---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.202 | B-22 | 0.02 |  $66^{\circ}$ $\pm$ $5^{\circ}$ | A | 186 |  $51^{\circ}$ $\pm$ $9^{\circ}$ | 13.95 $\pm$ 0.02 | 52 $\pm$ 3 | 90 $\pm$ 40 | None
|  |  |  | B | 103 |  $82^{\circ}$ $\pm$ $9^{\circ}$ | 17.58 $\pm$ 0.10 | 16 $\pm$ 1 | -470 $\pm$ 40 |  C ii-iii, O vi, Si ii-iii
|  |  |  | B |  |  | 15.03 $\pm$ 0.03 | 43 $\pm$ 2 | +110 $\pm$ 40 |  C iii, O vi
|  |  |  | B |  |  | 14.96 $\pm$ 0.08 | 18 $\pm$ 1 | +240 $\pm$ 40 |  C iii, O vi
|  |  |  | C | 431 |  $30^{\circ}$ $\pm$ $9^{\circ}$ | 13.96 $\pm$ 0.05 | 43 $\pm$ 6 | -10 $\pm$ 40 | None
|  |  |  | C |  |  | 14.65 $\pm$ 0.05 | 64 $\pm$ 3 | +280 $\pm$ 40 | None
| 25962 | 0.07 |  $81^{\circ}$ $\pm$ $1^{\circ}$ | A | 295 |  $65^{\circ}$ $\pm$ $1^{\circ}$ |  |  | (+100) |
|  |  |  | B | 516 |  $42^{\circ}$ $\pm$ $1^{\circ}$ |  |  |  |
|  |  |  | C | 895 |  $53^{\circ}$ $\pm$ $1^{\circ}$ |  |  |  |
| 31704 | 0.03 |  $41^{\circ}$ $\pm$ $4^{\circ}$ | A | 457 |  $16^{\circ}$ $\pm$ $8^{\circ}$ |  |  | (+280) |
|  |  |  | B | 360 |  $20^{\circ}$ $\pm$ $8^{\circ}$ |  |  |  |
|  |  |  | C | 246 |  $79^{\circ}$ $\pm$ $8^{\circ}$ |  |  |  |
| 31787 | 0.08 |  $62^{\circ}$ $\pm$ $2^{\circ}$ | A | 387 |  $69^{\circ}$ $\pm$ $2^{\circ}$ |  |  | (+100) |
|  |  |  | B | 230 |  $69^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
|  |  |  | C | 226 |  $56^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
| 32497 | 1.1 |  $85^{\circ}$ $\pm$ $2^{\circ}$ | A | 394 |  $62^{\circ}$ $\pm$ $1^{\circ}$ |  |  | (-150) |
|  |  |  | B | 226 |  $77^{\circ}$ $\pm$ $1^{\circ}$ |  |  |  |
|  |  |  | C | 215 |  $54^{\circ}$ $\pm$ $1^{\circ}$ |  |  |  |
| 32778 | 0.4 |  $43^{\circ}$ $\pm$ $1^{\circ}$ | A | 535 |  $40^{\circ}$ $\pm$ $2^{\circ}$ |  |  | (+300) |
|  |  |  | B | 412 |  $10^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
|  |  |  | C | 195 |  $63^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
|  |  |  | Impact params (kpc) |  |  |  |
|  |  |  | A | B | C |  |  |  |
| (24419) | 0.01 |  | 628 | 719 | 1118 |  |  | (-430) |
| (27581) | 0.03 |  | 724 | 930 | 1048 |  |  | (-380) |
| (28913) | 0.02 |  | 679 | 536 | 814 |  |  | (-50) |
| (30488) | 0.1 |  | 848 | 973 | 949 |  |  | (+420) |
| (31439) | 1.0 |  | 1082 | 889 | 1036 |  |  | (+100) |
| (31978) | 0.1 |  | 697 | 439 | 478 |  |  | (-50) |
| (32021) | 0.1 |  | 678 | 415 | 438 |  |  | (0) |
| (32308) | 0.8 |  | 597 | 331 | 355 |  |  | (-10) |
| (32591) | 0.03 |  | 856 | 598 | 586 |  |  | (-50) |
| (32755) | 0.7 |  | 784 | 531 | 572 |  |  | (+130) |
| (33166) | 0.03 |  | 759 | 493 | 344 |  |  | (+30) |
| (33268) | 0.9 |  | 820 | 556 | 505 |  |  | (+350) |
| (33496) | 0.1 |  | 1100 | 844 | 799 |  |  | (-490) |
| (34292) | 0.03 |  | 895 | 809 | 500 |  |  | (-320) |
| (34719) | 0.3 |  | 834 | 590 | 259 |  |  | (-420) |
| (34992) | 2.9 |  | 818 | 551 | 456 |  |  | (+150) |
| (35102) | 0.5 |  | 1163 | 1132 | 861 |  |  | (-330) |
| (36229) | 0.1 |  | 1106 | 856 | 528 |  |  | (+430) |
| (38978) | 0.1 |  | 1488 | 1305 | 903 |  |  | (-310) |

###  A.2 G-383

A-48 and A-49 are a close pair of small, star-forming galaxies near to the
line-of-sight to QSO-A, at z $\approx$ 0.38. They are separated by $\approx$ 5
kpc and aligned such that they lie approximately along each other’s minor
axis. A-48 is larger and closer to the QSO. They are illustrated in Figure 11
and detailed in Table 11. Other larger galaxies exist 1-1.5 Mpc from the
sightlines, but outside the HST image so do not have measured position angles.
Whilst emission-line velocities are measurable from the MUSE data for both
galaxies, neither show a clear velocity gradient.

Strong absorption exists in LOS-A, near the minor axis of both galaxies and
well within the virial radius, featuring C iii and O vi absorption. Both
indicate warm, metal-enriched material that would be consistent with a
galactic wind (although, as the Ly$\alpha$ column densities of the other
absorbers are small, their C iii and O vi fractions are not constrained, and
could be higher). As both galaxies are very close to edge on, the required
outflow velocity in order to produce the 20 or 40 km s-1 line-of-sight
velocity difference is massively affected by both the uncertainty in the
velocity offset itself and the galaxy inclination. Using the measured values
of both requires a fast outflow of $\sim$ 500 km s-1 to reproduce the
observations, but any value of outflow velocity $\lesssim$ 1000 km s-1 can
approximate the observations within the 1$\sigma$ region of $i$ and
$\Delta\textrm{v}$.

Weak absorption in QSO-B is apparent at $\approx$ 320 km s-1 blueward of the
galaxy pair, as well as 150 km s-1 redward. Possible absorbers in A at -320
and -150 are attributed to higher-order Lyman lines from higher redshift gas
(Ly-$\beta$ from z $\approx$ 0.64, and Ly$\gamma$ from z $\approx$ 0.73). All
apparent transitions in C are attributed to molecular lines from the sub-DLA
at z $\approx$ 0.56.

Neither of the absorbers in B can be reproduced using a reasonable outflow, as
it would need a very large opening angle to get close to the major axis, and
this would produce a very large velocity spread on the absorption in A. As the
absorbers lie near the major axis, a disk-like structure could be considered.
This could be invoked for either or both absorbers in B, with potentially the
two galaxies rotating in different directions producing the -320 and +150 km
s-1 absorbers. Such a disk could not fit the absorption in A in addition to
that in B, as a disk thick enough to produce strong absorption along the minor
axis would also cover a wide range of line-of-sight velocities along LOS-B,
and therefore wider absorption than is observed.

Given the uncertainties in inclination and velocity offset, an outflow fitting
the absorption in A alongside a disk from either galaxy matching the redward
component of absorption in B seems plausible. The large velocity offset
required to match the blue component in B would require a larger disk velocity
than the virial velocity of either galaxy. It is therefore more likely to be
associated with the more distant galaxies, or not physically associated with
any of the galaxies in this group.

