolutionary corrections for $z=0.6$ from
Poggianti (1997), we obtain $M_{J}=-21.02$. Applying a bolometric correction
BC = 2.13 for the VISTA $J$-band frames from Choi et al. (2016) gives an
absolute magnitude $M_{bol}=-23.15$, equivalent to a luminosity
$L_{gal}=1.45\times 10^{11}L_{\odot}$.

###  3.3 Lens model

Figure 3: The left hand panel is from a $J$-band frame from the VISTA Science
Archive. The frame has been re-binned and shows image A to the right and image
B to the left. The small bulge extending to the left of image A is the
candidate lensing galaxy. In the right hand panel the quasar light has been
subtracted, clearly revealing the presence of a galaxy between the two quasar
images, the positions of which are indicated by superimposed contours. The
plots are approximately 10 arcsec on a side, and north is up the page, east to
the left.

To provide a consistent picture of the properties of the lensing galaxy, we
made use of the lens modelling software of Keeton (2001) to estimate the lens
mass. We used the GAIA astrometry package to measure the positions of the two
quasar images, obtaining a refined value for the image separation of 4.481
arcsec, and also the position of the lensing galaxy. $J$-band flux measures
for the two quasar images from the VISTA frame were also included in the lens
model. We then used routines from the software of Keeton (2001) to fit a
singular isothermal ellipsoid plus external shear $({\rm SIE}+\gamma)$ model
to the astrometric and photometric data, which are summarised in Table 1. We
obtained a good fit to the positions, and derived a value for the Einstein
radius of $\theta_{E}=2.019$ arcsec, close to that found by Hawkins et al.
(1997). We obtained $e=0.1$ for the lens ellipticity, in position angle -10∘
measured east of north. For the external shear we found $\gamma=0.1$ in
position angle 40∘. The overall goodness of fit gave $\chi^{2}=12.6$.

Table 1: Summary of the observations for Q2138-431 used in the lens modelling. Observations | A | B | G
---|---|---|---
Positions | RA (arsec) | $4.117\pm 0.02$ | 0 | $2.815\pm 0.02$
| Dec (arsec) | $1.558\pm 0.02$ | 0 | $1.259\pm 0.02$
Fluxes |  $J$ (mags) | $18.24\pm 0.03$ | $19.85\pm 0.07$ | $20.68\pm 0.09$

The mass $m$ of the lens depends mainly on $\theta_{E}$, but is also a
function of the quasar and galaxy redshifts. To be specific,

$m=\frac{c^{2}{\theta_{E}}^{2}}{4G}\frac{D_{LS}}{D_{L}D_{S}}$ (2)

where $D_{L}$, $D_{S}$ and $D_{LS}$ are the angular diameter distances to the
lens, to the source, and from the lens to the source respectively. Using the
lower limit $z=0.6$ derived above, this gives $m_{lens}=1.31\times
10^{12}M_{\odot}$. Combining this limit with the luminosity limit
$L_{gal}=1.45\times 10^{11}L_{\odot}$ derived above gives a mass-to-light
ratio $M_{\odot}/L_{\odot}=9.0$. To give an idea of the sensitivity of this
result to redshift, if we assume $z=1.0$ then $m_{lens}=2.88\times
10^{12}M_{\odot}$, $L_{gal}=3.53\times 10^{11}L_{\odot}$ and
$M_{\odot}/L_{\odot}=8.2$.

##  4 Light curves

The discovery of Q2138-431 was part of a long term survey to detect quasars on
the basis of their variability (Hawkins, 1996). The survey was based on an
extensive sequence of photographic plates in several colours taken with the UK
1.2m Schmidt Telescope of the ESO/SERC field 287, centred on $21^{\rm h}$
$28^{\rm m}$, $-45^{\circ}$ (1950). Of particular interest for the detection
of quasars is the $B_{j}$ passband, bounded by the GG395 filter in the blue,
and the IIIaJ emulsion cut-off in the red at about 5400 Å, and quite close to
the SDSS $g$-band. The plates were originally measured with the COSMOS
measuring machine at the Royal Observatory, Edinburgh (MacGillivray & Stobie,
1984), which after photometric calibration produced light curves covering some
16 years. Although these light curves were sufficient to detect a large number
of quasars from their variability, including Q2138-431, the machine measures
used a low detection threshhold above the sky, which merged the two quasar
images. This meant that the light curves of the two images could not be
measured separately. Since the discovery of Q2138-431, the original sequence
of plates in the $B_{j}$ passband has been extended to include at least one
plate, and usually 4 or more, every year between 1977 and 2001. These have now
all been measured with the SuperCOSMOS measuring machine (Hambly et al.,
2001), and the scans and catalogues form part of the SuperCOSMOS Science
Archive.

Although the SuperCOSMOS measures are superior in many ways to the earlier
COSMOS measures, the SSA catalogues are also based on measures with a low
detection threshhold in order to maximize the depth of the survey. This again
has the effect that the double quasar images are not resolved, and so we
retrieved the original digitized mapping mode scans with a view to using more
sophisticated photometric routines to measure the two images separately. Fig.
4 shows cut-outs of the double quasar images from two of the plates from the
quasar variability survey. The exposures were taken in 1977 and 1986, and
illustrate the change in magnitude difference between the two images over 9
years.

To measure the light curves we first extracted a square array from the
SuperCOSMOS scan of each plate of side 8 arcmin and centred on Q2138-431. We
then selected 18 stars in this field, spanning the magnitudes of the two
quasar images, with which to construct a local photometric sequence. These
stars were then identified in the $g$-band frame of the area from the DES
survey. Taking advantage of the linearity of the CCD frames as opposed to the
non-linear reciprocity failure of the photographic plates, we used the GAIA
photometric package to measure the magnitudes of the sequence stars. The DES
$g$-band and the photographic $B_{J}$-band are very similar, with an effective
wavelength of 4670Å, and we could detect no significant colour term. On this
basis we used the $g$-band measures of the sequence stars to directly
calibrate the $B_{J}$ photographic images.

Figure 4: The two frames are from photographic plates taken by the UK 1.2m
Schmidt telescope, and scanned by the SuperCOSMOS measuring machine at the
Royal Observatory, Edinburgh. They illustrate the material on which the quasar
light curves are based, and the change in magnitude difference between the two
quasar images in the centre, over a period of 9 years. Image A to the right
has decreased in brightness by about 0.5 magnitudes relative to image B on the
left. Also included near the bottom of the plot is a standard star. The plots
are approximately 40 arcsec on a side, and north is up the page, east to the
left. Figure 5: The top panel shows a 25 year light curve for image A (red)
and image B (blue) of the double quasar Q2138-431, from SuperCOSMOS measures
of a long series of UK 1.2m Schmidt plates in the $B_{J}$ passband. The middle
panel shows the same data, binned in yearly intervals. The bottom panel shows
the difference light curve for the A and B images, with the observed lag of 1
year for image A removed.

