11institutetext:  Smart Medical Imaging, Learning and Engineering (SMILE) Lab,
Medical UltraSound Image Computing (MUSIC) Lab,
School of Biomedical Engineering, Shenzhen University Medical School,
Shenzhen University, Shenzhen, China
11email<EMAIL_ADDRESS>

# ModeT: Learning Deformable Image Registration via Motion Decomposition
Transformer

Haiqiao Wang  Dong Ni  Yi Wang

###### Abstract

The Transformer structures have been widely used in computer vision and have
recently made an impact in the area of medical image registration. However,
the use of Transformer in most registration networks is straightforward. These
networks often merely use the attention mechanism to boost the feature
learning as the segmentation networks do, but do not sufficiently design to be
adapted for the registration task. In this paper, we propose a novel motion
decomposition Transformer (ModeT) to explicitly model multiple motion
modalities by fully exploiting the intrinsic capability of the Transformer
structure for deformation estimation. The proposed ModeT naturally transforms
the multi-head neighborhood attention relationship into the multi-coordinate
relationship to model multiple motion modes. Then the competitive weighting
module (CWM) fuses multiple deformation sub-fields to generate the resulting
deformation field. Extensive experiments on two public brain magnetic
resonance imaging (MRI) datasets show that our method outperforms current
state-of-the-art registration networks and Transformers, demonstrating the
potential of our ModeT for the challenging non-rigid deformation estimation
problem. The benchmarks and our code are publicly available at
https://github.com/ZAX130/SmileCode.

###### Keywords:

Deformable image registration Motion decomposition Transformer Attention
Pyramid structure.

##  1 Introduction

Deformable image registration has always been an important focus in the
society of medical imaging, which is essential for the preoperative planning,
intraoperative information fusion, disease diagnosis and follow-ups [22, 10].
The deformable registration is to solve the non-rigid deformation field to
warp the moving image, so that the warped image can be anatomically similar to
the fixed image. Let $I_{f},I_{m}\in\mathbb{R}^{H\times W\times L}$ be the
fixed and moving images ($H,W,L$ denote image size), in the deep-learning-
based registration paradigm, it is often necessary to employ a spatial
transformer network (STN) [12] to apply the estimated sampling grid
$G\in\mathbb{R}^{H\times W\times L\times 3}$ to the moving image, where $G$ is
obtained by adding the regular grid and the deformation field. For any
position $p\in\mathbb{R}^{3}$ in the sampling grid, $G(p)$ represents the
corresponding relation, which means that the voxel at position $p$ in the
fixed image corresponds to the voxel at position $G(p)$ in the moving image.
That is to say, image registration can be understood as finding the
corresponding voxels between the moving and fixed images, and converting this
into the relative positional relationship between voxels, which is very
similar to the calculation method of Transformer [8].

Transformers have been successfully used in the society of computer vision and
have recently made an impact in the field of medical image computing [16, 11].
In medical image registration, there are also several related studies that
employ Transformers to enhance network structures to obtain better
registration performance, such as Transmorph [5], Swin-VoxelMorph [25], Vit-V-
Net [6], etc. The use of Transformer in these networks, however, often merely
leverages the self-attention mechanism in Transformers to boost the feature
learning (the same as the segmentation tasks do), but does not sufficiently
design for the registration tasks. Some other methods use cross-attention to
model the corresponding relationship between moving and fixed images, such as
Attention-Reg [21] and Xmorpher [20]. The cross-attention Transformer (CAT)
module is used in the bottom layer of Attention-Reg [21] and each layer in
Xmorpher [20] to establish the relationship between the features of moving and
fixed images. However, the usage of Transformer in [20, 21] is still limited
to improving the feature learning, with no additional consideration given to
the relationship between the attention mechanism and the deformation
estimation. Furthermore, due to the large network structure of [20], only
small windows can be created for similarity calculation, which may result in
performance degradation. Few studies consider the relationship between
attention and deformation estimation, such as Coordinate Translator [17] and
Deformer [4]. Deformer [4] uses the calculation mode of multiplication of
attention map and Value matrix in transformer to weight the predicted basis to
generate the deformation field, but its attention map calculation is only the
concatenation and projection of moving and fixed feature maps, without using
similarity calculation part. Coordinate Translator [17] calculates the
matching score of the fixed feature map and the moving feature map. Then the
computed scores are employed to re-weight the deformation field. However, for
feature maps with coarse-level resolution, a voxel often has multiple
possibilities of different motion modes [24], which is not considered in [17].

In this study, we propose a novel motion decomposition transformer (ModeT) to
explicitly model multiple motion modalities by fully exploiting the intrinsic
capability of the Transformer structure for deformation estimation.
Experiments on two public brain magnetic resonance imaging (MRI) datasets
demonstrate our method outcompetes several cutting-edge registration networks
and Transformers. The main contributions of our work are summarized as
follows:

  * $\bullet$

We propose to leverage the Transformer structure to naturally model the
correspondence between images and convert it into the deformation field, thus
explicitly separating the two tasks of feature extraction and deformation
estimation in deep-learning-based registration networks in which to make the
registration procedure more sensible.

  * $\bullet$

The proposed ModeT makes full use of the multi-head neighborhood attention
mechanism to efficiently model multiple motion modalities, and then the
competitive weighting module (CWM) fuses multiple deformation sub-fields in a
competitive way, which can improve the interpretability and consistency of the
resulting deformation field.

  * $\bullet$

The pyramid structure is employed for feature extraction and deformation
propagation, and is beneficial to reduce the scope of attention calculation
required for each level.

##  2 Method

###  2.1 Network Overview

Figure 1: Illustration of the proposed deformable registration network. The
encoder takes the fixed image $I_{f}$ and moving image $I_{m}$ as input to
extract hierarchical features $F_{1}$-$F_{5}$ and $M_{1}$-$M_{5}$. The motion
decomposition transformer (ModeT) is used to generate multiple deformation
sub-fields and the competitive weighting module (CWM) fuses them. Finally the
decoding pyramid outputs the total deformation field $\phi$.

The proposed deformable registration network is illustrated in Fig. 1. We
employ a pyramidal registration structure, which has the advantage of reducing
the scope of attention calculation required at each decoding level and
therefore alleviating the computational consumption. Given the fixed image
$I_{f}$ and moving image $I_{m}$ as input, the encoder extracts hierarchical
features using a 5-layer convolutional block, which doubles the number of
channels in each layer. This generates two sets of feature maps
$F_{1},F_{2},F_{3},F_{4},F_{5}$ and $M_{1}$, $M_{2}$, $M_{3}$, $M_{4}$,
$M_{5}$. The feature maps $M_{5}$ and $F_{5}$ are sent into the ModeT to
generate multiple deformation sub-fields, and then the generated deformation
sub-fields are input into the CWM to obtain the fused deformation field
$\varphi_{1}$ of the coarsest decoding layer as the initial of the total
deformation field $\phi$. The moving feature map $M_{4}$ is deformed using
$\phi$, and the deformed moving feature map is fed into the ModeT along with
$F_{4}$ to generate multiple sub-fields, which are input into the CWM to get
$\varphi_{2}$. Then $\varphi_{2}$ is compounded with previous total
deformation field to generate the updated $\phi$. The feature maps $M_{3}$ and
$F_{3}$ go through the similar operations. As the decoding feature maps become
finer, the number of motion modes at position $p$ decreases, along with the
number of attention heads we need to model. At the $F_{2}/M_{2}$ and
$F_{1}/M_{1}$ levels, we no longer generate multiple deformation sub-fields,
i.e., the number of attention heads in ModeT is 1. Finally, the obtained total
deformation field $\phi$ is used to warp $I_{m}$ to obtain the registered
image.

To guide the network training, the normalized cross correlation
$\mathcal{L}_{\text{ncc}}$ [18] and the deformation regularization
$\mathcal{L}_{\text{reg}}$ [3] is used:

$\mathcal{L}_{\text{train}}=\mathcal{L}_{\text{ncc}}(I_{f},I_{m}\circ\phi)+\lambda\mathcal{L}_{\text{reg}}(\phi),$
(1)

where $\circ$ is the warping operation, and $\lambda$ is the weight of the
regularization term.

