# Long-time Quantum Information Scrambling and Emergent Mereology

Faidon Andreadakis<EMAIL_ADDRESS>Department of Physics and Astronomy, and
Center for Quantum Information Science and Technology, University of Southern
California, Los Angeles, California 90089-0484, USA  Emanuel Dallas
<EMAIL_ADDRESS>Department of Physics and Astronomy, and Center for Quantum
Information Science and Technology, University of Southern California, Los
Angeles, California 90089-0484, USA  Paolo Zanardi<EMAIL_ADDRESS>Department
of Physics and Astronomy, and Center for Quantum Information Science and
Technology, University of Southern California, Los Angeles, California
90089-0484, USA  Department of Mathematics, University of Southern California,
Los Angeles, California 90089-2532, USA

###### Abstract

Recent work by Zanardi et al. [1] has associated each possible partition of a
quantum system with an operational subalgebra and proposed that the short-time
growth of the algebraic out-of-time-order-correlator (“$\mathcal{A}$-OTOC”) is
a suitable criterion to determine which partition arises naturally from the
system’s unitary dynamics. We extend this work to the long-time regime.
Specifically, the long-time average of the $\mathcal{A}$-OTOC serves as our
metric of subsystem emergence. Under this framework, natural system partitions
are characterized by the tendency to minimally scramble information over long
time scales. We derive an analytic expression for the $\mathcal{A}$-OTOC long-
time average under the non-resonance condition. We then consider several
physical examples and perform minimization of this quantity both analytically
and numerically over relevant families of algebras. For simple cases subject
to the non-resonant condition, minimal $\mathcal{A}$-OTOC long-time average is
shown to be related to minimal entanglement of the Hamiltonian eigenstates
across the emergent system partition. Finally, we conjecture and provide
evidence for a general structure of the algebra that minimizes the average for
non-resonant Hamiltonians.

##  I Introduction

Information-theoretic properties of quantum dynamics provide insights
applicable to a wide range of physical systems. Quantum information scrambling
is a prominent such property and refers to the dynamical generation of
correlations among initially distinguishable degrees of freedom. A wide range
of phenomena, from thermalization in quantum many-body systems to black hole
physics and holography [2, 3, 4, 5, 6, 7, 8, 9, 10], have been linked to
scrambling dynamics, which are commonly studied using the out-of-time-order
correlator (OTOC) as a diagnostic tool [11, 12, 13, 14, 15, 16, 17, 18, 19,
20]. Given a specific background structure (e.g., in terms of spatial
locality, such as a collection of neighboring qubits) and physical system of
interest, suitable averages of OTOCs over the corresponding degrees of freedom
quantify its scrambling dynamics and reveal connections with other
information-theoretic concepts, such as operator entanglement and entropy
production [21, 22, 23], quantum coherence [24] and quasiprobabilities [25],
amongst others [26, 27, 28].

However, one can take a reversed approach: given a dynamics, different
partitionings of the system can be distinguished by their distinct scrambling
properties, revealing an emergent structure. This corresponds to an
information-theoretic approach to quantum mereology 111The term mereology is
borrowed from philosophy, where it has a rich history. For a succinct
overview, see [29]., the study of parthood relations in quantum systems [30,
1, 31].

Most recently, Zanardi et al. [1] use the short-time expansion of an algebraic
out-of-time-order correlator ($\mathcal{A}$-OTOC) [32] as a mereological
diagnostic criterion. The $\mathcal{A}$-OTOC serves as a metric of information
scrambling between an algebra of observables and its commutant under dynamics,
and provides a unified framework that incorporates operator entanglement [33,
26, 21] and coherence-generating power [34, 35] as special cases. The
algebraic approach to quantum mereology relies on two general mathematical
properties of Hilbert spaces. First, Hilbert spaces are not equipped with
unique tensor product structures, even if a physical system may have a
“natural” decomposition (e.g., a collection of qubits, or an obvious system
and bath). Second, a subalgebra of observables induces a specific
decomposition of its Hilbert space222See Section II for a formal statement..
This “structural” freedom was recognized in previous work on virtual quantum
subsystems [36, 37] and is relevant to decoherence-free subspaces [38, 39],
noiseless subsystems [40, 41], operator-error correction [42] and quantum
reference frames [43]. It is intuitive, as Zanardi et al. proposed, to
consider the minimization of the short-time $\mathcal{A}$-OTOC as a selection
of a preferred Hilbert space decomposition into parts which scramble their
informational identities most slowly.

In this work, we extend this approach to the long-time regime. This regime is
relevant when the timescale of interest far exceeds the characteristic quantum
(i.e., “microscopic”) system timescale and has been examined in studies of
quantum chaos [44, 45, 46] and quantum phase transitions [47]. In fact, for
the case of bipartite algebras, the scaling of the equilibration value of the
$\mathcal{A}$-OTOC distinguishes between the chaotic and integrable phases of
quantum many-body systems [21, 23]. As correlation functions do not have
infinite-time limits under unitary dynamics in finite-dimensional systems, we
probe the long-time behavior of the $\mathcal{A}$-OTOC via its long-time
average (LTA), and we interpret the minimization of the LTA over algebras as a
selection of a preferred decomposition of a system at “macroscopic”
timescales. This selection is a task of fundamental interest in understanding
emergent structure and phenomena in quantum systems, as well as comparing the
differences in structure between short and long time scales. It may also serve
as a practical diagnostic for the study of scrambling of operationally
accessible observables on quantum circuits [14]. In some cases, the
$\mathcal{A}$-OTOC minimization can be interpreted as fixing an algebra of
observables and identifying a least-scrambling unitary dynamics, which can be
a more meaningful interpretation in examples involving quantum control.

The $\mathcal{A}$-OTOC LTA becomes more analytically tractable under the
assumption that the system dynamics satisfy the no-resonance condition (NRC),
meaning the energy spectrum contains no degeneracies nor degenerate gaps. The
condition of nondegenerate energy gaps is generically satisfied by fully
interacting Hamiltonians [48], while the full NRC is expected to hold (either
exactly or approximately) for generic chaotic Hamiltonians [49]. Furthermore,
the addition of small random perturbations generally lifts any degeneracies so
that the NRC is satisfied, granted that the timescales for which the effect of
such perturbations is significant may be large.

In Section II, we formally introduce the $\mathcal{A}$-OTOC as well as the
decomposition of Hilbert space induced by an algebra and its commutant. In
Section III, we take the long-time average of the $\mathcal{A}$-OTOC and
derive expressions for the $\mathcal{A}$-OTOC LTA that hold in non-resonant
systems, showcasing their use numerically for a family of stabilizer algebras.
In Section IV we study the minimization of the LTA for unitary families of NRC
Hamiltonians, providing a simple analytical application in the context of
quantum reference frames, as well as exhibiting numerically the connection of
the bipartite $\mathcal{A}$-OTOC LTA with the average eigenstate mutual
information in certain quantum many-body systems. Lastly, Section V provides
conclusions and steps forward for future research.

##  II Preliminaries

Consider a finite-dimensional quantum system represented by a Hilbert space
$\mathcal{H}\cong\mathbb{C}^{d}$. Any physical observable is represented by a
linear operator and we denote as $\mathcal{L}(\mathcal{H})$ the space of all
linear operators on $\mathcal{H}$. $\mathcal{L}(\mathcal{H})$ is also a
Hilbert space equipped with the Hilbert-Schmidt inner product: $\langle
X,Y\rangle=\Tr(X^{\dagger}\,Y)$. For closed quantum systems in the Heisenberg
picture, the time evolution of a physical observable
$X\in\mathcal{L}(\mathcal{H})$ is given as
$\mathcal{U}_{t}(X)=U_{t}\,XU_{t}^{\dagger}$, where $U_{t}=\exp{it\,H}$ is the
unitary evolution generated by the system Hamiltonian $H$.