The first stage in the calibration was to transform the transmision values of
the SuperCOSMOS scans to $I$, the intensity of light incident on the
photographic plate, using the relation:

$I\propto\biggl{(}{\frac{T_{c}}{T}}\biggr{)}^{1/\gamma}$ (3)

where $T$ is the transmission through exposed emulsion, and $T_{c}$ through
clear plate. $\gamma$ is the gradient of the linear part of the response curve
which in the first instance we set to 2.5. However, $\gamma$ is basically a
scaling factor which becomes redundant with the sequence star calibration.

The next step was to measure the magnitudes of the 18 sequence stars and two
quasar images on each of the 79 plate scans from the Supercosmos archive which
were available for incorporation into the light curves. To do this, the
transmision values were transformed to intensity using equation (3), and the
resulting arrays converted to FITS files for analysis with the GAIA photometry
package. Then for each plate the sequence star and quasar images were measured
with the aperture photometry routine to give a quasi-linear relative flux, and
thence converted to instrumental magnitudes. Using an aperture of radius 2
arcsec, the two quasar images were easily resolved on all plates. The final
step was to transform these instrumental magnitudes to the sequence star
magnitudes using a least squares third order polynomial fit. This
transformation was then used to evaluate the true magnitudes of the two quasar
images in the $B_{J}$ passband.

The top panel of Fig. 5 shows the light curves for the two quasar images from
1977 to 2001. The error bars are derived from the dispersion about the
calibration curve, and are typically about 0.05 magnitudes, with little
dependence on brightness. The long term trend of the light curves is better
seen in the middle panel of Fig. 5 where the observations for each year are
averaged with a weighted mean. Little is lost by this procedure, as in most
years the plates were taken within a period of about three months, and are not
useful for measuring changes within the year. It will be seen that until 1985
the difference between the two light curves averaged around 1.5 magnitudes.
After this image A dimmed, and the difference decreased by about 0.3
magnitudes. In the event that the double quasar is a gravitational lens
system, we would expect intrinsic variations in the quasar to show up in both
light curves, separated by a time interval corresponding to the difference in
light travel time to the two quasar images. In addition, it is clear from
large scale monitoring programmes of gravitational lenses that in most lensed
systems the quasar images also vary independently of each other, which is
generally accepted to be the result of microlensing by stellar mass bodies
along the line of sight (Tewes et al., 2013). To look for a time lag between
variations in the two images, and minimise the effects of microlensing, we
split the light curves into two sections defined by the larger and smaller
average magnitude differences, and measured the correlation coefficient for
different time lags. The results are displayed in Table 2, and show for both
time intervals a strong correlation between the two light curves when image B
leads image A by one year. Given the poor time resolution of the light curves,
this result should not be taken too seriously, but the existence of a time lag
provides confirmation that Q2138-431 is a gravitational lens, and is a
starting point for a more precise measurement.

Table 2: Correlation coefficients for time lags between images A and B of Q2138-431. B leads A by (years) | 2 | 1 | 0 | -1 | -2
---|---|---|---|---|---
1977 – 1985 | +0.04 | +0.70 | +0.04 | -0.69 | -0.59
1986 – 1994 | -0.20 | +0.61 | +0.31 | +0.04 | +0.18

The bottom panel of Fig. 5 shows the difference between the two light curves
in the middle panel, with the curve for image B shifted back one year. The
zero-point for the difference curve can be calculated from the magnitude
difference of the two images in the infrared, where due to the size of the
emitting region the effects of microlensing are negligible (Blackburne et al.,
2011). From the $J$-band frames used for the detection of the lensing galaxy
we find for the magnitude difference $J_{\rm A}-J_{\rm B}=-1.77$, which we use
as the zero-point for the difference curve in the bottom panel of Fig. 5. This
now shows the true differential effect of microlensing.

##  5 Discussion

The idea behind this paper is to review the status of the double quasar
Q2138-431 by taking advantage of improved observational data since its
discovery in 1997. At the time the case for classifying it as a gravitational
lens was very strong. The spectra were almost identical, as illustrated in
Figures 1 and 2 of Hawkins et al. (1997), and for two randomly selected
quasars this is very unlikely as illustrated in Figure 4 of Hawkins et al.
(1997). There is of course the possibility that two quasars evolving in the
same cluster environment could aquire the same observational characteristics,
but the actual mechanism by which this could happen consistent with our
knowledge of cluster evolution is far from clear. The one sticking point in
the way of classifying the system as a gravitational lens was the failure to
find a lensing galaxy.

Attempts to find a lensing galaxy by Hawkins et al. (1997) were mainly
focussed on deep CCD observations in the $R$-band. They also reported
observations in the $K$-band, but their frame is of a similar depth to the
$K_{s}$-band frame in Fig. 2, and barely detects image B of the quasar. Their
$R$-band frame goes much deeper, and they derive a limit to the lensing galaxy
brightness of $R>23.8$. With this faint upper limit to the brightness and an
estimate of the lens mass they derive a very large lower limit to the mass-to-
light ratio of any lensing galaxy. Their lens mass is similar to the one we
derive from mass modelling, but for reasons which are not clear Hawkins et al.
(1997) appear to assume zero redshift luminosities, and do not take into
account the very large $K$ and evolutionary corrections associated with a
distant galaxy. When these are allowed for, it seems unlikely that the galaxy
we have detected in the $J$-band would have been detected by Hawkins et al.
(1997) in their $R$-band frame for redshifts $z\gtrsim 0.6$.

Based on measurements of random fluctuations in the sky background, the object
revealed after PSF subtraction in Fig. 3 is a $5\sigma$ detection. Its
appearance is similar to other faint galaxies in the surrounding 10 arcmin
field, where the presence of an optical counterpart confirms their identity as
galaxies. A good example is the faint galaxy to the south-east of the quasar
system, used by Hawkins et al. (1997) to put limits on the $R$-band magnitude
of a lensing galaxy. This galaxy is also clearly visible on deep $r$\- and
$i$-band frames from the DES survey, and is detected near the limit of the
$J$-band frame in Fig. 3. There is in addition some indication of other faint
galaxies in the vicinity of image A of the quasar. We have also searched the
field for artefacts which might be mistaken for faint galaxies, but these are
confined to single pixels, presumably cosmic ray tracks.