Figure 11: Details of the absorption and galaxy environment around group G-383, a galaxy pair at z $\sim 0.39$, with kinematics taken from the H$\alpha$ emission line seen in the MUSE data. The layout is identical to that shown in Figure 3, and the model shown in the lower panel combines an outflow with 600 km s-1velocity and $10^{\circ}$ half-opening angle, and a rotating disk around A-49 with rotation velocity 180 km s-1. Additional absorbers indicated by red ticks are identified as follows: (1), (4) and (5) are H2 molecular lines from the sub-DLA at z $\approx$ 0.56; (2) is Ly$\delta$ from z=0.73; and (3) is Ly$\beta$ from z=0.64. Note that the close separation between the galaxies in both velocity and projection makes them difficult to distinguish in the central panel.  Table 11: Summary of galaxy and absorber properties of group G-383. Any additional galaxies and metal absorbers have velocities shown relative to the first galaxy (A-48). Columns are identical to those in Table 4. z | Galaxy | Lum ($L_{\star}$) | Inc | LOS | Imp (kpc) | Azimuth | log(N H i) | b ( km s-1) |  $\Delta$v ( km s-1) | Other ions
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.383 | A-48 | 0.03 |  $87^{\circ}$ $\pm$ $3^{\circ}$ | A | 38 |  $87^{\circ}$ $\pm$ $4^{\circ}$ | 14.67 $\pm$ 0.02 | 53 $\pm$ 2 | +40 $\pm$ 40 |  C iii, O vi
|  |  |  | B | 414 |  $4^{\circ}$ $\pm$ $4^{\circ}$ | 13.22 $\pm$ 0.09 | 20 $\pm$ 7 | -320 $\pm$ 40 | (None)
|  |  |  | B | 414 |  $4^{\circ}$ $\pm$ $4^{\circ}$ | 13.11 $\pm$ 0.13 | 24 $\pm$ 11 | +150 $\pm$ 40 | (None)
|  |  |  | C | 932 |  $38^{\circ}$ $\pm$ $4^{\circ}$ | (None, limit $\sim 13.0$) |  |  |
| A-49 | 0.02 |  $80^{\circ}$ $\pm$ $5^{\circ}$ | A | 43 |  $81^{\circ}$ $\pm$ $4^{\circ}$ |  |  | (-20) |
|  |  |  | B | 411 |  $0^{\circ}$ $\pm$ $4^{\circ}$ |  |  |  |
|  |  |  | C | 927 |  $35^{\circ}$ $\pm$ $4^{\circ}$ |  |  |  |

###  A.3 G-517

Group G-517 is made up of only two galaxies. B-43 is a $\approx 0.6L_{*}$
spiral galaxy at z$\approx$0.52, with a far smaller ($\approx 0.02L_{*}$)
companion galaxy B-34, illustrated in Table 12 and Figure 12. A large group
exists at the same redshift, but this is more than 1.5 Mpc from the lines-of-
sight. B-34 is sufficiently faint that it is difficult to obtain a position
angle. The uncertainty on the B-43 position angle may also be underestimated
as the flux in the spiral arms dominates over the disk that we use to measure
the position angle.

There is no significant absorption in any of the lines-of-sight. This is
somewhat surprising for LOS-B, as it lies near the minor axis at a small
impact parameter to a fairly large star-forming galaxy. This either indicates
that such outflows are not ubiquitous around SF galaxies, that these outflows
are ‘patchy’ (with covering factor $<$1), or that any outflow around this
galaxy is heating the gas enough that no H i absorption is visible. Coverage
of several metal species including C iii and O vi is provided by our
observations at this redshift, but no metals are detected.

Table 12: Summary of galaxy and absorber properties for group G-517. Any additional galaxies and metal absorbers have velocities shown relative to the first galaxy (B-43). Columns are identical to those in Table 4. z | Galaxy | Lum ($L_{\star}$) | Inc | LOS | Imp (kpc) | Azimuth | log(N H i) | b ( km s-1) |  $\Delta$v ( km s-1) | Other ions
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.517 | B-43 | 0.6 |  $66^{\circ}$ $\pm$ $1^{\circ}$ | A | 501 |  $3^{\circ}$ $\pm$ $6^{\circ}$ | (None, limit $\approx$13.3) |  |  |
|  |  |  | B | 95 |  $86^{\circ}$ $\pm$ $6^{\circ}$ | (None, limit $\approx$13.3) |  |  |
|  |  |  | C | 683 |  $37^{\circ}$ $\pm$ $6^{\circ}$ | (None, limit $\approx$13.4) |  |  |
| (B-34) | 0.02 |  | A | 666 |  |  |  | (-20) |
|  |  |  | B | 238 |  |  |  |  |
|  |  |  | C | 841 |  |  |  |  |
Figure 12: Details of the absorption and galaxy environment of group G-517,
with kinematics from [O iii] as seen in the MUSE data. The layout is identical
to that shown in Figure 3, except that no model is shown.

###  A.4 G-536

A-36 and A-37 are a close pair of galaxies that are likely interacting (given
the asymmetry in A-36), with A-40 a more distant third galaxy. Details are
given in Table 13 and Figure 13. A substantial number of galaxies exist at
similar redshifts with larger impact parameters. All three MUSE galaxies are
star-forming, and the interacting pair have measurable kinematics. Strong
absorption is visible along all three lines of sight. In galaxy A-36, the
bright bulge appears to be off-centre with respect to the disk, likely
indicating interaction with A-37. However, GALFIT is able to fit both galaxies
reasonably well using a simple disk that is not strongly distorted.

In Figure 13, we show the Ly$\beta$ transition in absorption at the redshift
of this group. This is due to the Ly$\alpha$ lying in the lower-resolution FOS
spectra, whilst Ly$\beta$ produces strong absorption in the COS spectra, so
more structure in the Ly$\beta$ can be discerned.

Given several galaxies, there are multiple ways of generating the absorption
features in the observed spectra. A wide outflow around A-36 can approximately
reproduce the saturated Ly$\beta$ absorption in LOS-A, or a narrower outflow
could explain the redmost component of this absorption. An outflow from galaxy
A-37 with sufficient velocity to cover the galaxy-absorber offset seen to the
absorbing components in LOS-B produces absorption that is too broad and
therefore inconsistent with the observations. Depending on the orientation of
the galaxy, an outflow around A-40 can also reasonably match either the blue
or red end of the broad saturated absorption in LOS-A.

Disk models may also contribute. A disk around A-36 will intersect both the A
and B sightlines, and the galaxy velocity gradient acts in the direction such
that any co-rotating disk cannot match the absorption found blueward of the
galaxy. Given the saturation in A, such a disk cannot reproduce any
substantial fraction of this absorption without also being detected in LOS-B
redward of the galaxy. The observations therefore rule out this scenario. A
disk around A-37 can fit a part of the absorption in A, most likely the blue
end based on the galaxy kinematics, whilst a disk around A-40 can reproduce
the strong absorption in B and a portion of the absorption in A.

The model shown in Figure 13 includes this disk around A-40, with a rotation
velocity of $\approx$ 110 km s-1. An outflow from galaxy A-36 produces the
other two components in A. However, the absorption in A could instead be
produced by a disk-like structure around A-37 or an outflow around A-40 in
combination with a weaker outflow from A-36.

Several metal features are detected at this redshift in sightlines A and B.
LOS-A exhibits C iii, O iv and O vi, whilst LOS-B features O ii alongside C
ii, C iii and C iv. The metals in A appear to form two components, rather than
the three components detected in H i, with similar column densities in the
three detected ions. This is consistent with the two components being part of
the same structure, an outflow from A-36 being our best model fit. The
absorption in LOS-B has a lower C iii content and a wider range of low ions
detected, which also seems consistent with our models suggesting that this is
primarily accretion onto A-40 (although the presence of some metals requires
that this accretion is not pristine).

We therefore cannot rule out a superposition of disks and outflows in this
group, although some combinations are ruled out. Given their close proximity
and the distortions visible in the HST image, we would expect substantial
interactions in the CGM of these galaxies, likely including tidal stripping,
but cannot show this using the current observations of these galaxies. The
absorption that is blueshifted by 300-500 km s-1 is not fit by our models, and
may originate from these galaxies or some of the more distant galaxies at this
redshift.