###  2.2 Motion Decomposition Transformer (ModeT)

In deep-learning-based registration networks, a position $p$ in the low-
resolution feature map contains semantic information of a large area in the
original image and therefore may often have multiple possibilities of
different motion modalities. To model these possibilities, we employ a multi-
head neighborhood attention mechanism to decompose different motion modalities
at low-resolution level. The illustration of the motion decomposition is shown
in Fig. 2.

Let $F,M\in\mathbb{R}^{c\times h\times w\times l}$ stand for the fixed and
moving feature maps from a specific level of the hierarchical encoder, where
$h,w,l$ denote feature map size and $c$ is the channel number. The feature
maps $F$ and $M$ go through linear projection ($proj$) and LayerNorm ($LN$)
[2] to get $Q$ ($query$) and $K$ ($key$):

$\begin{split}Q=LN(&proj(F)),\quad K=LN(proj(M)),\\\
Q=&\\{Q^{(1)},Q^{(2)},\dots,Q^{(S)}\\},\\\
K=&\\{K^{(1)},K^{(2)},\dots,K^{(S)}\\},\end{split}$ (2)

where the projection operation is shared weight, and the weight initialization
is sampled from $N(0,1e^{-5})$, the bias is initialized to 0. The $Q$ and $K$
are then divided according to channels, and $S$ represents the number of
divided heads.

Figure 2: Illustration of the proposed motion decomposition transformer, which
employs the multi-head neighborhood attention mechanism to decompose different
motion modalities. ($S=3$ in this illustration)

We then calculate the neighborhood attention map. We use $c(p)$ to denote the
neighborhood of voxel $p$. For a neighborhood of size $n\times n\times n$,
$||c(p)||=n^{3}$. The neighborhood attention map of multiple heads is obtained
by:

$NA(p,s)=softmax(Q_{p}^{(s)}\cdot K^{(s)T}_{c(p)}+B^{(s)}),$ (3)

where $B\in\mathbb{R}^{S\times n\times n\times n}$ is a learnable relative
positional bias, initialized to all zeros. We pad the moving feature map with
zeros to calculate boundary voxels because the registration task sometimes
requires voxels outside the field-of-view to be warped. Equation (3) shows how
the neighborhood attention is computed for the $s$-th head at position $p$, so
that the semantic information of voxels on low resolution can be decomposed to
compute similarity one by one, in preparation for modeling different motion
modalities. Moreover, the neighborhood attention operation narrows the scope
of attention calculation to reduce the computational effort, which is very
friendly to volumetric processing.

The next step is to obtain the multiple sub-fields at this level by computing
the regular displacement field weighted via the neighborhood attention map:

$\varphi_{p}^{(s)}=NA(p,s)V,$ (4)

where $\varphi^{(s)}\in\mathbb{R}^{h\times w\times l\times 3}$,
$V\in\mathbb{R}^{n\times n\times n}$, and $V$ ($value$) represents the
relative position coordinates for the neighborhood centroid, which is not
learned so that the multi-head attention relationship can be naturally
transformed into a multi-coordinate relationship. With the above steps, we
obtain a series of deformation sub-fields for this level:

$\varphi^{(1)},\varphi^{(2)},\dots,\varphi^{(S)}$ (5)

###  2.3 Competitive Weighting Module (CWM)

Figure 3: Illustration of the proposed competitive weighting module (CWM).

Multiple low-resolution deformation fields need to be reasonably fused when
deforming a high-resolution feature map. As shown in Fig. 3, we first upsample
these deformation sub-fields, then convolve them in three layers to get the
score of each sub-field, and use softmax to compete the motion modality for
each voxel. The convolution uses $3\times 3\times 3$ convolution rather than
direct projection because deformation fields often require correlation of
adjacent displacements to determine if they are reasonable. We formulate above
competitive weighting operation to obtain the deformation field $\varphi$ at
this level as follows:

$\begin{split}w^{(1)},w^{(2)},&\dots,w^{(S)}=WConv(cat(\varphi^{(1)},\varphi^{(2)},\dots,\varphi^{(S)})),\\\
\varphi&=w^{(1)}\varphi^{(1)}+w^{(2)}\varphi^{(2)}+,\dots,+w^{(S)}\varphi^{(S)},\end{split}$
(6)

where $w^{(s)}\in\mathbb{R}^{h\times w\times l}$, and $\varphi^{(s)}$ has
already been upsampled. $WConv$ represents the ConvBlock used to calculate
weights, as shown in the right part of Fig. 3.

##  3 Experiments

####  3.0.1 Datasets.

Experiments were carried on two public brain MRI datasets, including LPBA [19]
and Mindboggle [15]. For LPBA, each MRI volume contains 54 manually labeled
region-of-interests (ROIs). All volumes in LPBA were rigidly pre-aligned to
mni305. 30 volumes ($30\times 29$ pairs) were employed for training and 10
volumes ($10\times 9$ pairs) were used for testing. For Mindboggle, each
volume contains 62 manually labeled ROIs. All volumes in Mindboggle were
affinely aligned to mni152. 42 volumes ($42\times 41$ pairs from the NKI-RS-22
and NKI-TRT-20 subsets) were employed for training, and 20 volumes from OASIS-
TRT-20 ($20\times 19$ pairs) were used for testing. All volumes were pre-
processed by min-max normalization, and skull-stripping using FreeSurfer [9].
The final size of each volume was $160\times 192\times 160$ after a center-
cropping operation.

####  3.0.2 Evaluation Metrics.

To quantitatively evaluate the registration performance, Dice score (DSC) [7]
was calculated as the primary similarity metric to evaluate the degree of
overlap between corresponding regions. In addition, the average symmetric
surface distance (ASSD) [23] was evaluated, which can reflect the similarity
of the region contours. The quality of the predicted deformation $\phi$ was
assessed by the percentage of voxels with non-positive Jacobian determinant
(i.e., folded voxels). All above metrics were calculated in 3D. A better
registration shall have larger DSC, and smaller ASSD and Jacobian.

Table 1: The numerical results of different registration methods on two datasets. | Mindboggle (62 ROIs) | LPBA (54 ROIs)
---|---|---
| DSC ($\%$) | ASSD | $\%|J_{\phi}|\leq 0$ | DSC ($\%$) | ASSD | $\%|J_{\phi}|\leq 0$
SyN [1] | $56.7\pm 1.5$ | $1.38\pm 0.09$ | $\textless 0.00001\%$ | $70.1\pm 6.2$ | $1.72\pm 0.12$ | $\textless 0.0004\%$
VM [3] | $56.0\pm 1.6$ | $1.49\pm 0.11$ | $\textless 1\%$ | $64.3\pm 3.2$ | $2.03\pm 0.21$ | $\textless 0.7\%$
TM [5] | $60.7\pm 1.5$ | $1.35\pm 0.10$ | $\textless 0.9\%$ | $67.0\pm 3.0$ | $1.90\pm 0.20$ | $\textless 0.6\%$
I2G [17] | $59.8\pm 1.3$ | $1.30\pm 0.07$ | $\textless 0.03\%$ | $71.0\pm 1.4$ | $1.64\pm 0.10$ | $\textless 0.01\%$
PR++ [13] | $61.1\pm 1.4$ | $1.34\pm 0.10$ | $\textless 0.5\%$ | $69.5\pm 2.2$ | $1.76\pm 0.17$ | $\textless 0.2\%$
XM [20] | $53.6\pm 1.5$ | $1.46\pm 0.09$ | $\textless 1\%$ | $66.3\pm 2.0$ | $1.92\pm 0.15$ | $\textless 0.1\%$
DMR [4] | $60.6\pm 1.4$ | $1.34\pm 0.09$ | $\textless 0.7\%$ | $69.2\pm 2.4$ | $1.79\pm 0.18$ | $\textless 0.4\%$
Ours | $62.8\pm 1.2$ | $1.22\pm 0.07$ | $\textless 0.03\%$ | $72.1\pm 1.4$ | $1.58\pm 0.11$ | $\textless 0.007\%$
Figure 4: Visualized registration results from different methods on Mindboggle
(top row) and LPBA (bottom row).