The central mathematical structures in this paper are hermitian-closed, unital
subalgebras $\mathcal{A}\subset\mathcal{L}(\mathcal{H})$ that are used to
describe the relevant degrees of freedom of interest. The symmetries of
$\mathcal{A}$ constitute the commutant algebra
$\mathcal{A}^{\prime}=\\{Y\in\mathcal{A}^{\prime}\,\lvert\,[Y,X]=0\;\forall
X\in\mathcal{A}\\}$ and correspond to degrees of freedom initially
uncorrelated with $\mathcal{A}$. Due to the double commutant theorem,
$(\mathcal{A}^{\prime})^{\prime}=\mathcal{A}$ [50], and therefore these
algebras can be considered as pairs $(\mathcal{A},\mathcal{A}^{\prime})$.
Under time evolution, information is scrambled between $\mathcal{A}$ and
$\mathcal{A}^{\prime}$, quantified by the $\mathcal{A}$-OTOC [32]:

######  Definition 1.

The $\mathcal{A}$-OTOC of algebra $\mathcal{A}$ and unitary $\mathcal{U}_{t}$
is defined as:

$G_{\mathcal{A}}(\mathcal{U}_{t})=\frac{1}{2d}{{\mathbb{E}}}_{X_{\mathcal{A}},Y_{\mathcal{A}^{\prime}}}\left[\left\lVert
X_{\mathcal{A}},\mathcal{U}_{t}(Y_{\mathcal{A}^{\prime}})\right\rVert_{2}^{2}\right].$
(1)

$\mathcal{A}$$\mathcal{Z(A)}$$\mathcal{A^{\prime}}$Time
Evolution$\mathcal{A}$$\mathcal{Z(A)}$$\mathcal{A^{\prime}}(t)$Information
scrambling between $\mathcal{A^{\prime}}$ and $\mathcal{A}$ after time $t$

Figure 1: A visual representation of information scrambling induced by
Hamiltonian time evolution. The “component” of $\mathcal{A^{\prime}}(t)$ that
“leaks” into $\mathcal{A}$ corresponds to the scrambling; this is represented
by the dotted component of $\mathcal{A^{\prime}}(t)$.

$\mathbb{E}_{X_{\mathcal{A}},Y_{\mathcal{A}^{\prime}}}$ denotes the Haar
average over the unitaries $X_{\mathcal{A}}\in\mathcal{A}$ and
$Y_{\mathcal{A}^{\prime}}\in\mathcal{A}^{\prime}$. The evolution of operators
in $\mathcal{A}^{\prime}$ under $\mathcal{U}_{t}$ leads to potential non-
commutativity with operators in $\mathcal{A}$, which is intuitively
interpreted as information scrambling between the corresponding degrees of
freedom. This is operator spreading in the space of algebras, where locality
is defined relative to $\mathcal{A}$. The $\mathcal{A}$-OTOC thus provides an
information scrambling measure with respect to a generalized locality
structure (see Eq. 2), independent of a specific choice of operators
$X_{\mathcal{A}}$ and $Y_{\mathcal{A}^{\prime}}$.

Importantly, the subalgebra of observables $\mathcal{A}$ induces a
decomposition of the Hilbert space $\mathcal{H}$ into a direct sum of virtual
quantum subsystems [36, 37], referred to as generalized tensor product
structure (gTPS) [1]. Specifically, denoting the center of the algebra as
$\mathcal{Z}(\mathcal{A})\coloneqq\mathcal{A}\cap\mathcal{A}^{\prime}$ with
dimension $\dim{\mathcal{Z}(\mathcal{A})}=d_{\mathcal{Z}}$, we have

$\mathcal{H}=\bigoplus_{J=1}^{d_{Z}}\mathcal{H}_{J},\quad\mathcal{H}_{J}\cong\mathbb{C}^{n_{J}}\otimes\mathbb{C}^{d_{J}}.$
(2)

where $\mathcal{A}$ and $\mathcal{A}^{\prime}$ act irreducibly on the
$\mathbb{C}^{d_{J}}$ and $\mathbb{C}^{n_{J}}$ factors respectively:

$\mathcal{A}\cong\bigoplus_{J=1}^{d_{Z}}\mathbb{I}_{n_{J}}\otimes\mathcal{L}(\mathbb{C}^{d_{J}}),\quad\mathcal{A}^{\prime}\cong\bigoplus_{J=1}^{d_{Z}}\mathcal{L}(\mathbb{C}^{n_{J}})\otimes\mathbb{I}_{d_{J}}.$
(3)

From Eqs. 2 and 3, we have $d=\sum_{J}n_{J}d_{J}$, $\dim{\mathcal{A}}\coloneqq
d(\mathcal{A})=\sum_{J}d_{J}^{2}$ and $\dim{\mathcal{A}^{\prime}}\coloneqq
d(\mathcal{A}^{\prime})=\sum_{J}n_{J}^{2}$.

For any given algebra $\mathcal{A}$, there is an orthogonal completely
positive projection map
$\mathbb{P}_{\mathcal{A}}:\mathcal{L}(\mathcal{H})\rightarrow\mathcal{A}$,
such that $\mathbb{P}_{\mathcal{A}}=\mathbb{P}_{\mathcal{A}}^{2}$,
$\mathbb{P}_{\mathcal{A}}=\mathbb{P}_{\mathcal{A}}^{\dagger}$,
$\imaginary\mathbb{P}_{\mathcal{A}}=\mathcal{A}$. In terms of Eq. 3, we have
$\mathbb{P}_{\mathcal{A}}(\cdot)=\oplus_{J}\frac{\mathds{1}_{n_{J}}}{n_{J}}\otimes\Tr_{n_{J}}(\cdot)$
and, similarly,
$\mathbb{P}_{\mathcal{A}^{\prime}}(\cdot)=\oplus_{J}\Tr_{d_{J}}(\cdot)\otimes\frac{\mathds{1}_{d_{J}}}{d_{J}}$.
Using appropriate orthogonal bases
$\\{f_{\gamma}\\}_{\gamma=1}^{d(\mathcal{A}^{\prime})}$ of
$\mathcal{A}^{\prime}$ and $\\{e_{\alpha}\\}_{\alpha=1}^{d(\mathcal{A})}$ of
$\mathcal{A}$, the projection maps have the Kraus representations
$\mathbb{P}_{\mathcal{A}}(\cdot)=\sum_{\gamma}f_{\gamma}(\cdot)f_{\gamma}^{\dagger}$,
$\mathbb{P}_{\mathcal{A}^{\prime}}(\cdot)=\sum_{\alpha}e_{\alpha}(\cdot)e_{\alpha}^{\dagger}$
(see Eq. 24 for an explicit expression).

##  III Algebra scrambling in the long-time limit

We begin with a Hamiltonian with a spectral decomposition
$H=\sum_{k=1}^{M}E_{k}\Pi_{k}$, where $M$ is the number of distinct energy
levels. The scrambling properties of the unitary dynamics $U_{t}=\exp{it\,H}$
over algebras $\mathcal{A}$ in the long-time regime can be quantified by
utilizing the infinite-time average of the $\mathcal{A}$-OTOC, hereafter
referred to as $\mathcal{A}$-OTOC long-time time average (LTA):

$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}\coloneqq\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}G_{\mathcal{A}}(\mathcal{U}_{t})\,dt.$
(4)

Typical quantum systems reach equilibrium [51, 52, 53], whence the
$\mathcal{A}$-OTOC relaxes to an equilibration value given by the LTA.

In [1], the short-time expansion of the $\mathcal{A}$-OTOC, referred to as the
“Gaussian scrambling rate” $\tau_{S}^{-1}$, was introduced by Zanardi et al.
as a mereological criterion for any algebra $\mathcal{A}$. Specifically, it
was shown that $G_{\mathcal{A}}(\mathcal{U}_{t})=2(t/\tau_{S})^{2}+O(t^{3})$
where

$\tau_{S}^{-1}=\left\lVert\left(1-\mathbb{P}_{\mathcal{A}+\mathcal{A}^{\prime}}\right)\frac{H}{\sqrt{d}}\right\rVert_{2}$
(5)

At short time scales, this quantity describes the rate at which information
scrambles from $\mathcal{A}$ to its commutant $\mathcal{A^{\prime}}$ and is
related to the distance of the Hamiltonian $H$ from the “no-scrambling”
operator subspace $\mathcal{A}+\mathcal{A}^{\prime}$. Given the association
between algebras and virtual partitions of systems (Eq. 2), the Gaussian
scrambling rate describes how quickly a collection of parts loses its
“informational identity.” Equipped with this interpretation, Zanardi et al.
argue that it is natural to consider partitions which retain their identity
the longest to be dynamically “preferred.” Thus, for a family of algebras
$\left\\{\mathcal{A_{\theta}}\right\\}_{\theta}$ and Hamiltonian $H$, the
algebra $\mathcal{A}_{min}\in\left\\{\mathcal{A_{\theta}}\right\\}_{\theta}$
with the smallest $\tau_{S}^{-1}$ “emerges” as the dynamically preferred
decomposition of the system under the action of $H$. Then, the authors use
this framework to perform minimization of $\tau_{S}^{-1}$ for various
Hamiltonians and families of algebras.