Figure 13: Details of the absorption and galaxy environment around G-536. The layout is identical to that shown in Figure 3, although the lower panel shows the relevant Ly$\beta$ absorption, and kinematics shown are from the [O ii] emission line seen in the MUSE data. The model shown in the lower panel combines an outflow with velocity 140 km s-1and half-opening angle $55^{\circ}$ around A-36 and a disk with rotation velocity 110 km s-1around A-40. Absorption originating from another redshift is labelled with red ticks. Component (2) is Ly$\alpha$ from z $\approx$ 0.295, whilst all of the other labelled components are molecular lines originating from the sub-DLA in LOS-C.  Table 13: Summary of galaxy–absorber group G-536. Non-MUSE galaxies and metal absorbers have velocities shown relative to the first galaxy (A-36). Columns are identical to those in Table 4. z | Galaxy | Lum ($L_{\star}$) | Inc | LOS | Imp (kpc) | Azimuth | log(N H i) | b ( km s-1) |  $\Delta$v (km/s) | Other ions
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.536 | A-36 | 0.77 |  $62^{\circ}$ $\pm$ $2^{\circ}$ | A | 120 |  $77^{\circ}$ $\pm$ $2^{\circ}$ | 15.61 $\pm$ 0.02 | 39 $\pm$ 2 | 40 $\pm$ 60 |  O iv & C iii(40 & -110)
|  |  |  | A | 120 |  $77^{\circ}$ $\pm$ $2^{\circ}$ | 15.30 $\pm$ 0.02 | 67 $\pm$ 4 | -90 $\pm$ 60 |  O vi (0 & -120)
|  |  |  | A | 120 |  $77^{\circ}$ $\pm$ $2^{\circ}$ | 14.41 $\pm$ 0.08 | 290 $\pm$ 50 | -140 $\pm$ 60 |
|  |  |  | B | 463 |  $28^{\circ}$ $\pm$ $2^{\circ}$ | 15.30 $\pm$ 0.04 | 24 $\pm$ 2 | -180 $\pm$ 60 |  C ii(-150, -230), O ii (-140)
|  |  |  | B | 463 |  $28^{\circ}$ $\pm$ $2^{\circ}$ | 14.36 $\pm$ 0.07 | 39 $\pm$ 7 | -290 $\pm$ 60 |  C iii(-180), C iv (0)
|  |  |  | B | 463 |  $28^{\circ}$ $\pm$ $2^{\circ}$ | 14.48 $\pm$ 0.05 | 28 $\pm$ 3 | -400 $\pm$ 60 |
|  |  |  | C | 1166 |  $5^{\circ}$ $\pm$ $2^{\circ}$ | 15.37 $\pm$ 0.07 | 53 $\pm$ 5 | -360 $\pm$ 40 | (No metals)
| A-37 | 0.31 |  $72^{\circ}$ $\pm$ $3^{\circ}$ | A | 123 |  $31^{\circ}$ $\pm$ $2^{\circ}$ |  |  | (-40) |
|  |  |  | B | 459 |  $75^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
|  |  |  | C | 1163 |  $52^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
| A-40 | 0.02 |  $65^{\circ}$ $\pm$ $8^{\circ}$ | A | 240 |  $63^{\circ}$ $\pm$ $10^{\circ}$ |  |  | (-60) |
|  |  |  | B | 398 |  $15^{\circ}$ $\pm$ $10^{\circ}$ |  |  |  |
|  |  |  | C | 1140 |  $30^{\circ}$ $\pm$ $10^{\circ}$ |  |  |  |
| (28106) | 0.3 |  | A | 897 |  |  |  | (-50) |
|  |  |  | B | 744 |  |  |  |  |
|  |  |  | C | 1421 |  |  |  |  |
| (29364) | 1.1 |  | A | 874 |  |  |  | (-450) |
|  |  |  | B | 606 |  |  |  |  |
|  |  |  | C | 1229 |  |  |  |  |
| (29521) | 0.4 |  | A | 1007 |  |  |  | (-450) |
|  |  |  | B | 681 |  |  |  |  |
|  |  |  | C | 1220 |  |  |  |  |

###  A.5 G-558

This group of galaxies lies at the redshift of the sub-DLA at z=0.558 in the
spectrum of QSO-C. Whilst the three galaxies lying in the MUSE field (A-72,
A-75 and A-77) are the closest to QSO-A, there are several other galaxies at
this redshift at similar impact parameters to QSOs B and C. As neither the
high-resolution imaging nor the IFU data cover QSO-C, no host galaxy has been
identified for the sub-DLA. A-77 is a very low-surface-brightness galaxy,
making it difficult for GALFIT to fit, so the position angle and inclination
are not well-determined. The details are given in Table 14 and the galaxies
and absorption at this redshift are shown in Figure 14. The three closest
galaxies are separated by $\approx$ 50 kpc. There may be some distortion
visible in the light from these galaxies (they do not appear to be smooth
disks), but this cannot be clearly attributed to interactions between the
galaxies.

We use Ly-$\beta$ for the fitting, as unlike Ly-$\alpha$ it lies in the
higher-resolution COS wavelength range, and can therefore better constrain the
models. The absorption in C is saturated across a large range of wavelengths,
even in Ly-$\beta$, but absorption is also visible in both the A and B
sightlines.

We note that the absorption in B at this redshift in both Ly$\alpha$ and
Ly$\beta$ is a blend of H i absorption lines from other redshifts, making it
difficult to identify which of the absorption features seen in Figure 14 are
Ly$\beta$ and therefore should be considered in our fitting. In particular,
the absorption component identified at $\approx$ -100 km s-1 in LOS-B and
marked with a vertical line in the figure is apparent in Ly$\alpha$ but not
clearly visible in Ly$\beta$. The absorption in LOS-B that is clearly visible
at the wavelength of Ly$\beta$ was not clearly identified as such, and appears
to be a blend of Ly$\alpha$ from z $\approx$ 0.32, Ly$\beta$ from this
redshift, and Ly$\delta$ from z $\approx$ 0.64. However, the Ly$\beta$ does
contribute, as the column densities estimated for the other H i systems making
up this blend are not sufficient to produce the observed absorption.

Note that the velocity offset to galaxy 30142 is $\approx$ 300 km s-1, such
that any outflow with sufficient velocity to match this would also produce a
larger range of line-of-sight velocities than seen in the observed absorption.
This also applies to galaxy 29812, despite a somewhat smaller velocity offset
(as A and B lie nearer the minor axis). A disk around 30142 also fails to
reproduce the column density ratio between the two lines-of-sight, as the
smaller impact parameter to LOS-B leads to absorption stronger than observed.
Although a disk around 29812 is possible and can reproduce the absorption in
LOS-B, such a disk would need a radius of $\approx$ 1 Mpc in order to reach
the sightline. The absorption near this group of galaxies is therefore likely
due to the trio of galaxies closest to the sightlines. A-77 does not have
position angle and inclination measurements, so we can only model these
structures around A-72 and A-75.

Neither a disk nor an outflow can simultaneously match both of the clear
absorption components seen in sightlines A and B. For both galaxies with
measured position angles, A lies at a smaller impact parameter and nearer to
the major axis, so our disk models produce much larger column densities at A
than at B (even accounting for the other contributors to the blended
absorption in B). Any outflow with large enough opening angle to cover LOS-A
and fast enough velocity to match the galaxy-absorber offset will also produce
much wider absorption in B than is observed. A combination of models is
therefore required.

We find that the absorption in A can be approximately reproduced by a halo or
disk around A-75, whilst the absorption in B could be attributed to an outflow
from A-72 or A-75 (or a disk from 29812, but this is less likely due to the
larger extent required). We do detect O vi absorption in LOS-A but not in
LOS-B. However, our detection limits allow the absorption in B to have a
similar O vi to H i ratio to that in A. We therefore do not consider this non-
detection to make an outflow origin for the absorption in B any less
plausible.

The orientation (which side of the disk is inclined away from the observer,
denoted by the $S_{Nr}$ parameter) required of A-75 for an outflow to reach
the absorption in B is not consistent with the disk model that fits A, so the
combination of A-75 disk and outflow is not able to fit the observations. As
A-72 is the most luminous and has the highest SFR of the two galaxies, we
prefer the combination of outflow from A-72, and a disk or halo from A-75. A
combination of outflow and halo from A-75 is also consistent with the
observations and, as we note throughout this work, stripped material is
difficult to rule out using these observations.

The model shown in Figure 14 consists of an outflow around galaxy A-72 and a
disk around A-75. The outflow has half-opening angle $\approx 40^{\circ}$ and
velocity 150 km s-1, whilst the disk has rotational velocity of 100 km s-1and
an infall component of $\approx$ 50 km s-1.

