# PolarRec: Radio Interferometric Data Reconstruction with Polar Coordinate
Representation

Ruoqi Wang, Zhuoyang Chen, Jiayi Zhu
HKUST(GZ)
Qiong Luo
HKUST(GZ) & HKUST  Feng Wang
GZHU

###### Abstract

In radio astronomy, visibility data, which are measurements of wave signals
from radio telescopes, are transformed into images for observation of distant
celestial objects. However, these resultant images usually contain both real
sources and artifacts, due to signal sparsity and other factors. One way to
obtain cleaner images is to reconstruct samples into dense forms before
imaging. Unfortunately, existing reconstruction methods often miss some
components of visibility in frequency domain, so blurred object edges and
persistent artifacts remain in the images. Furthermore, the computation
overhead is high on irregular visibility samples due to the data skew. To
address these problems, we propose PolarRec, a transformer-encoder-conditioned
reconstruction pipeline with visibility samples converted into the polar
coordinate representation. This representation matches the way in which radio
telescopes observe a celestial area as the Earth rotates. As a result,
visibility samples distribute in the polar system more uniformly than in the
Cartesian space. Therefore, we propose to use radial distance in the loss
function, to help reconstruct complete visibility effectively. Also, we group
visibility samples by their polar angles and propose a group-based encoding
scheme to improve the efficiency. Our experiments demonstrate that PolarRec
markedly improves imaging results by faithfully reconstructing all frequency
components in the visibility domain while significantly reducing the
computation cost in visibility data encoding. We believe this high-quality and
high-efficiency imaging of PolarRec will better facilitate astronomers to
conduct their research.

##  1 Introduction

In radio astronomy, _visibility_ refers to radio signal data from celestial
objects, obtained by radio telescopes. These data are represented as complex
values in the _uv-plane_ , a geometric plane defined for interferometric
observations. Visibility data are subsequently converted into images through
_imaging_ for further analysis. However, these images, known as _dirty images_
, are often dominated by artifacts [22]. This phenomenon is due to limitations
in telescope configurations, under which not the entire uv-plane is sampled.
Therefore, visibility data normally have to be reconstructed before being
utilized in scientific analysis. In this paper, we propose a visibility
reconstruction method, aiming to reconstruct the real sky by recovering all
visibility components in the uv-plane effectively and efficiently.

(a) Traditional method: imaging followed by reconstruction

(b) Recent method: reconstruction followed by imaging.

Figure 1: Two visibility data processing flows.

Traditional methods first transfer the sparse visibility data into dirty
images through the imaging process and then reconstruct the dirty images to
_clean images_ [13, 1, 3, 25, 9, 29]. The process is shown in Figure 1 (a). In
contrast, some recent deep-learning-based studies [22, 30] have proposed to
first do inpainting on the visibility data to reconstruct the sparse samples
to dense coverage and then perform imaging to obtain the clean image, as shown
in Figure 1 (b), showing better performance. In order to produce a high-
fidelity sharp image, it is crucial to densely sample the full visibility
domain [30]. In this paper, we adopt the reconstruction-and-imaging processing
flow.

Existing methods for reconstructing visibility data face challenges in both
effectiveness and efficiency. Specifically, the first challenge is to
effectively capture visibility components within the uv-plane. For example,
Radionets [22] and U-Nets [21] use a convolutional neural network that is
based on pixels and grids, resulting in discontinuities in reconstructed
visibility maps. In comparison, Wu et al. [30] use neural fields to address
the continuity problem, but their method can only restore the low-frequency
part of visibility (located near the center of the uv-plane), largely missing
the high-frequency portion (found far from the uv-plane center). Such
discontinuity and incompleteness in the visibility domain cause blurred edges
of observed objects, disappearance of faint astronomical sources, and
persistence of artifacts, in resultant images.

The second major issue is the inefficiency of current strategies. For
instance, Radionets [22] encode visibility data together with an excessive
amount of zero-inpainting, causing a large amount of unnecessary computation.
Wu et al. [30] embed each visibility sample point as a token in their
Transformer encoder [28] to attend to each other, which is of quadratic
computation cost in the number of sample points [11], too high for real
applications. These limitations force a compromise between computational cost
and quality of reconstruction in current methods, severely limiting the
application of deep learning techniques in radio astronomy imaging and
reconstruction.

(a) Cartesian Representation.

(b) Polar Representation.

Figure 2: Visibility data samples with different representations.

To address these challenges, we explore the observational nature of radio
telescopes and then propose PolarRec, which leverages the polar representation
of sample points to enhance the reconstruction quality and reduce the cost.
Our key observation is that visibility samples are obtained by telescopes as
the Earth rotates, and that the high- and low-frequency signals are
distributed on the uv-plane in accordance with their distances from the
center. As a result, Cartesian-based approaches cannot represent the
visibility points evenly [18] (see Figure 2). In contrast, the polar-based
representation has shown promising properties in vision tasks including
instance segmentation and object detection [18, 31, 32, 7]. Therefore, we
propose to adopt the polar representation, using the radial coordinate to
associate with the frequency information in the loss function and the angular
coordinate to group sample points.

Specifically, we first design a loss function for visibility data with a
weighting scheme based on the radial coordinates. Our weighted loss function,
Radial Visibility Loss, associates visibility data with radial coordinates in
the frequency domain, enabling effective reconstruction of visibility data, on
both high- and low-frequency components. In comparison, existing work recovers
mainly low-frequency components. Consequently, our method produces sharper,
more detailed imaging results. Furthermore, by introducing group-level
encoding of visibility points according to angular coordinates, our method is
more efficient than visibility-point-level encoding. This group encoding
improves computation efficiency, making the use of Transformer encoders for
visibility encoding more practical and scalable.

In summary, our main contributions are as follows:

  * •

We propose PolarRec to seamlessly integrate visibility data with a
Transformer-encoded reconstruction pipeline by using polar coordinate
representation, addressing two fundamental challenges in visibility
reconstruction.

  * •

We introduce the Radial Visibility Loss (RVL), which incorporates the radial
coordinates of visibility data, adeptly capturing both low- and high-frequency
visibility components.

  * •

As the first to study the granularity of visibility encoding, we innovatively
utilize angular coordinates to group sampled points for Transformer encoding,
markedly enhancing computational efficiency.

We have experimentally evaluated our proposed PolarRec on public datasets of
four galaxy morphologies, including an overall comparison with other state-of-
the-art methods, ablation studies, and tests on group sizes and grouping
techniques. The experimental results demonstrate the effectiveness and
efficiency of our method.

##  2 Background and Related Work

###  2.1 Very Long Baseline Interferometry (VLBI)

In radio astronomy, using radio interference signals to image distant
astronomical sources requires telescopes of very large aperture [4], because
the angular resolution of a telescope is inversely proportional to the
diameter. A major observation technique is the Very Long Baseline
Interferometry (VLBI), which uses multiple radio telescopes spreading over the
globe to form a virtual Earth-sized telescope. The radio waves from
astronomical sources are recorded separately at individual telescopes. Then,
these signals are cross-correlated for all pairs of antennas at a central
location, generating _visibility_ data. A VLBI observation is typically
performed for hours to measure as many points in the uv-plane as possible.
However, the measurement results remain sparse due to the limited number of
antennas [27, 4]. Consequently, sparse-to-dense reconstruction on visibility
data is necessary to improve the imaging quality for the following analysis.