In this paper, we follow the same framework as [1] but focus on long time
scales. We use the $\mathcal{A}$-OTOC LTA as our mereological criterion, which
we interpret as measuring the extent to which a decomposition of the Hilbert
space persists, on average, with informational “integrity.”. The remainder of
this paper thus relies on the following:

######  Principle.

Given a Hamiltonian $H$ and set of Hilbert space partitions represented by a
parametrized family of algebras
$\left\\{\mathcal{A_{\theta}}\right\\}_{\theta}$, the dynamically preferred
partition at long time scales,
$\mathcal{A}_{min}\in\left\\{\mathcal{A_{\theta}}\right\\}_{\theta}$, is the
one that minimizes $\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}$.

###  III.1 THE $\mathcal{A}$-OTOC UNDER THE NRC

We will now look to the non-resonant regime. There are two types of systems
that we consider. The first are those satisfying the NRC+ [21]: energy gaps
are non-degenerate but there exist energy level degeneracies. The second are
those with both non-degenerate energy gaps and energy levels, satisfying the
full NRC. The LTA for each is written below:

######  Proposition 1.

  1. i.

$\begin{split}&\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC^{+}}=\\\
&1-\frac{1}{d}\left[\sum\limits_{\gamma}\left(\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(D_{H}(f_{\gamma}))\rVert_{2}^{2}-\frac{1}{2}\sum\limits_{k}\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(\Pi_{k}f_{\gamma}\Pi_{k})\rVert_{2}^{2}\right)+\right.\\\
&\quad\quad+\left.\sum\limits_{\alpha}\left(\lVert\mathbb{P}_{\mathcal{A}}(D_{H}(e_{\alpha}))\rVert_{2}^{2}-\frac{1}{2}\sum\limits_{k}\lVert\mathbb{P}_{\mathcal{A}}(\Pi_{k}e_{\alpha}\Pi_{k})\rVert_{2}^{2}\right)\right],\end{split}$
(6)

where $D_{H}(\cdot)\coloneqq\sum_{k=1}^{M}\Pi_{k}(\cdot)\Pi_{k}$ is the
dephasing map with respect to the eigenprojectors of $H$ and $f_{\gamma}$,
$e_{\alpha}$ are given in Eq. 24.

  2. ii.

$\begin{split}\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}=1-\frac{1}{d}&\left(\sum_{\mathcal{X}=\\{\mathcal{A},\mathcal{A}^{\prime}\\}}\Tr(R^{(0),\mathcal{X}}R^{(1),\mathcal{X}^{\prime}})-\right.\\\
&\left.\quad\quad-\frac{1}{2}\Tr(R_{D}^{(0),\mathcal{X}}R_{D}^{(1),\mathcal{X}^{\prime}})\right),\end{split}$
(7)

where, for algebra $\mathcal{X}$, we define
$R_{lk}^{(0),\mathcal{X}}\coloneqq\left\lVert\mathbb{P}_{\mathcal{X}}\left(\ket{\phi_{k}}\bra{\phi_{l}}\right)\right\rVert_{2}^{2}$,
$R_{kl}^{(1),\mathcal{X}}\coloneqq\left\langle\mathbb{P}_{\mathcal{X}}\left(\Pi_{k}\right),\mathbb{P}_{\mathcal{X}}\left(\Pi_{l}\right)\right\rangle$,
and for matrix $M$, $M_{D}\coloneqq diag(M)$.

A few observations follow directly. First, the infinite-time average ensures
that these NRC LTAs are independent of any specific energy eigenvalues. Any
Hamiltonians sharing a fixed set of eigenstates and satisfying the NRC will
have the same $\mathcal{A}$-OTOC LTA. Second, the above NRC expressions are
generally not bounds on the exact LTA for systems not satisfing the NRC — the
net sum of terms discarded in these expressions can be either positive or
negative. A notable exception to this is for the important special case of
algebras that satisfy $n_{J}=\lambda\,d_{J}\;\forall\,J$ for some integer
$\lambda$, whence the $\mathcal{A}$-OTOC has a neat geometrical representation
and the NRC approximation provides an upper bound [54]. Finally, systems with
energy degeneracies do not necessarily have a unique NRC LTA. This is because
the NRC expression depends on a choice of eigenbasis within degenerate energy
levels, since all projectors in the formula are one-dimensional. These last
two observations are especially relevant for usage of the NRC LTA expression
as an approximation for the exact value.

######  Example 1 (Stabilizer algebras).

Let $G_{S}$ be the stabilizer group generated by the stabilizer operators
$\\{S_{l}\\}_{l=1}^{n-k}$ associated with a stabilizer code with $k$ logical
qubits. Let $\mathcal{A}_{st}\coloneqq\mathbf{C}[G_{S}]$ be the group algebra
of $G_{S}$. Then, the Hilbert space decomposes into $2^{k}$-dimensional
sectors $\mathcal{H}\cong\oplus_{J=1}^{2^{n-k}}\mathbb{C}^{2^{k}}$ that
correspond to the encoding and syndrome subspaces. In terms of Eq. 2, we have
$d_{J}=1$ and $n_{J}=2^{k}\,\forall\,J=1,\dots,2^{n-k}$.

Suppose that we have a family of such stabilizer algebras and some unitary
dynamics $\mathcal{U}$. The “optimal” stabilizer algebra, in terms of
minimizing the $\mathcal{A}$-OTOC LTA, will correspond to encoding and
syndrome subspaces that are (on average) less “scrambled" by the dynamics. In
general, this optimization is non-trivial, but we expect that in many cases
the Eqs. 6 and 7 may serve as “proxies” to the exact behavior of
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}$. To illustrate this consider
the family of algebras
$\mathscr{A}=\\{\exp(i\theta\sigma_{y})\mathcal{A_{\text{5}}}\exp(-i\theta\sigma_{y})\,|\,\theta\in[0,\pi/4]\\}$,
where $\mathcal{A_{\text{5}}}$ is the stabilizer algebra associated to the
stabilizer operators of the 5-qubit “perfect” code:
$\\{XZZXI,IXZZX,XIXZZ,ZXIXZ\\}$. For the unitary dynamics, we consider the
Heisenberg model with closed boundary conditions and a magnetic field term
$H=\sum_{i=1}^{5}\left(h\,\sigma_{z}^{i}+\vec{\sigma}^{i}\vec{\sigma}^{i+1}\right)$,
where $\vec{\sigma}^{6}\equiv\vec{\sigma}^{1}$ and $h$ is a coupling constant.

Using the chosen family of algebras and dynamics, we compute as a function of
$\theta$ the NRC and NRC+ approximations of the $\mathcal{A}$-OTOC LTA, as
well as the exact one obtained by numerical simulations of the time evolution.
We perform the above computations for $h=0$ and $h=1$ (see Fig. 2). We observe
that for $h=1$ the NRC and NRC+ approximations accurately capture the behavior
of the exact LTA, with minima occurring at $\theta=0$ and $\theta=\pi/4$,
which corresponds to a rotation of $X\rightarrow Z$, $Z\rightarrow-X$. On the
other hand, for $h=0$ the rotation $\exp(i\theta\sigma_{y})$ is a symmetry of
the Hamiltonian, so the exact $LTA$ is constant as a function of $\theta$;
this behavior is accurately captured by the NRC+ (that uses the similarly
invariant under the rotation eigenprojectors), whereas the NRC approximation
gives a non-constant value as the chosen (non-unique) eigenbasis is not
necessarily invariant under the Hamiltonian symmetry.