The value obtained for the mass-to-light ratio of the lens in Section 3 of
$M_{\odot}/L_{\odot}=9.0$ assumes a redshift of $z=0.6$ based on non-detection
of the lens in the $i$-band. As this redshift is essentially a lower limit, we
investigated the effect of increasing it to $z=1.0$ and derived
$M_{\odot}/L_{\odot}=8.2$ for the lensing galaxy. This small change in
$M_{\odot}/L_{\odot}$ is due to the fact that an increase in redshift results
in larger values for both luminosity and mass which tend to cancel out in the
mass-to-light ratio. Our value of $M_{\odot}/L_{\odot}$ lies quite close to
the relation between mass and mass-to-light found by Cappellari et al. (2006),
and thus adds to a consistent picture of the double quasar as a gravitational
lens.

The mass modelling of Q2138-431 turned out to be straightforward. This was not
unexpected, as the lensing galaxy lies close to the line of centres of the
quasar images, and roughly in the position one would expect given their
brightness ratio. This contrasts strongly with for example the asymmetric
Q0957+561 (Fadely et al., 2010), and more closely resembles HE1104-1805 (Sluse
et al., 2012), another wide separation binary. This simplicity of modelling is
important if Q2138-431 is to be used for measuring the Hubble constant, where
the accuracy with which light travel time to the two images can be modelled is
the most important factor limiting the accuracy of the result.

The software package of Keeton (2001) allows for the estimation of the time
delay between two images of the modelled system, assuming a value for the
Hubble constant. We applied this procedure to our model of Q2138-431 using a
canonical value $H_{0}=71$ km s-1 Mpc-1 which implied a time lag with image B
leading image A by 410 days. This is consistent with our crude estimate of 1
year from yearly observations, and hopefully can provide a starting point for
a well sampled photometric monitoring programe designed to measure the value
of $H_{0}$. For this calculation we assumed a lens redshift $z=0.6$, equal to
the lower limit derived in Section 3. Increasing the lens redshift results in
a larger predicted time lag. For example a lens redshift of $z=1.0$ implies a
time lag of 1130 days, which suggests that the true lens redshift is not much
greater than our lower limit of $z\gtrsim 0.6$.

In the discovery paper of the double quasar Q2138-431 (Hawkins et al., 1997),
the strongest evidence supporting the gravitational lens hypothesis was the
close similarity of the spectra of the two quasar images, and the very small
difference between their redshifts. However, the authors concluded that
without the detection of a lensing galaxy the claim that the system was a
gravitational lens was insecure. In this paper we have brought together a
number of new lines of argument to support the classification of Q2138-431 as
a gravitational lens. The detection of the lensing galaxy is perhaps the most
conclusive, but the simplicity of the mass model, the derivation of a
plausible mass-to-light ratio, and the detection of a time lag between the
light curves of the two quasar images in agreement with the mass model help to
provide a consistent picture of a gravitational lens system. The measurement
of the redshift of the lensing galaxy and the accurate determination of the
time lag between the two images should then provide an excellent basis for a
reliable estimate of the value of the Hubble constant.

##  6 Conclusions

In this paper we have re-examined the question of whether the double quasar
Q2138-431 is a gravitational lens system. The system was discovered as part of
a survey for quasars based on their variability, and elongated images were
included as possible detections of gravitational lenses. Early analysis of
Q2138-431 (Hawkins et al., 1997) showed it to comprise two quasar images
separated by 4.5 arcsec, with almost identical spectra and redshifts. This
provided strong evidence for a gravitational lens, but deep CCD photometry in
the $R$-band failed to reveal a lensing galaxy. The authors thus concluded
that there was insufficient evidence to definitively identify the system as a
gravitational lens.

With the advent of more recent deep photometric surveys, especially in the
infrared, we searched again for a lensing galaxy. In this case we successfully
detected a candidate lens on a deep $J$-band frame from the VISTA Science
Archive. The apparent magnitude of the $5\sigma$ detection was measured to be
$J=20.68$, and non-detection in the optical bands implied a redshift $z\gtrsim
0.6$.

The wide separation of the quasar images at 4.481 arcsec and the apparent
simplicity of the lens system make Q2138-431 an attractive candidate for the
measurement of the Hubble constant, which would require mass modelling of the
system. Based on our measurements of the positions of the quasar images and
lensing galaxy we were able to obtain a satisfactory fit with a ${\rm
SIE}+\gamma$ model, implying an estimated mass for the lens of
$m_{lens}=1.31\times 10^{12}M_{\odot}$. Combining this with the infrared
photometry in the $J$-band gives a mass-to-light ratio
$M_{\odot}/L_{\odot}=9.0$ for the lensing galaxy.

The estimation of the Hubble constant in a gravitational lens system also
requires a measurement of the time delay or difference in light travel time
between the quasar images. We were able to make a preliminary assessment of
this from an archival photographic monitoring programme covering 21 years.
Although the effective time resolution of the survey was only about 1 year, we
were able to show by cross-correlating the light curves of the two quasar
images that image B leads image A by about a year. It was possible to estimate
from our mass model that the expected time delay was about 410 days,
consistent with the rough value from the light curve analysis.

The overall conclusion of this paper is that the double quasar Q2138-431 is
confirmed as a gravitational lens system. The close similarity of the spectra,
the detection of the lensing galaxy, the plausible mass-to-light ratio, and
the measurement of a time delay between the two images in agreement with
predictions from the mass model provide a consistent picture of a
gravitational lens. Q2138-431 appears to be a system well suited for the
measurement of the Hubble constant. The wide separation of the quasar images
should make measuring the time delay from photometry of the light curves
straightforward, and the simplicity of the system should enable accurate
modelling to estimate the value of the Hubble constant.

## Acknowledgements

I am most grateful to Nigel Hambly for retrieving the SuperCOSMOS scans used
for constructing the light curves.

## Data Availability

The data upon which this paper is based are all publicly available and are
referenced in Section 2, with some additional comments in the text.