###  2.2 Interferometric Imaging

Visibility data, represented as complex values, is the result of a Fourier
transform of the sky’s brightness distribution [17]. _Imaging_ converts
visibility data into images, which can be analyzed to provide insights about
the observed celestial bodies [27]. In the imaging process, an inverse Fourier
transform maps the $(u,v)$ coordinates from the Fourier domain to $(l,m)$
coordinates in the image domain [30]. The transformation can be described as:

$\displaystyle I(l,m)$ $\displaystyle=$ $\displaystyle\int_{u}\int_{v}e^{2\pi
i(ul+vm)}V(u,v)dudv.$ (1)

In this equation, $V(u,v)$ is the visibility data in Fourier space, and
$I(l,m)$ represents the intensity distribution in the image domain.

###  2.3 Radio Interferometric Data Reconstruction

Radio interferometric data reconstruction is vital for converting sparse
visibility data into clear, artifact-free images. However, previous methods in
this domain [13, 1, 3, 25, 9, 29, 30, 22] have been limited by either their
inability to capture continuous and complete frequency components or
computational inefficiencies. To the best of our knowledge, no previous
studies have managed to reconstruct all components in visibility data or
increase the granularity of visibility encoding for efficiency.

Figure 3: Effects of band limiting.

##  3 Our Method

Figure 4: An overview of our method. Sparse visibility data
${V}\left(u_{s},v_{s}\right)$ are grouped by the angular coordinate and passed
through two encoding layers: intra-group encoding to generate group tokens and
inter-group encoding by a Transformer encoder. The encoded output $T$ then
conditions the predicted visibility generation in the neural field. The final
output is compared with the ground truth to compute the Radial Visibility
Loss.

In this section, we first investigate the relation between visibility
components and imaging results. After that, we present PolarRec, which adopts
polar coordinate representation in visibility reconstruction. In PolarRec, we
design Radial Visibility Loss to incorporate a weighting scheme based on the
radial coordinate within the uv-plane. Moreover, we group the visibility
points by their angular coordinates and then extract grouping tokens for the
subsequent Transformer encoding. An overview of our method is presented in
Figure 4.

###  3.1 Imaging Results of Visibility

In radio interferometry, visibility data plays a vital role in generating
high-fidelity images of celestial objects. The imaging process is the inverse
Fourier transformation of visibility data to construct the brightness
distribution of the observed sky. Consequently, missing or incomplete
visibility components can significantly degrade the imaging result, leading to
artifacts and loss of crucial information about the object’s structure.
Therefore, we first investigate the impact of missing visibility components on
the final imaging output.

We explore the impact of visibility components in various frequency regions by
applying standard band-limiting operations [16] and analyzing the effects of
individual components on the imaging results. As shown in Figure 3, loss of
high frequency data due to a low-pass filter (Column low-pass-1 and low-
pass-2) results in blur and artifacts and causes the vanishing of weak sources
in the imaging results. Comparing low-pass-1 and low-pass-2, we can see that
the larger the radius of the retained portion in the frequency domain, the
clearer the resulting image is, and the fewer artifacts there are. In
comparison, when low frequency data are absent due to a high-pass filter
(Column high-pass-1 and high-pass-2), the overall quality of the image
declines, but the object edges are clear, and dim or small sources around the
main observed object are retained. Last, the band-and-stop filter (Column
Band-stop) also causes artifacts and blur in imaging results.

In summary, different frequency regions cause distinct imaging effects. This
observation suggests that recovering all missing visibility components could
enhance the quality of imaging results. More specifically, if a resultant
image has blurred edges or misses dim light sources, it is probably due to
poor reconstruction of the high-frequency components of the visibility data.

###  3.2 Polar Coordinate Representation

Our utilization of the polar coordinate representation is to represent the
visibility more uniformly. It is because visibility sampling is based on
Earth’s rotation, and high- and low-frequency visibility components are
distributed on the uv-plane according to their distances from the origin of
the plane.

In the uv-plane, we convert $(u,v)$ coordinates to polar coordinates, denoted
as $(r(u,v),\theta(u,v))$:

$r(u,v)=\sqrt{u^{2}+v^{2}}$ (2)

$\theta(u,v)=\text{arctan2}(v,u)$ (3)

Where $r(u,v)$ represents the radial distance from the origin, and
$\theta(u,v)$ represents the angle of the vector from the positive $u$-axis.

###  3.3 Radial Visibility Loss

We propose Radial Visibility Loss (RVL), incorporating the radial coordinates
of visibility data to capture both low- and high-frequency visibility.

To compute RVL, we first compute the weight matrix $w_{1}(u,v)$ to down-weight
easy visibility components (components whose predicted values are close to the
ground truth) based on Focal Frequency Loss (FFL) [16]:

$w_{1}(u,v)=|V_{r}(u,v)-V_{p}(u,v)|^{\alpha}$ (4)

where $V_{r}(u,v)=A_{r}+iB_{r}$ and $V_{p}(u,v)=A_{p}+iB_{p}$ are the ground
truth and predicted visibilities in complex form respectively, and $\alpha$ is
a scaling factor.

We then introduce an additional weight $w_{2}(u,v)$, computed from $r(u,v)$,
to make our model pay more attention to high-frequency components of the
visibility during reconstruction. The weight term $w_{2}(u,v)$ is calculated
as follows:

$w_{2}(u,v)=\left(\frac{r(u,v)}{\max(r(u,v))}+1\right)^{\beta}$ (5)

Where $\frac{r(u,v)}{\max(r(u,v))}$ normalizes the radial coordinate, ensuring
the weight is more significant for points farther from the center,
corresponding to higher frequencies in the visibility data. Adding 1 prevents
any weights from becoming zero, maintaining the influence of all visibility
components during the learning process. $\beta$ is a scaling factor.

The final weight $w(u,v)$ is computed as:

$w(u,v)=\left(\frac{r(u,v)}{\max(r(u,v))}+1\right)^{\beta}|V_{r}(u,v)-V_{p}(u,v)|^{\alpha}$
(6)

Figure 5: Example of visibility distance.

A thorough visibility distance measure must consider both amplitude and phase
as they hold distinct aspects of imaging information [16]. The complex
visibility value can be represented as a vector within a two-dimensional
plane. Specifically, we map ground truth visibility $V_{r}(u,v)=A_{r}+iB_{r}$
and the corresponding predicted value $V_{p}(u,v)=A_{p}+iB_{p}$ to vectors
$\vec{p_{r}}$ and $\vec{p_{p}}$ (as shown in Figure 5). According to the
definition of 2D discrete Fourier transform, the vector magnitude relates to
the amplitudes $|\vec{p_{r}}|$ and $|\vec{p_{p}}|$, and the angles
$\theta_{r}$ and $\theta_{p}$ reflect the phase. Consequently, following FFL
[16], we define the single-point visibility distance considering both
magnitude and direction as:

$d(\vec{p_{r}},\vec{p_{p}})=\|\vec{p_{r}}-\vec{p_{p}}\|^{2}_{2}=|V_{r}(u,v)-V_{p}(u,v)|^{2}.$
(7)

The final visibility distance is computed as the average of these individual
distances over all visibility points:

$d(V_{r},V_{p})=\frac{1}{MN}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}|V_{r}(u,v)-V_{p}(u,v)|^{2}.$
(8)

The final expression for the Radial Visibility Loss is derived by executing
the Hadamard product between the weight matrix $w(u,v)=w_{1}(u,v)w_{2}(u,v)$
and the visibility distance matrix $d(V_{r},V_{p})$, as follows:

$\text{RVL}=\frac{1}{MN}\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}w(u,v)|V_{r}(u,v)-V_{p}(u,v)|^{2}$
(9)

###  3.4 Encoding by Angular-Coordinate Groups

Using Transformers for encoding visibility samples causes quadratic
computation cost in the number of sample points [11]. Such high costs
constrain practical applications, especially as the number of visibility
points in each image expands with more telescope arrays or extended
observation duration. To solve the problem, we propose a group-based encoding
scheme to improve the efficiency.