(a)

(b)

Figure 2: The $\mathcal{A}$-OTOC LTA for the family of stabilizer algebras
$\mathscr{A}=\\{\exp(i\theta\sigma_{y})\mathcal{A_{\text{5}}}\exp(-i\theta\sigma_{y})\,|\,\theta\in[0,\pi/4]\\}$
of the 5-qubit “perfect” code with the dynamics given by the Heisenberg model.
(a) For $h=0$ the exact $LTA$ is constant for all the stabilizer algebras
since the rotation $\exp{i\theta\sigma_{y}}$ is a symmetry of the Hamiltonian;
the NRC+ approximation accurately captures this (since the eigenprojectors are
similarly invariant), however, the NRC approximation gives a non-constant
value as the chosen eigenbasis is not necessarily invariant under the
rotation. (b) For $h=1$ both the NRC and NRC+ approximation accurately capture
the behavior of the exact LTA, with minima occuring at $\theta=0$ and
$\theta=\pi/4$.

##  IV $\mathcal{A}$-OTOC long-time-average Minimization

We can utilize Eq. 7 to analytically minimize the $\mathcal{A}$-OTOC LTA over
unitary families of NRC Hamiltonians. Note that, in general, for Eq. 1,
$G_{\mathcal{A}}(\mathcal{W}\,\mathcal{U}_{t}\,\mathcal{W}^{\dagger})=G_{\mathcal{W}^{\dagger}(\mathcal{A})}(\mathcal{U}_{t})$,
where $\mathcal{W}$ is some unitary channel. This means that optimizing over
unitary families of Hamiltonians is dual to optimizing over unitary families
of algebras.

######  Proposition 2.

Let
$\mathcal{A}\cong\oplus_{J=1}^{d_{\mathcal{Z}}}\mathds{1}_{n_{J}}\otimes\mathcal{L}(\mathbb{C}^{d_{J}})$
be an algebra of observables and $H_{f}$ be a family of NRC Hamiltonians that
respect the superselection structure of the gTPS induced by $\mathcal{A}$,
i.e. $H\in H_{f}\Leftrightarrow H=\oplus_{J}H_{J}$ and $H$ satisfies NRC.
Then:

  1. i.

The Hamiltonian with eigenstates that are product states over the virtual
quantum subsystems, i.e. $\\{\ket{\psi_{J}}\otimes\ket{\phi_{J}}\,|\,J=1,\dots
d_{\mathcal{Z}};\,\psi_{J}=1,\dots,n_{J};\,\phi_{J}=1,\dots,d_{J}\\}$,
minimizes the $\mathcal{A}$-OTOC LTA NRC. The minimum value is

$\overline{G_{\mathcal{A}}}^{NRC}_{min}=1-\frac{1}{d}\left(\sum_{J}d_{J}+\sum_{J}n_{J}-d_{\mathcal{Z}}\right)$
(8)

  2. ii.

In particular, if $\mathcal{A}$ is a bipartite algebra ($d_{\mathcal{Z}}=1$),
the Hamiltonian with product eigenstates over the bipartition minimizes the
bipartite OTOC LTA NRC over all NRC Hamiltonians. The minimum value is

$\overline{G}^{NRC}_{min}=1-\frac{1}{d_{1}}-\frac{1}{d_{2}}+\frac{1}{d_{1}d_{2}}$
(9)

where $\mathcal{H}\cong\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$.

The above proposition shows that for NRC Hamiltonians that are block diagonal
with respect to the gTPS Eq. 2 induced by $\mathcal{A}$, the minimum of the
$\mathcal{A}$-OTOC LTA NRC is achieved exactly by Hamiltonians with
eigenstates that have zero entanglement across the virtual quantum subsystems
of the gTPS. Note that these eigenstates are unique up to local unitaries on
each virtual bipartition.

In addition, if $\mathcal{A}$ or $\mathcal{A}^{\prime}$ is abelian (i.e. if
$n_{J}=1\,\forall\,J$ or $d_{J}=1\,\forall\,J$), there is always an NRC block
diagonal Hamiltonian that belongs in $\mathcal{A}^{\prime}$ or $\mathcal{A}$,
respectively. Indeed, the NRC minimum of Eq. 8 is
$\overline{G_{\mathcal{A}}}^{NRC}_{min}=0$, i.e. there is no information
scrambling. A special case is when $\mathcal{A}$ is a maximally Abelian
algebra $\mathcal{A}=\\{\ket{k}\bra{k}\\}_{k=1}^{d}$, whence the
$\mathcal{A}$-OTOC is equal to the coherence-generating power of $U$ in the
basis $B=\\{\ket{k}\\}_{k=1}^{d}$[54, 32]. In this case, minimal scrambling
corresponds to Hamiltonians that have $B$ as the eigenbasis.

Based on the above results, we conjecture that, for a given algebra
$\mathcal{A}$, the NRC Hamiltonians with eigenstates of the form
$\\{\ket{\psi_{J}}\otimes\ket{\phi_{J}}\,|\,J=1,\dots
d_{\mathcal{Z}};\,\psi_{J}=1,\dots,n_{J};\,\phi_{J}=1,\dots,d_{J}\\}$ minimize
the $\mathcal{A}$-OTOC LTA over the set of all NRC Hamiltonians. Note that the
conjectured Hamiltonian is diagonal in the basis333The distinguished basis
corresponding to $\mathcal{A}$ consists of vectors
$\ket{p_{J}}\otimes\ket{k_{J}}$ $(p_{J}=1,\dots n_{J},\,k_{J}=1,\dots,d_{J})$
and is unique up to local unitaries in each $J$-block.  naturally selected by
the algebra $\mathcal{A}$. Intuitively, such an eigenstate structure has the
least coherence and entanglement over the additive and multiplicative
components of the gTPS Eq. 2. Using the duality between optimizing over
unitary families of Hamiltonians and algebras, searching over all possible NRC
Hamiltonians is equivalent to searching over the equivalence class of
isomorphic algebras, so the conjecture can be stated as:

######  Conjecture.

Let $H$ be an NRC Hamiltonian and $\mathcal{A}_{f}$ be the equivalence class
of isomorphic algebras of the form
$\mathcal{A}\cong\oplus_{J=1}^{d_{\mathcal{Z}}}\mathds{1}_{n_{J}}\otimes\mathcal{L}(\mathbb{C}^{d_{J}})$.
Then, the algebra corresponding to a gTPS (Eq. 2) for which the Hamiltonian
eigenstates have the form
$\\{\ket{\psi_{J}}\otimes\ket{\phi_{J}}\,|\,J=1,\dots
d_{\mathcal{Z}};\,\psi_{J}=1,\dots,n_{J};\,\phi_{J}=1,\dots,d_{J}\\}$
minimizes the $\mathcal{A}$-OTOC LTA over the family of isomorphic algebras.
The minimum value is

$\overline{G_{\mathcal{A}}}^{NRC}_{min}=1-\frac{1}{d}\left(\sum_{J}d_{J}+\sum_{J}n_{J}-d_{\mathcal{Z}}\right).$
(10)

######  Example 2 (Quantum reference frames).

Let us consider the possible quantum reference frame relativity of the
$\mathcal{A}$-OTOC LTA minimization. Before applying it to our framework, we
provide a brief overview of the perspective-neutral approach to quantum
reference frames (QRFs) [55, 56, 57]. Intuitively, using a quantum subsystem
as a reference frame corresponds to describing the state of the rest of the
system in relation to the state of the quantum frame. The reference frames we
consider are internal to the system, meaning they correspond to quantum
subsystems of the full Hilbert space $\mathcal{H}_{tot}$. For simplicity, we
shall focus on a full space consisting of two internal QRFs ($R_{1}$ and
$R_{2}$) and a system of interest $S$, so that
$\mathcal{H}_{tot}\cong\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\mathcal{H}_{S}$.

