# A Novel State Space Model with Local Enhancement and State Sharing for Image
Fusion

Zihan Cao  UESTCChengduChina<EMAIL_ADDRESS>,  Xiao Wu11footnotemark: 1
UESTCChengduChina<EMAIL_ADDRESS>,  Liang-Jian Deng  UESTCChengduChina
<EMAIL_ADDRESS>and  Yu Zhong  UESTCChengduChina
<EMAIL_ADDRESS>
(2024)

###### Abstract.

In image fusion tasks, images from different sources possess distinct
characteristics. This has driven the development of numerous methods to
explore better ways of fusing them while preserving their respective
characteristics. Mamba, as a state space model, has emerged in the field of
natural language processing. Recently, many studies have attempted to extend
Mamba to vision tasks. However, due to the nature of images different from
casual language sequences, the limited state capacity of Mamba weakens its
ability to model image information. Additionally, the sequence modeling
ability of Mamba is only capable of spatial information and cannot effectively
capture the rich spectral information in images. Motivated by these
challenges, we customize and improve the vision Mamba network designed for the
image fusion task. Specifically, we propose the local-enhanced vision Mamba
block, dubbed as LEVM. The LEVM block can improve local information perception
of the network and simultaneously learn local and global spatial information.
Furthermore, we propose the state sharing technique to enhance spatial details
and integrate spatial and spectral information. Finally, the overall network
is a multi-scale structure based on vision Mamba, called LE-Mamba. Extensive
experiments show the proposed methods achieve state-of-the-art results on
multispectral pansharpening and multispectral and hyperspectral image fusion
datasets, and demonstrate the effectiveness of the proposed approach. Code
will be made available.

Multispectral pansharpening, multispectral and hyperspectral image fusion,
Mamba, local-enhanced network

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
Make sure to enter the correct conference title from your rights confirmation
email; October 28 - November 1, 2024; Melbourne, Australia.††isbn:
978-1-4503-XXXX-X/18/06††ccs: Computing methodologies Reconstruction

##  1\. Introduction

Figure 1. Memory consumption and FLOPs using different operators on the same
U-Net architecture. Our LE-Mamba has linear memory consumption compared with
quadratic consumption of self-attention (Attention) (Dosovitskiy et al., 2021)
and lower FLOPs compared with the attention of Swin Transformer (Liu et al.,
2021).

In the field of image fusion, there are two flourishing tasks that are applied
in subsequent applications, i.e., multispectral pansharpening, and
multispectral and hyperspectral image fusion. While multispectral
pansharpening aims to fuse high-resolution panchromatic images and low-
resolution multispectral images to yield high-resolution multispectral images
(HRMS), multispectral and hyperspectral image fusion involves high-resolution
RGB images with low-resolution hyperspectral images to yield high-resolution
hyperspectral images.

Many deep-learning methods have been widely proposed for image fusion, such as
convolution neural networks (CNNs) and vision Transformers. The CNN-based
methods can extract and learn local feature information in image fusion. More
importantly, vision Transformers have achieved excellent performance in image
fusion (Vivone, 2023; Deng et al., 2022). However, vision Transformers are
limited by the quadratic spatial complexity of the Attention mechanism, making
it challenging to process high-resolution images. Many works (Han et al.,
2023; Wang et al., 2021; Liu et al., 2021) aim to reduce the complexity of
Transformers, but they often come with performance degradation, increased
parameter count, and poor generalization capability.

Recently, an alternative approach has shown promising results benefiting from
efficiently modeling dependencies of sequence data based on the state space
model (SSM). In recent work, (Gu et al., 2021; Lu et al., 2023) introduces the
structured state space sequence model as a general sequence model, called
Mamba. Afterward, vision Mamba (Zhu et al., 2024; Liu et al., 2024a) further
verifies that it has great potential to be the next-generation backbone for
vision foundation models. Compared with vision Transformers (Dosovitskiy et
al., 2021), vision Mambas adopt sequential visual representation (i.e., image
patches) and have subquadratic-time computation and near-linear memory
complexity. However, Mamba faces two challenges: 1) needs for non-
unidirectional modeling for 2D images in contrast to language sequence. 2) the
issue of information stored in the state being lost when the sequence length
increases, and 3) unable to characterize the spatial and spectral information
of images.

In addressing the first question, two pioneering works stand out: Vim (Zhu et
al., 2024), which introduced a bi-directional vision Mamba block tailored for
image processing, and VMamba (Liu et al., 2024a), which proposed a scanning
mechanism for images to tackle the causality issue inherent in Mamba’s
unsuitability for vision tasks. As for the rest of the issues, there has been
a lack of exploration of Mamba in the image fusion field. In this paper, we
propose two bespoke techniques named local-enhanced vision Mamba (LEVM) block
and state sharing to address issues 2) and 3), as its conception shown in Fig.
2, respectively enhancing Mamba’s perception of local information and fully
exploiting spatial and spectral information in the SSM’s state.

The proposed method is more suitable for image fusion tasks with two
advantages to the proposed methods. The first advantage is that the proposed
LEVM block can represent local and global spatial information. Local
information brings rich spatial details, while global information injects
long-range pixel dependencies, thus enhancing the detail recovery in the
fusion process. The second advantage is that the state sharing method allows
information to be shared between layers in the adjacent flow and skip-
connected flow. Furthermore, we have also proposed the spatial-spectral
learning (S2L) of SSM for the state sharing method. The S2L maps the input
image to the state space for spatial representation, and considers the state
space as the basis for spectral representation, thereby achieving interaction
between spatial and spectral information. To the best of our knowledge, there
have been no improvements made to vision Mamba from the perspective of spatial
and spectral learning specifically for image fusion tasks. Finally, we propose
the LE-Mamba, which is constructed based on the U-Net (Ronneberger et al.,
2015a) baseline with the inclusion of the LEVM block and the state sharing
technique. The proposed LE-Mamba can achieve superior fusion performance.

The contributions can be summarized as follows:

  1. 1)

We propose a local-enhanced vision mamba (LEVM) block for the existing vision
Mamba architecture to address the issue of spatial information representation.

  2. 2)

The state sharing technique is designed for the proposed LEVM block, which
customizes the image fusion tasks. Then, the state sharing can reduce
information loss and enable simultaneous learning of spatial and spectral
information within the state space model (SSM).

  3. 3)

The proposed LE-Mamba achieves state-of-the-art fusion performance for image
fusion on four widely-used multispectral pansharpening and hyperspectral
multispectral fusion datasets.

The rest of the paper will be organized as follows: In Sect. 2, we review the
related work on SSMs, then introduce deep learning-based methods on
multispectral pansharpening, as well as multispectral and hyperspectral image
fusion tasks. In Sect. 3, we revisit the background knowledge of SSM. Then, in
Sect. 4, we elaborate the proposed method with the overall architecture, the
local-enhanced vision Mamba block, and the state sharing technique in Sect.
4.1, Sect. 4.2 and Sect. 4.3, respectively. Afterward, Sect. 4.4 analyzes the
computational complexity of various common neural network operators. At last,
we conduct extensive experiments and ablation studies on the proposed
architecture to validate its effectiveness in Sect. 5.

##  2\. Related Work

###  2.1. State Space Models

State space models (SSMs) were initially proposed for sequence-to-sequence
transformation tasks in natural language processing (Gu et al., 2021; Lu et
al., 2023; Gu and Dao, 2023), adept at handling long-range dependencies. In
recent years, researchers have strived to address the computational and memory
bottlenecks of SSMs in practice, enabling higher efficiency and simplicity
when processing long sequences, such as structured state space (S4) (Gu et
al., 2021) introduces parameterization for state space, while Mamba (Gu and
Dao, 2023) incorporates high-efficient selection mechanism into the S4 based
on hardware optimization.

In vision tasks, S4ND method (Nguyen et al., 2022) was the first work to
introduce SSMs into the vision field, demonstrating their potential to achieve
performance on par with models like vision Transformer (Dosovitskiy et al.,
2021). VMamba (Liu et al., 2024a) and Vim (Zhu et al., 2024) further applied
the ideas of Mamba models to generic vision tasks, proposing mechanisms like
bi-directional scanning to better capture image information. Due to the recent
explosive growth in research on vision Mamba, we systematically summarize the
most recent relevant works in the supplementary for a detailed background.

###  2.2. Deep Learning Methods for Image Fusion

For the multispectral pansharpening task, many CNN-based methods have emerged
and obtained promising fused images, including DiCNN (He et al., 2019), PanNet
(Yang et al., 2017), and FusionNet (Deng et al., 2020). However, the local
feature representation obtained by CNNs hinders better fusion results. For the
multispectral and hyperspectral image fusion, there are also many outstanding
DL-based works including InvFormer (Zhou et al., 2022b) and MiMO-SST (Fang et
al., 2024). However, these methods bear a significant computational burden due
to the quadratic complexity of the Transformer. Despite many efforts to
mitigate this issue (Han et al., 2023; Wang et al., 2021; Liu et al., 2021),
these methods usually lead to a performance drop.

Recently, SSMs have emerged with near-linear memory consumption and relatively
low computational overhead, thus rapidly extending to various vision tasks.
Most relative to our work, PanMamba (He et al., 2024) adapted the
bidirectional vision Mamba block (Zhu et al., 2024) to the multispectral
pansharpening task. However, it fails to realize the issue of state
information loss and does not tailor state space representation to exploit the
spatial and spectral domain, leading to suboptimal fusion performance.

##  3\. Preliminary

State space models (SSMs) are a class of sequence models inspired by linear
systems, which aim to map a sequence $x(t)\in\mathbb{R}^{L}$ to
$y(t)\in\mathbb{R}^{L}$ through the hidden space
$h^{\prime}(t),h(t)\in\mathbb{R}^{N}$. A system matrics
$\bm{A}\in\mathbb{R}^{N\times N},\bm{B}\in\mathbb{R}^{N\times 1}$, and
$\bm{C}\in\mathbb{R}^{N\times 1}$ represents the dynamics of the system:

(1) $\displaystyle h^{\prime}(t)$ $\displaystyle=\bm{A}h(t)+\bm{B}x(t),$
$\displaystyle y(t)$ $\displaystyle=\bm{C}h(t).$

In practice, the above continuous system should be discretized using zero-
order hold assumption, converting the matrics $(\bm{A},\bm{B})$ to the
discretized forms by a timescale $\bm{\Delta}\in\mathbb{R}>0$:

(2) $\displaystyle\overline{\bm{A}}=\exp({\bm{\Delta A}})$
$\displaystyle\overline{\bm{B}}=(\bm{\Delta A})^{-1}(e^{\bm{\Delta
A}}-\bm{I})\cdot\bm{\Delta B}.$

The discretized operator $\Delta$ maps $A$ and $B$ from
$\Delta:\mathbb{R}^{N}\rightarrow\mathbb{R}^{L\times N}$ (decretize continous
system), where dimension $N$ is a higher dimensional latent state. This
discretization forms a discretized model written as:

(3) $\displaystyle h_{t}$ $\displaystyle=\overline{\bm{A}}\cdot
h_{t-1}+\overline{\bm{B}}\cdot x_{t},$ $\displaystyle y_{t}$
$\displaystyle=\bm{C}\cdot h_{t}.$

Furthermore, in higher-dimensional state space, we can rewrite the above
variables, that is input data $x:\mathbb{R}^{L\times
D}\rightarrow\mathbb{R}^{L\times D\times N}$ with $D$ channel. A timestep $t$
depends on the size of the sequence $L$ (i.e., $x_{t}\in\mathbb{R}^{D\times
N}$). Then, at each timestep $t$, system matrices
$\overline{\bm{A}}\in\mathbb{R}^{D\times N}$,
$\overline{\bm{B}}\in\mathbb{R}^{D\times N}$ are mapped into $N$-dimension
state space, and $\bm{C}\in\mathbb{R}^{D\times N}$ map $h_{t}$ back the
original space. Thus, a hidden state is $h_{t}\in\mathbb{R}^{D\times N}$ and
the output feature is $y_{t}\in\mathbb{R}^{D}$ (i.e., $y\in\mathbb{R}^{D\times
L}$). According to the implementation of Mamba (Gu and Dao, 2023), for
efficient computation, the previous computation process can be formulated as
the parallel convolution:

(4) $\displaystyle\bm{y}$
$\displaystyle=\bm{x}\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\ast}{\bigcirc}}\overline{\bm{K}}$
$\displaystyle\mathrm{with}\quad\overline{\bm{K}}$
$\displaystyle=(\bm{C}\overline{\bm{B}},\bm{C}\overline{\bm{A}\bm{B}},\cdots,\bm{C}\overline{\bm{A}}^{L-1}\overline{\bm{B}}),$

where $\stackMath\mathbin{\stackinset{c}{0ex}{c}{0ex}{\ast}{\bigcirc}}$
denotes convolution operation, and $\overline{K}$ is the structured
convolutional kernel.