As shown on the left in Figure 4, the sparsely sampled visibility data is in
the form of $\left\\{u_{s},v_{s},{V}\left(u_{s},v_{s}\right)\right\\}$, where
$(u_{s},v_{s})$ are the coordinates at which a measurement is sampled and
${V}\left(u_{s},v_{s}\right)$ is the complex value of the sample point.

First, we divide all sample points into $m$ groups according to their angular
coordinate $\theta(u_{s},v_{s})$. To integrate each visibility value and its
corresponding position, we encode each sample individually using positional
embedding ($\text{PE}(u_{s},v_{s})$ in Figure 4). Specifically, we encode the
positional information of a sample point using Random Fourier Embedding [26].
After that, the positional embedding $\text{PE}(u_{s},v_{s})$ and the
embedding of complex value of visibility ${V}\left(u_{s},v_{s}\right)$ are
concatenated to form visibility tokens $V^{\prime}$. Denote the sparse
visibility tokens as
$V^{\prime}=\left[v^{\prime}_{1};v^{\prime}_{2};v^{\prime}_{3};\ldots
v^{\prime}_{n}\right]$, where $V^{\prime}\in\mathbb{R}^{n\times d}$, $n$ is
the number of sample measurement points and $d$ is the number of dimensions of
visibility tokens.

Then, we apply a Multi-Layer Perceptron (MLP) mapping layer and averaging
aggregating for intra-group encoding. The encoding result is
$\hat{V}=\left[\hat{v}_{1};\hat{v}_{2};\hat{v}_{3};\ldots\hat{v}_{m}\right]$,
where $\hat{V}\in\mathbb{R}^{m\times d}$. For each group $i$, we compute the
group token $\hat{v}_{i}$ as follows:

$\displaystyle\hat{v}_{i}=\text{Avg}\left(\text{MLP}(v_{j})\right),\text{
}i=1\text{ to }m,\text{ }v_{j}\in\text{ group }i.$ (10)

In the implementation of intra-group encoding, we sort the tokens $V^{\prime}$
of sample points according to their angular coordinates. After mapping these
tokens through MLP, we use adaptive average pooling to compute the group
tokens $\hat{V}$. For data collected with the same telescope configuration,
the process of sorting based on angular coordinates needs to be performed only
once, as the locations of sample points remain unchanged. The intra-group
encoding process is as shown in Algorithm 1.

Algorithm 1 Intra-Group Encoding

1:for each $u_{si},v_{si},{V}(u_{si},v_{si})$ in set of sampled points do

2: $\text{pe}_{i}\leftarrow\text{PE}(u_{si},v_{si})$

3: $\text{le}_{i}\leftarrow\text{LinearEmbedding}({V}(u_{si},v_{si}))$

4: $v_{i}^{\prime}\leftarrow\text{concat}(\text{pe}_{i},\text{le}_{i})$

5: $\theta_{i}\leftarrow\text{arctan2}(u_{si},v_{si})$

6:end for

7:$V^{\prime}\leftarrow\left[v^{\prime}_{1};v^{\prime}_{2};\ldots;v^{\prime}_{n}\right],\Theta\leftarrow\left[\theta_{1};\theta_{2};\ldots;\theta_{n}\right]$

8:if sorted_indices not precomputed then

9: $\textit{sorted\\_indices}\leftarrow\text{argsort}(V^{\prime},\Theta)$

10:end if

11:$V^{\prime}\leftarrow V^{\prime}[\textit{sorted\\_indices}]$

12:$V^{\text{MLP}}\leftarrow\text{MLP}(V^{\prime\prime})$

13:$\hat{V}\leftarrow\text{AdaptiveAvgPooling}(V^{\text{MLP}},m)$

14:return $\hat{V}$

Finally, the group tokens $\hat{V}$ go through inter-group encoding by a
Transformer encoder. We base our encoder design on Transformer structures
similar to prior work [28, 11, 30]. The input group tokens are then
transformed into latent tokens by multi-headed self-attention layers.

###  3.5 Neural Field Conditioning

Our method follows the conditional neural field pipeline proposed by Wu et al.
[30]. Given the sparsely sampled visibility ${V}(u_{s},v_{s})$, our objective
is to determine a neural field $\Phi$, fulfilling a constraint set by the
function $F$:

$F(\Phi(u_{s},v_{s}),{V}(u_{s},v_{s}))=0.$ (11)

We approximate this implicit function $\Phi(u,v)$ with an MLP of $l$ layers
parameterized by weights $\Theta_{m}$.

We use the output tokens of inter-group encoding
$T=\left[t_{1};t_{2};t_{3};\ldots t_{l}\right]$ to extend the neural field
with a learning-based prior, with each token corresponding to an MLP layer.
Using the FiLM conditioning [20], the output tokens modulate the $i$th layer’s
activation $\mathbf{x_{i}}$ by:

$\operatorname{FiLM}(\mathbf{x_{i}})=\gamma(t_{i})\odot\mathbf{x_{i}}+\beta(t_{i}),i\in
1\text{ to }l,$ (12)

where $\gamma$ and $\beta$ are simple affine layers with non-linearities and
$\odot$ signifies a Hadamard product [15].

The MLP parameters $\Theta_{m}$ and the encoder parameters $\Theta_{e}$ are
jointly optimized during training:

$\begin{split}\min_{\Theta_{m},\Theta_{e}}\text{RVL}\left(\Phi\left(u_{d},v_{d};\left\\{T\right\\};\Theta_{m}\right),V_{\mathrm{gt}}(u_{d},v_{d})\right),\\\
\text{with
}\left\\{T\right\\}=\Psi\left(\left\\{u_{s},v_{s},V(u_{s},v_{s})\right\\};\Theta_{e}\right),\end{split}$
(13)

where $(u_{d},v_{d})$ are the dense coordinates in visibility plane and
${V_{gt}}\left(u_{d},v_{d}\right)$ is the ground truth of visibility
inpainting.

##  4 Experiments

In this section, we conduct a comprehensive evaluation of our method in
comparison with several classic and recent state-of-the-art methods to
demonstrate the overall improvement achieved by our approach. We also design
experiments to explore the effects of different grouping methods and group
sizes on the reconstruction results. In addition, we conduct ablation
experiments to study the effects of individual weighting techniques in RVL.

###  4.1 Experimental Setup

Platform. We conduct all experiments on a server with two AMD EPYC 7302 CPUs,
128GB main memory, and eight Nvidia RTX 3090 GPUs each with 24GB device
memory. The server is equipped with an NVME 2TB SSD and four 1TB SATA hard
disks. The operating system is Ubuntu 20.04. Our model is implemented in
PyTorch 1.8.1 [19].

Datasets. In the experiments, we evaluate our method on different public
datasets of four kinds of distinct galaxy morphologies [12, 8]: Merging
Galaxies (MG), In-between Round Smooth Galaxies (IRSG), Unbarred Tight Spiral
Galaxies (UTSG), and Edge-on Galaxies with Bulge (EGB). These data are derived
from sources under the DESI Legacy Imaging Surveys [10], integrating
contributions from the Beijing-Arizona Sky Survey (BASS) [34], the DECam
Legacy Survey (DECaLS) [2], and the Mayall z-band Legacy Survey [23]. The
visibility data are generated from these images using the eht-imaging toolkit
[6, 5], denoted by $\left\\{u_{s},v_{s},{V}\left(u_{s},v_{s}\right)\right\\}$.
The parameters for observation are adjusted to mirror an 8-telescope Event
Horizon Telescope (EHT) setup [30], with the EHT being one of the most
prominent arrays leveraging VLBI techniques.