## References

  * Abbott et al. (2018) Abbott T.M.C. et al., 2018, ApJS, 239, 18
  * Blackburne et al. (2011) Blackburne J.A., Pooley D., Rappaport S., Schechter P.L., 2011, ApJ, 729, 34
  * Cappellari et al. (2006) Cappellari M. et al., 2006, MNRAS, 366, 1126
  * Choi et al. (2016) Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B.D., 2016, ApJ, 823, 102
  * D’Souza et al. (2015) D’Souza R., Vegetti S., Kauffmann G., 2015, MNRAS, 454, 4027
  * Eigenbrod et al. (2005) Eigenbrod A. et al., 2005, A&A, 436, 25
  * Fadely et al. (2010) Fadely R., Keeton C.R., Nakajima R., Bernstein G.M., 2010, ApJ, 711, 246
  * Fohlmeister et al. (2008) Fohlmeister J. et al., 2008, ApJ, 676, 761
  * Hambly et al. (2001) Hambly N.C., Irwin M.J., MacGillivray H.T., 2001, MNRAS, 326, 1295
  * Hawkins (1996) Hawkins M.R.S., 1996, MNRAS, 415, 2744
  * Hawkins (1997) Hawkins M.R.S., 1997, A&A, 328, L25
  * Hawkins (2020) Hawkins M.R.S., 2020, A&A, 643, A10
  * Hawkins et al. (1997) Hawkins M.R.S, Clements D., Fried J.W., Heavens A.F., Véron P., Minty E.M., van der Werf P., 1997, MNRAS, 291, 811
  * Jordi et al. (2006) Jordi K., Grebel E.K., Ammon K., 2006, A&A, 460, 339
  * Keeton (2001) Keeton C.R., 2001, arXiv:astro-ph/0102340
  * MacGillivray & Stobie (1984) MacGillivray H.T., Stobie R.S., 1984, Vistas Astron., 27, 433
  * Mannucci et al. (2001) Mannucci F. et al., 2001, MNRAS, 326, 745
  * McKean et al. (2005) McKean J.P. et al., 2005, MNRAS, 356, 1009
  * McKean et al. (2010) McKean J.P. et al., 2010, MNRAS, 404, 749
  * Oguri et al. (2008) Oguri M. et al., 2008, ApJ, 676, L1
  * Poggianti (1997) Poggianti B.M., 1997, A&AS, 122, 399
  * Popović et al. (2010) Popović L.C̆. et al., 2010, ApJ, 721, L139
  * Sluse et al. (2012) Sluse D., Chantry V., Magain P., Courbin F., Meylan G., 2012, A&A, 538, A99
  * Suyu et al. (2013) Suyu S.H. et al., 2013, ApJ, 766, 70
  * Tewes et al. (2013) Tewes M., Courbin F., Meylan G., 2013, A&A, 553, A120
  * Trujillo et al. (2001) Trujillo I., Aguerri J.A.L., Cepa J., Gutiérrez C.M., 2001, MNRAS, 328, 977
  * Walsh et al. (1979) Walsh D., Carswell R.F., Weymann R.J., 1979, Nature, 279, 381
  * Wong et al. (2017) Wong K.C. et al., 2017, MNRAS, 465, 4895


# Security of One-Way Entanglement Purification with Quantum Sampling Against
a Restricted Adversary

Cameron<EMAIL_ADDRESS>Physics Department, University of
Connecticut, Storrs, CT 06269, USA

Entanglement purification protocols promise to play a critical role in the
future of quantum networks by distributing entanglement across noisy channels.
However, only the security of two-way purification protocols have been closely
studied. To address this, we propose a one-way entanglement purification
protocol which utilizes quantum sampling and prove its security against an
adversary restricted to single qubit Pauli gates. This is done through
leveraging the equivalence of one-way entanglement purification protocols with
error-correcting codes. To prove the security of this protocol, we first use
the quantum sampling framework introduced by Bouman and Fehr to estimate the
Hamming weight of the qubits which passed through the channel and then use the
estimated relative Hamming weight $\omega$ to determine the amount of
interference that Eve has subjected to the quantum channel. Since Eve is
restricted to single qubit Pauli gates, the number of applied gates can be
directly estimated using the Hamming weight. Estimating the number of
adversarial single qubit gates, allows us to perform error correction and
disentangle the logical qubit from Eve with probability
$1-\epsilon_{qu}^{\delta}$. Since this protocol allows communication only in
one direction, the distance of the code must be decided before transmission,
and therefore Bob will be forced to abort the protocol if he finds that Eve
has applied more gates than the code can correct. One-way protocols may find
use when communication is limited, or when we desire to decrease latency
compared to the multiple rounds of communication needed in two-way protocols.
Further research may investigate the security of this protocol against
arbitrary single or multi-qubit gates to obtain security guarantees against a
more general adversary.

##  1 Introduction

Entanglement is a novel computational resource in quantum information science
with no similar classical resource. This resource has found numerous uses
throughout quantum information science. Two parties with entangled qubits can
transmit perfectly secure quantum information through quantum teleportation
[1, 2] or perfectly secure classical information through superdense coding
[3]. Parties with access to entanglement can use quantum correlations to
succeed at games such as ”magic squares” more than is classically possible
[4]. Quantum entanglement has also found use in quantum processes such as
distributed quantum computation [5] and quantum cookies [6]. Motivated by
these uses, the topic of how to securely distribute quantum states has
recently gained interest, including through investigation of quantum secure
direct communication [7, 8, 9] and secure quantum dialogues [10, 11].

In contrast to the quantum secure direct communication and quantum dialogue
protocols above which attempt to distribute entanglement through bell pairs,
the protocol proposed in this paper will attempt to distribute entanglement
through error correcting codes. Due to this focus on distributing
entanglement, we will allow Alice and Bob the classical resource of a shared
secret key prior to the start of the protocol. In this protocol, we will also
utilize entanglement purification protocols, which were historically brought
about to distill high fidelity entangled states quantum states travelling
through a noisy channel [12]. However, since an eavesdropper can be viewed as
a source of noise in the quantum channel, entanglement purification protocols
also work to remove the effect of an adversary on the transmitted message.
Entanglement purification protocols have been investigated for use in high
fidelity quantum communication [13], and previously explored two-way protocols
promise strong security for transferring quantum information [14].

###  1.1 Entanglement Purification Protocols

There are two different classes of entanglement purification protocols, one-
way and two-way protocols [6]. Two-way entanglement purification protocols
allow for communication between Alice and Bob after qubits pass through the
quantum channel. Since Alice and Bob can conditionally apply gates to their
systems based off each other’s measurement results, two-way entanglement
purification protocols are in general stronger than one-way protocols. For
example, two-way purification protocols can allow for Alice and Bob to purify
their qubits from a $50\%$ depolarizing channel, which one-way protocols
cannot correct [6]. Two-way purification protocols have been previously proven
secure through demonstrating that a two-way protocol can disentangle the
purified system from an external system or eavesdropper [14]. This posits the
question if one-way protocols can similarly be proven secure.

In comparison to two-way protocols, one-way entanglement purification
protocols do not allow for communication between parties after the message is
sent. Restriction to one-way communication incidentally makes these protocols
equivalent to quantum error correction since Alice and Bob can be time-like
separated [6]. Due to this, we can use quantum error-correcting codes to
evaluate the security of one-way entanglement purification protocols [15]. The
correspondence between one-way entanglement purification protocols and error-
correcting codes has a prior basis in the literature, as it has previously
been used by Shor and Preskill to prove the BB84 QKD protocol secure [16].
Investigating the security of one-way protocols may be useful in scenarios
where communication between two parties is limited, or when we desire to
decrease the latency of quantum communication compared to the multiple rounds
of classical communication needed between Alice and Bob in two-way protocols.