####  3.0.3 Implementation Details.

Our method was implemented with PyTorch, using a GPU of NVIDIA Tesla V100 with
32GB memory. The regularization term $\lambda$ and neighborhood size $n$ were
set as $1$ and $3$. For the encoder part, we used the same convolution
structure as [17]. In the pyramid decoder, from coarse to fine, the number of
attention heads were set as $8,4,2,1,1$, respectively. We used 6 channels for
each attention head. The Adam optimizer [14] with a learning rate decay
strategy was employed as follows:

$lr_{m}=lr_{init}\cdot(1-\frac{m-1}{M})^{0.9},m=1,2,...,M$ (7)

where $lr_{m}$ represents the learning rate of $m$-th epoch and $lr_{init}=1$
represents the learning rate of initial epoch. We set the batch size as $1$,
$M$ as $30$ for training.

####  3.0.4 Comparison Methods.

We compared our method with several state-of-the-art registration methods: (1)
SyN [1]: a classical traditional approach, using the $SyNOnly$ setting in
ANTS. (2) VoxelMorph(VM) [3]: a popular single-stage registration network. (3)
TransMorph(TM) [5]: a single-stage registration network with SwinTransformer
enhanced encoder. (4) PR++ [13]: a pyramid registration network using 3D
correlation layer. (5) XMorpher(XM) [20]: a registration network using CAT
modules for each level of encoder and decoder. (6) Im2grid(I2G) [17]: a
pyramid network using a coordinate translator. (7) DMR [4]: a registration
network using a Deformer and a multi-resolution refinement module.

####  3.0.5 Quantitative and Qualitative Analysis.

The numerical results of different methods on datasets Mindboggle and LPBA are
reported in Table 1. It can be observed that our method consistently attained
the best registration accuracy with respect to DSC and ASSD metrics. For the
DSC results, our method surpassed the second-best networks by $1.7\%$ and
$1.1\%$ on Mindboggle and LPBA, respectively. We further investigated the
statistical significance of our method over comparison methods on DSC and ASSD
metrics, by conducting the paired and two-sided Wilcoxon signed-rank test. The
null hypotheses for all pairs (our method $v.s.$ other method) were not
accepted at the 0.05 level. As a result, our method can be regarded as
significantly better than all comparison methods on DSC and ASSD metrics.
Table 1 also lists the percentage of voxels with non-positive Jacobian
determinant ($\%|J_{\phi}|\leq 0$). Our method achieved satisfactory
performance, which was the best among all deep-learning-based networks.

Figure 5: Visualization of the generated multi-level deformation fields
($\varphi_{1}$-$\varphi_{5}$) to register one image pair. At low-resolution
levels, multiple deformation sub-fields are decomposed to effectively model
different motion modalities.

Fig. 4 visualizes the registered images from different methods on two
datasets. Our method generated more accurate registered images, and internal
structures can be consistently preserved using our method. Fig. 5 takes the
registration of one image pair as an example to show the multi-level
deformation fields generated by our method. Our ModeT effectively modeled
multiple motion modalities and our CWM fused them together at low-resolution
levels. The final deformation field $\phi$ accurately warped the moving image
to registered with the fixed image.

##  4 Conclusion

We present a motion decomposition Transformer (ModeT) to naturally model the
correspondence between images and convert this into the deformation field,
which improves the interpretability of the deep-learning-based registration
network. The proposed ModeT employs the multi-head neighborhood attention
mechanism to identify various motion patterns of a voxel in the low-resolution
feature map. Then with the help of competitive weighting module and pyramid
structure, the motion modes contained in a voxel can be gradually fused and
determined in the coarse-to-fine pyramid decoder. The experimental results
have proven the superior performance of the proposed method.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China under Grants 62071305, 61701312, 81971631 and 62171290, in part by the
Guangdong Basic and Applied Basic Research Foundation under Grant
2022A1515011241, and in part by the Shenzhen Science and Technology Program
(No. SGDX 20201103095613036).

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While fitting the evolutionary path described by eq:ivp, besides $\alpha$ and $n$, we also introduced parameters $Y_0$, the initial value of at $\dprj=2\Rzoo$, and $\log f$, the factor of variance underestimation.
The posterior probability distributions of all four parameters are shown in <ref> and the median values are reported with $1\sigma$ uncertainties.
The probability-weighted values (orange lines) are similar to the corresponding median values that we used in science analyses.

We adopt such a prior probability distribution that $Y_0$ cannot result in a strippable gas fraction larger than 100%.
We also found that there is a second peak of posterior probability on $n$–$\alpha$ corresponding to an unphysically large $\alpha$, which would result in a much larger than the Hubble time out of and an abrupt stripping at $\num{\sim 0.5}\Rzoo$.
We set the prior probability within this region as $0$.
The region is bordered by a blue dashed–dotted line in <ref>.

§.§ The Uncertainty of the Fitted Evolution Path

The uncertainty of the fitted evolutionary path, shown as green shading.
The meanings of circles and semicircles are the same as those in <ref> and symbols with dashed edges represent galaxies with $\log\left(\ssfr/\unit{\per\yr}\right)\le -11$.
Galaxies without measurements are plotted as empty salmon symbols.
The evolutionary path with the median values of $\alpha$ and $n$ is given as solid line.
The $1\sigma$ uncertainty of the path is estimated using bootstrap method and is plotted as green shading.
The 95th percentiles and evolutionary path taking into account low-galaxies are plotted as narrow blue error bars and dashed blue line respectively.

The uncertainty of fitted $\alpha$ and $n$ would be propagated into the uncertainty of the evolutionary path itself.
We show the bootstrapped $1\sigma$ uncertainty as the green shading in <ref>.
The 95th percentiles of (with $\ssfr>\qty{1e-11}{\per\yr}$) and the path of median parameters fitted from them are shown as  orange data points and the solid green line respectively, as references.
The green shading covers the percentile points well.

§.§ Modified Models with Different Assumptions

The influence of model specification on the values of and .
Panels are similar to <ref>(a) and (b).
Four models are plotted with different line-styles and patches.
The main model we used is the solid red line.
For details of other three models, please refer to the text.
In (b) the criterion of gas depletion is reaching 1.06 below the mean –$M_*$ relation.

We test three different modifications to our fiducial gas-stripping empirical model described in <ref>, and present the changes in and they introduce.

One modification is to use the 95th percentiles of all galaxies for MCMC fitting, instead of only those with high .
In <ref> we show all galaxies, and their 95th percentiles are shown as narrow blue error bars.
They are similar to the percentiles of high-galaxies (orange error bars), and the fitted result (dashed line) does not deviate much, except that the plummet happens nearer to the group center.

Shown in <ref> as dashed green lines, in this model, the around group center is a bit longer, and the increases by ∼150 because of the delay of stripping.

Another modification is lowering the criterion of from $\Pram=\Fanc(r)$ to $\Pram/\Fanc(r)=\Scri{\rps}$, in the fashion of defining .
From <ref> we have $\Scri{\rps}=10^\numScrirpsshort$.

The change due to this modification (dashed–dotted purple lines) is shown in <ref>.
All 's consistently increase to 3 times the original values.
Correspondingly, the stripping timescale increases by ∼2 toward the group center, where is strongest, since the extent of disk for stripping is larger.
At the outskirts, however, decreases a bit.
Because of the flatter relation of with under this model, shows a larger slope with respect to the initial .

The third modification considers the combined effect of SFE decreasing toward low-mass galaxies [Huang et al., 2012] and the increase of stellar mass as a function of time because of star formation.
These two effects lead to the lower and higher decrease rates in respectively than the fiducial model would imply.
The combined effect can be summarized into multiplying the slope term related to SFE by a factor of ∼3.

This modified model (dashed–dotted–dotted brown lines in <ref>) does not change much, but decreases the of galaxies with a low at .
The reason is that, for these initially gas-poor galaxies, enhanced star-formation consumption quickly depletes the before the stripping takes over.

§ COLORS OF HIGH-MASS TS GALAXIES

The same as <ref>, but only -detected high-mass galaxies are plotted.
The bisector fitting results of <ref> are plotted as references.
The $x$-axis range is the same as those in <ref>.

We plotted the four high-mass TS galaxies in <ref>.
Compared with low-mass ones, they do not have a very high .
They scatter around the linear fittings we get in <ref>.