Methods under Comparison. We compare our method with three other methods for
radio interferometry reconstruction, including the classic method CLEAN [13],
which is for dirty image reconstruction, and two latest deep learning-based
approaches for visibility data reconstruction – Radionets [22] and Neural
Interferometry [30]. We use the original code of these methods and follow the
parameter setting in the original code for the best performance. All these
methods are implemented on PyTorch. In addition, we test the magnetic
resonance imaging (MRI) reconstruction U-Net [33, 21] to reconstruct
visibility data as supplementary baselines.

Table 1: Overall performance comparison (mean and standard deviation) on
different datasets. All comparison methods have significant difference with
our method ($p<0.01$).

Models | Img | Vis | MG | IRSG | UTSG | EGB
---|---|---|---|---|---|---
$LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$ | $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$ | $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$ | $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$
Dirty |  |  | $N/A$ |  10.453 (1.256) |  0.680 (0.055) | $N/A$ |  10.875 (1.293) |  0.711 (0.050) | $N/A$ |  10.655 (1.092) |  0.690 (0.047) | $N/A$ |  10.158 (1.216) |  0.674 (0.055)
CLEAN [13] | ✓ |  | $N/A$ |  18.708 (2.336) |  0.767 (0.031) | $N/A$ |  20.974 (2.192) |  0.795 (0.025) | $N/A$ |  17.200 (2.465) |  0.750 (0.035) | $N/A$ |  20.175 (2.411) |  0.779 (0.033)
U-Net [33] |  | ✓ |  1.171 (0.198) |  18.126 (1.906) |  0.818 (0.021) |  1.235 (0.230) |  19.770 (1.925) |  0.828 (0.024) |  1.256 (0.227) |  14.877 (1.701) |  0.786 (0.032) |  1.219 (0.234) |  19.232 (2.078) |  0.822 (0.024)
Radionets [22] |  | ✓ |  1.580 (0.202) |  19.687 (1.897) |  0.836 (0.022) |  1.499 (0.206) |  21.369 (1.887) |  0.854 (0.022) |  1.419 (0.231) |  20.828 (2.003) |  0.844 (0.022) |  1.545 (0.220) |  20.322 (1.809) |  0.836 (0.023)
Neu_Int [30] |  | ✓ |  1.229 (0.282) |  20.560 (2.287) |  0.875 (0.028) |  0.916 (0.302) |  24.099 (2.760) |  0.898 (0.027) |  1.038 (0.313) |  21.323 (2.303) |  0.880 (0.027) |  1.146 (0.343) |  21.276 (2.675) |  0.879 (0.031)
PolarRec(Ours) |  | ✓ | 0.924 (0.284) | 24.206 (2.547) | 0.889 (0.029) | 0.666 (0.238) | 26.540 (2.653) | 0.911 (0.025) | 0.707 (0.241) | 25.430 (2.565) | 0.905 (0.025) | 0.707 (0.241) | 25.880 (2.529) | 0.906 (0.025)

Evaluation Metrics. To measure differences in frequency data, we use the Log
Frequency Distance (LFD) [16], which is defined as follows:

$\displaystyle\text{LFD}=\log\left[\frac{1}{MN}\left(\sum_{u=0}^{M-1}\sum_{v=0}^{N-1}\left|V_{r}(u,v)-V_{p}(u,v)\right|^{2}\right)+1\right]$
(14)

where $V_{r}(u,v)$ and $V_{p}(u,v)$ represent the real and predicted
visibility respectively. A lower LFD is better.

To evaluate the quality of images produced from the reconstructed visibility,
we employ two common metrics: Peak Signal-to-Noise Ratio (PSNR) and Structural
Similarity Index Measure (SSIM). PSNR quantifies the overall image quality of
the resultant images, and SSIM quantifies the perceptual similarity to the
ground truth images. We compute these two metrics using the scikit-image
package [24], which follows the formulas presented by Hore et al. [14]. A
higher PSNR and SSIM is better.

To evaluate the efficiency, we use inference time and Floating Point
Operations (FLOPs). The inference time in the experiments is tested on a
single Nvidia RTX 3090. We compare time performance between our method and
Neural Interferometry only, because only these two are transformer-encoder
based.

Figure 6: Visual comparison of visibility and image. The reconstructed images,
visibility data and their corresponding error maps are provided. The small
panels to the left of each image are the corresponding visibility data, with
the real part on top and the imaginary part at the bottom.

###  4.2 Overall Comparison

We calculate the LFD, PSNR, and SSIM values for all test images reconstructed
using our method and other methods, presenting both mean values and standard
deviations. These results are listed in Table 1. The results show that
PolarRec consistently outperforms the other methods in all three measures and
four datasets, underscoring the effectiveness of our reconstruction method.

We also illustrate some representative reconstructed visibility and the
corresponding images on the four datasets in Figure 6, including all deep-
learning-based methods under comparison as well as the dirty images and the
ground truth images of the real sky. Comparing the dirty images in Figure 6
(a) and the ground truth in Figure 6 (f), we find there are many artifacts and
distortion of object structure in dirty images because of the sparsity of the
visibility.

As shown in Figure 6 (b), reconstructions of the visibility by MRI
reconstruction U-Net [33] are dominated by the artifacts in resultant images.
In Figure 6 (c), Radionets [22] are able to reconstruct more visibility
content than the U-Net. However, the reconstruction is discontinuous. Although
Radionets reduce artifacts in the imaging results, it cannot distinguish
between separate sources that are in close proximity. Furthermore, many faint
astronomical sources are missing in the reconstructed images. In contrast, as
shown in Figure 6 (d), Neural Interferometry [30] (denoted Neu-Int) can
continuously and realistically reconstruct the low-frequency components of the
visibility, but misses much information from the high-frequency components,
leading to a loss of details in the reconstruction.

The results of PolarRec (Figure 6 (e)) show that our method can effectively
reconstruct more complete and continuous visibility data than others. The
resultant imaging results not only eliminate artifacts but also restore the
true structure of astronomical sources while preserving details and small
faint sources.

###  4.3 Computational Cost Comparison

To demonstrate the effectiveness of the group-granularity encoding in
PolarRec, we conduct a comparative analysis against the Neural Interferometry
[30] method with point-granularity encoding due to both methods are built upon
the Transformer architecture. We fix the number of visibility points as 1660
and the size of the reconstructed visibility map as 128 $\times$ 128, and
compare the computational efficiency of the encoders with batch size from 4 to
32. The comparison focuses on two main metrics: the number of Floating Point
Operations (GFLOPs) and inference time (Latency (ms)).

Table 2 shows the results of this comparison. The latency of group granularity
encoding in PolarRec is significantly lower than point granularity encoding in
Neural Interferometry [30]. Moreover, as the batch size increases, the rise in
latency for PolarRec is much smaller compared to Neural Interferometry [30].
These results indicate that PolarRec has greater advantages in handling large
batches of data, which has important implications for the processing and
analysis of massive astronomical observational data.