Figure 14: Details of the Ly$\beta$ absorption and galaxy environment around group G-558 at the redshift of the sub-DLA. The layout is identical to that shown in Figure 3. The model shown is an outflow around galaxy A-72 (with half-opening angle $\approx 40^{\circ}$ and velocity 150 km s-1) and a disk around A-75 (with a rotational velocity of 100 km s-1and an infall component of $\approx$ 50 km s-1). Additional features labelled (1) and (2) are H2 molecular lines at the same redshift as the sub-DLA (z$\approx$0.558), whilst (3) is the Lyman 7-0 transition from z=0.718.  Table 14: Summary of galaxy–absorber group G-558. Any additional galaxies and metal absorbers have velocities shown relative to the first galaxy (A-72). Columns are identical to those in Table 4. z | Galaxy | Lum ($L_{\star}$) | Inc | LOS | Imp (kpc) | Azimuth | log(N H i) | b ( km s-1) |  $\Delta$v ( km s-1) | Other ions
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.558 | A-72 | 0.10 |  $63^{\circ}$ $\pm$ $3^{\circ}$ | A | 148 |  $3^{\circ}$ $\pm$ $4^{\circ}$ | 14.81 $\pm$ 0.03 | 37 $\pm$ 2 | +50 $\pm$ 40 | O vi
|  |  |  | B | 426 |  $57^{\circ}$ $\pm$ $4^{\circ}$ | 14.26 $\pm$ 0.05 | 260 $\pm$ 40 | -60 $\pm$ 40 | None
|  |  |  | C | 1033 |  $14^{\circ}$ $\pm$ $4^{\circ}$ | 18.0 $\pm$ 0.9 | 64 $\pm$ 12 | +10 $\pm$ 40 |  C ii-iii, N iii, O vi, Si ii-iii
| A-75 | 0.03 |  $69^{\circ}$ $\pm$ $6^{\circ}$ | A | 183 |  $29^{\circ}$ $\pm$ $6^{\circ}$ |  |  | (+70) |
|  |  |  | B | 373 |  $75^{\circ}$ $\pm$ $6^{\circ}$ |  |  |  |
|  |  |  | C | 994 |  $30^{\circ}$ $\pm$ $6^{\circ}$ |  |  |  |
| A-77 | 0.04 |  | A | 205 |  |  |  | (+180) |
|  |  |  | B | 401 |  |  |  |  |
|  |  |  | C | 980 |  |  |  |  |
| (26644) | 0.5 |  | A | 857 |  |  |  | (+260) |
|  |  |  | B | 922 |  |  |  |  |
|  |  |  | C | 1684 |  |  |  |  |
| (27894) | 0.1 |  | A | 1025 |  |  |  | (-40) |
|  |  |  | B | 871 |  |  |  |  |
|  |  |  | C | 1544 |  |  |  |  |
| (29812) | 0.2 |  $78^{\circ}$ $\pm$ $12^{\circ}$ | A | 804 |  $83^{\circ}$ $\pm$ $8^{\circ}$ |  |  | (+150) |
|  |  |  | B | 442 |  $47^{\circ}$ $\pm$ $8^{\circ}$ |  |  |  |
|  |  |  | C | 1054 |  $6^{\circ}$ $\pm$ $8^{\circ}$ |  |  |  |
| (30142) | 0.6 |  $74^{\circ}$ $\pm$ $2^{\circ}$ | A | 842 |  $6^{\circ}$ $\pm$ $2^{\circ}$ |  |  | (+300) |
|  |  |  | B | 419 |  $35^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |
|  |  |  | C | 960 |  $87^{\circ}$ $\pm$ $2^{\circ}$ |  |  |  |

###  A.6 G-907

A-16 and B-19 are both small, star-forming galaxies at z $\sim$0.9,
illustrated in Table 15 and Figure 15. Both lines-of-sight show moderately
strong absorption slightly blueward of both galaxies. The galaxies have
sufficient signal-to-noise for kinematics to be measured, but are not well-
resolved by MUSE so do not show a clear velocity gradient. They are separated
by $\approx$600 kpc, so it is unsurprising that there is no sign of
interaction in the HST image.

There are no clear constraints from the kinematics, and the absorption
features in the FOS spectra are unresolved. Ly$\beta$ also lies in FOS at this
redshift, so cannot be used to resolve these features. Each galaxy lies
$\approx$ 200 kpc from one sightline, and $\approx$ 700 kpc from the other.
Our models therefore cannot produce both absorption features with near-
identical strengths and velocities using a disk and outflow from the same
galaxy. However, the parameters can easily be tuned such that a disk or
outflow from each galaxy can reproduce the absorption in the nearest line-of-
sight.

In Figure 15 we show an outflow with velocity 150 km s-1 and half-opening
angle $60^{\circ}$ from A-16, and a disk with scale height ratio 20:1 with
circular and inward velocities of 130 and 30 km s-1 respectively around B-19.
However, an outflow from B-19 and disk around A-16 can similarly reproduce the
observations. In this case, the outflow would require only 80 km s-1 velocity,
and a similar opening angle, and the disk would require $v_{\phi}\approx 150$
km s-1 and $v_{r}\approx 50$ km s-1. This scenario also produces more model
absorption in the more distant sightline than the A-16 outflow, which better
constrains the two disk velocity components and the outflow opening angle.
Either scenario is possible with parameters within the bounds suggested by
Paper 2 and other works.

Although our models cannot easily discriminate between these two scenarios, we
find that the absorption around these galaxies is consistent with plausible
CGM structures around the individual galaxies. Tidal material is not ruled
out, but given a separation between the galaxies that is substantially larger
than the virial radius, this is unlikely to dominate the observed absorption.

Table 15: Summary of galaxy–absorber group G-907. Any additional galaxies and metal absorbers have velocities shown relative to the first galaxy (A-16). Note that this group is beyond the redshift of QSO-C, so no absorption could be detected. Columns are identical to those in Table 4. z | Galaxy | Lum ($L_{\star}$) | Inc | LOS | Imp (kpc) | Azimuth | log(N H i) | b ( km s-1) |  $\Delta$v ( km s-1) | Other ions
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
0.907 | A-16 | 0.17 |  $53^{\circ}$ $\pm$ $12^{\circ}$ | A | 180 |  $72^{\circ}$ $\pm$ $20^{\circ}$ | 14.16 $\pm$ 0.05 | 170 $\pm$ 20 | -50 $\pm$ 40 |
|  |  |  | B | 635 |  $30^{\circ}$ $\pm$ $20^{\circ}$ | 14.15 $\pm$ 0.03 | 157 $\pm$ 11 | -60 $\pm$ 40 |
| B-19 | 0.10 |  $58^{\circ}$ $\pm$ $7^{\circ}$ | A | 631 |  $44^{\circ}$ $\pm$ $12^{\circ}$ |  |  | (-40) |
|  |  |  | B | 170 |  $57^{\circ}$ $\pm$ $12^{\circ}$ |  |  |  |
Figure 15: Details of the absorption and galaxy environment around G-907, a
galaxy pair at z $\sim 0.91$. The layout is identical to that shown in Figure
3, with kinematics from the [O ii] emission line seen in the MUSE data, and
the model shown in the lower panel combines an outflow with velocity 150 km
s-1and half-opening angle $60^{\circ}$ around A-16 and a disk with rotation
velocity 130 km s-1around B-19.


# Entangled state distillation from single copy mixed states beyond LOCC

Indranil Biswas<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Calcutta, 92, A.P.C. Road, Kolkata- 700009, India  Atanu Bhunia
<EMAIL_ADDRESS>Department of Applied Mathematics, University of
Calcutta, 92, A.P.C. Road, Kolkata- 700009, India  Indrani Chattopadhyay
<EMAIL_ADDRESS>Department of Applied Mathematics, University of
Calcutta, 92, A.P.C. Road, Kolkata- 700009, India  Debasis Sarkar
<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Applied
Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata- 700009, India

###### Abstract

No pure entangled state can be distilled from a $2\otimes 2$ or $2\otimes 3$
mixed state by separable operations. In $3\otimes 3$, pure entanglement can be
distilled by separable operation but not by LOCC. In this letter, we proved
the conjecture [PRL. 103, 110502 (2009)] that it is possible to distill pure
entanglement for $2\otimes 4$ system by LOCC and further improve these in
higher dimensions to distill a pure entangled state of Schmidt rank $d$ from a
$m\otimes n$ mixed state by separable operation when $m+n\geqslant 3d$. We
found results for tripartite systems with target state $d$-level GHZ-type
state. These results provide a class of systems where separable operation is
strictly stronger than LOCC.