§ ATLAS OF -RESOLVED GALAXIES

The atlas of five -resolved galaxies (see also <ref>).
RPS and TS galaxies are plotted on the left and right respectively.
For each galaxy, the DECaLS DR9 optical image and the WALLABY -column-density map are placed adjacently.
Of the intensity map, only pixels with values $\mathord{>}5\sigma$ are shown.
The optical image is overlaid with the contour of $N\sbHi$, with levels of [list-exponents=combine,print-unity-mantissa=true,list-units=single]1e20;2e20;4e20;8e20^-2.
The lowest and the highest level are plotted as dotted and dashed lines respectively for clarity.
The ID, beam size (;;∼30), 5kpc scale bar, stellar mass, and the relevant values of $f$ and $S$ are given at each panel.
The direction of the group center is indicated by a cyan arrow for RPS galaxies, with the direction of the major tidal perturber by a magenta arrow for TS ones.

We provide the optical images and -column-density maps of five galaxies resolved by WALLABY as <ref>.
Two of them (left column) are among the sample, and the other three (right) belong to the TS sample.
All of them have a stellar mass larger than [parse-numbers=false]10^8.5, and thus the irregularity of the disk, if any, is less likely due to internal supernovae feedback than to external perturbations.

Both galaxies undergoing have a low ([quantity-product=]<10%), and do not show strong directional irregularities.
Galaxy 𝕀67 has the largest , and shows a compressed feature in the southeast, backward to the group center (cyan arrow).
It is possible that 𝕀67, with $\dprj=1.05\Rzoo$ and $\inc v=0.93\sigma_v$, is at the backsplash stage, moving outward from the group.

Two of three TS galaxies (𝕀23 and 𝕀78) have high values of .
Compared with the other three galaxies with weaker tidal interaction, a striking feature is that both galaxies have a half-ring-like structure in the direction of the major tidal perturber, which provides the largest portion of .
It is a tentative result worth future investigation when more resolved images are available.

§ GALAXY PROPERTIES

Basic properties and the and optical measurements of our sample are listed in <ref>.
-related measurements of our sample (and the merging pair) are given in <ref>.

/Rotate 90

Galaxy Properties and Non-Measurements

pdf: put @thispage <</Rotate 90>>                                           3cColor At

ID     Name     R.A.     Decl.     $v\tsb{hel}$     Refs     $\log\sfr$          1p2emSFR Note     1p3emImage Source     $\log\Pram$     $\log\Stid$     $g$     $r$     Center          2     $\log M_*$

()     ()     ()         []     ()             []     ()     ()     ()     ()     ()     ()     []     ()     ()     ()

Column (1): ID.
Column (2): NGC/UGC/IC ID.
Columns (3)–(4): R.A. and decl. in decimal degrees (J2000).
Column (5): heliocentric radial velocity.
Column (6): source of $v\tsb{hel}$; FAST (F), ALFALFA (A), [Kourkchi & Tully, 2017] (K), SDSS DR16  table (S), and H = HIPASS Bright Galaxy Catalog (BGC; H).
Column (7): star formation rate.
Column (8): offset from SFMS.
Column (9): note on SFR measurement; estimated solely from UV data (u), from W4-band data (w), and UV and W4 data combined (c); if UV data are used, FUV (f, preferred), and NUV (n, when FUV is unavailable).
Column (10): source of optical images; SDSS (S), and DECaLS (D).
Column (11): ram pressure estimated using projected parameters.
Column (12): tidal strength parameter.
Columns (13)–(14): total $g$- and $r$-bands AB magnitude using GC method.
Columns (15)–(17): the color at the galaxy center, , and $2\Rso$.
Column (18): stellar mass; the uncertainty from distance is not included.
Column (19): the radius at which the $g$-band SB drops to 25.arcsec^-2, assuming a precise distance of 16.2.
Columns (20)–(21): the radii enclosing $50\%$ and $90\%$ of $r$-band fluxes.
This table is also available in machine-readable form.
aThis -nondetected galaxy (𝕀20) has a distinctly red color ($\gr=\qty{1.767}{\mag}$) and we assigned it the 95th percentile of among our samples (0.744) for stellar mass calculation instead.
Since most of our discussion is focused on -detections, this color assignment does not change our major results.

-related Properties and Measurements

ID     FAST     ALFALFA     HIPASS     WALLABY     $\log\MHi$     Refs          $\log\Srps$     $\log\frps$     $\log\ftid$     $\log\fstr$     1p4emStripping Status

[]         ()     ()     ()     ()     ()

Column (1): ID.
Column (2): ID in Zuo et al., 2022.
Column (3): ALFALFA catalog number.
Column (4): HIPASS BGC name.
Column (5): WALLABY identifier.
Column (6): mass; the uncertainties in flux and the calibration are considered here.
Column (7): source of flux; FAST (F), ALFALFA (A), HIPASS BGC (H), and WALLABY Pilot Survey (W).
Column (8): offset from the mean relation.
Column (9): strength parameter; galaxies with too little gas for the calculation of are omitted.
Columns (10)–(12): fraction of gas under , TS, and total stripping; values smaller than $1\%$ are omitted.
Column (13): stripping status; RPS sample (R), TS sample (T), non-stripping galaxies (N), and merging pair (M).

This table is also available in machine-readable form.
aGalaxy 𝕀110 and 𝕀111 are merging, and their fluxes are inseparable.
The total of them is reported here and should be considered as the upper limit.

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# On the Alberti-Uhlmann Condition for Unital Channels

Sagnik Chakraborty<EMAIL_ADDRESS>Institute of Physics, Faculty of
Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziądzka 5/7, 87-100 Toruń, Poland  Dariusz Chruściński
<EMAIL_ADDRESS>Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziądzka 5/7, 87-100 Toruń,
Poland  Gniewomir Sarbicki<EMAIL_ADDRESS>Institute of Physics,
Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziądzka 5/7, 87-100 Toruń, Poland  Frederik vom Ende  frederik.vom-
<EMAIL_ADDRESS>Department of Chemistry, Technische Universität München, 85747
Garching, Germany  Munich Centre for Quantum Science and Technology (MCQST),
Schellingstr. 4, 80799 München, Germany

###### Abstract

We address the problem of existence of completely positive trace preserving
(CPTP) maps between two sets of density matrices. We refine the result of
Alberti and Uhlmann and derive a necessary and sufficient condition for the
existence of a unital channel between two pairs of qubit states which
ultimately boils down to three simple inequalities.

##  1 Introduction

A fundamental aspect of resource theories is finding conditions which
characterize the possibility of state-conversion via “allowed” operations. In
quantum thermodynamics, for example, one usually asks whether a state $\tau$
can be generated from an initial state $\rho$ via a Gibbs-preserving channel,
that is, a completely positive and trace-preserving (cptp) map which leaves
the Gibbs-state of the system invariant [16, 5, 12, 10, 24]. Sometimes one
limits the problem from Gibbs-preserving channels to the smaller set of
thermal operations which are of a specific form compliant with the underlying
resource theory. However, in what follows we focus on general cptp maps.
Recall that complete positivity of a linear map $T:M_{n}(\mathbb{C})\to
M_{n}(\mathbb{C})$ is equivalent to positivity of ${\rm id}\otimes
T:M_{n}(M_{n}(\mathbb{C}))\to M_{n}(M_{n}(\mathbb{C}))$ [21, 23].