Table 2: Comparative Analysis of Computational Efficiency Method | BatchSize | Latency (ms) | GFLOPs
---|---|---|---
| 4 |  29.53 (3.33) | 44.02
Neural | 8 |  59.02 (3.19) | 88.04
Interferometry | 16 |  115.62 (7.30) | 176.08
| 32 |  232.04 (5.64) | 352.16
| 4 |  3.72 (1.84) | 3.05
PolarRec | 8 |  3.89 (3.23) | 6.10
Group Size = 32 | 16 |  4.20 (2.00) | 12.20
| 32 |  5.37 (3.05) | 24.40
| 4 |  3.47 (0.10) | 2.39
PolarRec | 8 |  3.90 (2.94) | 4.78
Group Size = 64 | 16 |  4.02 (2.11) | 9.57
| 32 |  5.11 (1.91) | 19.14

###  4.4 Effect of Group Size

In this experiment, we vary the group size in number of visibility points and
investigate its impact on the reconstruction results. We also record the
inference latency of the encoding process, and use Floating Point Operations
(FLOPs) to measure the computation cost. LFD, PSNR and SSIM are used to assess
the reconstruction quality.

As illustrated in Figure 7, there is a sharp drop in both FLOPs and inference
latency as the group size increases from 1 to 16. In contrast, the image
quality in SSIM and PSNR is almost constant under most settings. Only the PSNR
value of MG dataset decreases slightly. Between a group size of 32 to 128,
there is a slight decrease in PSNR and SSIM, implying that increasing group
size beyond 32 might compromise the output quality. Moreover, when the group
size is set to 1, it is the same as encoding at the granularity of individual
points. The results indicate that encoding with group granularity as input to
the transformer encoder is significantly more efficient than encoding at the
point granularity even when the group size is small.

Figure 7: Effects of group size. The experiment compares PSNR and SSIM metrics
(lines) against GFLOPs and latency time (bars) across different group sizes
for four datasets.

###  4.5 Ablation Experiment

This ablation experiment aims to examine the significance of two weight
matrices $w_{1},w_{2}$ in the Radial Visibility Loss by omitting them one at a
time. The results in Table 3 and Figure 8 show that the full Radial Visibility
Loss is the best. Removing either component $w_{1}$, or $w_{2}$ results in
reduced performance across all metrics.

Table 3: Ablation study performance comparison (mean and standard deviation) on different datasets. | MG | IRSG
---|---|---
| $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$ | $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$
w/o $w_{2}$ | 1.088 (0.309) | 22.498 (2.578) | 0.884 (0.030) | 0.876 (0.305) | 24.435 (2.577) | 0.904 (0.026)
w/o $w_{1}$ | 1.075 (0.317) | 22.706 (2.797) | 0.882 (0.032) | 0.818 (0.283) | 25.213 (2.641) | 0.907 (0.027)
RVL | 0.924 (0.284) | 24.206 (2.547) | 0.889 (0.029) | 0.666 (0.238) | 26.540 (2.653) | 0.911 (0.025)
| UTSG | EGB
| $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$ | $LFD\downarrow$ | $PSNR\uparrow$ | $SSIM\uparrow$
w/o $w_{2}$ | 0.908 (0.299) | 23.477 (2.574) | 0.897 (0.027) | 0.933 (0.310) | 23.793 (2.507) | 0.897 (0.026)
w/o $w_{1}$ | 0.854 (0.281) | 24.393 (2.627) | 0.901 (0.028) | 0.864 (0.287) | 24.569 (2.500) | 0.901 (0.026)
RVL | 0.707 (0.241) | 25.430 (2.565) | 0.905 (0.025) | 0.707 (0.241) | 25.880 (2.529) | 0.906 (0.025)

Figure 8: Visual examples of ablation experiments.

Figure 9: Comparison of grouping method over various group sizes and different
datasets.

###  4.6 Effect of grouping method

We also vary the grouping method and measure its performance impact on
PolarRec. We implemented three grouping strategies: (1) grouping by clustering
visibility points based on their positions, (2) grouping by the radial
coordinate, and (3) grouping by the angular coordinate. Each of these
strategies was tested under various group sizes on different datasets. The LFD
and PSNR results are presented in Figure 9. Regardless of the group size,
grouping the visibility points by angular coordinates always has the best
performance.

##  5 Conclusion and Future Work

We have presented PolarRec, reconstructing interferometric visibility with
sample points represented in the polar coordinate system. By adopting angular
and radial coordinates of visibility points, our method addresses two critical
challenges in visibility reconstruction: reconstruction quality and
computational efficiency. Our results show that PolarRec markedly improves
imaging outcomes while significantly reducing the computation cost,
demonstrating a new, feasible, and extendable approach for cross-disciplinary
progress in computer vision and radio astronomy.

While PolarRec is designed for radio astronomical data reconstruction, its
underlying principles may find utility in other domains requiring
sophisticated frequency component analysis and reconstruction, such as
magnetic resonance imaging or seismic imaging. Our proposed structural sparse
point encoding method may also be used in point cloud data analysis. Future
work could explore these potential applications and generalize the work to
other research fields.

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# Lindbladian-Induced Alignment in Quantum Measurements

R. Englman and A. Yahalom
Ariel University, Ariel 40700,Israel

###### Abstract

An expression of the Lindbladian form is proposed that ensures an unambiguous
time-continuous reduction of the initial system-pointer wave-packet to one in
which the readings and the observable’s values are aligned, formalized as the
transition from an outer product to an inner product of the system’s and
apparatus’ density matrices. The jump operators are in the basis of the
observables, with uniquely determined parameters derived from the measurement
set-up (thereby differing from S. Weinberg’s Lindbladian resolution of wave-
packet formalism) and conforming to Born’s probability rules. The novelty lies
in formalising the adaptability of the surroundings (including the measuring
device) to the mode of observation. Accordingly, the transition is of finite
duration (in contrast to its instantaneousness in the von Neumann’s
formulation). This duration is estimated for a simple half-spin-like model.

Keywords: Quantum measurement theory, Density matrix evolution, Quantum state
resolution, Lindblad operators, Quantum speed limit.

##  1 Introduction

In the century-run of quantum physics (plus 4 years, if one marks its
beginning with the award of a Nobel Prize in 1918 to Max Planck for ”his
discovery of quanta”) a single shadow of non-sequitur has darkened its
glorious achievements, one that goes variously under the names of wave-
function collapse, reduction of the wave-packet, quantum measurement,
einselection, etc. Aspects of the problem (or its articulations) were
manifold, such as the breakdown of the predicted time-development in
accordance with the Schrödinger equation, the abruptness of change in a
measurement (”natura non facit saltum”, where art thou?), the apparent non-
applicability of quantum rules to macroscopic systems, imputed arbitrariness
of Born’s probability rules, the requirement of ”infinite regress” for the
measuring apparatus and others. Numerous papers enlarged on these issues [2,
3] and various proposals for resolution of the problem were put forward. These
include the observer’s cognition [4], stochastic effects [5], in particular
spontaneous localization [3, 6, 7], a many world scenario [8], non-linearity
addition to the Schrödinger equation [9], Poincaré recurrent state [10],
gravitationally induced collapse [11, 12, 13], etc.