The basic ingredient of the formalism is a group $\mathcal{G}$, taken to be
Abelian here, that constitutes the space of frame orientations frame
orientations, in a similar manner that the Lorentz group SO+(3,1) is the space
of classical frame orientations in special relativity [57, 55]. For ideal
QRFs, the Hilbert spaces
$\mathcal{H}_{i}\cong\mathbb{C}^{\lvert\mathcal{G}\rvert}$ are spanned by the
frame configuration states $\ket{g}_{i}$, $g\in\mathcal{G}$, and furnish a
regular unitary representation of $\mathcal{G}$,
$U_{i}^{g^{\prime}}\ket{g}=\ket{g^{\prime}g}$. External frame reorientations
are given by elements of a unitary tensor product representation of
$\mathcal{G}$, $U_{12S}^{g}\coloneqq U_{1}^{g}\otimes U_{2}^{g}\otimes
U_{S}^{g}$. Relational states belong to the physical Hilbert space
$\mathcal{H}_{phys}\subset\mathcal{H}_{tot}$, obtained via a coherent group
averaging

$\begin{split}&\Pi_{phys}:\mathcal{H}_{tot}\rightarrow\mathcal{H}_{phys}\\\
&\Pi_{phys}\coloneqq\frac{1}{\lvert\mathcal{G}\rvert}\sum_{g\in\mathcal{G}}U_{12S}^{g}\end{split}$
(11)

and are invariant under the action of $U_{12S}^{g}$. $\mathcal{H}_{phys}$ is
the gauge-invariant Hilbert space, such that
$U_{12S}^{g}\ket{\psi_{phys}}=\ket{\psi_{phys}}\;\forall\,g\in\mathcal{G},\,\ket{\psi}_{phys}\in\mathcal{H}_{phys}$.
Jumping into the perspective of a QRF corresponds to gauge-fixing. For
example, we can fix the $R_{1}$ frame to be in a given orientation $g$. This
is achieved via the reduction map

$\begin{split}&\mathcal{R}_{1}^{g}:\mathcal{H}_{phys}\rightarrow\mathcal{H}_{2}\otimes\mathcal{H}_{S}\\\
&\mathcal{R}_{1}^{g}=\sqrt{\lvert\mathcal{G}\rvert}\left(\bra{g}_{1}\otimes\mathds{1}_{2}\otimes\mathds{1}\right)\Pi_{phys}.\end{split}$
(12)

This is a unitary map with the inverse given as [58, Lemma 21]

$(\mathcal{R}_{1}^{g})^{-1}=(\mathcal{R}_{1}^{g})^{\dagger}=\sqrt{\lvert\mathcal{G}\rvert}\,\Pi_{phys}(\ket{g}_{1}\otimes\mathds{1}_{2}\otimes\mathds{1}_{S}).$
(13)

Here, $\mathcal{H}_{2}\otimes\mathcal{H}_{S}$ is the perspectival Hilbert
space as “seen” from the $R_{1}$ point of view. Utilizing the reduction maps,
changing from the description of the full system relative to $R_{1}$ in
orientation $g$ to the one relative to $R_{2}$ in orientation $g^{\prime}$ is
obtained as

$V_{1\rightarrow
2}^{g,g^{\prime}}=\mathcal{R}_{2}^{g^{\prime}}\,(\mathcal{R}_{1}^{g})^{\dagger}$
(14)

$V_{1\rightarrow 2}^{g,g^{\prime}}$ is unitary and defines an isomorphism
between the perspective Hilbert spaces $\mathcal{H}_{2}\otimes\mathcal{H}_{S}$
and $\mathcal{H}_{1}\otimes\mathcal{H}_{S}$, while the adjoint action
$\hat{V}_{1\rightarrow 2}^{g,g^{\prime}}(\cdot)\equiv V_{1\rightarrow
2}^{g,g^{\prime}}(\cdot)(V_{1\rightarrow 2}^{g,g^{\prime}})^{\dagger}$ is an
algebra isomorphism between
$\mathcal{L}(\mathcal{H}_{2}\otimes\mathcal{H}_{S})$ and
$\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{S})$.

Let $g_{1}$ and $g_{2}$ be the orientations of $R_{1}$ and $R_{2}$ and assume
that the dynamics from the perspective of $R_{1}$ are given by a Hamiltonian
$H_{\overline{1}}\in\mathcal{L}(\mathcal{H}_{2}\otimes\mathcal{H}_{S})$. The
associated dynamics from the perspective of $R_{2}$ are, then, given by the
Hamiltonian $H_{\overline{2}}=\hat{V}_{1\rightarrow
2}^{g_{1},g_{2}}(H_{\overline{1}})\in\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{S})$.
We distinguish two cases of information scrambling minimization based on the
set of algebras of interest:

Perspective-neutral minimization\- Consider a set of algebras of relational
observables expressed as
$\mathscr{A}_{\overline{1}}=\\{\mathcal{A}_{2S}^{\mu}\subset\mathcal{L}(\mathcal{H}_{2}\otimes\mathcal{H}_{S})\\}_{\mu}$
in $R_{1}$-frame and $\mathscr{A}_{\overline{2}}=\\{\hat{V}_{1\rightarrow
2}^{g_{1},g_{2}}(\mathcal{A}_{2S}^{\mu})\subset\mathcal{L}(\mathcal{H}_{1}\otimes\mathcal{H}_{S})\\}_{\mu}$
in $R_{2}$-frame. The minimization of the $\mathcal{A}$-OTOC LTA can then be
performed equivalently either over $\mathscr{A}_{\overline{1}}$ with dynamics
given by $H_{\overline{1}}$ or over $\mathscr{A}_{\overline{2}}$ and dynamics
given by $H_{\overline{2}}$ and is, in this sense, perspective-neutral. The
algebras obtained in this way are simply related by $\hat{V}_{1\rightarrow
2}^{g_{1},g_{2}}$, but will, in general, differ in terms of their locality
structure with respect to the corresponding perspective Hilbert spaces
$\mathcal{H}_{2}\otimes\mathcal{H}_{S}$ and
$\mathcal{H}_{1}\otimes\mathcal{H}_{S}$ [57].

Perspectival minimization\- Let
$\\{\mathcal{A}_{S}^{\mu}\subset\mathcal{L}(\mathcal{H}_{S})\\}_{\mu}$ be a
set of observable algebras of interest in the system $S$. From the perspective
of $R_{1}$, these degrees of freedom are described by
$\mathscr{A}_{\overline{1}}=\\{\mathds{1}_{2}\otimes\mathcal{A}_{S}^{\mu}\\}_{\mu}$,
while from the perspective of $R_{2}$ by
$\mathscr{A}_{\overline{2}}=\\{\mathds{1}_{1}\otimes\mathcal{A}_{S}^{\mu}\\}_{\mu}$.
Although we use the same notation as before for the sets of algebras
$\mathscr{A}_{\overline{1}}$, $\mathscr{A}_{\overline{2}}$, we note that they
are no longer simply related by $\hat{V}_{1\rightarrow 2}^{g_{1},g_{2}}$; they
are now distinct and the $\mathcal{A}$-OTOC LTA minimization will depend on
the “chosen” perspective.

To more concretely illustrate the above observations, let us consider a toy
example where both the frames $R_{1}$ and $R_{2}$ and the system $S$ are
qubits. In this case, the frame configuration group is simply the cyclic group
of order $2$,
$\mathcal{G}=\mathbb{Z}_{2}=\\{e,g\,|\,e^{2}=e,\,eg=g,\,g^{2}=e\\}$. Also,
$\mathcal{H}_{i}\cong\mathbb{C}^{2}$ and we identify the configuration states
with the computational basis of the $R_{1}$, $R_{2}$ qubits, namely
$\ket{e}_{i}\coloneqq\ket{0}_{i},\,\ket{g}_{i}\coloneqq\ket{1}_{i}$ and, so,
$U_{i}^{e}=\mathds{1}_{i}$, $U_{i}^{g}=\sigma_{i}^{x}$, where we will denote
as $\sigma_{\alpha}^{x,y,z}$ the Pauli operators acting on the $\alpha$
subsystem. For simplicity, let the group action on $S$ to be the same, namely
$U_{S}^{e}=\mathds{1}_{S}$, $U_{S}^{g}=\sigma_{S}^{x}$. Assuming that both
QRFs are in the $e$ orientation, we have $V_{1\rightarrow
2}^{e,e}=\ket{0}_{1}\otimes\bra{0}_{2}\otimes\mathds{1}_{S}+\ket{1}_{1}\otimes\bra{1}_{2}\otimes\sigma_{S}^{x}$,
$V_{2\rightarrow
1}^{e,e}=\bra{0}_{1}\otimes\ket{0}_{2}\otimes\mathds{1}_{S}+\bra{1}_{1}\otimes\ket{1}_{2}\otimes\sigma_{S}^{x}$.