##  4\. Method

Figure 2. Illustration of (a) the main conception of the proposed LEVM block,
(b) the composition of the vision Mamba block, (c) the proposed LEVM block,
and (d) the overall architecture of LE-Mamba. “SS2D” in (b) indicates 2D
selective scan in (Liu et al., 2024a).

In this section, we will first illustrate the overall network design of LE-
Mamba in Sect. 4.1, including the network’s input and output, as well as its
structure. Subsequently, each local-enhanced vision Mamba (LEVM) block within
the network will be delineated in detail in Sect. 4.2. Finally, the
implementation details of the proposed state sharing technique are elucidated
in Sect. 4.3. In Sect. 4.4, we analyze the complexity of the LEVM block.

###  4.1. Overall Architecture

In some high-level tasks (e.g., classification (Liu et al., 2024a; Zhu et al.,
2024) and segmentation (Liu et al., 2024b; Ma et al., 2024; Ruan and Xiang,
2024; Liu et al., 2024a)), architectures similar to Meta-formers (Yu et al.,
2022) are commonly used, where plain backbones can offer lower parameter
counts, smaller computational loads, and lower latency. However, for the image
fusion task, we opted for a multi-scale architecture akin to U-Net
(Ronneberger et al., 2015b), which has been proven effective (Cao et al.,
2024). The designed network is illustrated in Fig. 2(d), featuring an encoder-
decoder architecture. The input comprises LRMS and PAN images, while the
output is fused images. The encoder and decoder are composed of multiple LEVM
layers, with each LEVM layer comprising several LEVM blocks.

For the LEVM block in the encoder, denoted as $f_{enc}(\cdot)$, the input
consists of the SSM hidden state $\bm{h}^{l-1}$ from the previous block, LRMS,
and PAN:

(5)
$\displaystyle\bm{x}^{l},\bm{h}^{l}=f_{enc}(\bm{x}^{l-1},PAN,\bm{h}^{l-1}),$

where $l$ represents the $l$-th layer, and $\bm{x}^{0}$ denotes LRMS in the
first encoder layer (i.e., $l=0$), then outputs features and hidden states
into the next layer, denoted as $\bm{x}^{l}$ and $\bm{h}^{l}$, respectively.
Different from the previous vision Mambas, the state $\bm{h}^{l-1}$ is
incorporated. Its rationale will be explained in Sect. 4.3.

For the LEVM blocks in the decoder, their inputs are the concatenation of the
output from the corresponding encoder at the same resolution and the output
from the previous decoder layer, denoted as $f_{dec}(\cdot)$. Similar to the
LEVM block in the encoder, by incorporating the corresponding encoder’s state
$\bm{h}_{enc}^{l}$, a better fusion performance can be achieved, which is akin
to the encoder-decoder skip connections commonly employed in U-Nets. Our
intuition is that the encoder’s state contains abundant low-level information,
which can complement the semantic information in the decoder’s state space.
This process can be formulated as:

(6)
$\displaystyle\bm{x}^{l},\bm{h}^{l}=f_{dec}(concat(\bm{x}^{l-1},\bm{x}_{enc}^{l}),\bm{h}^{l-1},\bm{h}_{enc}^{l}).$

where $\bm{x}^{l-1}$ and $\bm{h}^{l-1}$ are input features and hidden states
of the $(l-1)$-th layer in the decoder, respectively. Finally, the output of
the last LEVM block is mapped back to the pixel space by a linear layer.
Following the high-frequency learning strategy proposed in (Deng et al.,
2020), we add the LRMS to the network’s output to obtain the fused image. To
supervise training, we set the loss function as:

(7)
$\displaystyle\mathcal{L}=\|\bm{x}^{L}-GT\|_{1}+\lambda\mathcal{L}_{ssim}(\bm{x}^{L},GT),$

where $GT$ is the ground truth, $L$ denotes the total number of layers of the
network, and $\mathcal{L}_{ssim}$ represents loss function based on the SSIM
(Wang et al., 2004). In practice, we set $\lambda=0.1$ to balance the two
losses.

###  4.2. Local-enhanced Vision Mamba (LEVM)

The current vision Mamba models (Zhu et al., 2024; Liu et al., 2024a) directly
extend the Mamba (Gu and Dao, 2023), originally designed for language
sequences, to 2D images by patchifying the images into image tokens. According
to Eq. (3), each input $x_{t}$ shares the same system matrices
$\overline{\bm{A}},\overline{\bm{B}},\overline{\bm{C}}$ to extract global
information. This inspires us to study the representation ability of the Mamba
model. As shown in Fig. 2(c), we design a local enhancement vision Mamba
(LEVM) block to cope with local and global information. The LEVM block has a
local and a global VMamba block (i.e., VMambaBlock function, see Fig. 2(b)).

In the local information representation, inspired by Swin Transformer, we
partition the image into windows $\bm{W}\in\mathbb{R}^{p_{h}\times p_{w}\times
h\times w\times D}$, then feed them into the local VMamba block to extract
local information, where $h,w$ denote the window size and there exists
$p_{h}=H/h$, $p_{w}=W/w$, which can be formulated as:

(8) $\displaystyle\bm{W}^{l-1}=WindowPartition(\bm{x}^{l-1},(h,w)),$ (9)
$\displaystyle\bm{W}^{l-1},\bm{h}_{local}^{l}=VMambaBlock(\bm{W}^{l-1},\bm{h}^{l-1}_{local}).$

In the global information representation, we can gather the local windows to
further represent features from the local VMamba block. Specifically, we
utilize the WindowMerge function that merges these windows, yielding the
entire images, i.e., $\bm{x}^{l-1}=[\bm{W}^{l-1}_{1},\bm{W}^{l-1}_{2},\\\
\cdots,\bm{W}^{l-1}_{p_{h}\times p_{w}}]\in\mathbb{R}^{(p_{h}\times
p_{w})\times h\times w\times D}$. For the $l$-th layer, the global VMamba
block can be formulated as follows:

(10) $\displaystyle\bm{x}^{l-1}=WindowMerge(\bm{W}^{l-1},(h,w))+\bm{x}^{l-1},$
(11)
$\displaystyle\bm{x}^{l},\bm{h}_{global}^{l}=VMambaBlock(\bm{x}^{l-1},\bm{h}^{l-1}_{global}).$

The overall LEVM block can generate local and global state space. The local
VMamba block learns local information using SSM and then outputs local windows
into the global VMamba block. The global VMamba block combines local
information and global information to improve the local representation ability
of the original VMamba block.

However, due to the computation limitations of the VMamba block, these SSMs
are unable to extract spatial and spectral information separately, resulting
in insufficient information fusion. Therefore, we introduce the state sharing
technique to achieve interaction between spatial and spectral information. The
state sharing technique will be stated in the next section.

###  4.3. State Sharing Technique

In this section, the state sharing technique considers layer-wise information
representation and spatial-spectral learning (S2L) of SSM. Specifically, our
LE-Mamba uses efficient SSM based on convolution representation to compute
image tokens. Then, in a layer of SSM, $\overline{\bm{A}}$ and
$\overline{\bm{B}}$ are linear transformations, and $\bm{C}$ sums these states
to generate output features. Therefore, the state has a layer-wise information
representation. Also, the layer-wise information becomes increasingly semantic
from shallow to deep layers.

Inspired by RevCol (Cai et al., 2023), we aim to retain low-level information
across LEVM layers. In contrast to RevCol, which employs deep supervision for
each column, we leverage the hidden state of SSM for two information flows
propagated across layers, which is a direct and convenient approach. As shown
in Fig. 2(d), we build an adjacent flow (denoted as a blue dotted line) and a
skip-connected flow (denoted as a green dotted line) to transfer the current
state into the next layer and the corresponding decoder layer, respectively.
These two information flows are summarized as follows:

Figure 3. Illustration of input feature and corresponding state share feature
as Eq. (14). Adjacent flow helps to learn more semantic information and skip-
connected flow maintains more low-level details and helps fusion. More
examples are shown in the supplementary.

  1. (1)

The adjacent flow: We can get the local and global hidden states from the LEVM
at the $(l-1)$-th layer, i.e., $\bm{h}^{l-1}_{local}$ and
$\bm{h}^{l-1}_{global}$. Then $\bm{h}^{l-1}_{local}$ and
$\bm{h}^{l-1}_{global}$ are fed into the next layer.

  2. (2)

The skip-connected flow: Instead of the skip-connected method of U-Net, the
$(l-1)$-th layer’s global hidden state $\bm{h}^{l-1}_{global}$ is fed into the
corresponding decoder layer.

In Fig. 3, we further show feature maps of two information flows containing
semantic information and low-level details, respectively. Also, there are some
ablation studies (see Tab. 4) to demonstrate the information of adjacent flow
and skip-connected can improve fusion performance.

Table 1. Complexity of various vision models. The input image has batch size
($B$), pixel count ($L$), and input channel ($D$) (window-based pixel count
$L^{\prime}$ with $P$ windows), and the hidden dimension is $N$ (that is, the
output channel). According to (Gu et al., 2021), we show parameter counts
(Params) and FLOPs, space requirements (Space) for the image fusion task.

|  $k\times k$ Conv | Attention | Swin | VMamba | LEVM
---|---|---|---|---|---
Params | $k^{2}DN$ | $5DN$ | $5DN$ | $5DN$ | $10DN+D^{2}+DN$
FLOPs | $BLk^{2}DN$ | $B(L^{2}+5LDN)$ | $BP(L^{\prime 2}+5L^{\prime}DN)$ | $4BLDN+2LDN$ | $10BLN+2LD+2DN$
Space | $BLD$ | $B(L^{2}+LD)$ | $BP(L^{\prime 2}+L^{\prime}D)$ | $BLD$ | $BLD$

Now, we improve the LEVM block with state sharing. Specifically, we perform a
residual connection between the current input features and spatial-spectral
learning of SSM (S2L). In S2L, two different linear blocks are conducted to
project state space and input features, which can be formulated as follows:

(12) $\displaystyle\bm{h}^{l-1}=Linear(\bm{h}^{l-1}),$ (13)
$\displaystyle\bm{g}=Linear(\bm{x}^{l-1}),$

where the first linear projects $\bm{h}^{l-1}$ along the channel dimension,
and the second linear projects $\bm{x}^{l-1}:\mathbb{R}^{D\times
L}\rightarrow\mathbb{R}^{N\times L}$.

Although the VMamba block is capable of capturing pixel dependencies well to
achieve spatial fidelity, it does not fully exploit spectral information and
overlooks the interaction between spatial and spectral information. Based on
this, for the $l$-th layer, our S2L decomposes
$\bm{x}^{l-1}\in\mathbb{R}^{D\times L}$ into state space
$\bm{h}^{l-1}\in\mathbb{R}^{D\times N}$ and feature space
$\bm{g}\in\mathbb{R}^{N\times L}$. The proposed state sharing can be
formulated as:

(14) $\bm{x}^{l-1}=\bm{x}^{l-1}+\alpha\bm{h}^{l-1}\bm{g},$

where the learnable parameter $\alpha\in\mathbb{R}^{D}$ is used to balance the
information between the S2L and input features. In S2L, the state space is
regarded as the basis for spectral representation, and the input image is
mapped into the feature space with rich spatial information. Then, we use
matrix multiplication to establish the interaction between spectral
information and spatial information in the state space. In contrast to the SSM
which focuses on processing spatial information, the S2L method takes into
account the representation of hidden state in the spectral domain and the
spatial characteristics of the image. This allows for extracting and learning
two features that simultaneously possess spatial and spectral properties,
making it more suitable for image fusion tasks.

Finally, our state sharing technique is summarized in Algorithms 1 and 2. To
the best of our knowledge, there haven’t been improvements made to the vision
Mamba that specifically addresses image fusion tasks. The proposed state
sharing technique can be viewed as an independent contribution as it can
adapted to any SSMs.

Input: Input feature tokens $\bm{x}$:  (B, D, L), parameters for
$\text{Param}^{\bm{A}}$:  (K, D, N) and $\text{Param}^{\bm{\Delta}}$:  (K, D).