The above is just a special case of the following more general question: given
two pairs of quantum states $\\{\tau_{1},\tau_{2}\\}$,
$\\{\rho_{1},\rho_{2}\\}$ can one find a characterization (which is non-
trivial and possibly simple to verify) for the existence of a quantum channel
which transforms both $\rho_{j}$ into $\tau_{j}$? While this problem is fully
solved in the classical case (for an overview on the equivalent conditions see
[12, Ch. V] or [26, Prop. 4.2]), in the quantum realm this remains unanswered,
with the qubit case being the notable exception: In a seminal paper [1]
Alberti and Uhlmann derived a necessary and sufficient condition for such a
simultaneous state conversion. The Alberti-Uhlmann condition reads

$\|\rho_{1}-t\rho_{2}\|_{1}\geq\|\tau_{1}-t\tau_{2}\|_{1}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ t\geq 0,$ (1)

where $\|A\|_{1}=\operatorname{tr}\sqrt{A^{\dagger}A}$ denotes the trace-norm
[20]. Note that (1) is equivalent to

$\|p_{1}\rho_{1}-p_{2}\rho_{2}\|_{1}\geq\|p_{1}\tau_{1}-p_{2}\tau_{2}\|_{1},$
(2)

for all $p_{1},p_{2}\geq 0$ and $p_{1}+p_{2}=1$. Due to Helstrom theory [14]
condition (2) means that distinguishability of states
$\\{\rho_{1},\rho_{2}\\}$ given with probabilities $\\{p_{1},p_{2}\\}$ is not
less than distinguishability of states $\\{\tau_{1},\tau_{2}\\}$ (occurring
with the same probabilities). For the sake of computation (1) can be reduced
to finitely many inequalities via the formulae

$\displaystyle\|A\|_{1}^{2}$
$\displaystyle=\operatorname{tr}(A^{\dagger}A)+2|\det(A)|$
$\displaystyle\det(A+B)$
$\displaystyle=\det(A)+\det(B)+\operatorname{tr}(A^{\\#}B)$ (3)

for all $A,B\in M_{2}(\mathbb{C})$ where $(\cdot)^{\\#}$ denotes the adjugate
[15, Ch. 0.8] given by

$M^{\\#}=\begin{bmatrix}d&-b\\\
-c&a\end{bmatrix},\text{where}\leavevmode\nobreak\ M=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}.$

Thus the function resulting from (1) is piecewise quadratic in $t$ so non-
negativity reduces to certain conditions on the coefficients (with respect to
$t$).

It is well known that every positive trace-preserving (ptp) map is a
contraction in the trace-norm and hence the Alberti-Uhlmann condition (1) is
necessary for the existence of a ptp map $T$. Interestingly, this condition is
sufficient in two dimensions – even for the existence of a quantum channel –
and fails to be sufficient for dimension three and larger [13, Ch. VII.B].

In [9], Chefles et al. generalized the problem to input and output sets
$\\{\rho_{1},\ldots,\rho_{n}\\}$ and $\\{\tau_{1},\ldots,\tau_{n}\\}$,
respectively, with arbitrary dimension and arbitrary value of $n$, under the
constraint that at least one of the two sets must be a set of pure states.
They derived conditions for the existence of a cptp map between the sets in
terms of the Gram matrices of the two sets.

A result for arbitrary (non-pure) states was derived by Huang et al. [17]
where their characterization (of existence of a cptp map) goes via the
existence of some more abstract decomposition of the initial and target
states. More interestingly, they considered the case of qubit states
$\\{\rho_{1},\rho_{2},\rho_{3}\\}$, $\\{\tau_{1},\tau_{2},\tau_{3}\\}$ under
the generic assumption of the input states being pure (cf. footnote 111 For
arbitrary qubit states $\rho_{1},\rho_{2}$ there exists $c\in[0,1)$ such that
$\tilde{\rho}_{1}:=\frac{\rho_{1}-c\rho_{2}}{1-c}$ is a pure state. Thus the
problem $\\{\rho_{1},\ldots,\rho_{n}\\}\to\\{\tau_{1},\ldots,\tau_{n}\\}$ by
linearity of the desired channel can be reduced to
$\\{\tilde{\rho}_{1},\ldots,\tilde{\rho}_{n}\\}\to\\{\tilde{\tau}_{1},\ldots,\tilde{\tau}_{n}\\}$.
). However the characterization they derived, while verifiable with standard
software, seems to not generalize to a condition for arbitrary input states
and, moreover, seems to not lead to much physical insight.

In [13], Heinosaari et al. considered a slightly different version of the
problem and studied conditions for the existence of only cp maps between two
sets of quantum states. Moreover they gave a fidelity characterization for the
existence of a cptp transformation on pairs of qubit states, consisting of
only a finite number of conditions.

In a more recent paper [10], Dall’Arno et al. derived a condition for the
existence of a cptp map when the input set is a collection of qubit states
which can be, through a simultaneous unitary rotation, written as real
matrices. They study the problem from the perspective of quantum statistical
comparison and derive that if the testing region of the real input states
includes the testing region of the output states then there exists a cptp map
connecting them. More precisely, the main result of [10] says that a set of
qubit states represented by real density matrices
$\\{\rho_{1},\ldots,\rho_{n}\\}$ can be mapped via a quantum channel to a set
$\\{\tau_{1},\ldots,\tau_{n}\\}$ if and only if the corresponding testing
regions satisfy

$\mathcal{R}(\\{\tau_{k}\\})\subset\mathcal{R}(\\{\rho_{k}\\}),$ (4)

where $\mathcal{R}(\\{\rho_{k}\\})$ is a set of vectors
$\mathbf{x}\in\mathbb{R}^{n}$ which can be realized via

$x_{k}={\rm Tr}(\rho_{k}\pi)\ ,\ \ \ k=1,\ldots,n,$

for some $0\leq\pi\leq\mathbbm{1}$. Similarly one defines
$\mathcal{R}(\\{\tau_{k}\\})$. For further analysis on the relation of this
problem to quantum statistical comparison see [7, 6].

In this paper we refine the original Alberti-Uhlmann problem by asking about
the existence of a unital channel, that is, $T$ maps $\rho_{k}$ to $\tau_{k}$
and additionally $T(\mathbbm{1})=\mathbbm{1}$. Original condition (1)
guarantees the existence of a cptp map but does say nothing whether $T$ is
unital. Clearly, condition (1) is again necessary but no longer sufficient.
Note, that the map $T$ is uniquely defined only on an at most 3-dimensional
subspace $\mathcal{M}$ spanned by $\\{\mathbbm{1},\rho_{1},\rho_{2}\\}$ and
one asks whether this map can be extended to the whole algebra
$M_{2}(\mathbb{C})$ such that the extended map is cptp and unital. Extension
problems such as this one were already considered by many authors before [21].
The classical result of Arveson [3] says that if $\mathcal{M}$ is an operator
system in ${\mathcal{B}(\mathcal{H})}$, that is, $\mathcal{M}$ is a linear
subspace closed under hermitian conjugation and containing $\mathbbm{1}$, and
if $\Phi:\mathcal{M}\to{\mathcal{B}(\mathcal{H})}$ is unital and completely
positive, then it can be extended to a unital completely positive map
$\widetilde{\Phi}:{\mathcal{B}(\mathcal{H})}\to{\mathcal{B}(\mathcal{H})}$.
Note, however, that this result says nothing about trace-preservation. Hence,
even if the unital map $\Phi$ is trace-preserving the unital extension
$\widetilde{\Phi}$ need not be trace-preserving. Actually, unitality may be
relaxed by assuming that the hermitian subspace $\mathcal{M}$ contains a
strictly positive operator [13]. Interestingly, it was shown [18] that if
$\mathcal{M}$ is spanned by positive operators and $\Phi$ is completely
positive then there exists a completely positive extension $\widetilde{\Phi}$.

The main result of this paper reads as follows. Let
$\\{\rho_{1},{\rho_{2}}\\}$ and $\\{\tau_{1},\tau_{2}\\}$ be two pairs of
qubit states and consider the two subspaces of $M_{2}(\mathbb{C})$

$\displaystyle\mathcal{M}$ $\displaystyle:={\rm
span}\\{{\rho_{0}=}\mathbbm{1},\rho_{1},\rho_{2}\\},$
$\displaystyle\mathcal{N}$ $\displaystyle:={\rm
span}\\{{\tau_{0}=}\mathbbm{1},\tau_{1},\tau_{2}\\}.$

In what follows we assume w.l.o.g. that
$\operatorname{dim}\mathcal{M}=3$—indeed $\operatorname{dim}\mathcal{M}=1$ is
trivial and $\operatorname{dim}\mathcal{M}=2$ can be traced back to the
original result of Alberti and Uhlmann.

######  Theorem 1.

The following statements are equivalent:

  * (i)

There exists a unital quantum channel mapping $\\{\rho_{1},{\rho_{2}}\\}$ to
$\\{\tau_{1},\tau_{2}\\}$.

  * (ii)

There exists a unital ptp map which sends $\\{\rho_{1},{\rho_{2}}\\}$ to
$\\{\tau_{1},\tau_{2}\\}$.