Let the dynamics in the $R_{1}$ perspective be given by the Hamiltonian

$H_{\overline{1}}=J_{z}\,\sigma_{2}^{z}\otimes\sigma_{S}^{z}+J_{x}\,\sigma_{2}^{x}\otimes\sigma_{S}^{x}+J_{y}\,\sigma_{2}^{y}\otimes\sigma_{S}^{y}$
(15)

where $J_{x},J_{y},J_{z}$ are coupling constants. In $R_{2}$’s perspective,
the Hamiltonian is

$\begin{split}H_{\overline{2}}&=\hat{V}_{1\rightarrow
2}^{e,e}(H_{\overline{1}})=\\\
&=J_{z}\,\mathds{1}_{1}\otimes\sigma_{S}^{z}+J_{x}\,\sigma_{1}^{x}\otimes\mathds{1}_{S}-J_{y}\,\sigma_{1}^{x}\otimes\sigma_{S}^{z}\end{split}$
(16)

Using this setup we will now showcase examples for the two types of
minimization we conceptually described above.

  1. 1.

As an example of perspective-neutral optimization, we consider the so-called
natural bipartitions [57]. Intuitively, these bipartitions correspond to the
way the QRF observers divide the “rest” of the system into “other frame” and
$S$ subsystems. Expressing these bipartitions in the $R_{1}$ frame, we have

$\begin{split}&\mathscr{A}_{\overline{1}}=\\{\mathcal{A}^{1},\mathcal{A}^{2}\\},\text{
where}\\\
&\mathcal{A}^{1}=\mathds{1}_{2}\otimes\mathcal{L}(\mathcal{H}_{S})\\\ &\hskip
13.0pt=\langle\mathds{1}_{2S},\,\mathds{1}_{2}\otimes\sigma_{S}^{z},\,\mathds{1}_{2}\otimes\sigma_{S}^{x},\,\mathds{1}_{2}\otimes\sigma_{S}^{y}\rangle\\\
&\mathcal{A}^{2}=\hat{V}_{2\rightarrow
1}^{e,e}(\mathds{1}_{1}\otimes\mathcal{L}(\mathcal{H}_{S}))=\\\ &\hskip
13.0pt=\langle\mathds{1}_{2S},\,\sigma_{2}^{z}\otimes\sigma_{S}^{z},\,\mathds{1}_{2}\otimes\sigma_{S}^{x},\,\sigma_{2}^{z}\otimes\sigma_{S}^{y}\rangle\end{split}$
(17)

For $J_{x},\,J_{y},\,J_{z}$ all different to each other, the Hamiltonian Eq.
15 satisfies the NRC and the eigenstates are given by the Bell states
$\ket{\phi^{\pm}}=1/\sqrt{2}\left(\ket{0}_{2}\otimes\ket{0}_{S}\pm\ket{1}_{2}\otimes\ket{1}_{S}\right),\,\ket{\psi^{\pm}}=1\sqrt{2}\left(\ket{0}_{2}\otimes\ket{1}_{S}\pm\ket{1}_{2}\otimes\ket{0}_{S}\right)$.
Denoting the Bell states as
$\ket{\chi^{\lambda}},\,\chi=\phi,\,\psi,\;\lambda=+,\,-$, notice that
$\mathcal{A}^{2}$ is the algebra of observables that acts non-trivially only
on $\chi$ and thus corresponds to a virtual bipartition where the Bell states
are product states. Then, due to Proposition 2,
$\overline{G_{\mathcal{A}^{2}}(\mathcal{U}_{\overline{1},t})}^{t}=1/4$, where
$\mathcal{U}_{\overline{\alpha},t}(\cdot)=\exp{it\,H_{\overline{\alpha}}}(\cdot)\exp{-it\,H_{\overline{\alpha}}}$,
$\alpha=1,2$, and $\mathcal{A}^{2}$ minimizes the $\mathcal{A}$-OTOC LTA.
Also, $\overline{G_{\mathcal{A}^{1}}(\mathcal{U}_{\overline{1},t})}^{t}=3/4$,
which saturates the upper-bound
$\max\\{1-1/d(\mathcal{A}),\;1-1/d(\mathcal{A}^{\prime})\\}$ of the
$\mathcal{A}$-OTOC [32]. Therefore, $\mathcal{A}^{2}$ is the emergent
bipartition, corresponding to $R_{1}$ and $S$ being distinguishable systems
from the perspective of $R_{2}$, and $R_{2}$ and $S$ to be “entangled” from
the perspective of $R_{1}$. In this sense, $R_{2}$ remains “hidden” from
$R_{1}$, while $R_{1}$ is “visible" to $R_{2}$.

  2. 2.

As an example of perspectival minimization, consider a family of subalgebras
of
$\mathcal{A}_{S}\equiv\\{\langle\mathds{1}_{S},\sigma_{S}^{\vec{\eta}}\rangle\\}_{\vec{\eta}}$,
parametrized by the unit vector
$\vec{\eta}=\\{(\eta_{x},\eta_{y},\eta_{z})\in\mathbb{R}^{3}\,|\,\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}=1\\}$,
where
$\sigma_{S}^{\vec{\eta}}=\eta_{x}\,\sigma_{S}^{x}+\eta_{y}\,\sigma_{S}^{y}+\eta_{z}\,\sigma_{S}^{z}$.
These observables correspond to the spin of $S$ in the $\vec{\eta}$ direction
and relate to the distinct families of relational observable algebras
$\mathscr{A}_{1}=\\{\mathcal{A}_{1}^{\vec{\eta}}\equiv\langle\mathds{1}_{2S},\mathds{1}_{2}\otimes\sigma_{S}^{\vec{\eta}}\rangle\\}_{\vec{\eta}}$
and
$\mathscr{A}_{2}=\\{\mathcal{A}_{2}^{\vec{\eta}}\equiv\langle\mathds{1}_{1S},\mathds{1}_{1}\otimes\sigma_{S}^{\vec{\eta}}\rangle\\}_{\vec{\eta}}$.
In $R_{1}$’s perspective the dynamics are given by Eq. 15, while in $R_{2}$’s
perspective by Eq. 16, and the $\mathcal{A}$-OTOC LTA for
$\mathcal{A}_{1}^{\vec{\eta}}$, $\mathcal{A}_{2}^{\vec{\eta}}$ is given
respectively as a function of $\vec{\eta}$ as

$\displaystyle\overline{G_{\mathcal{A}_{1}^{\vec{\eta}}}(\mathcal{U}_{\overline{1},t})}^{t}=\frac{1}{2}-\frac{\eta_{x}^{4}+\eta_{y}^{4}+\eta_{z}^{4}}{8}$
(18)
$\displaystyle\overline{G_{\mathcal{A}_{2}^{\vec{\eta}}}(\mathcal{U}_{\overline{2},t})}^{t}=(1-\eta_{z}^{2})\,\frac{(3+5\eta_{z}^{2})}{8}$
(19)

The minimum value of Eq. 18 is $3/8$ and is achieved when any of
$\eta_{x},\eta_{y},\eta_{z}$ is equal to $\pm 1$; from the $R_{1}$ perspective
all three algebras that correspond to the spin directions
$\hat{x},\hat{y},\hat{z}$ of $S$ indistinguishably minimize the LTA. The
equivalence of the $\hat{x},\hat{y},\hat{z}$ directions is readily anticipated
by the fact that the eigenstates of Eq. 15 are invariant under any $\pi/2$
rotation around any coordinate axis $x,y,z$. The minimum value of Eq. 19 is
$0$ and is achieved when $\eta_{z}=\pm 1$; from the $R_{2}$ perspective, the
algebra corresponding to the spin direction $\hat{z}$ of $S$ is emergent,
maximally retaining its informational content. This is intuitive since the
spin of $S$ in the $z$-direction is a conserved quantity of the Hamiltonian in
Eq. 16. This shows that using the $\mathcal{A}$-OTOC minimization criterion,
the emergent partition of the system $S$ differs for observers in different
QRFs.

######  Example 3 (Many-body Emergent Bipartitions).

(a)

(b)

Figure 3: The dependence of the bipartite $\mathcal{A}$-OTOC LTA
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}$ and the average eigenstate
mutual information $\bar{I}$ on the choice of bipartition for (a) the
transverse field Ising model (TFIM) and (b) the XXZ Heisenberg model with
$N=6$ qubits. The qualitative behavior of
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}$ and $\bar{I}$ is identical,
showing that, for the chosen many-body models, the bipartite
$\mathcal{A}$-OTOC LTA probes the amount of correlations of the Hamiltonian
eigenstates across the bipartition A:B.