Common to these works, and with the specific purpose of providing a blue-print
for measurements compatible with the Copenhagen formulation of quantum theory,
was the need to give expression to the coupling of the microscopic system with
its macroscopic environment. Standing apart from these and belonging to the
field of non-equilibrium thermodynamics and to the establishment of
equilibrium, a general form for this interaction was given by Lindblad [14]
and by Gorin , Kossakowski and Sudarshan [15], satisfying some necessary
conditions. Constructing a merger between the two separately oriented fields,
S. Weinberg recently proposed a Lindblad-operator mechanism for the collapse
of the density matrix (DM) in the course of a complete measurement [16].
Notably, the mechanism was linear in the state’s DM. The collapsed state (Eq.
(1) in [16]) comprises the set of projection operators of the measurable item;
the system’s Hamiltonian is described by a spectral decomposition onto the
same operators (Eq. (16) in [16]) (although in the verbal discussion a more
general situation is considered): collapse is achieved ”independent[ly] of the
details of these [Lindblad] operators”. Decay between energy eigenstates had
earlier been treated by the Lindblad formalism (for a pedagogical presentation
the volume [17], Chapter 8 may be consulted) employing the interaction
representation. However, this is not convenient for treating measurements of
observables that do not commute with the Hamiltonian. Detailed theories relate
to the outcome (”mapping”) of quantum operations, including measurements; the
present work describes the process of these happening.(For a pedagogical
introduction to stochasticity-induced wave-packet- reduction, obviating
pointer reading, one may refer to [18].)

##  2 Overview of the Method and Terms

###  2.1 The leading idea, also in review

While the concept of unity of observer and observation had already featured in
Bohr’s view: ”The answer that we get is built up from the combined interaction
of [the observer’s] state and the object of interrogation.” [19], this was not
given a formal expression in the Copenhagen interpretation. It was more
emphatically asserted both by J. Bell: ”I meant that the ’apparatus’ should
not be severed from the rest of the world in boxes …[20]” and A. Peres: ” A
measurement both creates and records a property of the system [21]”. This
change in the course of a measurement affects also the environment outside the
observed system ; in the words of A. Leggett ”…under these conditions the
macroscopic apparatus, and more generally any part of the macro-world which
has suffered changes in the course of the measurement process, does not end up
in a state with definite macroscopic properties at all,… [2]”.

The same line of thought appears to motivate S. Weinberg, who wrote in his
preamble to a 2016 Lindbladian formulation of the masurement process[16], that
”We will instead [of the original formulation of the Copenhagen
interpretation, (which we will not dwell on here)] adopt the popular modern
view that the Copenhagen interpretation refers to open systems in which the
transition is driven by the ineraction of the microscopic system under study
(which may include an observer) chosen  to bring the transition about.” (Our
italics.)

These developments indicate the justification for a formulation in which the
effect of the apparatus is incorporated in the equation defining the evolution
of the system, rather than one in which the two entities are separate, barring
an interaction between them.

####  2.1.1 ”Alignment”

The process whereby the pointer readings become in correspondence with the
possible values of the observable. Formally, for $I$ possible values, the
combined density matrix reduces from comprising $I^{2}$ terms to one having
$I$ terms. (E.g., equation (2.5) in [2].)

####  2.1.2 ”Dissipator”

Added term (in the form of sums of appropriately weighted jump-operators) to
the standard time dependent Schrödinger equation, inducing non-unitary
evolution in the system, accompanied by changes of its information entropy.

###  2.2 Motivation for the choice of formalism

Thermalization of open systems can be described by a Lindbladian formalism in
which Gibbsian probabilities are so inserted as parameters, that the
”Dissipator” vanishes at these values of the density matrix. Replacement of
the Gibbsian probabilities by Born probabilities achieves alignment in a state
reduction and does so continuously.

Limitations: Born’s probability rules are assumed, not derived; the
interaction term is not traced to a microscopic mechanism.

The source of this interaction term, shown in Eqn. 6 below, incorporating the
coupling between the observed system and its surroundings (including the
measuring device) is an open question (also raised by a referee). In its
application to a thermalization process, the Lindbladian jump operators have
been derived, though with the aids of several approximations (e.g.,[22]), as
well as, more recently, for the dissipation in a Dicke system with a bosonic
background [23]. We are not in the position to provide such first principle
derivation for the Lindbladian jump-operators bringing about a transition and
incorporating the Born rules. It seems to be specific to the type of
measurement under consideration and it is clear that just any jump operator,
as in Weinberg’s Lindbladian formulation will not do the job . Likely, one
would need to include non-Markovian dynamics, so that the coupling to the
device and eventual pointer reading are two separate consecutive events.
Inclusion of such dynamics is outside the scope of the present work.

##  3 Assumptions

We explore the time (t)-development of the combined density matrix $\rho(t)$
of the measured system (S) and of the reading (pointer, dial, etc.) on the
measuring apparatus (A) for a complete and discrete measurement , expressing
the underlying assumptions by three propositions.

Proposition 1. In accord with the long-time historical approach, the measured
object S and the pointer of the measuring set-up A are treated on equal
footings as subject to microscopic quantum laws, and formally describable by
their respective Hamiltonians. Aware of the difficulties connected with an
”infinite regress”, the effects of the rest of the Universe on S+A are not
included in the formalism; instead, for a phenomenological, approximative
description, a Lindbladian term appears in the master equation.

Proposition 2. Prior to the measurement with A and S decoupled, and being free
of external influence for a long time, both are in energy quantum states, pure
or mixed. After the measurement, the state is not an energy eigenstate and
subsequently it will spread over to a superposition of energy eigenstates. The
fast decoherence case treated below in section 5 is akin to the Zeno effect
[24].

Proposition 3. Only those states of the reading apparatus (e.g., the right or
left positions of a pointer) that may be in direct correspondence with the
measured states of the system (e.g., spin up or down) are given expression in
the formalism. (At a beginning, the case treated is one in which there is a
one-to-one correspondence between the states of the system and the readings of
the apparatus; a generalization is given subsequently.) A discussion in
section 8 touches on the epistemological status of the Lindbladian terms in a
measurement process.

##  4 Analysis

Considering (for simplicity) a pure state for the system, its initial state-
vector written in the basis of the observed property $|S,i>$ takes the form

$\psi^{S}(t=0)=\sum_{i=1,..,I}c^{S}_{i}|S,i>$ (1)

Born’s rule for the probability of observing the $i$-component is
$|c^{S}_{i}]^{2}\equiv p_{i}$, summing to unity. Likewise, for the apparatus
readings $j$, numbering $J$, one has the superposition with (complex and
normalized) coefficients $c_{j}^{A}$

$\psi^{A}(t=0)=\sum_{j=1,..,J}c_{j}^{A}|A,j>$ (2)

We start with the one-to-one correspondence situation, for which $I=J$, and
the reading $j$ on A establishes uniquely the value $i=j$ for the system’s
measured property.

For the combined state-vector the density operator has the outer-product form
(where the stars denote complex conjugates):

$\sum_{i,j,i^{\prime},j^{\prime}}|S,i>|A,j>c^{S*}_{i}c^{A*}_{j}c^{A}_{j^{\prime}}c^{S}_{i^{\prime}}<A,j^{\prime}|<S,i^{\prime}|\equiv\sum_{i,j,i^{\prime},j^{\prime}}|i,j>C_{iji^{\prime}j^{\prime}}<i^{\prime},j^{\prime}|$
(3)

the right hand side written in an obvious shortened notation, in which
$C_{iji^{\prime}j^{\prime}}=c^{S*}_{i}c^{A*}_{j}c^{A}_{j^{\prime}}c^{S}_{i^{\prime}}$.
After collapse, the density operator takes the aligned, single-sum form

$\sum_{i}|S,i>|A,i>|c^{S}_{i}|^{2}<S,i|<A,i|$ (4)

It will be now shown that this is the time-asymptotic solution of the
Lindbladian master equation properly parametrized.