In one-way entanglement purification, Bob is forbidden from sending messages
to Alice. This presents a problem for quantum sampling, as this restriction
requires Bob to know the states in which the sampling qubits were prepared in
order to perform sampling and determine the Hamming weight of the sampling
qubits. If Alice naively announces the prepared sampling states over a
classical channel, then Eve can simply intercept all the qubits and use this
information to resend identical qubits to Bob, circumventing the quantum
sampling procedure. Fundamentally, this problem occurs because in one-way
protocols the actors Bob and Eve are symmetric [6]. To solve this problem, we
will assume that Alice and Bob share a secret classical key, breaking this
symmetry.

A shared classical key will allow Alice to send a secure classical message to
Bob. This message will contain many critical pieces of information for the
protocol, including the prepared sampling states, the permutation which Alice
applied to the transmitted qubits, and the distance of the error-correcting
code which Alice employed. At this point in the protocol, Bob will be able to
perform quantum sampling and estimate the relative Hamming weight $\omega$ of
Eve’s attack. Bob will then be able to use the relative Hamming weight to
estimate the number of gates Eve has applied to the logical qubit, if we
assume Eve is restricted to single qubit Pauli gates. Finally, Bob can
calculate if the error-correcting code distance is large enough to disentangle
Eve. If the code is sufficiently large, then Bob performs error correction and
keeps the resulting logical qubit. Otherwise, the code distance is too small
and Bob aborts.

##  2 Sampling

Before presenting the protocol we should first familiarize ourselves with
quantum sampling. The quantum sampling framework used here was introduced by
Bouman and Fehr [17], and we refer to their paper for a more rigorous
introduction. This section is intended to be a broad overview to the framework
put forth there.

Generically, sampling allows for an individual to learn information about a
population through measuring a subset of that population. Classical sampling
has been well studied [18]. However, due to entanglement in quantum systems it
is not obvious how classical sampling strategies could be extended into
quantum systems. Bouman and Fehr’s quantum sampling framework addresses this
by allowing for classical sampling strategies to be used in quantum systems.
The sampling framework outputs an estimate of the Hamming weight of a quantum
system. The definition of the Hamming weight is extended in this framework to
entangled states and states in superposition [17].

From Bouman and Fehr’s quantum sampling framework we will be interested in two
quantities: the estimated relative Hamming weight $\omega$ of the message
qubits and the error bound of the quantum sampling process
$\epsilon_{qu}^{\delta}$. In Sections 5 and 6 we will use the relative Hamming
weight $\omega$ to estimate the number of gates applied by an eavesdropper,
and we will use the error bound $\epsilon_{qu}^{\delta}$ to estimate the
probability that the sampling strategy has failed and Eve has gained access to
the transmitted logical qubit without detection.

###  2.1 Classical Sampling

As an introduction to sampling, let us consider a classical string
$q=(q_{1},q_{2},...q_{n})\in\\{0,1\\}^{n}$. The Hamming weight of this string
is defined as $wt(q)=|\\{i|q_{i}\neq 0\\}|$, or in simpler terms, the Hamming
weight is the number of nonzero characters in this string. The role of
sampling is to use a substring $q_{t}$ to estimate the Hamming weight of the
remaining string $q_{\bar{t}}$. The sampling strategy employed will select a
completely random subset of $q$ to determine $q_{t}$. With the sampled
substring $q_{t}$, the relative Hamming weight
$\omega(q_{t})=\frac{wt(q_{t})}{N}$ will be used as an estimate of the
relative Hamming weight of the remaining string
$\omega(q_{\bar{t}})=\frac{wt(q_{\overline{t}})}{M}$, where $N$ is the number
of sampling qubits and $M$ is the number of message qubits.

To find the failure probability of the sampling procedure, we will start by
considering the set of all substrings that would output a relative Hamming
weight which is $\delta$-close to the true relative Hamming weight.

$B_{t}^{\delta}=\\{\textbf{q}\in\mathcal{A}^{n}:\absolutevalue{\omega(q_{\overline{t}})-\omega(q_{t})}<\delta\\}$
(1)

This is the set of all substrings for which the sampling procedure will
succeed. Through considering this set, it is clear that the probability that
the sampling procedure fails, producing an estimate greater than $\delta$ from
the true Hamming weight, which is equal to the probability that the string is
not in the set $B_{t}^{\delta}$.

$\epsilon_{cl}^{\delta}=\max_{q\in\mathcal{A}^{n}}Pr[q\notin B_{t}^{\delta}]$
(2)

Assuming we are randomly sampling $k$ entries [17], we find,

$\epsilon_{cl}^{\delta}<4\exp{-\frac{1}{3}\delta^{2}k}$ (3)

Therefore, classical sampling is able to estimate the relative Hamming weight
$\omega(q_{t})$ of a string which is $\delta$-close to the true relative
Hamming weight $\omega(q_{\bar{t}})$, with probability
$1-\epsilon_{cl}^{\delta}$. From this point on we will simply refer to the
estimated relative Hamming weight as $\omega$.

###  2.2 Quantum Sampling

The quantum version of sampling naturally extends from classical sampling in
Bouman and Fehr’s framework, except that sampling is performed in both the $X$
and $Z$ bases. Classical sampling along these two non-orthogonal bases allows
for us to estimate the Hamming weight of the quantum system while still using
well-studied classical sampling methods. The main result of Bouman and Fehr’s
paper shows that the error bound of a quantum sampling protocol is simply the
square root of the error bound of the underlying classical sampling protocol
utilized [17].

$\epsilon_{qu}^{\delta}=\sqrt{\epsilon_{cl}^{\delta}}$ (4)

Using this, we find the quantum error bound for a randomly sampled substring
to be,

$\epsilon_{qu}^{\delta}<2\exp{-\frac{1}{6}\delta^{2}k}$ (5)

Through this equation, Bouman and Fehr’s framework allows for classical
sampling methods to be applied to quantum systems. This finding has allowed
this framework to aid in proving the security of Quantum Key Distribution [17]
and Quantum Random Number Generators [19, 20], as well as in deriving lower
bounds on the quantum conditional min entropy of high dimensional systems
[21].

Figure 1. Sampling and message qubits sent over a quantum channel. Eve applies
Pauli gates randomly to the message and sampling qubits. Bob can estimate the
number of Pauli gates applied to the message qubits through the sampling
procedure.