###### pacs:

03.67.Mn.; 03.65.Ud.; Keywords: Entanglement, LOCC, Separable operations,
distillation;

The phenomenon of entanglement in composite quantum systems is a striking
feature without any analogue in classical physics. Although initially studied
to unreveal the foundational aspects, the progress made in the last few
decades elevated it to a resource in quantum information theory. Quantum
teleportation, superdense coding, quantum key distribution, etc., are some of
the examples of quantum information processing tasks where entanglement plays
a crucial role. The precision of such tasks depend on the presence of a pure
entangled state, preferably a maximally entangled state of a bipartite system.
But such states are rare in nature. Due to imperfect quantum operations and
decoherence, we usually have a noisy state in the laboratory. Bennett,
Brassard et. al. 1  in 1996 first considered this problem and developed a
protocol to distill $k$ copies of nearly singlet states from $n(>k)$ copies of
noisy entangled states. However, ‘useful’ entanglement cannot be distilled
from every noisy entangled state. There are examples of bound entangled states
2  from which pure entangled states cannot be distilled even in the asymptotic
regime.

Pure entangled states are very fragile in nature. Entanglement is lost
whenever the state is exposed to excessive environmental noise. A clever way
to perform quantum operations is to measure one party at a time. The outcome
is then broadcasted globally by that party through classical means of
communications. Depending on the outcome, some other parties will then measure
his/her local systems and again broadcast the outcomes globally. This
continues until the termination of that particular protocol. Such measurement
schemes are acronymed by LOCC (Local Operations along with Classical
Communications) 3  in the literature. Despite having a simple physical
description, mathematical characterization of LOCC is yet to discover.
Nevertheless, a larger class of operations than LOCC, called separable
operations have a rather compact mathematical structure.

Every physically realizable quantum operation is described by a completely
positive superoperator $\Omega$. When applied on a physical state $\rho$, such
superoperators map $\rho$ by the rule
$\Omega(\rho)=\sum_{i=1}^{n}{E_{i}}{\rho}{E_{i}}^{\dagger}$ satisfying
$\sum_{i=1}^{n}{E_{i}}^{\dagger}{E_{i}}=\mathbb{I}$, where
$\\{{{E_{i}\\}}_{i=1}^{n}}$ is called a complete set of Kraus operators 4 . If
each ${E_{i}}$’s are of the form
${E_{i}}={A_{i}}^{1}\otimes{A_{i}}^{2}\otimes........\otimes{A_{i}}^{k}$ for a
$k$-partite system, then the operation is called a separable operation (SEP).
It is evident from the mathematical description of separable operation that
LOCC forms a strict subclass of the former. Although, LOCC $\subsetneq$ SEP,
it is still intriguing to know exactly where LOCC differs from separable
operation. Several distinction has been found in terms of distinguishability
of states 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19
, of these two classes of operation. Another arena where this distinction
exists is deterministic transformation of single copy mixed states into pure
entangled states 21 . Kent 20  and Chitamber et. al. 21  have shown that both
these two classes of operation are equally incompetent in distilling pure
entangled states from one copy of a mixed state for $2\otimes 2$ and $2\otimes
3$ systems. But for $3\otimes 3$ systems, separable operation outperforms
LOCC. These results inspired us to ask a very relevant question: Does there
exist any relation between distillation of pure entangled states by separable
operations and dimensions of the subsystems? Also, how LOCC and separable
operations differ under general distillation scenario? We have found some
affirmative answers to these questions for bipartite quantum systems which are
considered as long standing problem in quantum information theory. We have
further improved these results upto some extent for tripartite systems. Our
result generalizes the possibility of distilling a pure entangled state of
Schmidt rank $d$ from a bipartite $m\otimes n$ system by separable operation
when $m+n\geqslant 3d$. We have constructed a generic class of mixed states
and a class of separable operators that can distill a pure maximally entangled
state when the equality holds. Furthermore, when one of the subsystem is of
dimension $2d$ (and the other is of dimension at least $d$), the
transformation is even possible by LOCC and thereby achieving a class of
bipartite systems where separable operation is strictly stronger than LOCC.
Note that in this letter we are only concerned about deterministic
distillation 21 . Unless otherwise stated, the term distillation would simply
mean here distillation in deterministic sense.

For a tripartite $p\otimes q\otimes r$ system, we have upgraded the bipartite
bound to $p+q+r\geqslant 4d$. However, this bound exists only for certain
classes of mixed states and when the target state is a $d$-level Greenberger-
Horne-Zeilinger (GHZ) type state where
$\ket{GHZ_{d}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i}\otimes\ket{i}\otimes\ket{i}$.
A distinction of separable and LOCC classes, similar to that of bipartite
system has also been found. We now move to present the main results of this
letter below.

In 21 , it is conjectured that there exist mixed states in $2\otimes n$
$(n>3)$ systems from which pure entangled states can be distilled. In fact,
LOCC is sufficient to distill such a state. We find that this is indeed the
case. Consider the state of a $2\otimes 4$ system of the form
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$ shared
between Alice and Bob, where
$\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ and
$\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{02}+\ket{13}).$ Now, to distill the
pure entangled state $\ket{\psi_{1}}$, Bob will perform a local measurement
$\\{B_{1},B_{2}\\}$ on his subsystem such that
$B_{1}=\ket{0}\bra{0}+\ket{1}\bra{1}$, $B_{2}=\ket{0}\bra{2}+\ket{1}\bra{3}$.

Next, we observe that although LOCC and separable operations do not coincide
in $3\otimes 3$ systems, the entangled state distilled by the latter class of
operations is only of Schmidt rank $2$. Thus we have succeeded in
characterization of bipartite systems for arbitrary (but finite) dimensions.

######  Theorem 1.

There exist $m\otimes n$ mixed states that can be deterministically converted
into a maximally entangled state of Schmidt rank $d$ by a suitable separable
operation if and only if $m+n\geqslant 3d$ where $\min\\{m,n\\}\geqslant d$.
Additionally, if $\max\\{m,n\\}\geqslant 2d$ the task can also be achieved by
LOCC.

###### Proof.

Without loss of generality, we assume that $n\geqslant m\geqslant d$ and
$m=d+k_{1},n=d+k_{2}$ such that $k_{1}+k_{2}=d$ where $k_{1}\geqslant
1,k_{2}\geqslant 1$ are integers so that $m+n=3d$. Consider the states
$\ket{\psi_{1}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i}$ and
$\ket{\psi_{2}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{k_{1}+i,d\oplus_{n}i}$
where $\oplus_{k}$ denotes the binary operation addition modulo $k$ and
$\ket{i,j}=\ket{i}\otimes\ket{j}$. Set,
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$ where
$p\in(0,1)$. Since $\innerproduct{\psi_{1}}{\psi_{2}}=0$, $\rho$ is a valid
mixed state of rank two. Then the separable operation with Kraus operators
$\\{E_{1},E_{2}\\}$ defined by

$E_{1}=\big{(}{\sum_{i=0}^{d-1}\eta_{i}\ket{i}\bra{i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\nu_{i}\ket{i}\bra{i}}\big{)},$
$E_{2}=\big{(}{\sum_{i=0}^{d-1}\eta_{i}{{}^{\prime}}\ket{i}\bra{{k_{1}}+i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\nu_{i}{{}^{\prime}}\ket{i}\bra{d\oplus_{n}i}}\big{)}$

where

$\displaystyle\eta_{i}=\begin{cases}1\hskip 11.38109pt;\>0\leqslant i\leqslant
k_{1}-1\\\ \frac{1}{\sqrt{2}}\hskip 2.84526pt;\>k_{1}\leqslant i\leqslant
d-1\end{cases}\nu_{i}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{1}-1\\\ 1\hskip
11.38109pt;\>k_{1}\leqslant i\leqslant d-1\end{cases}$
$\displaystyle\eta_{i}{{}^{\prime}}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{2}-1\\\ 1\hskip
11.38109pt;\>k_{2}\leqslant i\leqslant
d-1\end{cases}\nu_{i}{{}^{\prime}}=\begin{cases}1\hskip
11.38109pt;\>0\leqslant i\leqslant k_{2}-1\\\ \frac{1}{\sqrt{2}}\hskip
2.84526pt;\>k_{2}\leqslant i\leqslant d-1\end{cases}$

can convert $\rho$ into the pure maximally entangled state $\ket{\psi_{1}}$ of
Schmidt rank $d$.