We recall Lindblad’s equation for the time varying density of states operator
$\rho\equiv\rho(t)$, as being of the following general form:

$\frac{\partial\rho}{\partial
t}=-\frac{i}{\hbar}[H,\rho]+\sum_{n}\gamma_{n}\langle L_{n}\rho
L_{n}^{\dagger}-\frac{1}{2}(L_{n}^{\dagger}L_{n}\rho+\rho
L_{n}^{\dagger}L_{n})\rangle$ (5)

The second term, here named the ”Lindblad term” [14, 15] though in different
contexts also referred to as the Dissipator [25], contains $L_{n}$’s that are
Lindblad jump-operators. We shall consistently work in the observable $+$
pointer’s basis (i.e., not in an energy basis). In this basis, neither the
density operator $\rho=\rho(t)$, nor the A+S Hamiltonian $H$ is diagonal at
the beginning or in the course of the development. But, as will be
demonstrated, the Lindbladian formalism, by a proper choice of its form,
drives A+S to the desired diagonal form for the combined observable $+$pointer
basis. We postulate just one single term in the previous n-sum, as well as
off-diagonal forms, namely $|i,j><i^{\prime},j^{\prime}|,~{}(i,j\neq
i^{\prime},j^{\prime})$, for the jump-operators in the observable basis,
leading to the following parametrized form of the Lindblad term

$\displaystyle{\cal L}\rho$ $\displaystyle\equiv$
$\displaystyle\Gamma\Omega\sum_{i^{\prime},j^{\prime}\neq
i,j}\frac{r(i,j)}{r(i^{\prime},j^{\prime})}\langle|i,j><i^{\prime},j^{\prime}|\rho|i^{\prime},j^{\prime}><i,j|$
(6) $\displaystyle-$
$\displaystyle\frac{1}{2}(|i^{\prime},j^{\prime}><i,j|i,j><i^{\prime},j^{\prime}|\rho+\rho|i^{\prime},j^{\prime}><i,j|i,j><i^{\prime},j^{\prime}|\rangle$

Here a circular frequency $\Omega$ is inserted, so as to make $\Gamma$ , that
quantifies the strength of the system-environment coupling, dimensionless. One
notes that in the pre-factor appear the parameters
$r(i,j),r(i^{\prime},j^{\prime})(i,j,i^{\prime},j^{\prime}=1,...,I)$ whose
significance will be clear by deriving the matrix elements of the above
operator. These are

$\displaystyle{\cal L}\rho_{i,j,i^{\prime},j^{\prime}}$ $\displaystyle=$
$\displaystyle\delta_{i,i^{\prime}}\delta_{j,j^{\prime}}r(i,j)\sum_{k,l}r^{-1}(k,l)\rho_{k,l,k,l}$
(7) $\displaystyle-$
$\displaystyle\frac{1}{2}[r^{-1}(i,j)+r^{-1}(i^{\prime},j^{\prime})]\rho_{i,j,i^{\prime},j^{\prime}}\sum_{k,l}r(k,l)$

It can be seen that the trace of the above vanishes and that each matrix
element vanishes upon the substitution

$\rho_{i,j,i^{\prime},j^{\prime}}=\delta_{i,i^{\prime}}\delta_{j,j^{\prime}}r^{2}(i,j)$
(8)

While these properties hold for any arbitrary $r(ij)$, the observable-pointer
alignment is achieved by identifying the $r$ parameters with the system’s
superposition coefficient: $r(ij)=|c^{S}_{i}|\delta_{i,j}$, or

$r(i,j)^{2}=|c^{S}_{i}|^{2}\equiv p_{i}\delta_{i,j}$ (9)

the last being the Born probabilities appearing in the collapsed state. As
already noted, this identification of probabilities relates to the well known
procedure for the Lindblad-induced thermalization of open systems, for which
detailed balance imposes the relation between the pre-factors $\gamma(\delta
E)$/$\gamma(-\delta E)=e^{-\beta\delta E}/Z$, the latter being the canonical
probabilities (with $\beta=1/k_{B}T$, $k_{B}$ the Boltzmann constant, $T$ the
ambient temperature and $Z$ the partition function [25, 26, 22]) .

[It also seems fair to point out that also in the standard (Copenhagen, or von
Neumannian) description of the alignment stage, as appears in e.g. Eq.(2.5) of
[2], this development is summarily stated, without specification of the
underlying mechanism.]

##  5 Fast Decoherence Limit

We now consider the case that the time development in the state is
predominantly due to the coupling to the environment, rather than to the
unitary change induced by the Hamiltonian, meaning that the second term on the
right hand side in Eq. 5 dominates the first. Quantitatively:
$\Gamma>>||H||/\hbar\Omega$. Neglecting the commutator we now form matrix
elements of the Lindblad term in Eq. 5 in the observable+pointer basis.
Because of the approximation made, the off-diagonal matrix elements are
decoupled from the diagonal ones. The master equation of the off-diagonal
terms reads (with a notation simplified by writing for the index pairs
$i,j\rightarrow r$, $i^{\prime},j^{\prime}\rightarrow s$ and consequently for
$\rho_{i,j,i^{\prime},j^{\prime}}\rightarrow\rho_{rs}\equiv\rho_{rs}(t)$

$\frac{d\rho_{rs}}{dt}=-\Gamma\Omega\left[\frac{\sqrt{p_{r}}+\sqrt{p_{s}}}{2}\rho_{rs}\sum_{m}\sqrt{p_{m}}\right],~{}r\neq
s$ (10)

This shows that off-diagonal matrix elements decay exponentially in time
(decohere), maintaining their real character that they had initially. Had we
kept the (imaginary) commutator term, we would have found that the decay is
modulated by the eigen-energies of the Hamiltonian.

For the diagonal matrix elements we find,

$\frac{d\rho_{rr}}{dt}=\Gamma\Omega\left[\sqrt{p_{r}}\sum_{m}\frac{\rho_{mm}}{\sqrt{p_{m}}}-\frac{\rho_{rr}}{\sqrt{p_{r}}}\sum_{m}\sqrt{p_{m}}\right]$
(11)

Again, it can be seen that the trace of the last expression vanishes, and so
does the right-hand side under the substitution $\rho_{rr}\rightarrow p_{r}$.
With these taking the values as in Eq. 9, one arrives at the aligned form
(written out in the original, system-pointer indexes)

$\rho(t\rightarrow\infty)=\sum_{i}|\psi^{A}_{i}>|\psi^{S}_{i}>|c^{S}_{i}|^{2}<\psi^{S}_{i}|<\psi^{A}_{i}|$
(12)

###  5.1 Illustrative example for a two-way experiment

Exemplifying the foregoing for a two-valued system (such as a
$\frac{1}{2}$-spin electron), prepared as an eigenstate of a Zeeman-field with
the magnetic field inclined at an angle $2\alpha^{S}$ to the vertical, in
conjunction with an apparatus pointer, represented as being likewise in an
eigenstate of a quasi-Zeeman field inclined at an angle $2\alpha^{A}$ to the
vertical. The eigenstates are linear superpositions of their $z$\- spins;
these are the observables that are to be determined by the measurement.
Initially, the system and the pointer are in the superposition states as shown
above in Eqs. 1 and 2 and whose superposition coefficients $c^{S}_{i}$ and
$c^{A}_{j}$ now have the values, $\sin/\cos(\alpha^{S})$ and
$\sin/\cos(\alpha^{A})$, respectively. The DM in the observable basis is now a
$4$x$4$ matrix, in which appear all the combinations of the products of the
above circular functions. As the outcome of the application of the Lindblad
operator in the rate equation, at long times the matrix becomes reduced to the
diagonal form discussed earlier. In these, $\cos^{2}(\alpha^{S})=p_{1}$ and
$\sin^{2}(\alpha^{S})=p_{4}$ belonging to the aligned observable lie on the
diagonal and are non-zero; the other two diagonal entries for the anti-aligned
situations are zero.