To illustrate quantum sampling with an example, consider the situation in
Figure 1. Focusing on the first qubit, Alice prepared a sampling qubit in the
$\ket{1}$ state and sent it to Bob over a quantum channel. However, Eve
tampered with the qubit in the channel, and when Bob measured this qubit he
found it in the $\ket{0}$ state. In this case, Bob can deduce that an error
has occurred. Now considering all the sampling qubits sent, Bob can obtain a
more accurate estimate of the influence of the quantum channel. With this
information, when Alice sends additional message qubits along with these
sampling qubits, Bob can use the quantum sampling to estimate the total error
of this quantum message in the form of the Hamming weight, $M\omega$.

When analyzing the proposed protocol in Section 5, $\omega$ will be used to
determine the number of gates Eve has applied, and $\epsilon_{qu}^{\delta}$
will be used to determine the failure probability of the protocol. The value
of $\delta$ will be determined in Sections 5 and 6 by the Hamming weight, code
distance, and gate set given to Eve.

##  3 Estimating Eve’s Interference

We will now use the Hamming weight, along with restrictions on Eve’s available
gate set, to estimate the number of gates Eve has applied. Let us first
consider the example of Eve tampering with the quantum sampling procedure in
Figure 1. Alice began by preparing sampling qubits in the states $\ket{0}$,
$\ket{1}$, $\ket{+}$, and $\ket{-}$ and sent these sampling qubits along with
some message qubits to Bob. After Bob received these qubits, he measured the
sampling qubits in the same basis Alice prepared and estimated the relative
Hamming weight $\omega$ of the remaining qubits. Bob can use this relative
Hamming weight $\omega$ to gain insight into the possible attacks Eve could
have performed as follows.

The Hamming distance is defined as the number of qubits which would be
measured by Bob in an orthogonal state as compared to the state which Alice
prepared. This change of state is caused by Eve’s interference in the quantum
channel, shown below by the operator $E$. This implies a definition of the
relative Hamming weight as follows,

$\omega=\frac{1}{4}\absolutevalue{\bra{1}E\ket{0}}^{2}+\frac{1}{4}\absolutevalue{\bra{0}E\ket{1}}^{2}+\frac{1}{4}\absolutevalue{\bra{+}E\ket{-}}^{2}+\frac{1}{4}\absolutevalue{\bra{-}E\ket{+}}^{2}$
(6)

Extending this equation to multiple qubits gives the Hamming weight for the
sampling qubits as,

$N\omega=\sum_{i}^{N}\frac{1}{4}\absolutevalue{\bra{1}E_{i}\ket{0}}^{2}+\frac{1}{4}\absolutevalue{\bra{0}E_{i}\ket{1}}^{2}+\frac{1}{4}\absolutevalue{\bra{-}E_{i}\ket{+}}^{2}+\frac{1}{4}\absolutevalue{\bra{+}E_{i}\ket{-}}^{2}$
(7)

By multiplying the relative Hamming weight by the number of message qubits
$M$, we can determine the estimated Hamming weight of the message qubits,

$M\omega=\frac{M}{N}\sum_{i}^{N}\frac{1}{4}\absolutevalue{\bra{1}E_{i}\ket{0}}^{2}+\frac{1}{4}\absolutevalue{\bra{0}E_{i}\ket{1}}^{2}+\frac{1}{4}\absolutevalue{\bra{-}E_{i}\ket{+}}^{2}+\frac{1}{4}\absolutevalue{\bra{+}E_{i}\ket{-}}^{2}$
(8)

Given $\omega$, we will use this equation to gain insight into the operators
$E_{i}$ in Sections 5 and 6. In these sections we will find that knowledge of
the gate set and the Hamming weight of the message qubits $M\omega$ can be
used to estimate the total number of gates applied.

##  4 The Protocol

With quantum sampling and the Hamming weight of quantum systems better
understood, we are now in a position to state the proposed one-way
entanglement purification protocol, which is as follows,

  1. 1.

Alice chooses a code distance $d$ and encodes a logical qubit with this
distance. She entangles this logical qubit with a local qubit, which she will
keep.

  2. 2.

Alice then concatenates many sampling qubits to this logical qubit and
permutes her quantum registers. She sends all these qubits through the quantum
channel.

  3. 3.

Alice sends a classical message to Bob using a previously distributed shared
secret key, informing him of the permutation, sampling states, and code
distance.

  4. 4.

Bob receives both the classical message and qubits from their respective
channels.

  5. 5.

Bob uses the classical information given by Alice to locate the sampling
qubits and perform quantum sampling, obtaining an estimate of the relative
Hamming weight $\omega$.

  6. 6.

Using the estimated Hamming weight $M\omega$, Bob can additionally estimate
the number of operations applied to the logical qubit. If the number of
operations is less than that allowed by the code distance, then Bob performs
error correction and keeps the logical qubit. Otherwise, Bob aborts.

Bob’s prediction is correct with probability $1-\epsilon_{qu}^{\delta}$. Bob
can determine the value of $\delta$ as is shown in the next section.

##  5 Security Against Pauli Gates

We will now prove the security of this protocol when Eve’s gate set is
restricted to single qubit Pauli gates. Starting from equation 8, with Eve’s
attacks restricted to single qubit Pauli gates
$E_{i}\in\\{X_{i},Y_{i},Z_{i}\\}$,

$M\omega=\frac{M}{N}\sum_{i}^{N}\frac{1}{4}\absolutevalue{\bra{1}E_{i}\ket{0}}^{2}+\frac{1}{4}\absolutevalue{\bra{0}E_{i}\ket{1}}^{2}+\frac{1}{4}\absolutevalue{\bra{-}E_{i}\ket{+}}^{2}+\frac{1}{4}\absolutevalue{\bra{+}E_{i}\ket{-}}^{2}$
(9)

From this equation, we find that the estimated Hamming weight of the sampled
substring is at least equal to half of the number of applied gates. For
example, applying Pauli $X$ gates to all qubits gives,

$M\omega=\frac{M}{N}\sum_{i}^{N}\frac{1}{4}\absolutevalue{\bra{1}X\ket{0}}^{2}+\frac{1}{4}\absolutevalue{\bra{0}X\ket{1}}^{2}+\frac{1}{4}\absolutevalue{\bra{-}X\ket{+}}^{2}+\frac{1}{4}\absolutevalue{\bra{+}X\ket{-}}^{2}$
(10)

From this we find that the relative Hamming weight of these applied $X$-gates
is,

$\omega=\frac{1}{2}$ (11)

We obtain the same results from applying $Z$-gates as well. $Y$-gates can be
detected from every sampling qubit and give a larger relative Hamming weight
of $\omega=1$. Therefore, given the adversary is restricted to Pauli gates, we
can estimate the number of gates applied to the message qubits as twice the
estimated Hamming weight, $2M\omega$. Given that the error-correcting code can
correct up to $\frac{d-1}{2}$ errors, the code can remove Eve’s influence if,

$2M\omega\leq\frac{d-1}{2}$ (12)

Figure 2. The proposed protocol using the distance 5 surface code and some
sampling qubits. In this depiction, Eve applies four X-gates in the quantum
channel. Bob uses the sampling qubits to estimate the number of gates applied
to the logical qubit and uses the surface code to remove Eve’s influence.