Output: Output parameters $\bm{A}$:  (K, D, N), $\bm{B}$:  (B, D, N),
$\bm{C}$:  (B, D, N), $\bm{\Delta}$:  (B, D, L)

1

2$\bm{B}\leftarrow$ Linear${}^{\bm{B}}(\bm{x})$;

3 $\bm{C}\leftarrow$ Linear${}^{\bm{C}}(\bm{x})$;

4
$\bm{\Delta}\leftarrow\log(1+\exp(\text{Linear}^{\bm{\Delta}}(\bm{x})+\text{Param}^{\bm{\Delta}}))$;

5 $\bm{A}\leftarrow-\exp(\text{Param}^{\bm{A}})$;

return $\bm{A},\bm{B},\bm{C},\bm{\Delta}$

Algorithm 1 Mamba parametrization function (ParamFn)

Input: Last SSM block state $\bm{h}^{l-1}$:  (B, D, N), input feature
$\bm{x}^{l-1}$:  (B, D, L).

Output: Mamba block output $\bm{x}^{l}$, SSM state $\bm{h}^{l}$:  (B, D, N).

1

2

$\triangleright$ Init ParamA, ParamΔ and $\bm{D}$.

3 ParamA:  (K, D, N), ParamΔ:  (L, D) $\leftarrow$ initA,Δ();

4

5def  _forwardFunction(_$\bm{x}^{l-1},\bm{h}^{l-1}$_)_:

6 $K\leftarrow 4$;

$\triangleright$ “CrossScan” refers to (Liu et al., 2024a).

7 $\bm{x}^{l-1}$:  (B, K, D, L) $\leftarrow$ CrossScan($\bm{x}^{l-1}$);

8

$\triangleright$ State sharing. “groups” is the group of convolution.

9 $\bm{h}^{l-1}$:  (B, K, D, N)$\leftarrow$Linear($\bm{h}^{l-1}$, groups=$K$);

10

11 $\bm{g}$:  (B, K, N, L) $\leftarrow$ Linear($\bm{x}^{l-1}$, groups=$K$);

12

$\triangleright$ Spatial-spectral learning (S2L) of SSM.

13 $\bm{x}^{l-1}\leftarrow$ $\bm{x}^{l-1}+$
einsum(‘BKDN,BKNL$\to$BKDL’,$\bm{h}^{l-1}$,$\bm{g}$);

14

$\triangleright$ Parameterize $\bm{A,B,C}$ and $\bm{\Delta}$. See Algo. 1.

15 $\bm{A,B,C,\Delta}\leftarrow$ ParamFn($\bm{x}^{l-1}$, ParamA, ParamΔ);

16

$\triangleright$ 2D selective scan. Ref to (Liu et al., 2024a).

17 $\bm{x}^{l},\bm{h}^{l}\leftarrow$ SS2D($\bm{x}^{l-1},\bm{A,B,C,\Delta}$);

18

$\triangleright$ “CrossMerge” refers to (Liu et al., 2024a).

19 $\bm{x}^{l}\leftarrow$ CrossMerge($\bm{x}^{l}$);

20

$\triangleright$ Feed forward network with residual connection.

21 $\bm{x}^{l}\leftarrow$ FFN($\bm{x}^{l}$) $+\bm{x}^{l-1}$;

22

23 return $\bm{x}^{l},\bm{h}^{l}$;

24 end

25

return forwardFunction(_$\bm{x}^{l-1},\bm{h}^{l-1}$_);

Algorithm 2 LEVM block with state sharing.

###  4.4. Complexity Analysis

In this section, we analyze the parameter counts and computational complexity
of the proposed LEVM with the state sharing method. The parameter count of the
local VMamba block and global VMamba block is both $O(5DN)$, respectively. The
floating point operations (FLOPs) of these two blocks are
$O(5BL^{\prime}N+2L^{\prime}D)$ and $O(5BLN+2LD)$, respectively. For the state
sharing method, the parameter count is $O(D^{2}+DN)$ and the FLOPs is
$O(DLN)$. The total parameter count is $O(10DN+D^{2}+DN)$. The total FLOPs is
$O(10BLN+2LD+2DN)$. Then, the space requirement is $O(BLD)$. Furthermore, we
list these complexity metrics in Tab. 1. In addition, Fig. 1 illustrates that
although our LE-Mamba has some increasing parameters and FLOPs than classical
methods, we have near-linear memory consumption compared with convolution,
self-attention, and attention of Swin Transformer.

##  5\. Experiments

This section presents extensive experiments on the proposed LE-Mamba,
describing the used datasets, and implementation details, and comparing it
with previous methods. We analyze the reduced-resolution and full-resolution
performances in the multispectral pansharpening task, as well as the
performances in the multispectral and hyperspectral image fusion task.
Ablation studies are conducted on the proposed LEVM and the state sharing
technique to validate their effectiveness.

Table 2. Quantitative results of all competing methods. The best results are
in red and the second best results are in blue. Upper panel indicates WV3 (8
bands) dataset and lower panel is GF2 (4 bands) dataset.

Methods | Reduced-Resolution (RR): Avg$\pm$std | Full-Resolution (FR): Avg$\pm$std |  #Params |  #FLOPs
---|---|---|---|---
SAM ($\downarrow$) | ERGAS ($\downarrow$) | Q2n ($\uparrow$) | SCC ($\uparrow$) |  $D_{\lambda}$ ($\downarrow$) |  $D_{s}$ ($\downarrow$) | HQNR ($\uparrow$)
MTF-GLP-FS (Vivone et al., 2018) | 5.32$\pm$1.65 | 4.65$\pm$1.44 | 0.818$\pm$0.101 | 0.898$\pm$0.047 | 0.021$\pm$0.008 | 0.063$\pm$0.028 | 0.918$\pm$0.035 | — | —
BT-H (Lolli et al., 2017) | 4.90$\pm$1.30 | 4.52$\pm$1.33 | 0.818$\pm$0.102 | 0.924$\pm$0.024 | 0.057$\pm$0.023 | 0.081$\pm$0.037 | 0.867$\pm$0.054 | — | —
LRTCFPan (Wu et al., 2023) | 4.74$\pm$1.41 | 4.32$\pm$1.44 | 0.846$\pm$0.091 | 0.927$\pm$0.023 | 0.018$\pm$0.007 | 0.053$\pm$0.026 | 0.931$\pm$0.031 | — | —
DiCNN (He et al., 2019) | 3.59$\pm$0.76 | 2.67$\pm$0.66 | 0.900$\pm$0.087 | 0.976$\pm$0.007 | 0.036$\pm$0.011 | 0.046$\pm$0.018 | 0.920$\pm$0.026 | 0.23M | 0.19G
FusionNet (Deng et al., 2020) | 3.33$\pm$0.70 | 2.47$\pm$0.64 | 0.904$\pm$0.090 | 0.981$\pm$0.007 | 0.024$\pm$0.009 | 0.036$\pm$0.014 | 0.941$\pm$0.020 | 0.047M | 0.32G
LAGConv (Jin et al., 2022) | 3.10$\pm$0.56 | 2.30$\pm$0.61 | 0.910$\pm$0.091 | 0.984$\pm$0.007 | 0.037$\pm$0.015 | 0.042$\pm$0.015 | 0.923$\pm$0.025 | 0.15M | 0.54G
Invformer (Zhou et al., 2022a) | 3.25$\pm$0.64 | 2.39$\pm$0.52 | 0.906$\pm$0.084 | 0.983$\pm$0.005 | 0.055$\pm$0.029 | 0.068$\pm$0.031 | 0.882$\pm$0.049 | 0.14M | 3.89G
DCFNet (Wu et al., 2021) | 3.03$\pm$0.74 | 2.16$\pm$0.46 | 0.905$\pm$0.088 | 0.986$\pm$0.004 | 0.078$\pm$0.081 | 0.051$\pm$0.034 | 0.877$\pm$0.101 | 2.77M | 3.46G
HMPNet (Tian et al., 2023) | 3.06$\pm$0.58 | 2.23$\pm$0.55 | 0.916$\pm$0.087 | 0.986$\pm$0.005 | 0.018$\pm$0.007 | 0.053$\pm$0.006 | 0.929$\pm$0.011 | 1.09M | 2.00G
PanDiff (Meng et al., 2023) | 3.30$\pm$0.60 | 2.47$\pm$0.58 | 0.898$\pm$0.088 | 0.980$\pm$0.006 | 0.027$\pm$0.012 | 0.054$\pm$0.026 | 0.920$\pm$0.036 | 45.33M | 14.83G
PanMamba (He et al., 2024) | 2.94$\pm$0.54 | 2.24$\pm$0.51 | 0.916$\pm$0.090 | 0.985$\pm$0.006 | 0.020$\pm$0.007 | 0.042$\pm$0.014 | 0.939$\pm$0.020 | 0.48M | 1.31G
Proposed | 2.76$\pm$0.52 | 2.02$\pm$0.43 | 0.921$\pm$0.080 | 0.988$\pm$0.003 | 0.016$\pm$0.006 | 0.031$\pm$0.003 | 0.954$\pm$0.007 | 0.74M | 3.58G
MTF-GLP-FS (Vivone et al., 2018) | 1.68$\pm$0.35 | 1.60$\pm$0.35 | 0.891$\pm$0.026 | 0.939$\pm$0.020 | 0.035$\pm$0.014 | 0.143$\pm$0.028 | 0.823$\pm$0.035 | — | —
BT-H (Lolli et al., 2017) | 1.68$\pm$0.32 | 1.55$\pm$0.36 | 0.909$\pm$0.029 | 0.951$\pm$0.015 | 0.060$\pm$0.025 | 0.131$\pm$0.019 | 0.817$\pm$0.031 | — | —
LRTCFPan (Wu et al., 2023) | 1.30$\pm$0.31 | 1.27$\pm$0.34 | 0.935$\pm$0.030 | 0.964$\pm$0.012 | 0.033$\pm$0.027 | 0.090$\pm$0.014 | 0.881$\pm$0.023 | — | —
DiCNN (He et al., 2019) | 1.05$\pm$0.23 | 1.08$\pm$0.25 | 0.959$\pm$0.010 | 0.977$\pm$0.006 | 0.041$\pm$0.012 | 0.099$\pm$0.013 | 0.864$\pm$0.017 | 0.23M | 0.19G
FusionNet (Deng et al., 2020) | 0.97$\pm$0.21 | 0.99$\pm$0.22 | 0.964$\pm$0.009 | 0.981$\pm$0.005 | 0.040$\pm$0.013 | 0.101$\pm$0.013 | 0.863$\pm$0.018 | 0.047M | 0.32G
LAGConv (Jin et al., 2022) | 0.78$\pm$0.15 | 0.69$\pm$0.11 | 0.980$\pm$0.009 | 0.991$\pm$0.002 | 0.032$\pm$0.013 | 0.079$\pm$0.014 | 0.891$\pm$0.020 | 0.15M | 0.54G
Invformer (Zhou et al., 2022a) | 0.83$\pm$0.14 | 0.70$\pm$0.11 | 0.977$\pm$0.012 | 0.980$\pm$0.002 | 0.059$\pm$0.026 | 0.110$\pm$0.015 | 0.838$\pm$0.024 | 2.77M | 3.46G
DCFNet (Wu et al., 2021) | 0.89$\pm$0.16 | 0.81$\pm$0.14 | 0.973$\pm$0.010 | 0.985$\pm$0.002 | 0.023$\pm$0.012 | 0.066$\pm$0.010 | 0.912$\pm$0.012 | 2.77M | 3.46G
HMPNet (Tian et al., 2023) | 0.80$\pm$0.14 | 0.56$\pm$0.10 | 0.981$\pm$0.030 | 0.993$\pm$0.003 | 0.080$\pm$0.050 | 0.115$\pm$0.012 | 0.815$\pm$0.049 | 1.09M | 2.00G
PanDiff (Meng et al., 2023) | 0.89$\pm$0.12 | 0.75$\pm$0.10 | 0.979$\pm$0.010 | 0.989$\pm$0.002 | 0.027$\pm$0.020 | 0.073$\pm$0.010 | 0.903$\pm$0.021 | 45.33M | 14.83G
PanMamba (He et al., 2024) | 0.68$\pm$0.12 | 0.64$\pm$0.10 | 0.982$\pm$0.008 | 0.985$\pm$0.006 | 0.016$\pm$0.008 | 0.045$\pm$0.009 | 0.939$\pm$0.010 | 0.48M | 1.31G
Proposed | 0.60$\pm$0.11 | 0.52$\pm$0.09 | 0.987$\pm$0.007 | 0.994$\pm$0.001 | 0.018$\pm$0.009 | 0.027$\pm$0.008 | 0.955$\pm$0.011 | 0.74M | 3.58G

###  5.1. Dataset and Implementation Details

To evaluate the proposed LE-Mamba, we conduct experiments on two remote
sensing multispectral pansharpening datasets: WV3 and GF2 datasets with super-
resolved scale 4, and further on indoor hyperspectral-multispectral fusion
datasets: CAVE and Harvard datasets with scale 8. More details on the datasets
can be found in the supplementary.