Also, if $m=d,\;n=2d$, then consider
$\ket{\psi_{1}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i}$ and
$\ket{\psi_{2}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,d+i}$ such that
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$. Bob
now applies a local measurement $\\{B_{1},B_{2}\\}$ on his subsystem and
converts $\rho$ into $\ket{\psi_{1}}$ where
$B_{1}=\sum_{i=0}^{d-1}\ket{i}\bra{i}$ and
$B_{2}=\sum_{i=0}^{d-1}\ket{i}\bra{d+i}$.

Conversely, we assume that $k_{1}+k_{2}<d$. If possible, suppose there exists
a mixed state $\rho$ that can be converted into a pure entangled state of
Schmidt rank $d$, by a suitable separable operation with Kraus operators
$\\{A_{i}\otimes B_{i}\\}_{i}$. Without loss of generality, we assume that
rank of $\rho$ to be two and $\ket{\psi_{1}},\ket{\psi_{2}}$ are orthogonal
eigenstates of $\rho$. The linear nature of separable operators implies that
$\ket{\psi_{1}},\ket{\psi_{2}}$ must both be of Schmidt rank at least $d$. So,
there exists $A\otimes B\in\\{A_{i}\otimes B_{i}\\}_{i}$ such that $(A\otimes
B)\ket{\psi_{1}}=a\ket{\phi}$ and $(A\otimes B)\ket{\psi_{2}}=b\ket{\phi}$
where $\ket{\phi}(a,b$ are scalars) is a pure entangled state of Schmidt rank
$d$ and either (i) $a\neq 0,b\neq 0$ or (ii) only one of $a,b$ is non-zero.

For the case (i), $(A\otimes B)(b\ket{\psi_{1}}-a\ket{\psi_{2}})=0$ implies
that $b\ket{\psi_{1}}-a\ket{\psi_{2}}$ is a vector with Schmidt rank atmost
$d-1$, which is impossible, since $(A\otimes
B)(x\ket{\psi_{1}}+y\ket{\psi_{2}})=(ax+by)\ket{\phi},\forall
x,y\in\mathbb{C}$. For the case (ii), suppose $a\neq 0$. Then both $A$ and $B$
are operators of rank at least $d$ and therefore, for $b$ to be zero the sum
of dimensions of the null spaces of $A$ and $B$ together is of at least $d$.
But this contradicts our primary assumption that $k_{1}+k_{2}<d$.

Next for the case of LOCC, we follow the mathematical technique of 21 . We
assume that $\max\\{k_{1},k_{2}\\}<d$. Without loss of generality, let $\rho$
be a mixed state of rank two and $\ket{\psi_{1}},\ket{\psi_{2}}$ are its
orthogonal eigenstates of Schmidt rank at least $d$. If possible, suppose
there exists some LOCC protocol $\Omega$ such that
$\Omega(\rho)=\ket{\phi}\bra{\phi}$ where $\ket{\phi}$ is a pure entangled
state of Schmidt rank $d$. In fact, the LOCC protocol must transform
$\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ simultaneously into $\ket{\phi}$, i.e.,
$\Omega(\ket{\psi_{1}}\bra{\psi_{1}})=\Omega(\ket{\psi_{2}}\bra{\psi_{2}})=\ket{\phi}\bra{\phi}$.

Now, any bipartite LOCC protocol consists of alternating rounds of
measurements and broadcasting of outcomes where one of the parties, say, Alice
makes a measurement on her subsystem and communicates the result to the other
party, say, Bob. He then chooses a measurement on his subsystem depending on
the outcomes of Alice and reports back the outcome to Alice and the cycle
continues in this way. These operations therefore create a tree of all
possible outcomes where each branch represents a series of operators
implemented by the two parties. Suppose $N_{i}$ index the $i$-th branch after
$N$ rounds of measurement so that $A^{N_{i}}\otimes B^{N_{i}}$ describes the
net operation throughout all $N$ rounds along $N_{i}$. We now prove the
following claim: For any branch $N_{i}$, $A^{N_{i}}\otimes
B^{N_{i}}\ket{\psi_{1}}\neq 0$ if and only if $A^{N_{i}}\otimes
B^{N_{i}}\ket{\psi_{2}}\neq 0$.

By induction we assume that it is to be true and suppose any of Alice or Bob,
say Bob, makes the next measurement. For any $B^{j}$ let $(N+1)_{ji}$
correspond to the branch $(\mathbb{I}_{A}\otimes B^{j})(A^{N_{i}}\otimes
B^{N_{i}})$. Since the conversion is deterministic, $A^{N_{i}}\otimes
B^{N_{i}}\ket{\psi_{1}}$ and $A^{N_{i}}\otimes B^{N_{i}}\ket{\psi_{2}}$ are
both states of Schmidt rank at least $d$. Thus if the measurement operator
$B^{j}$ is applied, it must be of rank at least $d$ and therefore cannot
annihilate any state with Schmidt rank $d$ or more. Hence, $A^{N_{i}}\otimes
B^{N_{i}}\ket{\psi_{1}}\neq 0$ if and only if $A^{N_{i}}\otimes
B^{N_{i}}\ket{\psi_{2}}\neq 0$. This argument also holds true when $N=0$ and
therefore the claim is justified.

Next, consider a branch in the protocol that transforms $\ket{\psi_{1}}$ to
$\ket{\phi}$. Clearly, $\ket{\psi_{2}}$ must also be converted into
$\ket{\phi}$ along this branch. The states $\ket{\psi_{1}}$ and
$\ket{\psi_{2}}$ are linearly independent, but at the end they become the same
state. So there exist some round $N$ on this branch such that
$\ket{\psi_{1}^{N-1}}$ and $\ket{\psi_{2}^{N-1}}$ are linearly independent but
$\ket{\psi_{2}^{N}}$ is a scalar multiple of $\ket{\psi_{1}^{N}}$. Here,
$\ket{\psi_{1(2)}^{N}}$ is the resultant state after $N$ rounds, which was
originally $\ket{\psi_{1(2)}}$. Without loss of generality, we assume that
$N$-th round measurement is performed by Bob. Then $(\mathbb{I}_{A}\otimes
B)\ket{\psi_{1}^{N-1}}=c\ket{\phi}$ and $(\mathbb{I}_{A}\otimes
B)\ket{\psi_{2}^{N-1}}=d\ket{\phi},$ where $c,d$ are non-zero scalars. This
implies that $(\mathbb{I}_{A}\otimes
B)(d\ket{\psi_{1}^{N-1}}-c\ket{\psi_{2}^{N-1}})=0$ where
$d\ket{\psi_{1}^{N-1}}-c\ket{\psi_{2}^{N-1}}$ is a pure entangled state of
Schmidt rank at least $d$. But this contradicts the assumption that $k_{2}<d$.
∎

Notice that the class of separable operations can transform the class of mixed
states into a maximally entangled state. Clearly this result is somewhat
stronger than previous result in 21 . However, even if the bound is satisfied
by some bipartite system, there exist mixed states for which we could not find
a suitable separable operation to convert it into a pure entangled state.
Consider the state
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$ where
$\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})$ and
$\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$. This is indeed one
such example. We now proceed further to analyze the case of tripartite quantum
systems.

In tripartite scenario, we will only be concerned about distilling GHZ-type
states. We find that for a certain class of mixed states, it is possible to
distill GHZ-type states with unit probability. A bound similar to bipartite
systems also exist for this case.

######  Theorem 2.

There exists mixed states in $p\otimes q\otimes r$ system that can be
deterministically converted into a pure $d$-level GHZ-type state by a suitable
separable operation if and only if $p+q+r\geqslant 4d$ where
$\min\\{p,q,r\\}\geqslant d$. Additionally, if $\max\\{p,q,r\\}\geqslant 2d$
the task can also be achieved by LOCC.