For a specific example, suppose that Alice prepared the logical qubit in the
distance 5 surface code as is depicted in Figure 2. She sends this logical
qubit through a quantum channel along with some sampling qubits. After passing
through the quantum channel, Bob is able to use the sampling qubits to
estimate the Hamming weight of the logical qubit as $M\omega=1$. In this
scenario, we can use this Hamming weight to estimate that Eve has applied 2
Pauli gates to the logical qubit. Through this, Bob can determine that the
distance 5 surface code prepared by Alice will be able to remove these two
gates applied by Eve.

###  5.1 Using the Error Bound $\epsilon_{qu}^{\delta}$

We must now address the probability that the sampling procedure has failed and
Eve has by chance avoided the sampling qubits. This failure would allow for
the true relative Hamming weight to be greater than $\omega+\delta$ with
probability $\epsilon_{qu}^{\delta}$. This would in turn imply that Eve has
applied greater than $2M(\omega+\delta)$ Pauli gates to the logical qubit. To
calculate this failure probability we must first determine the value of
$\delta$ which would cause the protocol to fail. As we are removing Eve’s
influence through an error-correcting code, the greatest value of $\delta$ we
can allow is the value which saturates the error-correcting code. In this way,
$\delta$ is set such that,

$2M(\omega+\delta)=\frac{d-1}{2}$ (13)

The true relative Hamming weight is less than this estimate of $\omega+\delta$
with probability $1-\epsilon_{qu}^{\delta}$. Recall from Section 2 that
$\epsilon_{qu}^{\delta}=2\exp{-\frac{1}{6}\delta^{2}k}$, where $k$ is the
number of sampling qubits, giving,

$1-\epsilon_{qu}^{\delta}=1-2\exp{-\frac{1}{6M^{2}}(\frac{d-1}{4}-M\omega)^{2}k}$
(14)

Let us illustrate this with a numerical example. Alice sends the distance 5
surface code, along with 20000 sampling qubits to Bob. Bob then performs
quantum sampling and estimates the Hamming weight of the message qubits as
$M\omega=\frac{1}{2}$. Since the employed code can correct up to
$\frac{5-1}{2}=2$ errors, Bob can now perform error correction, keep the
logical qubit, and state that the protocol has succeeded with probability,

$1-2\epsilon_{qu}^{\delta}=1-2\exp{-\frac{1}{12(25)^{2}}*20000}=86.1\%$ (15)

As can be seen through this example, this one-way protocol requires
significantly more resources to distribute a single entangled qubit than two-
way purification protocols. However, only one round of communication is
necessary, decreasing the latency of the protocol at the cost of requiring
more qubits.

##  6 Security Against Qubit Measurements

Now that the security of the proposed protocol has been established against
single qubit Pauli gates, we can explore the security of the protocol if we
additionally allow Eve the ability to perform qubit measurements.

For example, consider a measurement resulting in finding the qubit in the
$\ket{0}$ state,

$E_{i}=M_{\ket{0}}=\ket{0}\bra{0}$ (16)

We obtain from Equation 7,

$N\omega=\sum_{i}^{N}\frac{1}{4}(\absolutevalue{\bra{1}\ket{0}\bra{0}\ket{0}}^{2}+\absolutevalue{\bra{0}\ket{0}\bra{0}\ket{1}}^{2}+\absolutevalue{\bra{-}\ket{0}\bra{0}\ket{+}}^{2}+\absolutevalue{\bra{+}\ket{0}\bra{0}\ket{-}}^{2})$
(17)

This leads to the Hamming weight,

$\omega=\frac{1}{4}$ (18)

This similarly follows for measurements $M_{\ket{1}}$ which result in
$\ket{1}$. Qubit measurements therefore increase the Hamming weight by
$\frac{1}{4}$. This makes measurements more difficult to detect with quantum
sampling compared to a Pauli gate.

Equation 17 indicates that if we restrict Eve to measurements she can now
affect twice as many qubits while retaining the same Hamming weight as
compared to Section 4. For a code with distance $d$, we must ensure that the
distance is large enough to correct double the number of tampered qubits. This
changes the requirement in step 6 of the protocol from
$2M\omega\leq\frac{d-1}{2}$ to $4M\omega\leq\frac{d-1}{2}$ and Bob’s
calculation of $\delta$ to,

$4M(\omega+\delta)=\frac{d-1}{2}$ (19)

This changes the probability of success to,

$1-2\epsilon_{qu}^{\delta}=1-2\exp{-\frac{1}{6M^{2}}(\frac{d-1}{8}-M\omega)^{2}k}$
(20)

By changing this constraint in the protocol, Alice and Bob will now be able to
additionally guarantee security against qubit measurements.

##  7 Conclusion and Future Directions

In this paper, we have proposed a one-way entanglement purification protocol
with quantum sampling and proved its security against a restricted adversary
with access to Pauli gates and measurements. The proof of security of this
protocol is straightforwardly proven from the properties of quantum error-
correcting codes and the sampling framework utilized.

This protocol breaks the symmetry between Bob and Eve which is typically
present in one-way entanglement purification protocols by utilizing a shared
secret key. This secret key allows Bob to securely conduct quantum sampling
without requiring him to send message back to Alice. This protocol may allow
for lower latency as compared to two-way error-correcting protocols, as less
communication is needed between the two parties. However, more qubits are
necessary in one-way protocols to achieve similar performance to two-way
protocols.

The protocol proposed in this paper has so far only been proven secure to a
significantly restricted adversary. Further research directions include
examining the security of this protocol with respect to an adversary with
access to arbitrary single or multi-qubit gates. At first glance, this
approach may seem futile when Eve is given access to infinitesimally small
gates, as Eve could apply infinitely small rotation gates to each qubit while
maintaining approximately zero Hamming weight. However, two facts help to
mitigate the effectiveness of this approach for Eve. First, error-correcting
codes discretize errors, and therefore would discretize the small gates Eve
has applied, suppressing them from affecting the logical qubit. Second, the
Eastin Knill theorem states that an operator made of infinitesimally small
transverse gates cannot be a fault-tolerant logical operator. Due to this, any
single qubit gates Eve employs must be finite sized to manipulate the logical
qubit sent by Alice [22]. This indicates that there may be a lower limit to
the size of single qubit gates which Eve can apply to improve her likelihood
of eavesdropping.