During the training process, we use the AdamW (Loshchilov and Hutter, 2017)
optimizer and set the base learning rate to 1e-3, which is decreased to 1e-4
at 300 epochs, and further decreased to 1e-5 at 600 epochs. After that, the
training continues till 1000 epochs for WV3 and GF2 datasets and 1600 epochs
for CAVE and Harvard datasets. The weight decay is set to 1e-6. All
experiments are conducted on two RTX 3090 GPUs. Network configurations on
different datasets can be found in the supplementary.

###  5.2. Benchmark

For the multispectral pansharpening datasets, we compare the proposed method
with the recently SOTA traditional methods: MTF-GLP-FS (Vivone et al., 2018),
BT-H (Lolli et al., 2017) and LRTCFPan (Wu et al., 2023), and DL-based
methods: DiCNN (He et al., 2019), LAGConv (Jin et al., 2022), DCFNet (Wu et
al., 2021), HMPNet (Tian et al., 2023), PanDiff (Meng et al., 2023) and
PanMamba (He et al., 2024). For the multispectral and hyperspectral datasets,
we also choose some recent traditional methods including CSTF-FUS (Li et al.,
2018), LTTR (Dian et al., 2019), LTMR (Dian and Li, 2019) and IR-TenSR (Xu et
al., 2022). Moreover, most competitive DL-based methods are compared:
ResTFNet (Liu et al., 2020), SSRNet (Zhang et al., 2020), HSRNet (Hu et al.,
2021), MogDCN (Dong et al., 2021), Fusformer (Hu et al., 2022), DHIF (Huang et
al., 2022), PSRT (Deng et al., 2023), 3DT-Net (Ma et al., 2023), DSPNet (Sun
et al., 2023), and MIMO-SST (Fang et al., 2024).

###  5.3. Main Results

Table 3. The average and standard deviation calculated for all the compared
approaches on 11 CAVE examples and 10 Harvard examples simulating a scaling
factor of 8. The best results are in red and the second best results are in
blue.

Methods | CAVE $\times 8$ | Harvard $\times 8$ | $\\#$Params | $\\#$FLOPs
---|---|---|---|---
PSNR ($\uparrow$) | SAM ($\downarrow$) | ERGAS ($\downarrow$) | SSIM ($\uparrow$) | PSNR ($\uparrow$) | SAM ($\downarrow$) | ERGAS ($\downarrow$) | SSIM ($\uparrow$)
Bicubic | 29.96$\pm$3.54 | 5.89$\pm$2.32 | 5.56$\pm$3.08 | 0.887$\pm$0.066 | 33.18$\pm$6.85 | 3.10$\pm$0.90 | 3.83$\pm$1.84 | 0.894$\pm$0.0732 |  $-$ |  $-$
CSTF-FUS (Li et al., 2018) | 38.44$\pm$4.25 | 7.00$\pm$2.65 | 2.11$\pm$1.15 | 0.959$\pm$0.033 | 39.84$\pm$6.51 | 4.49$\pm$1.52 | 2.40$\pm$1.84 | 0.932$\pm$0.092 |  $-$ |  $-$
LTTR (Dian et al., 2019) | 37.92$\pm$3.59 | 5.37$\pm$1.96 | 2.44$\pm$1.05 | 0.972$\pm$0.018 | 42.09$\pm$4.56 | 3.62$\pm$1.34 | 1.80$\pm$0.96 | 0.960$\pm$0.048 |  $-$ |  $-$
LTMR (Dian and Li, 2019) | 38.41$\pm$3.57 | 5.04$\pm$1.70 | 2.24$\pm$0.97 | 0.974$\pm$0.017 | 42.09$\pm$4.56 | 3.62$\pm$1.34 | 1.80$\pm$0.92 | 0.959$\pm$0.060 |  $-$ |  $-$
IR-TenSR (Xu et al., 2022) | 36.79$\pm$3.64 | 12.87$\pm$4.98 | 2.68$\pm$1.41 | 0.944$\pm$0.031 | 40.04$\pm$3.89 | 5.40$\pm$1.76 | 4.75$\pm$1.55 | 0.958$\pm$0.016 |  $-$ |  $-$
ResTFNet (Liu et al., 2020) | 43.77$\pm$5.34 | 3.49$\pm$0.94 | 1.38$\pm$1.25 | 0.992$\pm$0.006 | 43.50$\pm$3.96 | 3.53$\pm$1.11 | 1.74$\pm$0.93 | 0.979$\pm$0.012 | 2.387M | 1.75G
SSRNet (Zhang et al., 2020) | 46.23$\pm$4.19 | 3.13$\pm$0.97 | 1.05$\pm$0.73 | 0.993$\pm$0.004 | 45.76$\pm$3.34 | 2.99$\pm$0.98 | 1.34$\pm$0.74 | 0.983$\pm$0.010 | 0.027M | 0.11G
HSRNet (Hu et al., 2021) | 46.69$\pm$4.48 | 2.91$\pm$0.86 | 0.93$\pm$0.63 | 0.994$\pm$0.003 | 44.02$\pm$4.89 | 3.64$\pm$1.79 | 1.49$\pm$0.81 | 0.980$\pm$0.013 | 1.09M | 2.00G
MogDCN (Dong et al., 2021) | 49.21$\pm$4.99 | 2.44$\pm$0.74 | 0.76$\pm$0.63 | 0.996$\pm$0.003 | 45.14$\pm$5.41 | 3.19$\pm$1.45 | 1.75$\pm$1.66 | 0.980$\pm$0.019 | 6.840M | 47.48G
Fusformer (Hu et al., 2022) | 47.96$\pm$7.79 | 2.75$\pm$1.30 | 1.45$\pm$2.69 | 0.990$\pm$0.022 | 44.93$\pm$5.65 | 3.63$\pm$2.40 | 1.49$\pm$0.96 | 0.979$\pm$0.017 | 0.504M | 9.83G
DHIF (Huang et al., 2022) | 48.46$\pm$4.89 | 2.50$\pm$0.79 | 0.83$\pm$0.67 | 0.996$\pm$0.003 | 45.00$\pm$4.13 | 3.70$\pm$1.68 | 1.32$\pm$0.61 | 0.983$\pm$0.011 | 22.462M | 54.27G
PSRT (Deng et al., 2023) | 47.86$\pm$7.53 | 2.73$\pm$0.80 | 1.52$\pm$3.02 | 0.994$\pm$0.005 | 45.10$\pm$4.06 | 2.90$\pm$0.84 | 1.37$\pm$0.84 | 0.985$\pm$0.009 | 0.247M | 1.14G
3DT-Net (Ma et al., 2023) | 49.41$\pm$5.83 | 2.26$\pm$0.66 | 0.83$\pm$1.07 | 0.996$\pm$0.003 | 44.41$\pm$5.38 | 2.93$\pm$0.88 | 1.55$\pm$0.89 | 0.983$\pm$0.010 | 3.464M | 68.07G
DSPNet (Sun et al., 2023) | 49.18$\pm$4.84 | 2.57$\pm$0.79 | 0.75$\pm$0.62 | 0.996$\pm$0.003 | 45.84$\pm$3.62 | 2.97$\pm$0.75 | 1.33$\pm$0.64 | 0.984$\pm$0.010 | 6.064M | 6.81G
MIMO-SST (Fang et al., 2024) | 48.31$\pm$5.04 | 2.88$\pm$0.86 | 0.89$\pm$0.79 | 0.995$\pm$0.004 | 46.59$\pm$3.34 | 2.91$\pm$0.75 | 2.29$\pm$1.03 | 0.985$\pm$0.009 | 4.983M | 1.58G
Proposed | 49.86$\pm$4.77 | 2.31$\pm$0.69 | 0.70$\pm$0.56 | 0.997$\pm$0.002 | 46.84$\pm$3.82 | 2.75$\pm$0.71 | 1.16$\pm$0.53 | 0.986$\pm$0.009 | 3.158M | 9.80G

The fusion performance of LE-Mamba on the WV3 and GF2 datasets is provided in
Tab. 2, indicating the SOTA performance of our method in reduced-resolution
metrics. Traditional approaches lag significantly behind DL-based methods in
terms of reduced-resolution metrics. Among DL-based methods, our approach
demonstrates good fidelity for both spectral modality (i.e., SAM metric) and
spatial modality (i.e., ERGAS metric). In terms of full-resolution metrics,
LE-Mamba exhibits SOTA performance on the GF2 test set, indicating its good
generalization capability. We present fusion images on the WV3 test set in
Fig. 4, showcasing LE-Mamba’s minimal errors, particularly evident in the
fusion of high-frequency components such as the edges of buildings and roads.

Tab. 3 provides widely used numerical fusion metrics on CAVE and Harvard
datasets. Our LE-Mamba outperforms all previous traditional and DL-based
methods contributed by the multi-scale backbone and proposed LEVM equipped
with state sharing technique. Error maps are depicted in Fig. 5 which shows
that LE-Mamba owns fewer errors and better fusion performances. More visual
results can be found in the supplementary.

###  5.4. Ablation Study

Figure 4. Error maps of LE-Mamba and compared previous SOTA methods on WV3
test set. Our LE-Mamba shows fewer errors on GT. Some close-ups are depicted
at the corner. Figure 5. Error maps of LE-Mamba and compared previous SOTA
methods on Harvard ($\times$ 8) test set. Our LE-Mamba shows fewer errors on
GT. Some close-ups are depicted at the corner.

####  5.4.1. LEVM and State Sharing Technique can Boost Fusion Performance

We conduct ablation studies on the proposed local-enhanced vision Mamba block
and state sharing technique and report the fusion performance on Tab. 4.
Starting at the basic multi-scale network baseline which only contains
NAFBlock (Chen et al., 2022), we observe the performance is poor whose SAM
only has 3.52. Then we replace the NAFBlock with the LEVM block, the
performance is boosted to 2.86. Based on it, we further share the SSM state at
adjacent flow and skip-connected flow and conduct spatial-spectral learning
(S2L), one can notice the fusion performance is better and the SAM metric
reaches 2.76. From the consistent performance improvement in the above
experiments, we can summarize that our proposed LEVM and state sharing
technique are effective in boosting the performance of the image fusion task.
We will conduct ablation studies on the S2L method in the next round.

Table 4. Ablation study on proposed multi-scale architecture, LEVM block, and
state sharing technique (including adjacent flow and skip-connected flow) on
WV3 dataset.

Composition | SAM ($\downarrow$) | ERGAS ($\downarrow$) | Q2n ($\uparrow$) | SCC ($\uparrow$)
---|---|---|---|---
Multi-scale baseline | 3.52 | 2.66 | 0.902 | 0.977
\+ VMamba | 2.93 | 2.15 | 0.910 | 0.987
\+ LEVM block | 2.86 | 2.17 | 0.914 | 0.986
\+ adjacent flow | 2.80 | 2.08 | 0.918 | 0.987
\+ skip-connected flow | 2.76 | 2.02 | 0.921 | 0.988

####  5.4.2. State Sharing Beats Non-state Residual Connection

From Eq. (14), we can observe that the state sharing technique is a residual
connection that adds input feature $\bm{x}^{l-1}$ with S2L (i.e.,
$\bm{h}^{l-1}\bm{g}$). To verify the effectiveness of using the S2L (whether
the performance gain comes from residual connections and additional
convolutional parameters), we design another residual variant that does not
use the SSM state:

(15) $\bm{x}^{l-1}=\bm{x}^{l-1}+\alpha Linear(\bm{x}^{l-1}).$

We validate its fusion performance on the WV3 dataset, as shown in Tab. 5. Our
state sharing outperforms the residual variant. From this ablation, we can
conclude that: using S2L to share state across different blocks can bring
performance gain.

Table 5. Discussion on the S2L of the proposed LE-Mamba on the WV3 dataset.