  * (iii)

For all $\alpha,\beta,\gamma\in\mathbb{R}$

$\displaystyle\lVert\alpha\mathbbm{1}+\beta\tau_{1}+\gamma\tau_{2}\rVert_{1}\leq\lVert\alpha\mathbbm{1}+\beta\rho_{1}+\gamma\rho_{2}\rVert_{1}\
\ \ \ $ (5)

  * (iv)

For all $\beta,\gamma\in\mathbb{R}$

$\displaystyle\lVert\tfrac{\mathbbm{1}}{2}+\beta\tau_{1}+\gamma\tau_{2}\rVert_{1}\leq\lVert\tfrac{\mathbbm{1}}{2}+\beta\rho_{1}+\gamma\rho_{2}\rVert_{1}$
(6)

  * (v)

For all $t\in\mathbb{R}$ one has $\tau_{1}-t\tau_{2}\prec\rho_{1}-t\rho_{2}$.

Here $\prec$ denotes classical matrix majorization which is usually defined
via the comparison of eigenvalues but can be characterized as follows (e.g.,
[2, Ch. 7]): for any two Hermitian $n\times n$ matrices one has $B\prec A$ iff
there is a unital quantum channel $T$ such that $B=T(A)$. Interestingly [2,
Thm. 7.4] $B\prec A$ iff for every real $t$ one has
$\|\mathbbm{1}-tB\|_{1}\leq\|\mathbbm{1}-tA\|_{1}$ which may be considered the
order-free characterization of majorization.

Clearly, the original Alberti-Uhlmann condition (1) provides now only the
necessary condition corresponding to $\alpha=0$ in (5).

While the conditions in Theorem 1 give conceptional insight, one can also
reduce the problem to three easy-to-verify inequalities.

######  Theorem 2.

There exists a unital quantum channel mapping qubit states
$\\{\rho_{1},\rho_{2}\\}$ to $\\{\tau_{1},\tau_{2}\\}$ if and only if
$\det(\tau_{j})\geq\det(\rho_{j})$ for $j=1,2$ as well as

$\begin{split}&\big{(}\operatorname{tr}(\rho_{1}^{\\#}\rho_{2})-\operatorname{tr}(\tau_{1}^{\\#}\tau_{2})\big{)}^{2}\\\
&\leq
4\big{(}\det(\tau_{1})-\det(\rho_{1})\big{)}\big{(}\det(\tau_{2})-\det(\rho_{2})\big{)}.\end{split}$
(7)

##  2 Proof of Alberti-Uhlmann conditions for unital maps

Consider a linear map $T:\mathcal{M}\to\mathcal{N}$ mapping $\rho_{k}$ to
$\tau_{k}$. By construction this map is unital and preserves the trace as well
as hermiticity.

######  Proposition 1.

The map $T$ is unitarily equivalent to a Pauli diagonal map
$S:\Sigma\to\Sigma$ where, here and henceforth,
$\Sigma=\operatorname{span}\\{\mathbbm{1},\sigma_{x},\sigma_{y}\\}$.

###### Proof.

Since $\mathcal{M}$ is a 3-dimensional operator system in $M_{2}(\mathbb{C})$
there exists a hermitian operator orthogonal to $\mathcal{M}$ w.r.t. the
Hilbert-Schmidt inner product. Being orthogonal to $\mathbbm{1}$ it is
traceless so after choosing an appropriate basis it equals $\sigma_{z}$.
Analogously, choosing an appropriate basis on the output Hilbert space makes
$\mathcal{N}$ orthogonal to $\sigma_{z}$. More precisely this yields unitary
$\tilde{U},\tilde{V}\in M_{2}(\mathbb{C})$ such that

$\tilde{U}^{\dagger}\,\mathcal{M}\,\tilde{U}\perp\sigma_{z}\ ;\ \ \
\tilde{V}^{\dagger}\,\mathcal{N}\,\tilde{V}\perp\sigma_{z}\,.$

Now, define the map
$\tilde{S}(\rho):=\tilde{V}^{\dagger}T(\tilde{U}\rho\tilde{U}^{\dagger})\tilde{V}$.
Since $\tilde{S}$ preserves hermiticity and the trace one has

$\tilde{S}(\sigma_{x})=\left[\begin{array}[]{cc}0&z_{x}\\\
{z^{*}_{x}}&0\end{array}\right],\qquad\tilde{S}(\sigma_{y})=\left[\begin{array}[]{cc}0&z_{y}\\\
{z^{*}_{y}}&0\end{array}\right],$

with $z_{x},z_{y}\in\mathbb{C}$. Hence the action of $\tilde{S}$ can be
represented via

$\begin{bmatrix}\mathbbm{1}\\\ \sigma_{x}\\\ \sigma_{y}\end{bmatrix}\ \to\
\begin{bmatrix}1&0&0\\\ 0&a_{11}&a_{12}\\\
0&a_{21}&a_{22}\end{bmatrix}\begin{bmatrix}\mathbbm{1}\\\ \sigma_{x}\\\
\sigma_{y}\end{bmatrix},$

for some $A=(a_{jk})_{j,k=1}^{2}\in\mathbb{R}^{2\times 2}$. Applying the
singular value decomposition to $A$ one finds orthogonal matrices
$W_{1},W_{2}\in M_{2}(\mathbb{R})$ such that
$W_{1}^{T}AW_{2}=\operatorname{diag}(a,b)$ with $a,b\geq 0$ [15, Thm. 7.3.5].
But every orthogonal $2\times 2$ matrix is a rotation matrix (possibly up to a
composition with $\sigma_{z}$) [15, p. 68]. Hence, the channel
$S_{\phi}(\rho):=U_{\phi}^{\dagger}\rho U_{\phi}$ with
$U_{\phi}=\operatorname{diag}(1,e^{i\phi})$, $\phi\in\mathbb{R}$ leaves
$\operatorname{span}\\{\mathbbm{1},\sigma_{x},\sigma_{y}\\}$ invariant and
acts on $(\mathbbm{1},\sigma_{x},\sigma_{y})$ as

$\begin{bmatrix}\mathbbm{1}\\\ \sigma_{x}\\\ \sigma_{y}\end{bmatrix}\ \to\
\begin{bmatrix}1&0&0\\\ 0&\cos\phi&-\sin\phi\\\
0&\sin\phi&\cos\phi\end{bmatrix}\begin{bmatrix}\mathbbm{1}\\\ \sigma_{x}\\\
\sigma_{y}\end{bmatrix}\ \,.$

Now, there exist $\phi,\theta\in\mathbb{R}$ such that
$S_{\phi}\circ\tilde{S}\circ S_{-\theta}$ corresponds to the diagonal matrix
$\operatorname{diag}(1,a,b)$ or $\operatorname{diag}(1,a,-b)$. Thus one has

$S(\rho)=V^{\dagger}T(U\rho U^{\dagger})V,$ (8)

where $U=\tilde{U}U_{\theta}$, $V=\tilde{V}U_{\phi}$, that is, the map $T$ is
unitarily equivalent to the Pauli diagonal map $S$ defined on $\Sigma$. ∎

Therefore it is clear that the unital map $T:\mathcal{M}\to\mathcal{N}$ can be
extended to a unital cp map $\mathbf{T}:M_{2}(\mathbb{C})\to
M_{2}(\mathbb{C})$ if and only if the unital Pauli diagonal map
$S:\Sigma\to\Sigma$ can be extended to a unital cp map
$\mathbf{S}:M_{2}(\mathbb{C})\to M_{2}(\mathbb{C})$. Now,
$T:\mathcal{M}\to\mathcal{N}$ is a positive map if and only if
$S:\Sigma\to\Sigma$ is positive. However, for Pauli diagonal maps the
positivity conditions are well-known [22]: $S$ is positive if and only if
$|a|,|b|\leq 1$. Now, if $S:\Sigma\to\Sigma$ is positive it can be always
extended to a unital ptp map $\mathbf{S}$ on $M_{2}(\mathbb{C})$. Indeed, one
may use the trivial extension defining

$\mathbf{S}(\sigma_{z})=0.$ (9)

######  Remark 1.

Actually it was shown [8] that if $\mathcal{M}$ is a 3-dimensional operator
system, then there exists a unital ptp projector
$\Pi:M_{2}(\mathbb{C})\to\mathcal{M}$. Hence $\mathbf{S}:=S\circ\Pi$ defines a
unital ptp extension of $S$. Interestingly, however, it was proven [8] that
there is no cptp projector $\Pi:M_{2}(\mathbb{C})\to\mathcal{M}$.