Plotted in Figure 1 are computed DM eigenvalues as functions of time (in red
and blue), normalized to their respective Born probabilities, showing their
asymptotic convergence. In green, the typical decohering tendency of an off-
diagonal element is demonstrated. Figure 2 depicts the entropy
$S(t)=-\sum_{r}P_{r}(t)\log P_{r}(t)$ of the system and apparatus-pointer, (in
which $P_{r}(t)$ are computed eigenvalues of the DM.) The non-monotonic
behavior is characteristic of of the Lindblad formalism, in which the
environment’s entropy change is not taken into account.

[In numerical work, based on forward integration, putting zeros for some of
the $p_{i}$’s introduces singularities, eventually algebraically cancelling
out, but preventing flow of computation. Therefore, instead, one puts
arbitrarily small values for these and obtains for the aligned DM one that is
arbitrarily close to, but not exactly equal to the true one.]

Figure 1: Density matrix eigenvalues normalized to their asymptotic (pointer-
aligned) values for the two aligned terms in the illustrative example (in red
and blue), plotted against time in inverse circular frequency unit. In green
is shown a decohering off-diagonal matrix element. Lindblad coupling strength
$\Gamma=5$, $\alpha$ angles $.37~{}\pi$ and $.65~{}\pi$.

##  6 Eigenvalue analysis

An alternative to the numerical solution of the differential rate equation is
eigenvalue analysis, already treated in [16], based on the Landbladian term
being a linear function of the diagonals in the density matrix. Thereby, the
resulting rate equations have solution of the form

$\rho_{nn}(t)=\sum_{k}v_{n,k}e^{\lambda_{k}t}$ (13)

in which $\lambda_{k}$ and $v_{n,k}$ are the diagonalized eigenvalues and
eigenvectors of the Lindbladian matrix diagonals in Eq. 11. Calculation shows
that for the $4$ x $4$ matrix considered above there are three negative
eigenvalues and one zero eigenvalue, which alone is of interest at the long
term behavior. Belonging to this eigenvalue, the (transposed) eigenvector is
found to be
$\\{p_{1},p_{2},p_{3},p_{4}\\}\approx\\{c^{S}_{1},0,0,c^{S}_{4}\\}$, as
required for the alignment between the quantum states and the reading in the
measuring apparatus.

Figure 2: Entropy of the combined system plus apparatus. Noteworthy is the
initial peak common to the Lindblad formalism .

###  6.1 Measurement speed

Figure 1 shows that alignment is achieved for the model with the chosen
strength parameter ($\Gamma=5$) by a time of cca. $0.1/\Omega$. By varying the
strength in the computed model, we find a shortening of this time that is
inversely proportional to the strength. This is expected from the quantum
speed limit (QSL) results that border quantum transition times $\tau$ from
below.

Essentially, QSL is the ratio of two norms [27, 28], that of the ”quantum
distance” [29] and of the speed of the state evolution. Formally

$\tau>\frac{||\rho(t\rightarrow\infty)-\rho(t=0)||}{||\frac{d\rho(t)}{dt}||}=\frac{||\rho(t\rightarrow\infty)-\rho(t=0)||}{||[L\rho(t)]||}$
(14)

Ways of calculating the norms vary, e.g., [30, 31]. Recently, for a system
developing due to a Lindbalian operator, three contributions to the speed were
discerned [25]. To estimate $||\rho(t\rightarrow\infty)-\rho(t=0)|$, we have
used the ”Trace Distance ”defined as

$T(\rho,\sigma)=\frac{1}{2}Tr[\sqrt{(}\rho-\sigma)]=\frac{1}{2}\sum_{i}|\mu_{i}|$
(15)

[32], where $\mu_{i}$ are the eigenvalues of the matrix differences. The DM
velocity, as defined above , changes (decreases) with time, ultimately
vanishing at the fulfilment of alignment; we have taken the root-mean-square
sum of the rate of the diagonal matrix elements at initial times. These yield
a very low limit of

$\tau>\frac{1}{2}\frac{1.41}{5*69.03}=.0045/\Omega$ (16)

to be compared with the actually computed value, about 20 times longer. Better
(higher) limits of transition times may be generated by different ways of
forming the norm for the DM velocity (e.g. not at the beginning).

###  6.2 Multiple Reading-System correspondence

A simple generalization of the foregoing applies when each (eigen-)value of
the observable is in correspondence with not just one reading of the pointer,
but with several (say, R) readings, all of the same significance for the
outcome. Then one simply inserts $p_{i}/R$ into the corresponding Lindblad
term, in place of just $p_{i}$. In the more complex case, that not all
readings have the same likelihood, $p_{i}$ would have to be weighted by a
probability factot, rather than by a constant denominator.

##  7 The Lindbladian, ”Who ordered this?”

Historically, Lindblad terms were introduced as the most general forms that
maintain complete positivity of the DM’s and preserve their trace [14, 15].
The various derivations that have been presented (and among these a recent one
by [33]), involve several approximations for the coupling between the system
and its environment. Insomuch that the derivation involves also tracing over
the degrees of freedom of the environment, much detail of the latter is lost
and of course it is impossible to work backwards from the Lindbladian to the
environment. What is remarkable is that for special purposes the appropriate
Lindbladian operators take a very special, practically unique form. Such is
the case for the accepted description of thermalization [22, 25] by a Lindblad
formalism. The parametrization of the Lindblad term employed in the present
work, though it may appear arbitrary and particular for each case, is in fact
identical to the one used for thermalization subject to the relabelling of the
Gibbsian thermal distribution function as (the Born) probabilities
($p_{1},p_{2},...$),with the proviso of working in the observable, rather than
in the energy basis. (This contrasts with the different approach in [16],
which claims attainment of collapse for any Lindbladian operator.) At the same
time, it needs to be noted that the analog of detailed balance is missing in
wave-function collapse. How come to have such a specific Lindbladian, whose
source may be any measurement device and procedure? One is left to wonder
about the possibility of a special meta-physical status of the Lindblad terms,
or query with Wheeler ”Who ordered this?”

##  8 Conclusion

The well-known Lindbladian extension to the quantum theory of motion to
environmental effects is here adapted to establish the resolution of a wave-
packet in a measurement as a smooth process. This is enabled by an unambiguous
parametrization of the jump-operators describing the interaction of the broad
environment with the observed system, both regarded as quantal entities.

Above, in section 2.1, a brief historically oriented preview has been provided
for the distinct approach in this work, namely, one based on the wholeness of
the entities (observed system and observing device), through the (Lindbladian)
equation yielding the evolution of the system.

A main result emerging from the formalism, and capable of experimental
verification, is the finitely temporal variation of the system, and this in a
deterministic way rather than just statistically, on the average, contrasting
also with the instantaneous collapse description by (e.g.) von Neumann. Such
temporal variation in continuous-thermalization processes has been proposed
quite recently [34, 35], also by employment of a Lindbladian formalism and
within a Markovian framework.

Experimentally, verification of the time dependence of the transition in any
particular measurement, implicit in our formulae, could be observed by
repeated observation performed on the system subject to non-demolition
transitions. These observations would be akin to the Zeno-effect measurement,
such as has been achieved in the form of quasi-periodic oscillation of the
result for a Superconducting flux cubit[36]. Further work is needed for
quantifying the information-entropy change in the environment [32, 37].
Acknowledgement

The authors thank the referee for meticulous reading and insightful
questioning of a previous version of this paper.

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