Variants | SAM ($\downarrow$) | ERGAS ($\downarrow$) | Q2n ($\uparrow$) | SCC ($\uparrow$)
---|---|---|---|---
$\bm{x}^{l-1}$ | 2.86 | 2.17 | 0.914 | 0.986
$\bm{x}^{l-1}=\bm{x}^{l-1}+\alpha Linear(\bm{x}^{l-1})$ | 2.84 | 2.13 | 0.917 | 0.986
$\bm{x}^{l-1}=\bm{x}^{l-1}+\alpha\bm{h}^{l-1}\bm{g}$ | 2.76 | 2.02 | 0.921 | 0.988

####  5.4.3. LEVM Outperforms Previous Attentions

To compare the proposed LEVM with previously common Attention mechanisms, we
select common attention and its variants including traditional Attention
(Dosovitskiy et al., 2021), PVT Attention (Wang et al., 2021), Swin Attention
(Liu et al., 2021), and Linear Attention (Han et al., 2023). We replace the
LEVM block with the corresponding Attention block (including FFN) and train
them uniformly on the WV3 dataset until convergence. Their fusion performance
is shown in Tab. 6. It can be seen that the traditional Attention performs the
worst, which is because using Attention in the first layer of the network
occupies an extremely large GPU memory, leading to the inability to train.
Therefore, most methods based on traditional Attention perform patch embedding
in the first layer to downsample the image size, which is disastrous for tasks
like image fusion that require low-level information. The performance of
Linear Attention is also relatively poor, which is caused by the approximation
errors introduced by its approximation of the Attention mechanism. Next are
PVT Attention and Swin Attention, both of which impose restrictions on
Attention. PVT downsamples both the keys and values, resulting in information
loss, while Swin limits the global information to the local regions. The
proposed LEVM outperforms previous attention operations.

Table 6. Results of different self-attention (attention) types on WV3 dataset.

Attention types | SAM ($\downarrow$) | ERGAS ($\downarrow$) | Q2n($\uparrow$) | SCC ($\uparrow$)
---|---|---|---|---
Self-attention (Dosovitskiy et al., 2021) | 3.09 | 2.33 | 0.906 | 0.917
PVT attention (Wang et al., 2021) | 2.89 | 2.16 | 0.919 | 0.986
Swin attention (Liu et al., 2021) | 2.86 | 2.15 | 0.919 | 0.986
Linear attention (Han et al., 2023) | 2.94 | 2.21 | 0.914 | 0.984
Proposed | 2.76 | 2.02 | 0.921 | 0.988

####  5.4.4. Enlarging SSM State Capacity Does not Solve Issue 2)

To solve issue 2) mentioned in Sect. 1, one simple and straightforward idea is
to enlarge the SSM state capacity $N$. Usually, $N$ is set to 16 and suits
most high-level tasks. We design a series of enlarged SSM state variants with
$N=32$, $N=64$ based on Tab. 4 “+VMamba” model. One can observe that even when
setting $N=64$ with a large state capacity, the performance gain is limited.
Moreover, large $N$ brings additional memory consumption and FLOPs. It is
unacceptable for resource-constraint scenarios.

Table 7. The performance of LE-Mamba with different SSM states capacity on WV3 datasets. Variants | SAM ($\downarrow$) | ERGAS ($\downarrow$) | Q2n ($\uparrow$) | SCC ($\uparrow$)
---|---|---|---|---
$N=16$ | 2.93 | 2.15 | 0.910 | 0.987
$N=32$ | 2.90 | 2.14 | 0.914 | 0.987
$N=64$ | 2.88 | 2.13 | 0.915 | 0.987

##  6\. Conclusion

Based on the observation that sequence modeling in the Mamba network doesn’t
fully consider the characteristics of the image fusion task, this paper
designs a multi-scale network architecture equipped with the local-enhanced
vision Mamba (LEVM) block and the state sharing technique, called LE-Mamba.
The LEVM mitigates the information loss caused by limited states in the Mamba
network by combining local and global information to enhance fused image
details. Additionally, the state sharing technique shares the state between
layers in adjacent and skip-connected flows, enabling the deep layers of the
network to contain richer spatial information. Then, the fused image exhibits
improved spatial and spectral details by incorporating spatial-spectral
learning (S2L). The proposed network architecture achieves state-of-the-art
performance in widely used image fusion datasets.

## References

  * (1)
  * Cai et al. (2023) Yuxuan Cai, Yizhuang Zhou, Qi Han, Jianjian Sun, Xiangwen Kong, Jun Li, and Xiangyu Zhang. 2023.  Reversible Column Networks. In _ICLR_.
  * Cao et al. (2024) Zihan Cao, Shiqi Cao, Liang-Jian Deng, Xiao Wu, Junming Hou, and Gemine Vivone. 2024.  Diffusion model with disentangled modulations for sharpening multispectral and hyperspectral images.  _Inf. Fus._ 104 (2024), 102158.
  * Chen et al. (2022) Liangyu Chen, Xiaojie Chu, Xiangyu Zhang, and Jian Sun. 2022.  Simple baselines for image restoration. In _ECCV_. Springer, 17–33.
  * Deng et al. (2020) Liang-Jian Deng, Gemine Vivone, Cheng Jin, and Jocelyn Chanussot. 2020.  Detail injection-based deep convolutional neural networks for pansharpening.  _IEEE Trans. Geosci. Remote Sens._ 59, 8 (2020), 6995–7010.
  * Deng et al. (2022) Liang-jian Deng, Gemine Vivone, Mercedes E. Paoletti, Giuseppe Scarpa, Jiang He, Yongjun Zhang, Jocelyn Chanussot, and Antonio Plaza. 2022.  Machine Learning in Pansharpening: A benchmark, from shallow to deep networks.  _IEEE Geosci. Remote Sens. Mag._ 10, 3 (2022), 279–315.
  * Deng et al. (2023) Shang-Qi Deng, Liang-Jian Deng, Xiao Wu, Ran Ran, Danfeng Hong, and Gemine Vivone. 2023.  PSRT: Pyramid shuffle-and-reshuffle transformer for multispectral and hyperspectral image fusion.  _IEEE Trans. Geosci. Remote Sens._ 61 (2023), 1–15.
  * Dian and Li (2019) Renwei Dian and Shutao Li. 2019.  Hyperspectral Image Super-Resolution via Subspace-Based Low Tensor Multi-Rank Regularization.  _IEEE Trans. Image Process._ 28, 10 (2019), 5135–5146.
  * Dian et al. (2019) Renwei Dian, Shutao Li, and Leyuan Fang. 2019.  Learning a Low Tensor-Train Rank Representation for Hyperspectral Image Super-Resolution.  _IEEE Trans. Neural Netw. Learn. Syst._ 30, 9 (2019), 2672–2683.
  * Dong et al. (2021) Weisheng Dong, Chen Zhou, Fangfang Wu, Jinjian Wu, Guangming Shi, and Xin Li. 2021.  Model-guided deep hyperspectral image super-resolution.  _IEEE Trans. Image Process._ 30 (2021), 5754–5768.
  * Dosovitskiy et al. (2021) Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. 2021.  An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. In _ICLR_.
  * Fang et al. (2024) Jian Fang, Jingxiang Yang, Abdolraheem Khader, and Liang Xiao. 2024.  MIMO-SST: Multi-Input Multi-Output Spatial-Spectral Transformer for Hyperspectral and Multispectral Image Fusion.  _IEEE Trans. Geosci. Remote Sens._ 62 (2024), 1–20.
  * Gu and Dao (2023) Albert Gu and Tri Dao. 2023.  Mamba: Linear-time sequence modeling with selective state spaces.  _arXiv preprint arXiv:2312.00752_ (2023).
  * Gu et al. (2021) Albert Gu, Karan Goel, and Christopher Ré. 2021.  Efficiently modeling long sequences with structured state spaces.  _arXiv preprint arXiv:2111.00396_ (2021).
  * Han et al. (2023) Dongchen Han, Xuran Pan, Yizeng Han, Shiji Song, and Gao Huang. 2023.  Flatten transformer: Vision transformer using focused linear attention. In _CVPR_. 5961–5971.
  * He et al. (2019) Lin He, Yizhou Rao, Jun Li, Jocelyn Chanussot, Antonio Plaza, Jiawei Zhu, and Bo Li. 2019.  Pansharpening via detail injection based convolutional neural networks.  _IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens._ 12, 4 (2019), 1188–1204.
  * He et al. (2024) Xuanhua He, Ke Cao, Keyu Yan, Rui Li, Chengjun Xie, Jie Zhang, and Man Zhou. 2024.  Pan-Mamba: Effective pan-sharpening with State Space Model.  _arXiv preprint arXiv:2402.12192_ (2024).
  * Hu et al. (2022) Jin-Fan Hu, Ting-Zhu Huang, Liang-Jian Deng, Hong-Xia Dou, Danfeng Hong, and Gemine Vivone. 2022.  Fusformer: A transformer-based fusion network for hyperspectral image super-resolution.  _IEEE Geoscience and Remote Sensing Letters_ 19 (2022), 1–5.
  * Hu et al. (2021) Jin-Fan Hu, Ting-Zhu Huang, Liang-Jian Deng, Tai-Xiang Jiang, Gemine Vivone, and Jocelyn Chanussot. 2021.  Hyperspectral image super-resolution via deep spatiospectral attention convolutional neural networks.  _IEEE Trans. Neural Netw. Learn. Syst._ 33, 12 (2021), 7251–7265.
  * Huang et al. (2022) Tao Huang, Weisheng Dong, Jinjian Wu, Leida Li, Xin Li, and Guangming Shi. 2022.  Deep hyperspectral image fusion network with iterative spatio-spectral regularization.  _IEEE Transactions on Computational Imaging_ 8 (2022), 201–214.
  * Jin et al. (2022) Zi-Rong Jin, Tian-Jing Zhang, Tai-Xiang Jiang, Gemine Vivone, and Liang-Jian Deng. 2022.  LAGConv: Local-Context Adaptive Convolution Kernels with Global Harmonic Bias for Pansharpening.  _AAAI_ 36, 1 (Jun. 2022), 1113–1121.
  * Li et al. (2018) Shutao Li, Renwei Dian, Leyuan Fang, and José M Bioucas-Dias. 2018.  Fusing hyperspectral and multispectral images via coupled sparse tensor factorization.  _IEEE Transactions on Image Processing_ 27, 8 (2018), 4118–4130.
  * Liu et al. (2024b) Jiarun Liu, Hao Yang, Hong-Yu Zhou, Yan Xi, Lequan Yu, Yizhou Yu, Yong Liang, Guangming Shi, Shaoting Zhang, Hairong Zheng, et al. 2024b.  Swin-umamba: Mamba-based unet with imagenet-based pretraining.  _arXiv preprint arXiv:2402.03302_ (2024).
  * Liu et al. (2020) Xiangyu Liu, Qingjie Liu, and Yunhong Wang. 2020.  Remote sensing image fusion based on two-stream fusion network.  _Inf. Fus._ 55 (2020), 1–15.
  * Liu et al. (2024a) Yue Liu, Yunjie Tian, Yuzhong Zhao, Hongtian Yu, Lingxi Xie, Yaowei Wang, Qixiang Ye, and Yunfan Liu. 2024a.  VMamba: Visual State Space Model.  _arXiv preprint arXiv:2401.10166_ (2024).
  * Liu et al. (2021) Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. 2021.  Swin transformer: Hierarchical vision transformer using shifted windows. In _ICCV_. 10012–10022.
  * Lolli et al. (2017) Simone Lolli, Luciano Alparone, Andrea Garzelli, and Gemine Vivone. 2017.  Haze correction for contrast-based multispectral pansharpening.  _IEEE Geosci. Remote Sens. Lett._ 14, 12 (2017), 2255–2259.
  * Loshchilov and Hutter (2017) Ilya Loshchilov and Frank Hutter. 2017.  Decoupled weight decay regularization.  _arXiv preprint arXiv:1711.05101_ (2017).
  * Lu et al. (2023) Chris Xiaoxuan Lu, Yannick Schroecker, Albert Gu, Emilio Parisotto, Jakob Nicolaus Foerster, Satinder Singh, and Feryal M. P. Behbahani. 2023.  Structured State Space Models for In-Context Reinforcement Learning.  _ArXiv_ abs/2303.03982 (2023).
  * Ma et al. (2024) Jun Ma, Feifei Li, and Bo Wang. 2024.  U-mamba: Enhancing long-range dependency for biomedical image segmentation.  _arXiv preprint arXiv:2401.04722_ (2024).
  * Ma et al. (2023) Qing Ma, Junjun Jiang, Xianming Liu, and Jiayi Ma. 2023.  Learning a 3D-CNN and transformer prior for hyperspectral image super-resolution.  _Inf. Fus._ 100 (2023), 101907.
  * Meng et al. (2023) Qingyan Meng, Wenxu Shi, Sijia Li, and Linlin Zhang. 2023.  PanDiff: A Novel Pansharpening Method Based on Denoising Diffusion Probabilistic Model.  _IEEE Trans. Geosci. Remote Sens._ 61 (2023), 1–17.
  * Nguyen et al. (2022) Eric Nguyen, Karan Goel, Albert Gu, Gordon Downs, Preey Shah, Tri Dao, Stephen Baccus, and Christopher Ré. 2022.  S4nd: Modeling images and videos as multidimensional signals with state spaces.  _Neurips_ 35 (2022), 2846–2861.
  * Ronneberger et al. (2015a) Olaf Ronneberger, Philipp Fischer, and Thomas Brox. 2015a.  U-net: Convolutional networks for biomedical image segmentation. In _Medical Image Computing and Computer-Assisted Intervention (MICCAI)_. Springer, 234–241.
  * Ronneberger et al. (2015b) Olaf Ronneberger, Philipp Fischer, and Thomas Brox. 2015b.  U-net: Convolutional networks for biomedical image segmentation. In _MICCAI_. Springer, 234–241.
  * Ruan and Xiang (2024) Jiacheng Ruan and Suncheng Xiang. 2024.  Vm-unet: Vision mamba unet for medical image segmentation.  _arXiv preprint arXiv:2402.02491_ (2024).
  * Sun et al. (2023) Yucheng Sun, Han Xu, Yong Ma, Minghui Wu, Xiaoguang Mei, Jun Huang, and Jiayi Ma. 2023.  Dual Spatial-spectral Pyramid Network with Transformer for Hyperspectral Image Fusion.  _IEEE Trans. Geosci. Remote Sens._ (2023).
  * Tian et al. (2023) Xin Tian, Kun Li, Wei Zhang, Zhongyuan Wang, and Jiayi Ma. 2023.  Interpretable Model-Driven Deep Network for Hyperspectral, Multispectral, and Panchromatic Image Fusion.  _IEEE Trans. Neural Netw. Learn. Syst._ (2023), 1–14.
  * Vivone (2023) Gemine Vivone. 2023.  Multispectral and hyperspectral image fusion in remote sensing: A survey.  _Information Fusion_ 89 (2023), 405–417.
  * Vivone et al. (2018) Gemine Vivone, Rocco Restaino, and Jocelyn Chanussot. 2018.  Full scale regression-based injection coefficients for panchromatic sharpening.  _IEEE Trans. Image Process._ 27, 7 (2018), 3418–3431.
  * Wang et al. (2021) Wenhai Wang, Enze Xie, Xiang Li, Deng-Ping Fan, Kaitao Song, Ding Liang, Tong Lu, Ping Luo, and Ling Shao. 2021.  Pyramid vision transformer: A versatile backbone for dense prediction without convolutions. In _ICCV_. 568–578.
  * Wang et al. (2004) Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. 2004.  Image quality assessment: from error visibility to structural similarity.  _IEEE Transactions on Image Processing_ 13, 4 (2004), 600–612.
  * Wu et al. (2021) Xiao Wu, Ting-Zhu Huang, Liang-Jian Deng, and Tian-Jing Zhang. 2021.  Dynamic cross feature fusion for remote sensing pansharpening. In _ICCV_. 14687–14696.
  * Wu et al. (2023) Zhong-Cheng Wu, Ting-Zhu Huang, Liang-Jian Deng, Jie Huang, Jocelyn Chanussot, and Gemine Vivone. 2023.  LRTCFPan: Low-Rank Tensor Completion Based Framework for Pansharpening.  _IEEE Transactions on Image Processing_ 32 (2023), 1640–1655.  https://doi.org/10.1109/TIP.2023.3247165
  * Xu et al. (2022) Ting Xu, Ting-Zhu Huang, Liang-Jian Deng, and Naoto Yokoya. 2022.  An Iterative Regularization Method Based on Tensor Subspace Representation for Hyperspectral Image Super-Resolution.  _IEEE Trans. Geosci. Remote Sens._ 60 (2022), 1–16.
  * Yang et al. (2017) Junfeng Yang, Xueyang Fu, Yuwen Hu, Yue Huang, Xinghao Ding, and John Paisley. 2017.  PanNet: A Deep Network Architecture for Pan-Sharpening. In _ICCV_. 1753–1761.
  * Yu et al. (2022) Weihao Yu, Mi Luo, Pan Zhou, Chenyang Si, Yichen Zhou, Xinchao Wang, Jiashi Feng, and Shuicheng Yan. 2022.  Metaformer is actually what you need for vision. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_. 10819–10829.
  * Zhang et al. (2020) Xueting Zhang, Wei Huang, Qi Wang, and Xuelong Li. 2020.  SSR-NET: Spatial–spectral reconstruction network for hyperspectral and multispectral image fusion.  _IEEE Trans. Geosci. Remote Sens._ 59, 7 (2020), 5953–5965.
  * Zhou et al. (2022a) Man Zhou, Xueyang Fu, Jie Huang, Feng Zhao, Aiping Liu, and Rujing Wang. 2022a.  Effective Pan-Sharpening With Transformer and Invertible Neural Network.  _IEEE Trans. Geosci. Remote Sens._ 60 (2022), 1–15.
  * Zhou et al. (2022b) Man Zhou, Jie Huang, Yanchi Fang, Xueyang Fu, and Aiping Liu. 2022b.  Pan-sharpening with Customized Transformer and Invertible Neural Network. In _AAAI_.
  * Zhu et al. (2024) Lianghui Zhu, Bencheng Liao, Qian Zhang, Xinlong Wang, Wenyu Liu, and Xinggang Wang. 2024.  Vision Mamba: Efficient Visual Representation Learning with Bidirectional State Space Model.  _arXiv preprint arXiv:2401.09417_ (2024).