Yet one can find an extension of $S$ which is not only ptp but even cptp:

######  Proposition 2.

If a Pauli diagonal map $S:\Sigma\to\Sigma$ is positive it can always be
extended to unital cptp map $\mathbf{S}$ on $M_{2}(\mathbb{C})$.

###### Proof.

We show that there exists $\mathbf{S}$ which is also a Pauli diagonal map,
that is,

$\mathbf{S}(\sigma_{z})=c\sigma_{z}.$

The Pauli diagonal map $\mathbf{S}$ is cptp if and only if the well-known
Fujiwara-Algoet conditions are satisfied [11, 22]

$|a+b|\leq 1+c\ ,\ \ |a-b|\leq 1-c\,,$ (10)

that is,

$|a+b|-1\leq c\leq 1-|a-b|\,.$

Clearly, there exists a non-trivial solution for $c$ if and only if

$2\geq|a-b|+|a+b|=2\max\\{|a|,|b|\\},$

which concludes the proof. ∎

It turns out that the above Proposition also has a simple geometric
representation:

######  Remark 2.

It is well known [22, 19, 4] that the parameters $(a,b,c)$ of the Pauli
diagonal unital cptp maps form a tetrahedron defined by Fujiwara-Algoet
conditions (10) with vertices $\mathbf{x}_{0}=(1,1,1)$,
$\mathbf{x}_{1}=(1,-1,-1)$, $\mathbf{x}_{2}=(-1,1,-1)$ and
$\mathbf{x}_{3}=(-1,-1,1)$. These four vertices correspond to four unitary
Pauli channels —
$\mathcal{U}_{\alpha}(\rho)=\sigma_{\alpha}\rho\sigma_{\alpha}$, with
$\sigma_{0}=\mathbbm{1}$, and $\\{1,2,3\\}=\\{x,y,z\\}$. Projecting the
tetrahedron along the $z$-axis onto the $xy$-plane one obtains the unit square
of parameters $a,b$ satisfying $|a|,|b|\leq 1$, cf. Figure 1. Hence it is
clear that starting from any point in the unit square and going along the
$z$-axis one eventually ends up inside the tetrahedron, thus realizing a cptp
extension of a ptp map from the unit square.

$(-1,-1,-1)$$(1,-1,-1)$$(-1,1,-1)$$(-1,-1,1)$$(1,1,-1)$$(-1,1,1)$$(1,-1,1)$$(1,1,1)$$a$$b$$c$
Figure 1: The tetrahedron represents Pauli diagonal channels with $a,b$ and
$c$ being the respective eigenvalues for eigenvectors $\sigma_{x},\sigma_{y}$
and $\sigma_{z}$. The horizontal plane represents Pauli maps with $c=0$.

Now, we are ready to prove the main theorem.

###### Proof of Theorem 1.

(i) $\Rightarrow$ (ii) is obvious and (ii) $\Rightarrow$ (iii) follows from
the well-known fact that the trace-norm is contractive under the action of ptp
maps. To show (iii) $\Rightarrow$ (i) let us observe that the very condition
(5) is equivalent to

$\displaystyle\lVert\alpha\mathbbm{1}+\beta a\sigma_{x}$ $\displaystyle+\gamma
b\sigma_{y}\rVert_{1}$
$\displaystyle\leq\lVert\alpha\mathbbm{1}+\beta\sigma_{x}+\gamma\sigma_{y}\rVert_{1}$
(11)

for all $\alpha,\beta,\gamma\in\mathbb{R}$ with $a,b$ being the parameters of
the Pauli diagonal channel $S$ as above. Indeed, assume (5), i.e. for all
$\alpha,\beta,\gamma$

$\displaystyle\|\alpha\mathbbm{1}+\beta V^{\dagger}\tau_{1}V+\gamma$
$\displaystyle
V^{\dagger}\tau_{2}V\|_{1}=\|\alpha\mathbbm{1}+\beta\tau_{1}+\gamma\tau_{2}\|_{1}$
$\displaystyle\leq\|\alpha\mathbbm{1}+\beta\rho_{1}+\gamma\rho_{2}\|_{1}$
$\displaystyle=\|\alpha\mathbbm{1}+\beta U^{\dagger}\rho_{1}U+\gamma
U^{\dagger}\rho_{2}U\|_{1}$

with $U,V$ being the unitary matrices from Proposition 1, using unitary
equivalence of the trace norm. Writing

$U^{\dagger}\rho_{j}U=\frac{\mathbbm{1}}{2}+r_{jx}\sigma_{x}+r_{jy}\sigma_{y}$

for some $r_{jx},r_{jy}\in\mathbb{R}$ yields

$V^{\dagger}\tau_{j}V=\frac{\mathbbm{1}}{2}+ar_{jx}\sigma_{x}+br_{jy}\sigma_{y}\,.$

Thus the trace norm condition (5) becomes:

$\displaystyle\lVert\alpha^{\prime}\mathbbm{1}+\beta^{\prime}a\sigma_{x}$
$\displaystyle+\gamma^{\prime}b\sigma_{y}\rVert_{1}$
$\displaystyle\leq\lVert\alpha^{\prime}\mathbbm{1}+\beta^{\prime}\sigma_{x}+\gamma^{\prime}\sigma_{y}\rVert_{1},$
(12)

where

$\begin{bmatrix}\alpha^{\prime}\\\ \beta^{\prime}\\\
\gamma^{\prime}\end{bmatrix}=\begin{bmatrix}1&\frac{1}{2}&\frac{1}{2}\\\
0&r_{1x}&r_{2x}\\\ 0&r_{1y}&r_{2y}\end{bmatrix}\begin{bmatrix}\alpha\\\
\beta\\\ \gamma\end{bmatrix}.$

Now, observe that

$\det\begin{bmatrix}1&\frac{1}{2}&\frac{1}{2}\\\ 0&r_{1x}&r_{2x}\\\
0&r_{1y}&r_{2y}\end{bmatrix}=\det\begin{bmatrix}r_{1x}&r_{2x}\\\
r_{1y}&r_{2y}\end{bmatrix}\neq 0$

because $\operatorname{dim}(\mathcal{M})=3$. Hence condition (5) for all
$\alpha,\beta,\gamma$ is equivalent to (2) for all
$\alpha^{\prime},\beta^{\prime},\gamma^{\prime}$, that is, it is equivalent to
(2) for all $\alpha,\beta,\gamma$.

Now, as we have shown (iii) implies (2), the last step is to show that (2)
implies (i). Choosing $(\alpha,\beta,\gamma)=(0,1,0)$ and
$(\alpha,\beta,\gamma)=(0,0,1)$ in (2) implies $|a|\leq 1$ and $|b|\leq 1$ so
there exists a cptp extension of $S$ (and hence also of $T$) due to
Proposition 2.

(iii) $\Leftrightarrow$ (iv) is readily verified using homogeneity and
continuity of the norm.

(v) $\Leftrightarrow$ (iv): By [2, Thm. 7.1] and (1) condition (v) is
equivalent to
$\|\tfrac{\mathbbm{1}}{2}-s(\tau_{1}-t\tau_{2})\|_{1}\leq\|\tfrac{\mathbbm{1}}{2}-s(\rho_{1}-t\rho_{2})\|_{1}$
for all $s,t\in\mathbb{R}$ which by the same argument as before is equivalent
to (iv). ∎

######  Remark 3.

Note that an input set $\\{\rho_{0}=\mathbbm{1}/2,\rho_{1},\rho_{2}\\}$ can be
always represented by real matrices and hence it fits the scenario of [10].
Therefore, condition (4) is equivalent to all conditions from Theorem 1.