We now focus on the important, special case of a bipartite algebra
$\mathcal{A}$, with $\mathcal{H}\cong\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
and $\mathcal{A}\cong\mathds{1}_{A}\otimes\mathcal{L}(\mathcal{H}_{B})$. Here,
$G_{\mathcal{A}}(\mathcal{U}_{t})$ coincides with the operator entanglement of
$U_{t}$ across the A:B bipartition [21], where
$\mathcal{U}_{t}(\cdot)=U_{t}(\cdot)U_{t}^{\dagger}$.

For a bipartite algebra, the Gaussian scrambling rate Eq. 5 that dictates the
short-time behavior of the $\mathcal{A}$-OTOC is [1]

$\tau_{S}^{-1}=\frac{1}{\sqrt{d}}\left\lVert{H-\frac{\mathds{1}_{A}}{d_{A}}\otimes\Tr_{A}(H)-\Tr_{B}(H)\otimes\frac{\mathds{1}_{B}}{d_{B}}}\right\rVert_{2},$
(20)

assuming for simplicity that $Tr(H)=0$. Eq. 20 shows that the short-time
behavior of the bipartite $\mathcal{A}$-OTOC depends exactly on the strength
of the interaction part of the Hamiltonian between A and B.

For the bipartite $\mathcal{A}$-OTOC NRC LTA of Eq. 7 we see that [21]

$\begin{split}&d^{2}\left(1-\overline{G(\mathcal{U}_{t})}^{NRC}\right)=\\\
&\hskip
5.0pt=\sum_{X=\\{A,B\\}}\left(\sum_{k,l=1}^{d}\langle\rho_{k}^{X},\rho_{l}^{X}\rangle^{2}-\frac{1}{2}\sum_{k=1}^{d}\langle\rho_{k}^{X},\rho_{k}^{X}\rangle^{2}\right)\end{split}$
(21)

where $R^{(X)}_{kl}\coloneqq\langle\rho_{k}^{X},\rho_{l}^{X}\rangle$ is the
Gram matrix of the reduced Hamiltonian eigenstates on the $X$ subsystem. Eq.
21 shows that the bipartite $\mathcal{A}$-OTOC LTA is intimately related to
the entanglement structure of the Hamiltonian eigenstates that depends on the
full details of the Hamiltonian operator, well beyond simply the interaction
strength.

In what follows, we consider a background tensor product structure (TPS)
$\mathcal{H}\cong\bigotimes_{i}\mathcal{H}_{i}$ and compare the short- and
long-time behavior of the $\mathcal{A}$-OTOC for algebras that act non-
trivially on a subset of the subsystems $\mathcal{H}_{i}$. Specifically, we
consider a spin-chain of $N$ qubits,
$\mathcal{H}\cong\left(\mathbb{C}^{2}\right)^{N}$ with open boundary
conditions and dynamics given by: i) the XXZ Heisenberg model
$H_{XXZ}=\sum_{i=1}^{N-1}\left(J_{X}\left(\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}\right)+J\sigma_{i}^{z}\sigma_{i+1}^{z}\right)$
with $J_{X}=-0.4,\,J=-1$, ii) the transverse field Ising model (TFIM)
$H_{I}=\sum_{i=1}^{N}\left(h\sigma_{i}^{z}\sigma_{i+1}^{z}+g\sigma_{i}^{x}\sigma_{i+1}^{x}\right)-\sum_{i=1}^{N-1}\sigma_{i}^{z}\sigma_{i+1}^{z}$
with $h=-0.5,\,g=1.05$. Let $N$ be even and $\mathscr{A}$ be the set of
algebras that act non-trivially on exactly $N/2$ (not necessarily contiguous)
qubits. We are interested in the emergent spatial bipartition for the short-
and long-time limits based on the principle of minimal scrambling. Clearly,
the bipartition that minimizes the interaction Hamiltonian, and thus the
Gaussian scrambling rate, for either of the models is the contiguous one, i.e.
$\mathcal{H}_{A}=\bigotimes_{i=1}^{N/2}\mathcal{H}_{i}$,
$\mathcal{H}_{B}=\bigotimes_{i=N/2+1}^{N}\mathcal{H}_{i}$.

Note that for the above coupling constants, the XXZ model satisfies NRC+ as
there are no gap degeneracies, while the Ising model is chaotic, satisfying
NRC [21]. Using Eq. 6 for the XXZ model and Eq. 7 for the Ising model, we
calculate for $N=6$ the $\mathcal{A}$-OTOC LTA for all possible choices of
algebras in $\mathscr{A}$. We find that the bipartition that minimizes the
$\mathcal{A}$-OTOC LTA coincides with the short-time one
$\mathcal{H}_{A}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\mathcal{H}_{3}$
for the Ising model, while for the XXZ model there are two non-contiguous
long-time emergent bipartitions,
$\mathcal{H}_{A}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\mathcal{H}_{6}$
and
$\mathcal{H}_{A}=\mathcal{H}_{1}\otimes\mathcal{H}_{5}\otimes\mathcal{H}_{6}$.

As it is clear from Eq. 21 the $\mathcal{A}$-OTOC LTA is related to the
eigenstate correlations between the subsystems of each bipartition. In order
to make this relation more concrete, we compute for the models above an
average eigenstate mutual information

$\bar{I}\equiv\frac{1}{M}\sum_{i=1}^{M}I\left(\rho_{E_{i}}\right)$ (22)

where $\rho_{E_{i}}\coloneqq\Pi_{i}/\Tr(\Pi_{i})$ is the uniform pure state
ensemble of eigenstates in a given energy $E_{i}$ and $I(\rho)\coloneqq
S(\rho^{A})+S(\rho^{B})-S(\rho)$ is the mutual information of $\rho$ across
the $A:B$ bipartition. As shown in Fig. 3 the qualitative behavior of the
$\mathcal{A}$-OTOC as a function of the chosen bipartition is identical to
that of $\bar{I}$. Quite intuitively, for the many-body systems considered
here, the information scrambling between $A$ and $B$ in long times is
controlled by the amount of eigenstate mutual information, and its
minimization over $\mathscr{A}$ is related to the subsystem emergence in the
long-time limit.

###  IV.1 Numerical Evidence for Conjecture

We make use of the algebra-Hamiltonian duality of the NRC LTA by fixing an
algebra $\mathcal{A}$ and searching over the space of Hamiltonians for a
violation of the conjecture. As seen in Eq. 7, the NRC LTA depends only on the
Hamiltonian eigenbasis, which can be represented by a unitary matrix with
eigenstates as the columns. Given an underlying basis, this representation is
unique up to permutations of the columns. Succinctly, for Hamiltonian
eigenbasis (unitary) $E$ and algebra $\mathcal{A}$, the LTA is a function
$f_{\mathcal{A}}(E)$.

For any arbitrary unitary $U$, $UE$ is also a unitary corresponding to the
eigenbasis of some other class of Hamiltonians, and for any eigenbasis $Q$,
there exists a $U$ such that $Q=UE$. Fixing $E=I$ for convenience, which
corresponds to a Hamiltonian eigenbasis satisfying the conjecture3, the LTA is
a function $f_{\mathcal{A}}(U)$444It is important to recognize that $U$ here
is distinct from the time-evolution unitary $U_{t}$ in previous formulae.. We
can now search over all unitaries to try to find a $U$ such that
$f_{\mathcal{A}}(U)$ is less than the conjectured minimum. To do this, for a
fixed algebra, we use an algorithm devised by Abrudan et al. [59] to perform
gradient descent of the LTA on the space of unitary matrices (see Section A.4
for elaboration). The algorithm is run until convergence, beginning with a
random unitary. At this point, we compare the observed LTA value with the
conjectured minimum.

Now, to systematically search for a violation of the conjecture, we need to
perform this gradient descent for all algebras of a given Hilbert space
dimension. Recall that each algebra has a structure given by Eq. 3. This
allows us to characterize each algebra by $d_{Z}$ and the set of pairs
$\\{(n_{J},d_{J})\\}_{J=1}^{d_{Z}}$. Note that these values correspond to an
equivalence class of algebras that are identical up to intra-block unitary
conjugation, which can be thought of as just performing a change of basis
within a block. Because we are already performing a search over unitaries, it
is sufficient to select just one representative algebra from each equivalence
class on which to run the algorithm.