# Reverse quantum speed limit and minimum Hilbert space norm

Mark A. Rubin

<EMAIL_ADDRESS>

###### Abstract

The reverse quantum speed limit (RQSL) gives an upper limit to the time for
evolution between initial and final quantum states. We show that, in
conjunction with the existence of a minimum time scale, the RQSL implies a
lower limit to the norm of the change in a quantum state, and confirm that
this limit is satisfied in two-state and ideal-measurement models. Such a
lower limit is of relevance for interpretational issues in probability and for
understanding the meaning of probability in Everett quantum theory.

Key words: reverse quantum speed limit, minimum Hilbert space norm, minimum
time scale, discrete Hilbert space, probability, preclusion, Everett
interpretation

##  1 Introduction

The spacetime that provides the mathematical framework within which both
classical and quantum fields are defined is a continuum. Nevertheless, it has
been argued for decades that the combination of quantum mechanics and general
relativity implies the existence of a minimum length scale at or near the
Planck length.111A device-independent argument for such a minimum length scale
is presented in [1]. For a review of this subject, see [2]. Relativistic
covariance then implies the existence of a corresponding minimum time scale
comparable in magnitude to the Planck time [3, 4].

Buniy, Hsu and Zee [3] have argued that the existence of a minimum length
scale implies a minimum scale to the norms of states in quantum-mechanical
Hilbert space, such that components of the wavefunction with norm below this
scale can be removed from the wavefunction.222Recently Calmet and Hsu [5] have
presented an argument for such a Hilbert-space minimum norm based on a minimum
scale to angular measurement coming from quantum mechanics and general
relativity. The existence of such a minimum Hilbert-space norm provides a
solution to the problem of “maverick” states[6] in Everett quantum theory[7,
8, 9, 10, 11, 12, 13, 14, 15] and thus allows for an understanding of the
origin and nature of probability in that theory [16, 17, 18, 19, 20].

Hsu[19] has presented an estimate for the norm of the change in a quantum
state in the presence of a minimum time scale. In the present note we employ
the reverse quantum speed limit (RQSL) of Mohan and Pati [21] to derive a
lower limit to this norm.

##  2 Reverse quantum speed limit

The RQSL gives an upper limit to the time $T$ for an initial state
$|\psi(0)\rangle$ to evolve to a final state $|\psi(T)\rangle$. The expression
for this limit involves the reference section $|\chi(t)\rangle$, a function of
the state vector $|\psi(t)\rangle$ and the initial time (here taken to be
$t=0$) defined as

$|\chi(t)\rangle=\frac{\langle\psi(t)|\psi(0)\rangle}{|\langle\psi(t)|\psi(0)\rangle|}|\psi(t)\rangle.$
(1)

The length of the reference section for evolution from time $t=0$ to time
$t=T$ is defined to be

$l(\chi(t))|_{0}^{T}=\int_{0}^{T}\langle\dot{\chi}(t)|\dot{\chi}(t)\rangle^{\frac{1}{2}}dt.$
(2)

In terms of the length (2), the RQSL is

$T\leq\frac{\hbar l(\chi(t))|_{0}^{T}}{\Delta H},$ (3)

where $\Delta H$ is the square root of the expectation value of the variance
of the Hamiltonian $\widehat{H}$ in the state $|\psi(t)\rangle$:

$\Delta
H=\left(\langle\psi(t)|\widehat{H}^{2}|\psi(t)\rangle-\langle\psi(t)|\widehat{H}|\psi(t)\rangle^{2}\right)^{\frac{1}{2}}.$
(4)

Here we will only consider the case of time-independent Hamiltonians,333For
the generalization to time-dependent Hamiltonians, see [21]. in which case
$\Delta H$ is time-independent and the formula (4) can be evaluated at any
convenient time, in particular at $t=0$.

##  3 Minimum Hilbert space norm

Consider the time evolution to occur over a time interval $0\leq t\leq\Delta
t$ short enough so that we can approximate the derivative of the reference
section (1) during this interval as

$|\dot{\chi}(t)\rangle=\frac{|\chi(\Delta t)\rangle-|\chi(0)\rangle}{\Delta
t}.$ (5)

Clearly the range of applicability of subsequent results will depend on the
range of validity of the approximation (5); we will return to this issue
below.

Using (5) in (2) with $T=\Delta t$,

$\displaystyle l(\chi(t))|_{0}^{\Delta t}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{\langle\chi(\Delta t)|-\langle\chi(0)|}{\Delta
t}\right)\left(\frac{|\chi(\Delta t)\rangle-|\chi(0)\rangle}{\Delta
t}\right)\right]^{\frac{1}{2}}\Delta t$ (6) $\displaystyle=$
$\displaystyle\left[\left(\langle\chi(\Delta
t)|-\langle\chi(0)|\right)\left(|\chi(\Delta
t)\rangle-|\chi(0)\rangle\right)\right]^{\frac{1}{2}}$

Using (1) in (6), along with the normalization of $|\psi(t)\rangle$, yields

$l(\chi(t))|_{0}^{\Delta t}=\left[\langle\psi(\Delta t)|\psi(\Delta
t)\rangle-2|\langle\psi(\Delta t)|\psi(0)\rangle|+1\right]^{\frac{1}{2}}$ (7)

Expressing $|\psi(\Delta t)\rangle$ in terms of $|\Delta\psi(\Delta
t)\rangle$, the change in $|\psi(t)\rangle$, i.e.