###### Proof.

Without loss of generality, we assume that $r\geqslant q\geqslant p\geqslant
d$ and $p=d+k_{1},q=d+k_{2},r=d+k_{3}$ such that $k_{1}+k_{2}+k_{3}=d$, where
$k_{1},k_{2},k_{3}$ are non-negative integers so that $p+q+r=4d$. Consider a
mixed state $\rho$ of this system defined by
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$ where
$\ket{\psi_{1}},\ket{\psi_{2}}$ are two orthogonal $d$-level(or more) GHZ-type
state.

(I) If $k_{1}=0$ and $k_{2},k_{3}$ are non-zero and suppose

$\ket{\psi_{1}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i,i},\ket{\psi_{2}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,k_{2}+i,d\oplus_{r}i}.$

The separable operation $\\{E_{1},E_{2}\\}$ defined by

$E_{1}=\frac{1}{\sqrt{2}}\mathbb{I}_{A}\otimes\big{(}{\sum_{i=0}^{d-1}\eta_{i}\ket{i}\bra{i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\nu_{i}\ket{i}\bra{i}}\big{)},$
$E_{2}=\frac{1}{\sqrt{2}}\mathbb{I}_{A}\otimes\big{(}{\sum_{i=0}^{d-1}\eta_{i}{{}^{\prime}}\ket{i}\bra{{k_{2}}+i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\nu_{i}{{}^{\prime}}\ket{i}\bra{d\oplus_{r}i}}\big{)}$

where

$\displaystyle\eta_{i}=\begin{cases}1\hskip 17.07164pt;\>0\leqslant i\leqslant
k_{2}-1\\\ \frac{1}{\sqrt{2}}\hskip 8.53581pt;\>k_{2}\leqslant i\leqslant
d-1\end{cases}\nu_{i}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
8.53581pt;\>0\leqslant i\leqslant k_{2}-1\\\ 1\hskip
17.07164pt;\>k_{2}\leqslant i\leqslant d-1\end{cases}$
$\displaystyle\eta_{i}{{}^{\prime}}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
8.53581pt;\>0\leqslant i\leqslant k_{3}-1\\\ 1\hskip
17.07164pt;\>k_{3}\leqslant i\leqslant
d-1\end{cases}\nu_{i}{{}^{\prime}}=\begin{cases}1\hskip
17.07164pt;\>0\leqslant i\leqslant k_{3}-1\\\ \frac{1}{\sqrt{2}}\hskip
8.53581pt;\>k_{3}\leqslant i\leqslant d-1\end{cases}$

(II) If all $k_{i}$’s are non-zero, let

$\ket{\psi_{1}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i,i},$
$\ket{\psi_{2}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{k_{1}+i,\overline{k_{1}+k_{2}}\oplus_{q}i,d\oplus_{r}i}.$

The separable operation $\\{E_{1},E_{2}\\}$ defined by

$E_{1}=\big{(}{\sum_{i=0}^{d-1}\alpha_{i}\ket{i}\bra{i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\beta_{i}\ket{i}\bra{i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\gamma_{i}\ket{i}\bra{i}}\big{)},$
$E_{2}=\big{(}{\sum_{i=0}^{d-1}\alpha_{i}{{}^{\prime}}\ket{i}\bra{{k_{1}}+i}}\big{)}\otimes\big{(}{\sum_{i=0}^{d-1}\beta_{i}{{}^{\prime}}\ket{i}\bra{\overline{k_{1}+k_{2}}\oplus_{q}i}}\big{)}$

$\otimes\big{(}{\sum_{i=0}^{d-1}\gamma_{i}{{}^{\prime}}\ket{i}\bra{d\oplus_{r}i}}\big{)}$,
where,

$\displaystyle\alpha_{i}=\begin{cases}1\hskip 11.38109pt;\>0\leqslant
i\leqslant k_{1}-1\\\ \frac{1}{\sqrt{2}}\hskip 2.84526pt;\>k_{1}\leqslant
i\leqslant
d-1\end{cases}\alpha_{i}{{}^{\prime}}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{2}+k_{3}-1\\\ 1\hskip
11.38109pt;\>k_{2}+k_{3}\leqslant i\leqslant d-1\end{cases}$
$\displaystyle\beta_{i}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{1}-1\\\ 1\hskip
11.38109pt;\>k_{1}\leqslant i\leqslant k_{1}+k_{2}-1\\\
\frac{1}{\sqrt{2}}\hskip 2.84526pt;\>k_{1}+k_{2}\leqslant i\leqslant
d-1\end{cases}$
$\displaystyle\beta_{i}{{}^{\prime}}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{3}-1\\\ 1\hskip
11.38109pt;\>k_{3}\leqslant i\leqslant k_{2}+k_{3}-1\\\
\frac{1}{\sqrt{2}}\hskip 2.84526pt;\>k_{2}+k_{3}\leqslant i\leqslant
d-1\end{cases}$ $\displaystyle\gamma_{i}=\begin{cases}\frac{1}{\sqrt{2}}\hskip
2.84526pt;\>0\leqslant i\leqslant k_{1}+k_{2}-1\\\ 1\hskip
11.38109pt;\>k_{1}+k_{2}\leqslant i\leqslant d-1\end{cases}$

$\displaystyle\gamma_{i}{{}^{\prime}}=\begin{cases}1\hskip
11.38109pt;\>0\leqslant i\leqslant k_{3}-1\\\ \frac{1}{\sqrt{2}}\hskip
2.84526pt;\>k_{3}\leqslant i\leqslant d-1\end{cases}$

(III) Finally for the LOCC case, let $p=d,q=d,r=2d$ and

$\ket{\psi_{1}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i,i},\ket{\psi_{2}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i,d+i}.$

Now, Charlie can perform a local measurement $\\{C_{1},C_{2}\\}$ on his
subsystem to distill the state $\ket{\psi_{1}}$ from $\rho$ where
$C_{1}=\sum_{i=0}^{d-1}\ket{i}\bra{i}$,
$C_{2}=\sum_{i=0}^{d-1}\ket{i}\bra{d+i}$.

The converse part of the theorem is very similar to that of bipartite case. If
interested, the reader may find it in the appendix. ∎

Unlike bipartite scenario, some entanglement can be distilled with unit
probability even for the simplest case of a three qubit system by LOCC.
However, the entangled state may not be genuine.

For example, consider the $2\otimes 2\otimes 2$ mixed state
$\rho=p\ket{\psi_{1}}\bra{\ket{\psi_{1}}}+(1-p)\ket{\psi_{2}}\bra{\ket{\psi_{2}}}$
where,

$\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(\ket{000}+\ket{111}),~{}~{}\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{001}+\ket{110}).$

First, we write Charlie’s system in $\\{\ket{+}_{{}_{C}},\ket{-}_{{}_{C}}\\}$
basis where,

$\ket{+}_{{}_{C}}=\frac{1}{\sqrt{2}}(\ket{0}_{{}_{C}}+\ket{1}_{{}_{C}}),~{}\ket{-}_{{}_{C}}=\frac{1}{\sqrt{2}}(\ket{0}_{{}_{C}}-\ket{1}_{{}_{C}}).$

Then, $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ can be written as

$\ket{\psi_{1}}=\frac{1}{\sqrt{2}}(\ket{\phi^{+}}_{{}_{AB}}\ket{+}_{{}_{C}}+\ket{\phi^{-}}_{{}_{AB}}\ket{-}_{{}_{C}}),$
$\ket{\psi_{2}}=\frac{1}{\sqrt{2}}(\ket{\phi^{+}}_{{}_{AB}}\ket{+}_{{}_{C}}-\ket{\phi^{-}}_{{}_{AB}}\ket{-}_{{}_{C}})$

where,

$\ket{\phi^{+}}_{{}_{AB}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11})_{{}_{AB}},~{}\ket{\phi^{-}}_{{}_{AB}}=\frac{1}{\sqrt{2}}(\ket{00}-\ket{11})_{{}_{AB}}.$

Charlie now measures his local system in the basis $\\{C_{1},C_{2}\\}$ where
$C_{1}=\ket{+}\bra{+}$, $C_{2}=\ket{-}\bra{-}$. If the outcome is $C_{1}$, an
EPR state $\ket{\phi^{+}}$ is generated between $A$ and $B$ by tracing out
subsystem of Charlie from the system. If $C_{2}$ occurs, Charlie communicates
this result to Bob. He then performs a unitary measurement $\sigma_{z}$ on his
subsystem ($\sigma_{z}=\ket{0}\bra{0}-\ket{1}\bra{1}$). Tracing out $C$, the
EPR $\ket{\phi^{+}}$ is again generated between $A$ and $B.$