###### Proof of Theorem 2.

The eigenvalue formula for $2\times 2$ matrices directly yields that if
$A,B\in M_{2}(\mathbb{C})$ are hermitian and of same trace then $A\prec B$ iff
$\det(A)\geq\det(B)$. Using (3), condition (v) from Theorem 1 is equivalent to

$\displaystyle t^{2}\big{(}\det(\tau_{2})-\det(\rho_{2})\big{)}$
$\displaystyle-t\big{(}\operatorname{tr}(\rho_{1}^{\\#}\rho_{2})-\operatorname{tr}(\tau_{1}^{\\#}\tau_{2})\big{)}$
$\displaystyle+\big{(}\det(\tau_{1})-\det(\rho_{1})\big{)}\geq 0$

for all $t\in\mathbb{R}$. But a parabola $t\mapsto at^{2}+bt+c$ is non-
negative iff $a,c\geq 0$ and $b^{2}\leq 4ac$ which concludes the proof. ∎

##  3 Example

Let us now present an example which hopefully demonstrates the power of our
results. Consider the linear map

$\left[\begin{array}[]{cc}\rho_{11}&\rho_{12}\\\
\rho_{21}&\rho_{22}\end{array}\right]\mapsto\left[\begin{array}[]{cc}\rho_{11}+(1-p)\,\rho_{22}&\kappa\rho_{12}\\\
\kappa\rho_{21}&p\,\rho_{22}\end{array}\right],$ (13)

which is cptp whenever $1\geq p\geq\kappa^{2}$. Choose $p=\kappa=\frac{1}{2}$
and recall that every qubit state can be represented as

$\left[\begin{array}[]{cc}c&\gamma\sqrt{c(1-c)}\\\
\gamma^{*}\sqrt{c(1-c)}&1-c\end{array}\right]$

for some $c\in[0,1]$ and some $\gamma\in\mathbb{C}$, $|\gamma|\leq 1$. Thus we
may consider the two valid density matrices:

$\displaystyle\rho_{1}$
$\displaystyle=\left[\begin{array}[]{cc}c&\frac{3i}{4}\sqrt{c(1-c)}\\\
-\frac{3i}{4}\sqrt{c(1-c)}&1-c\end{array}\right],\quad$
$\displaystyle\rho_{2}$ $\displaystyle=\left[\begin{array}[]{cc}0.2&0.4\\\
0.4&0.8\end{array}\right]$

with $c\in[0,1]$ arbitrary which are mapped to

$\displaystyle\tau_{1}$
$\displaystyle=\left[\begin{array}[]{cc}\frac{1}{2}(1+c)&\frac{3i}{8}\sqrt{c(1-c)}\\\
-\frac{3i}{8}\sqrt{c(1-c)}&\frac{1}{2}(1-c)\end{array}\right],\quad$
$\displaystyle\tau_{2}$ $\displaystyle=\left[\begin{array}[]{cc}0.6&0.2\\\
0.2&0.4\end{array}\right]$

by means of (13). The Alberti-Uhlmann condition is obviously satisfied since
by constriction the $\tau_{k}$ are related to the $\rho_{k}$ via a cptp map.
Now let us use Theorem 2 to check for which $c$ one can find a unital
extension:

  * •

$\det(\tau_{1})\geq\det(\rho_{1})$: one readily computes that this is
satisfied iff $(1-c)(\frac{1}{4}-\frac{21}{64}c)\geq 0$. Together with the
base assumption $c\in[0,1]$ this leads to $c\in[0,\frac{16}{21}]$.

  * •

$\det(\tau_{2})=0.2\geq 0=\det(\rho_{2})$ $\checkmark$

  * •

Condition (7) after a straightforward computation becomes $(0.7c-0.3)^{2}\leq
0.8(1-c)(\frac{1}{4}-\frac{21}{64}c)$ which together with $c\in[0,1]$ means
$c\leq 0.6082\ldots$

While this fully solves the one-parameter problem let us look at two
interesting and hopefully illuminating special cases.

######  Example 1.

Choosing $c=0$ we know there exists a unital channel which simultaneously maps
$\\{\rho_{1}=|1\rangle\langle 1|,\rho_{2}\\}$ to $\\{\tau_{1}=\mathbbm{1}/2$,
$\tau_{2}\\}$ with $\rho_{2},\tau_{2}$ from above. Interestingly, it can also
be found that a unital cptp map giving rise to this transformation is given by

$\displaystyle\left[\begin{array}[]{cc}\rho_{11}&\rho_{12}\\\
\rho_{21}&\rho_{22}\end{array}\right]\,\mapsto\,\left[\begin{array}[]{cc}\frac{\rho_{11}+\rho_{22}}{2}+\frac{\rho_{12}+\rho_{21}}{8}&\frac{\rho_{12}+\rho_{21}}{4}\\\
\frac{\rho_{12}+\rho_{21}}{4}&\frac{\rho_{11}+\rho_{12}}{2}-\frac{\rho_{12}+\rho_{21}}{8}\end{array}\right].$

######  Example 2.

For the second example choose $c=\frac{2}{3}$ so by the above analysis no
unital cptp map simultaneously transforms

$\rho_{1}=\left[\begin{array}[]{cc}\frac{2}{3}&\frac{i\sqrt{2}}{8}\\\
-\frac{i\sqrt{2}}{8}&\frac{1}{3}\end{array}\right],\qquad\rho_{2}=\left[\begin{array}[]{cc}0.2&0.4\\\
0.4&0.8\end{array}\right]$

into

$\tau_{1}=\left[\begin{array}[]{cc}\frac{5}{6}&\frac{i\sqrt{2}}{16}\\\
-\frac{i\sqrt{2}}{16}&\frac{1}{6}\end{array}\right],\qquad\tau_{2}=\left[\begin{array}[]{cc}0.6&0.2\\\
0.2&0.4\end{array}\right]\,.$

However, $\tau_{1}\prec\rho_{1}$ and $\tau_{2}\prec\rho_{2}$ so by [2, Thm.
7.1] one can find unital channels $T_{1},T_{2}$ such that
$T_{j}(\rho_{j})=\tau_{j}$ for $j=1,2$. This is what makes this example
remarkable: while there exists a cptp map sending $\\{\rho_{1},\rho_{2}\\}$ to
$\\{\tau_{1},\tau_{2}\\}$ (i.e. (13) for $p=\kappa=\frac{1}{2}$) and while
there exist unital channels which map either $\rho_{1}$ to $\tau_{1}$ or
$\rho_{2}$ to $\tau_{2}$, there is no simultaneous unital cptp transformation.
Thus this example clearly emphasizes the necessity of Theorem 1.

##  4 Conclusions & Outlook

In this paper we derived necessary and sufficient conditions for the existence
of a unital quantum channel mapping a pair of qubit states
$\\{\rho_{1},\rho_{2}\\}$ to $\\{\tau_{1},\tau_{2}\\}$. These conditions
connect the problem to trace-norm inequalities (in the spirit of Alberti-
Uhlmann (1) which is reproduced by setting $\alpha=0$ in (2)) and majorization
on matrices. Moreover we reduced the infinite family of conditions to just
three inequalities which are simple enough to be verified with pen and paper.
We also provided an example of two pairs of qubit states
$\tau_{k}\prec\rho_{k}$ which satisfy the Alberti-Uhlmann condition, that is,
there exists a quantum channel mapping $\rho_{k}$ to $\tau_{k}$, but condition
(7) is violated meaning there is no single unital channel between $\rho_{k}$
and $\tau_{k}$. The authors of [13] provide an example of a qutrit system
(i.e. 3-dimensional) where pairs of states $\rho_{1},\rho_{2}$ and
$\tau_{1},\tau_{2}$ satisfy the Alberti-Uhlmann condition (1) but there is no
quantum channel mapping $\rho_{k}\to\tau_{k}$ (cf. [13], Sec. VII B).
Actually, the same example may be used to show that
$\\{\mathbbm{1}/2,\rho_{1},\rho_{2}\\}$ and
$\\{\mathbbm{1}/2,\tau_{1},\tau_{2}\\}$ satisfy (5) but there is no (unital)
quantum channel mapping $\rho_{k}\to\tau_{k}$.

We expect that our result will encourage more research in this direction and
shed light on finding more general closed form conditions for the existence of
channels between sets of quantum states. Possible next steps could focus on
the case of the input set consisting of any three linearly independent qubit
states or—in spirit of thermo- and general $D$-majorization [25]—on how to
modify Theorem 1 & 2 if the fixed point of the channel is not the identity but
an arbitrary Gibbs state (i.e. an arbitrary positive-definite state $D$).

## Acknowledgements

The authors are grateful to Teiko Heinosaari, Antonella De Pasquale, Namit
Anand and Francesco Buscemi for fruitful discussion as well as the anonymous
referees’ constructive comments. S.C., D.C. and G.S. were supported by the
Polish National Science Centre project 2018/30/A/ST2/00837. F.v.E. is
supported by the Bavarian excellence network enb via the International PhD
Programme of Excellence Exploring Quantum Matter (exqm).

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