$|\psi(\Delta t)\rangle=|\psi(0)\rangle+|\Delta\psi(\Delta t)\rangle,$ (8)

and defining

$z=1+\langle\Delta\psi(\Delta t)|\psi(0)\rangle$ (9)

we obtain from (7)

$\left(l(\chi(t))|_{0}^{\Delta t}\right)^{2}=\langle\Delta\psi(\Delta
t)|\Delta\psi(\Delta t)\rangle+2\left({\rm Re}(z)-|z|\right).$ (10)

From the RQSL, eq. (3),

$\left(l(\chi(t))|_{0}^{\Delta t}\right)^{2}\geq\left(\frac{\Delta t\Delta
H}{\hbar}\right)^{2},$ (11)

so, with (10), we have

$\langle\Delta\psi(\Delta t)|\Delta\psi(\Delta t)\rangle-\left(\frac{\Delta
t\Delta H}{\hbar}\right)^{2}\geq 2\left(|z|-{\rm Re}(z)\right).$ (12)

But for any complex number $z$

$|z|-{\rm Re}(z)\geq 0,$ (13)

so

$\langle\Delta\psi(\Delta t)|\Delta\psi(\Delta
t)\rangle^{\frac{1}{2}}\geq\frac{\Delta t\Delta H}{\hbar}.$ (14)

Now suppose that time is in fact discrete; i.e., there is a lower limit
$\Delta t_{min}$ during which time evolution can take place. Then

$\langle\Delta\psi(\Delta t_{min})|\Delta\psi(\Delta
t_{min})\rangle^{\frac{1}{2}}\geq{\rm NormLim_{RQSL}},$ (15)

where

${\rm NormLim_{RQSL}}=\frac{\Delta t_{min}\Delta H}{\hbar}.$ (16)

That is, there is no Hamiltonian with expected variance $\left(\Delta
H\right)^{2}$ in state $|\psi(0)\rangle$ which will in a time $\Delta t_{min}$
evolve $|\psi(0)\rangle$ into a state $|\psi(\Delta t_{min})\rangle$ closer in
norm to $|\psi(0)\rangle$ than ${\rm NormLim_{RQSL}}$.444Note that the
estimate given by Hsu [19, eq. (6)] for the norm of the change in the state,
which in our notation is $\Delta
t_{min}\langle\psi(0)|\widehat{H}^{2}|\psi(0)\rangle^{\frac{1}{2}}/\hbar$,
will always be greater than or equal to ${\rm NormLim_{RQSL}}$. Hsu argues
that by considering the estimates for all possible initial states in a finite-
dimensional Hilbert space one may arrive at a lower bound on the norm. The
minimum norm given in the present paper applies to all states with a specified
value of $\Delta H$.

##  4 Examples

Since time appears to us to be continuous rather than discrete, we will take
the approximation (5) and the conclusion (15) that follows from it be valid
provided $\Delta t_{min}$ is much smaller than the smallest time scale
characterizing changes in $|\psi(t)\rangle$. We consider two examples:

###  4.1 Two-state system

The Hilbert space ${\cal H}_{2}$ is spanned by two vectors, $|{\cal
S}_{1}\rangle$ and $|{\cal S}_{2}\rangle$, which we take to be normalized
eigenstates of the Hamiltonian $\widehat{H}_{{\cal H}_{2}}$:

$\widehat{H}_{{\cal H}_{2}}|{\cal S}_{i}\rangle=E_{i}|{\cal
S}_{i}\rangle,\hskip 28.45274pt\langle{\cal S}_{i}|{\cal
S}_{j}\rangle=\delta_{i,j},\hskip 28.45274pti,j=1,2.$ (17)

The normalized initial state vector is

$|\psi_{{\cal H}_{2}}(0)\rangle=c_{1}|{\cal S}_{1}\rangle+c_{2}|{\cal
S}_{2}\rangle,\hskip 28.45274pt|c_{1}|^{2}+|c_{2}|^{2}=1.$ (18)

The lower limit to the norm of the change in the state vector from $t=0$ to
$t=\Delta t_{min}$ is, from (4), (16), (17) and (18),

${\rm NormLim}_{{\rm RQSL}-{\cal
H}_{2}}=\left[|c_{1}|^{2}E_{1}^{2}+|c_{2}|^{2}E_{2}^{2}-\left(|c_{1}|^{2}E_{1}+|c_{2}|^{2}E_{2}\right)^{2}\right]^{\frac{1}{2}}\frac{\Delta
t_{min}}{\hbar}.$ (19)

The exact state vector for $t\geq 0$ is, from (17), (18) and the Schrödinger
equation,

$|\psi_{{\cal
H}_{2}}(t)\rangle=c_{1}\exp\left(\frac{-iE_{1}t}{\hbar}\right)|{\cal
S}_{1}\rangle+c_{1}\exp\left(\frac{-iE_{2}t}{\hbar}\right)|{\cal
S}_{2}\rangle.$ (20)

So, the shortest time scale characterizing changes in $|\psi_{{\cal
H}_{2}}(t)\rangle$ is

$\Delta t_{{\cal H}_{2}}=2\pi\hbar/\max_{i}(|E_{i}|),$ (21)

in that $|\psi(t)\rangle_{{\cal H}_{2}}$ will be approximately constant over
time intervals much smaller than (21).

Using (8), (18) and (20),

$\langle\Delta\psi_{{\cal H}_{2}}(\Delta t)|\Delta\psi_{{\cal H}_{2}}(\Delta
t)\rangle^{\frac{1}{2}}=\left[2-2|c_{1}|^{2}\cos\left(\frac{E_{1}\Delta
t}{\hbar}\right)-2|c_{2}|^{2}\cos\left(\frac{E_{2}\Delta
t}{\hbar}\right)\right]^{\frac{1}{2}}.$ (22)

If

$\Delta t\ll\Delta t_{{\cal H}_{2}}$ (23)

then

$\langle\Delta\psi_{{\cal H}_{2}}(\Delta t)|\Delta\psi_{{\cal H}_{2}}(\Delta
t)\rangle^{\frac{1}{2}}=\left[|c_{1}|^{2}E_{1}^{2}+|c_{2}|^{2}E_{2}^{2}\right]^{\frac{1}{2}}\frac{\Delta
t}{\hbar},$ (24)

so the norm of the change in the state vector during the minimum possible time
interval $\Delta t_{min}$ (assumed, as discussed above, to satisfy (23)) is
greater than or equal to the limit (19) deduced from the RQSL:

$\langle\Delta\psi_{{\cal H}_{2}}(\Delta t_{min})|\Delta\psi_{{\cal
H}_{2}}(\Delta t_{min})\rangle^{\frac{1}{2}}\geq{\rm NormLim}_{{\rm
RQSL}-{\cal H}_{2}}$ (25)

###  4.2 Ideal measurement

We consider the simplest ideal measurement model, perhaps more appropriately
described as a model of a “detector” rather than a “measuring device.” An
observer (a human or an apparatus) starts in a ready state $|{\cal
O}_{1}\rangle$ and measures a two-state system that starts and remains in its
initial state. If the observer finds that the measured system is in state
$|{\cal S}_{1}\rangle$ the observer remains in the ready state; if she finds
the measured system to be in state $|{\cal S}_{2}\rangle$ the observer
transitions to a state $|{\cal O}_{2}\rangle$.

In detail: The Hilbert space ${\cal H}_{D}$ is spanned by the four vectors
$|{\cal S}_{i}\rangle\otimes|{\cal O}_{j}\rangle,$ $i,j=1,2$, where

$\langle{\cal S}_{i}|{\cal S}_{j}\rangle=\langle{\cal O}_{i}|{\cal
O}_{j}\rangle=\delta_{i,j},\hskip 28.45274pti,j=1,2.$ (26)

The Hamiltonian is

$\widehat{H}_{{\cal H}_{D}}=\widehat{P}^{{\cal S}_{2}}\otimes\widehat{h}^{\cal
O}$ (27)

where

$\widehat{P}^{{\cal S}_{2}}=|{\cal S}_{2}\rangle\langle{\cal S}_{2}|$ (28)

and

$\widehat{h}^{\cal O}=i\kappa\left(|{\cal O}_{2}\rangle\langle{\cal
O}_{1}|-|{\cal O}_{1}\rangle\langle{\cal O}_{2}|\right)$ (29)

The state vector describing the observer and measured system at the initial
time $t=0$ has the observer in the ready state and uncorrelated with the
measured system:

$|\psi_{{\cal H}_{D}}(0)\rangle=\left(c_{1}|{\cal S}_{1}\rangle+c_{2}|{\cal
S}_{2}\rangle\right)\otimes|{\cal O}_{1}\rangle,\hskip
28.45274pt|c_{1}|^{2}+|c_{2}|^{2}=1.$ (30)

Using (4), (16) and (26)-(30), we find the lower limit on the norm of the
change in the state vector from $t=0$ to $t=\Delta t_{min}$ to be

${\rm NormLim}_{{\rm RQSL}-{\cal H}_{D}}=\frac{|c_{2}|\kappa\Delta
t_{min}}{\hbar}.$ (31)

The exact state vector for $t\geq 0$ is, from (27)-(30) and the Schrödinger
equation,555Note from (32) that at time $t_{meas}=\pi\hbar/(2\kappa)$ the
ideal measurement has completed, in that the observer and observed-system
states have become perfectly correlated. This confirms that the choice of
Hamiltonian (27) and initial state (30) indeed models an ideal measurement. Of
course $t_{meas}$ is much longer than $\Delta t_{min}$; see (33), (36).

$|\psi_{{\cal H}_{D}}(t)\rangle=c_{1}|{\cal S}_{1}\rangle|{\cal
O}_{1}\rangle+c_{2}|{\cal S}_{2}\rangle\left[\cos\left(\frac{\kappa
t}{\hbar}\right)|{\cal O}_{1}\rangle+\sin\left(\frac{\kappa
t}{\hbar}\right)|{\cal O}_{2}\rangle\right]$ (32)

From (32) we see that the time scale characterizing changes in $|\psi_{{\cal
H}_{D}}(t)\rangle$ is

$\Delta t_{{\cal H}_{D}}=\frac{2\pi\hbar}{|\kappa|}.$ (33)

We can arrive at the same conclusion by noting that the distinct eigenvalues
of $\widehat{H}_{{\cal H}_{D}}$ are $0,\kappa$ and $-\kappa$, so

$\Delta t_{{\cal H}_{D}}=2\pi\hbar/\max(|0|,|\kappa|,|-\kappa|).$ (34)

From (8), (26), (30) and (32),

$\langle\Delta\psi_{{\cal H}_{D}}(t)|\Delta\psi_{{\cal
H}_{D}}(t)\rangle^{\frac{1}{2}}=|c_{2}|\left[2-2\cos\left(\frac{\kappa
t}{\hbar}\right)\right]^{\frac{1}{2}}.$ (35)

For $\Delta t$ satisfying

$\Delta t\ll\Delta t_{{\cal H}_{D}}$ (36)

the norm of the change in the state vector, (35), becomes

$\langle\Delta\psi_{{\cal H}_{D}}(\Delta t)|\Delta\psi_{{\cal H}_{D}}(\Delta
t)\rangle^{\frac{1}{2}}=\frac{|c_{2}|\kappa\Delta t}{\hbar}.$ (37)

Taking $\Delta t$ to be the minimum possible time increment $\Delta t_{min}$,
we see from (31) and (37) that in this case the norm of the change in the
state vector is equal to the limit coming from the RQSL:

$\langle\Delta\psi_{{\cal H}_{D}}(\Delta t_{min})|\Delta\psi_{{\cal
H}_{D}}(\Delta t_{min})\rangle^{\frac{1}{2}}={\rm NormLim}_{{\rm RQSL}-{\cal
H}_{D}}.$ (38)

##  5 Discussion

While the arguments and results in [3, 5, 16, 17, 18, 19, 20] and the present
paper suggest that the usual continuous Hilbert space is not the the proper
arena in which to do quantum physics they do not make this case conclusively,
let alone point to a mathematical structure to serve as a replacement.
Nevertheless Hilbert-space discreteness is an idea that deserves continued
investigation, particularly for its potential to explain one of the most
recondite conceptual problems in physics and philosophy of science, that of
the meaning and nature of probability.

This issue is particularly salient for probability in the Everett
interpretation of quantum mechanics. In that theory, all outcomes of quantum
processes occur in parallel, and since the inception of the theory physicists
have struggled to understand the sense in which probabilities can be
associated with outcomes if everything that can happen does happen [7, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

For example, when the ideal measurement of Sec.4.2 is complete, at time
$t=t_{meas}=\pi\hbar/(2\kappa)$, the state vector (32) has the form

$|\psi_{{\cal H}_{D}}(t_{meas})\rangle=c_{1}|{\cal S}_{1}\rangle|{\cal
O}_{1}\rangle+c_{2}|{\cal S}_{2}\rangle|{\cal O}_{2}\rangle.$ (39)

In the standard (single-outcome, non-Everett) interpretation of quantum
mechanics commonly presented in textbooks, a post-measurement state such as
(39) is replaced stochastically by one of the states corresponding to the
possible outcomes of the measurement, either $|{\cal S}_{1}\rangle|{\cal
O}_{1}\rangle$ (“system is in state 1, observer is in the ready state 1”) or
$|{\cal S}_{2}\rangle|{\cal O}_{2}\rangle$ (“system is in state 2, observer is
in state 2”), with the respective probabilities for each of these outcomes
occurring being the absolute-squared amplitudes $|c_{1}|^{2}$, $|c_{2}|^{2}$.
This is termed “reduction of the state vector”[40, p. 236].