In conclusion, we have provided a bound on dimensions of bipartite systems for
which the distillation of pure entangled states with certain Schmidt ranks are
never possible and resolves the long standing problem. A similar bound is also
observed for tripartite systems when the original state is a mixture of GHZ-
type states and the target state is a pure entangled GHZ-type state.
Unfortunately, we could not find such bounds for distilling W-type states
which we leave as an open problem. Although the mathematical treatment is
sometimes very similar to 20 ; 21 , our results provide a complete answer to
the bipartite systems and up to some extent for tripartite systems regarding
the existence of distillable states. Also the separable operation is optimal
for the chosen class of states since it is of Kraus rank two and is effective
enough to extract a maximally entangled state for both bipartite and
tripartite systems. Moreover, these results prove the existence of a class of
both bipartite and tripartite systems where separable operation conclusively
outperforms LOCC. This class becomes increasingly large with the increment in
local dimensions. The results of tripartite systems can be generalised for
$n$-partite systems. Under similar restrictions, for a $n$-partite system
$\bigotimes_{i=1}^{n}a_{i}$, the bound would be
$\sum_{i=1}^{n}a_{i}\geqslant(n+1)d$ where the target state is a $d$-level
$n$-partite GHZ-type state
$\ket{GHZ_{n}^{d}}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{i,i,i,......(n\>times)}$.
An interesting progress of this work would be if someone will consider W-type
states under distillation scenario. Also, it seems that there are only certain
classes of mixed states that can be transformed into a pure entangled state.
One should naturally be curious to know the characterization of those classes.
Moreover, we hope that our work will be helpful in realizing $d$-level
entangled state in the distant lab scenario.
The authors I.C. and D.S. acknowledge the work as part of Quest initiatives by
DST, India. The authors A.B. and I.B. acknowledge the support from UGC, India.

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## Appendix : Proof of converse part of Theorem 2

First we consider the separable case. Let $p+q+r<4d$ where
$p=d+k_{1},\>q=d+k_{2},\>r=d+k_{3}$ satisfying $k_{1}+k_{2}+k_{3}<d$. If
possible, let there exist a mixed state
$\rho=p\ket{\psi_{1}}\bra{\psi_{1}}+(1-p)\ket{\psi_{2}}\bra{\psi_{2}}$ of rank
two where $\ket{\psi_{1}},\ket{\psi_{2}}$ are two $d$-level (or more)
orthogonal GHZ-type states, that can be converted into a pure $d$-level GHZ-
type entangled state $\ket{\phi}$ with certainty by some separable operation,
say $\\{A_{i}\otimes B_{i}\otimes C_{i}\\}_{i}$. Then there exist $A\otimes
B\otimes C\in\\{A_{i}\otimes B_{i}\otimes C_{i}\\}_{i}$ such that $(A\otimes
B\otimes C)\ket{\psi_{1}}=a\ket{\phi}$ and $(A\otimes B\otimes
C)\ket{\psi_{2}}=b\ket{\phi}$. (i) if both $a,b$ are non-zero, $(A\otimes
B\otimes C)(b\ket{\psi_{1}}-a\ket{\psi_{2}})=0$. Since $(A\otimes B\otimes
C)(x\ket{\psi_{1}}+y\ket{\psi_{2}})=(ax+by)\ket{\phi}\forall
x,y\in\mathbb{C}$, it must be that $b\ket{\psi_{1}}-a\ket{\psi_{2}}$ is at
least a $d$-level GHZ-type state. But the rank space of all $A,B,C$ are
$d$-dimensional, implies that the annihilator subspace is atmost of dimension
$k_{1}+k_{2}+k_{3}$, a contradiction. (ii) if $b=0,a\neq 0$ then $(A\otimes
B\otimes C)\ket{\psi_{1}}=a\ket{\phi}$ implies that all $A,B,C$ are of rank at
least $d$. Thus for $b$ to be zero, we must have $k_{1}+k_{2}+k_{3}\geqslant
d$, a contradiction. Therefore in either cases, no separable operation exist.
Next for LOCC case, we assume that $max\\{k_{1},k_{2},k_{3}\\}<d$. We again
assume that $\rho$ be the mixed state defined above. If possible, let there
exist some LOCC protocol $\Omega$ such that
$\Omega(\rho)=\ket{\phi}\bra{\phi}$. Any finite round LOCC protocol for
deterministic conversion in a tripartite system must be initiated by one of
the three parties Alice, Bob or Charlie through a measurement on his (her)
subsystem and reporting the outcome to some other party by classical
communications. Depending on the outcome, some other party picks a measurement
for its part of the system again reports the outcome to some other party and
the process continues according to the protocol. These operations therefore
create a tree of all possible outcomes where each branch represents a series
of operators implemented by the three parties. Let $N_{i}$ index the $i$-th
branch after $N$ rounds of measurement so that $A^{N_{i}}\otimes
B^{N_{i}}\otimes C^{N_{i}}$ describes the net operation throughout all $N$
rounds along $N_{i}$. We now proceed to prove the following claim:

For any branch $N_{i}$, $A^{N_{i}}\otimes B^{N_{i}}\otimes
C^{N_{i}}\ket{\psi_{1}}\neq 0$ if and only if $A^{N_{i}}\otimes
B^{N_{i}}\otimes C^{N_{i}}\ket{\psi_{2}}\neq 0$. By induction, we assume that
it is true and let Charlie makes the next measurement. For any $C^{j}$, let
$(N+1)_{ji}$ correspond to the branch $(\mathbb{I}\otimes\mathbb{I}\otimes
C^{j})(A^{N_{i}}\otimes B^{N_{i}}\otimes C^{N_{i}})$. Since the conversion is
deterministic, $A^{N_{i}}\otimes B^{N_{i}}\otimes C^{N_{i}}\ket{\psi_{1}}$ and
$A^{N_{i}}\otimes B^{N_{i}}\otimes C^{N_{i}}\ket{\psi_{2}}$ are both GHZ-type
states of at least level $d$. If $C^{j}$ is applied on either state, it must
at least be of rank $d$ and thus fails to eliminate any $d$-level GHZ-type
states. Consequently, $A^{(N+1)_{ji}}\otimes B^{(N+1)_{ji}}\otimes
C^{(N+1)_{ji}}\ket{\psi_{1}}\neq 0$ if and only if $A^{(N+1)_{ji}}\otimes
B^{(N+1)_{ji}}\otimes C^{(N+1)_{ji}}\ket{\psi_{2}}\neq 0$. This argument is
also applicable for the case $N=0$.

Next, we consider a branch in the protocol that converts $\ket{\psi_{1}}$ to
$\ket{\phi}$. The state $\ket{\psi_{2}}$ must also be converted into
$\ket{\phi}$ along this branch. The states $\ket{\psi_{1}}$ and
$\ket{\psi_{2}}$ are orthogonal but ultimately become the same the state
$\ket{\phi}$ in the end of the protocol. So there exist some round $N$ in this
branch such that $\ket{\psi_{1}^{N-1}}$ and $\ket{\psi_{2}^{N-1}}$ are
linearly independent but $\ket{\psi_{2}^{N}}$ is a scalar multiple of
$\ket{\psi_{1}^{N}}$. Here $\ket{\psi_{1(2)}^{N}}$ is the resultant state
after $N$ rounds, which originally was $\ket{\psi_{1(2)}}$. Without loss of
generality, we assume that $N$-th round measurement is performed by Charlie.
Then $(\mathbb{I}\otimes\mathbb{I}\otimes C)\ket{\psi_{1}^{N-1}}=a\ket{\phi}$
and $(\mathbb{I}\otimes\mathbb{I}\otimes C)\ket{\psi_{2}^{N-1}}=b\ket{\phi}$
where $a,b$ are non-zero scalars. Thus $(\mathbb{I}\otimes\mathbb{I}\otimes
C)(b\ket{\psi_{1}^{N-1}}-a\ket{\psi_{2}^{N-1}})=0$ where
$b\ket{\psi_{1}^{N-1}}-a\ket{\psi_{2}^{N-1}}$ is a pure entangled GHZ-type
state of level $d$, at least. But this contradicts the fact $k_{3}<d$. This
proves our claim. $\blacksquare$