However, as the Eastin-Knill theorem does not apply to multi-qubit non-
transverse gates, it may be more difficult to prove the security of one-way
entanglement purification against arbitrary multi-qubit attacks. This would
require research into the lowest Hamming weight multi-qubit operator that can
perform logical rotations on an error corrected qubit. However, there has been
relatively little research into constructing continuous logical operators on
error-corrected qubits [22, 23], since fault-tolerant quantum computation can
be achieved with finite sized gate sets such as Clifford+T [24]. Therefore,
further examination into constructing continuous logical operators may have
applications in the security of this protocol against arbitrary multi-qubit
attacks.

## References

  * [1] M.A. Bashar et al.  “A Review and Prospects of Quantum Teleportation”  In _2009 International Conference on Computer and Automation Engineering_ , 2009, pp. 213–217  DOI: 10.1109/ICCAE.2009.77
  * [2] Charles H. Bennett et al.  “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels”  In _Physical Review Letters_ 76.5 American Physical Society (APS), 1996, pp. 722–725  DOI: 10.1103/physrevlett.76.722
  * [3] Chuan Wang et al.  “Quantum secure direct communication with high-dimension quantum superdense coding”  In _Phys. Rev. A_ 71 American Physical Society, 2005, pp. 044305  DOI: 10.1103/PhysRevA.71.044305
  * [4] Felix Leditzky, Mohammad A. Alhejji, Joshua Levin and Graeme Smith  “Playing games with multiple access channels”  In _Nature Communications_ 11.1 Springer ScienceBusiness Media LLC, 2020  DOI: 10.1038/s41467-020-15240-w
  * [5] Marcello Caleffi et al.  “Distributed Quantum Computing: a Survey”  arXiv, 2022  DOI: 10.48550/ARXIV.2212.10609
  * [6] Charles H. Bennett, David P. DiVincenzo, John A. Smolin and William K. Wootters  “Mixed-state entanglement and quantum error correction”  In _Physical Review A_ 54.5 American Physical Society (APS), 1996, pp. 3824–3851  DOI: 10.1103/physreva.54.3824
  * [7] Kim Boström and Timo Felbinger  “Deterministic Secure Direct Communication Using Entanglement”  In _Phys. Rev. Lett._ 89 American Physical Society, 2002, pp. 187902  DOI: 10.1103/PhysRevLett.89.187902
  * [8] Jiawei Wu, Zaisheng Lin, Liuguo Yin and Gui-Lu Long  “Security of quantum secure direct communication based on Wyner’s wiretap channel theory” e26 que2.26  In _Quantum Engineering_ 1.4, 2019, pp. e26  DOI: https://doi.org/10.1002/que2.26
  * [9] Dong Pan et al.  “Single-Photon-Memory Two-Step Quantum Secure Direct Communication Relying on Einstein-Podolsky-Rosen Pairs”  In _IEEE Access_ 8, 2020, pp. 121146–121161  DOI: 10.1109/ACCESS.2020.3006136
  * [10] Tian-Yu Ye  “Quantum Secure Dialogue with Quantum Encryption”  In _Communications in Theoretical Physics_ 62.3 IOP Publishing, 2014, pp. 338–342  DOI: 10.1088/0253-6102/62/3/08
  * [11] Tian-Yu Ye and Li-Zhen Jiang  “Quantum dialogue without information leakage based on the entanglement swapping between any two Bell states and the shared secret Bell state”  In _Physica Scripta_ 89.1 IOP Publishing, 2013, pp. 015103  DOI: 10.1088/0031-8949/89/01/015103
  * [12] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu and Benjamin Schumacher  “Concentrating partial entanglement by local operations”  In _Physical Review A_ 53.4 American Physical Society (APS), 1996, pp. 2046–2052  DOI: 10.1103/physreva.53.2046
  * [13] Jian-Wei Pan, Christoph Simon, Časlav Brukner and Anton Zeilinger  “Entanglement purification for quantum communication”  In _Nature_ 410.6832 Springer ScienceBusiness Media LLC, 2001, pp. 1067–1070  DOI: 10.1038/35074041
  * [14] Hans Aschauer and Hans J. Briegel  “Security proof of quantum cryptography based entirely on entanglement purification”  In _Phys. Rev. A_ 66 American Physical Society, 2002, pp. 032302  DOI: 10.1103/PhysRevA.66.032302
  * [15] W Dür and H J Briegel  “Entanglement purification and quantum error correction”  In _Reports on Progress in Physics_ 70.8 IOP Publishing, 2007, pp. 1381–1424  DOI: 10.1088/0034-4885/70/8/r03
  * [16] Peter W. Shor and John Preskill  “Simple Proof of Security of the BB84 Quantum Key Distribution Protocol”  In _Physical Review Letters_ 85.2 American Physical Society (APS), 2000, pp. 441–444  DOI: 10.1103/physrevlett.85.441
  * [17] Niek J. Bouman and Serge Fehr  “Sampling in a Quantum Population, and Applications”  arXiv, 2009  DOI: 10.48550/ARXIV.0907.4246
  * [18] Steven Thompson  “Sampling”  John Wiley & Sons, 2012
  * [19] Keegan Yao, Walter O. Krawec and Jiadong Zhu  “Quantum Sampling for Optimistic Finite Key Rates in High Dimensional Quantum Cryptography”, 2020  arXiv:2012.04151 [quant-ph]
  * [20] Walter O. Krawec  “Quantum sampling and entropic uncertainty”  In _Quantum Information Processing_ 18.12, 2019, pp. 368  DOI: 10.1007/s11128-019-2481-5
  * [21] Walter O. Krawec  “A New High-Dimensional Quantum Entropic Uncertainty Relation with Applications”  In _2020 IEEE International Symposium on Information Theory (ISIT)_ , 2020, pp. 1978–1983  DOI: 10.1109/ISIT44484.2020.9174330
  * [22] Bryan Eastin and Emanuel Knill  “Restrictions on Transversal Encoded Quantum Gate Sets”  In _Physical Review Letters_ 102.11 American Physical Society (APS), 2009  DOI: 10.1103/physrevlett.102.110502
  * [23] Cameron Cianci  “Toward Constructing a Continuous Logical Operator for Error-Corrected Quantum Sensing”, 2023  arXiv:2305.00547 [quant-ph]
  * [24] Michael A. Nielsen and Isaac L. Chuang  “Quantum Computation and Quantum Information”  Cambridge University Press, 2000