In the Everett interpretation, on the other hand, there is no reduction. The
two terms on the right-hand side of (39) represents two Everett worlds. In one
of these worlds the outcome $|{\cal S}_{1}\rangle|{\cal O}_{1}\rangle$ has
occurred, in the other the outcome $|{\cal S}_{2}\rangle|{\cal O}_{2}\rangle$
has occurred.

What can it possibly mean to say that one outcome is more or less probable
than the other when in fact both have occurred?666It is precisely the
existence of multiple outcomes that is critical in enabling the Everett
interpretation to evade Bell’s theorem and the resulting conclusion that
quantum mechanics is nonlocal; see [41, Sec. 1.1] and references therein. This
question is referred to as the “incoherence problem”[14].

The incoherence problem presents itself in a particularly acute form if we
consider a situation in which one or more of the outcomes is a world
corresponding to a state of affairs that is assigned such small probability by
the standard quantum interpretation that we would feel justified in neglecting
even the possibility of its occurring. For example, such a world might be one
in which large violations of the second law of thermodynamics occur, with heat
flowing from colder to warmer bodies, gases and liquids spontaneously
unmixing, etc. Such worlds are termed “maverick worlds”[6]. Of course our
intuition about the nature of probability leads us to feel that we understand
why any outcome with exceedingly small probability can be
neglected.777Kolmogorov: “We apply the theory of probability to the actual
world of experiments in the following manner…If P($A$) is very small, one can
be practically certain that when conditions [for a repeatable experiment] are
realized only once, the event $A$ would not occur at all[42, pp. 3-4].” But,
in the absence of a meaning we can assign to probability small or large, we
are unable to explain why maverick outcomes are not features of our day-to-day
experience.

Incorporating Hilbert-space discreteness into Everett quantum mechanics
provides a resolution to the problem of maverick worlds. It implies that there
is a minimum norm to state vectors, and one can then argue that maverick
worlds, having norms below this minimum, are simply absent from any
superposition of outcomes[16].

This might seem to still leave open the issue of how to assign meaning to
probabilities larger than, say, those involving violation of the second law of
thermodynamics—i.e., the probabilities encountered in the vast majority of
physical situations. But in fact, preclusion of state-vector components with
norms below a certain minimum holds the potential of giving meaning to these
probabilities as well. For those probabilistic phenomena that are not
precluded but that in fact may or may not be observed to occur, probability
may be identified with subjective experience of a particular kind. In the
words of de Finetti, one of the most prominent proponents of this subjective
interpretation of probability: “What do we mean when we say, in ordinary
language, that an event is more or less probable? We mean that we would be
more or less surprised to learn that it has not happened[43, p. 174].”
Furthermore, there is a substantial body of experimental evidence (see [20,
Sec. 6.4] and references therein) that the subjective experience of
probability is at its base a result of biological evolution[44], [20, Sec.
6.1]. And it can be argued that the biological evolution, over generation upon
generation, of subjective-probabilistic expectations fit to maverick worlds
will be of sufficiently small norm to be precluded, leaving only organisms
with subjective expectations matching our familiar notions of probability and
statistics. In a nutshell, “preclusion explains evolution which then explains
subjective probability[20, p. 21].” A proof-of-concept quantum-mechanical
model of this process is presented in [20, Secs. 6.2, 6.3].

In addition to thus accounting for our experiences of probabilistic phenomena
within the completely-deterministic Everettian framework, Everett quantum
mechanics with preclusion of states below a minimum norm addresses
longstanding interpretational problems of probability per se[45]. Along with
purely-subjective theories of probability, this theory has the virtue of
leaving no ambiguity as to what is meant by probability; namely, the
subjective reactions of organisms to probabilistic situations. At the same
time, the objective feature of preclusion allows it to provide explanations
for physical phenomena in general and for biological evolution in particular,
something which purely subjective theories are incapable of doing. An
explanation should at the very least be “a statement of what is there in
reality, and how it behaves and how that accounts for the explicanda[46].” So,
the explanation of objectively-existing phenomena (including subjective
probability888Subjective reactions are themselves objectively-existing
phenomena. “The subjective opinion, as something known by the individual under
consideration is, at least in this sense, something objective and can be a
reasonable subject of a rigorous study”[47, p. 5].) must itself be something
that exists objectively (in this theory, preclusion and the norms of quantum
states).

For a detailed exposition of probability in the version of Everett quantum
theory with preclusion outlined here, see [20, Secs. 6.1, 7, 8]. Other
approaches to solving the Everett probability problem via discrete Hilbert
space are presented in [16, 17, 18, 19]. Much more work needs to be done to
place minimum Hilbert space norms on a concrete foundation, but the completion
of such a program should prove highly consequential.

## Acknowledgments

I would like to thank Jianbin Mao, Jacob A. Rubin and Allen J. Tino for
helpful discussions.

## References

  * [1] Calmet, X., Graesser, M. and Hsu, S. D. H., “Minimum length from quantum mechanics and classical general relativity,” Phys. Rev. Lett. 93, 211101 (2004);arXiv:hep-th/0405033.
  * [2] S. Hossenfelder, “Minimal length scale scenarios for quantum gravity,” Living Rev. Relativity, 16, 2 (2013).
  * [3] R. V. Buniy, S. D. H. Hsu and A. Zee, “Is Hilbert space discrete?” Phys. Lett. B 630, 68-72 (2005); arXiv:hep-th/0508039.
  * [4] M. Faizal, M. M. Khalil and S. Das, “Time crystals from minimum uncertainty,” Eur. Phys. J. C 76, 30 (2016); arXiv:1501.03111.
  * [5] X. Calmet and S. D. H. Hsu, “Fundamental limit on angular measurements and rotations from quantum mechanics and general relativity,” arXiv:2108.11990 (2021).
  * [6] B. S. DeWitt, “Quantum mechanics and reality,” Physics Today 23, 30-35 (1970).Reprinted in [8].
  * [7] H. Everett III, “ ‘Relative state’ formulation of quantum mechanics,” Rev. Mod. Phys. 29 454-462 (1957). Reprinted in [8].
  * [8] B. S. DeWitt and N. Graham, eds., The Many Worlds Interpretation of Quantum Mechanics (Princeton University Press, Princeton, NJ, 1973).
  * [9] M. C. Price, “The Everett FAQ,” https://www.hedweb.com/everett/ (1995).
  * [10] J. A. Barrett, The quantum mechanics of minds and worlds (Oxford University Press, Oxford,1999).
  * [11] L. Vaidman, “Many-worlds interpretation of quantum mechanics,” The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), E. N. Zalta, ed., https://plato.stanford.edu/archives/fall2018/entries/qm-manyworlds/.
  * [12] C. Hewitt-Horsman, “An introduction to many worlds in quantum computation,” Found. Phys. 39, 826-902 (2009); arXiv:0802.2504.
  * [13] S. Saunders, J. Barrett, A. Kent and D. Wallace, eds., Many Worlds? Everett, Quantum Theory & Reality, (Oxford University Press, Oxford, 2010).
  * [14] D. Wallace, The Emergent Multiverse: Quantum Theory According to the Everett Interpretation, (Oxford University Press, Oxford, 2012).
  * [15] D. Wallace, “The Everett interpretation,” in R. Batterman, ed., The Oxford Handbook of Philosophy of Physics (Oxford University Press, Oxford, 2013).
  * [16] R. V. Buniy, S. D. H. Hsu and A. Zee, “Discreteness and the origin of probability in quantum mechanics,” Phys.Lett. B 640, 219-223 (2006) ; arXiv:hep-th/0606062.
  * [17] S. D. H. Hsu, “On the origin of probability in quantum mechanics,” Mod. Phys. Lett. A 27, 1230014 (2012); arXiv:1110.0549.
  * [18] S. D. H. Hsu, “The measure problem in no-collapse (many worlds) quantum mechanics,” Int. J. Mod. Phys. D 26, 1730008 (2017); arXiv:1511.08881.
  * [19] S. D. H. Hsu, “Discrete Hilbert space, the Born rule, and quantum gravity,” Mod. Phys. Lett. A 36, 2150013 (2021); arXiv:2007.12938.
  * [20] M. A. Rubin, “Probability, preclusion and biological evolution in Heisenberg-picture Everett quantum mechanics,” Int. J. Mod. Phys. A 36, 2150117 (2021); arXiv:2011.10029.
  * [21] B. Mohan and A. K. Pati, “Reverse quantum speed limit: How slow quantum battery can discharge?” arXiv:2006:14523 (2020).
  * [22] B. S. DeWitt, “The many-universes interpretation of quantum mechanics,” in Proceedings of the International School of Physics “Enrico Fermi” Course IL: Foundations of Quantum Mechanics, (Academic Press, Inc., New York, 1972). Reprinted in [8]
  * [23] N. Graham, “The measurement of relative frequency,” in [8].
  * [24] Y. Okhuwa, “Decoherence functional and probability interpretation,” Phys. Rev.  D48, 1781-1784 (1993).
  * [25] A. Kent, “Against many-worlds interpretations,” Int. J. Mod. Phys. A5 1745 (1990); arXiv:gr-qc/9703089.
  * [26] D. Deutsch, “Quantum theory of probability and decisions,” Proc. Roy. Soc. Lond. A 455, 3129-3137 (1999); quant-ph/9906015.
  * [27] H. Barnum, C. M. Caves, J. Finkelstein, C. A. Fuchs and R. Schack, “Quantum probability from decision theory?” Proc. Roy. Soc. Lond. A456, 1175-1182 (2000); arXiv:quant-ph/9907024
  * [28] A. I. M. Rae, “Everett and the Born rule,” Studies in History and Philosophy of Modern Physics 40 243-250 (2009); arXiv:0810.2657.
  * [29] S. Saunders, “Chance in the Everett interpretation,” in [13], 181-205.
  * [30] D. Papineau, “A fair deal for Everettians,” in [13], 206-226.
  * [31] D. Wallace, “How to prove the Born rule,” in [13], 227-263.
  * [32] H. Greaves and W. Myrvold, “Everett and evidence,” in [13], 264-304.
  * [33] A. Kent, “One world versus many: The inadequacy of Everettian accounts of evolution, probability, and scientific confirmation,” in [13], 307-334.
  * [34] D. Albert, “Probability in the Everett interpretation,” in [13], 355-368.
  * [35] H. Price, “Decisions, decisions, decisions: Can Savage salvage Everettian probability?,” in [13], 369-390.
  * [36] W. H. Zurek, “Quantum jumps, Born’s rule, and objective reality,” in [13], 409-432.
  * [37] R. Schack, “The principal principle and probability in the many-worlds interpretation,” in [13], 467-475.
  * [38] L. Vaidman “Probability and the Many-Worlds interpretation of quantum theory, ” in Proceedings of the Conference ”Quantum Theory, Reconsideration of Foundations” : Växjö (Smaland), Sweden, 17-21 June, 2001 (Växjö University Press, Växjö, Sweden, 2001)); arXiv:quant-ph/0111072
  * [39] C. T. Sebens and S. M. Carroll, “Self-locating uncertainty and the origin of probability in Everettian quantum mechanics.” The British Journal for the Philosophy of Science, 69, 25-74 (2018);arXiv:1405.7577.
  * [40] L. E. Ballentine, Quantum Mechanics: A Modern Development (World Scientific Publishing Co. Pte. Ltd., Singapore, 1978).
  * [41] M. A. Rubin, “Observers and locality in Everett quantum field theory,” Found.Phys. 41, 1236-1262 (2011);arXiv:0909.2673.
  * [42] A. N. Kolmogorov, Foundations of the Theory of Probability, 2nd English ed. (Chelsea Publishing Company, New York, 1956).
  * [43] B. de Finetti, “Probabilism: A critical essay on the theory of probability and on the value of science,” Erkenntnis 31, 169-223 (1989).
  * [44] B. O. Koopmans, “The axioms and algebra of intuitive probability,” Annals of Mathematics, 2nd series, 41, 269-292 (1940).
  * [45] A. Hájek, “Interpretations of Probability,” The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), E. N. Zalta ed., https://plato.stanford.edu/archives/fall2019/entries/probability-interpret/.
  * [46] D. Deutsch, “The logic of experimental tests, particularly of Everettian quantum theory,” Studies in History and Philosophy of Modern Physics 55 24-33 (2016).
  * [47] B. de Finetti, Theory of Probability: A Critical Introductory Treatment (John Wiley & Sons Ltd., West Sussex, 2017).