To generate these algebras, given a Hilbert space dimension $d$, we generate
all unique sets of positive integers which sum to $d$. Each integer
corresponds to the dimension of a block in the direct sum Eq. 3, and $d_{Z}$
is the order of the set. The ordering of elements in a given set doesn’t
matter, since the rearrangement of blocks in the direct sum doesn’t affect the
algebra structure. Next, we decompose each integer into a product of two
factors, corresponding to $n_{J}$ and $d_{J}$; this set of pairs corresponds
to an equivalence class of algebras. We continue to perform this decomposition
until we generate all unique sets of pairs, i.e., every equivalence class of
algebras. Finally, we run the unitary gradient descent algorithm on each
algebra class.

We were able to generate all algebra classes for Hilbert spaces up to $d=40$.
For each class of algebras generated, the resultant LTA value equals the
conjectured minimum. This provides some numerical evidence that the
conjectured minimum is at least a local minimum for small systems.

$0$$10$$20$$30$$40$$0$$0.2$$0.4$$0.6$$0.8$$1$$\cdot 10^{7}$Hilbert space
dimensionNumber of algebra classesNumber of algebra classes vs. Hilbert space
dimension Figure 4: The number of algebra classes scales exponentially in
Hilbert space dimension, making higher dimensional tests of the conjecture
difficult.

##  V Conclusion

We have extended the use of the $\mathcal{A}$-OTOC as a criterion for quantum
mereology into the “macroscopic” regime by studying its long-time average
(LTA). We propose that, given a system Hamiltonian, the minimization of the
$\mathcal{A}$-OTOC LTA selects an algebra of observables which corresponds to
a preferred Hilbert space partition that best retains its informational
structure under the dynamics.

In systems with energy spectra that satisfy non-resonant conditions, we
derived simplified expressions for the $\mathcal{A}$-OTOC LTA in terms of the
Hamiltonian eigenprojectors and projections onto the algebras $\mathcal{A}$
and $\mathcal{A}^{\prime}$. We, then, used these formulas and a duality in
optimizing the $\mathcal{A}$-OTOC over unitary families of algebras and
unitary families of Hamiltonians to perform analytic minimizations of the
$\mathcal{A}$-OTOC LTA. For certain unitary families of NRC Hamiltonians,
including the family of all NRC Hamiltonians in a bipartite Hilbert space, we
showed that the $\mathcal{A}$-OTOC LTA is minimized when the eigenstates are
unentangled across the virtual bipartitions induced by $\mathcal{A}$. Based on
these analytic results and some numerical evidence, we conjectured that the
minimization of the $\mathcal{A}$-OTOC LTA over all isomorphic algebras with
fixed structural dimensions $d_{\mathcal{Z}},\\{d_{J}\\},\\{n_{J}\\}$ is
achieved by algebras that induce a gTPS (Eq. 2) for which the eigenstates
belong to a unique superselection sector and are unentangled across the
virtual bipartitions.

In addition, we showed how one can use the non-resonant approximations of the
$\mathcal{A}$-OTOC to perform the minimization of the LTA for physically
motivated systems and algebras. Specifically, we considered a unitary family
of stabilizer algebras and dynamics generated by a Heisenberg Hamiltonian and
illustrated how the non-resonant approximations can be used as a proxy for the
minimization of the exact LTA. We expect that such a procedure will be of
practical significance in determining “preferred” algebras of observables in
physical systems of interest, as computing the non-resonant approximations is
considerably less computationally complex than calculating the exact LTA
value. As a further application, we illustrated via a toy example how our
$\mathcal{A}$-OTOC LTA minimization framework depends on the choice of an
internal quantum reference frame (QRF) in the context of the perspective-
neutral approach to QRFs. Finally, we studied numerically the
$\mathcal{A}$-OTOC LTA in the case of bipartite algebras for certain quantum
many-body models and showed that its behavior with respect to the chosen
bipartition is connected to the average eigenstate mutual information between
the corresponding subsystems.

The work here can be extended in several paths. Further analytical and
numerical studies are needed to investigate the validity of the conjecture for
the algebra that minimizes the $\mathcal{A}$-OTOC for NRC Hamiltonian
dynamics. It is also natural to consider the long-time behavior of the
$\mathcal{A}$-OTOC in open quantum systems, where there is a competition
between the entropic contributions of information scrambling and decoherence
[22, 32].

##  VI Acknowledgments

FA and PZ acknowledge partial support from the NSF award PHY-2310227. FA
acknowledges partial support from the Gerondelis foundation. This research was
(partially) sponsored by the Army Research Office and was accomplished under
Grant Number W911NF-20-1-0075. The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of the Army
Research Office or the U.S. Government. The U.S. Government is authorized to
reproduce and distribute reprints for Government purposes notwithstanding any
copyright notation herein.

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*

## Appendix A

###  A.1 Proof of Proposition 1

Let $\mathcal{H}\cong\mathbb{C}^{d}$ be a Hilbert space. For a Hamiltonian
evolution $U_{t}=\exp(itH)$ the $\mathcal{A}$-OTOC can be expressed as [32]:

$G_{\mathcal{A}}(\mathcal{U}_{t})=1-\frac{1}{d}\Tr(S\,\Omega_{\mathcal{A}}\,\mathcal{U}_{t}^{\otimes
2}\left(\Omega_{\mathcal{A}^{\prime}}\right))$ (23)

where $\mathcal{U}_{t}(\cdot)=U_{t}(\cdot)U_{t}^{\dagger}$,
$\Omega_{\mathcal{A}}=\sum_{\alpha}e_{\alpha}\otimes e_{\alpha}^{\dagger}$,
$\Omega_{\mathcal{A}^{\prime}}=\sum_{\gamma}f_{\gamma}\otimes
f_{\gamma}^{\dagger}$ and $\\{e_{\alpha}\\}_{\alpha=1}^{d(\mathcal{A})}$,
$\\{f_{\gamma}\\}_{\gamma=1}^{d(\mathcal{A}^{\prime})}$ are appropriate
orthonormal bases of $\mathcal{A}$ and $\mathcal{A}^{\prime}$ repsectively. We
can choose these algebra bases in terms of the decomposition Eq. 2 as

$\begin{split}&f_{\gamma}=\outerproduct{p_{J}}{q_{J}}\otimes\frac{\mathds{1}_{d_{J}}}{\sqrt{n_{J}}},\,\gamma=(J,p_{J},q_{J});\,J=1,\dots,d_{\mathcal{Z}};\,p_{J},q_{J}=1,\dots,n_{J}\\\
&e_{\alpha}=\frac{\mathds{1}_{n_{J}}}{\sqrt{d_{J}}}\otimes\outerproduct{k_{J}}{l_{J}},\,\alpha=(J,k_{J},l_{J});\,J=1,\dots,d_{\mathcal{Z}};\,k_{J},l_{J}=1,\dots,d_{J}\end{split}$
(24)

where we used a basis
$\mathbb{B}=\\{\ket{p_{J}}\otimes\ket{k_{J}}\,|\,J=1,\dots,d_{\mathcal{Z}};\,p_{J}=1,\dots,n_{J};\,k_{J}=1,\dots,d_{J}\\}$
of $\mathcal{H}$. Using the spectral decomposition
$H=\sum_{k=1}^{M}E_{k}\Pi_{k}$, we have

$\begin{split}\overline{\mathcal{U}_{t}^{\otimes
2}(\cdot)}^{t}&=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}\sum_{k,l,m,n=1}^{M}\exp(it(E_{k}+E_{l}-E_{m}-E_{n}))\Pi_{k}\otimes\Pi_{l}(\cdot)\Pi_{m}\otimes\Pi_{n}\;dt\\\
&=\sum_{k,l,m,n=1}^{M}\delta_{E_{k}+E_{l},E_{m}+E_{n}}\,\Pi_{k}\otimes\Pi_{l}(\cdot)\Pi_{m}\otimes\Pi_{n}\end{split}$
(25)

Substituting the explicit expressions on Eq. 23 and using the identity

$\Tr(S\,A\otimes B)=\Tr(AB)$ (26)

we have

$\begin{split}\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{t}&=1-\frac{1}{d}\sum_{k,l,m,n=1}^{M}\delta_{E_{k}+E_{l},E_{m}+E_{n}}\Tr(\sum_{\alpha=1}^{d(\mathcal{A})}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}e_{\alpha}\Pi_{k}f_{\gamma}\Pi_{m}e_{\alpha}^{\dagger}\Pi_{l}f_{\gamma}^{\dagger}\Pi_{n})\\\
&=1-\frac{1}{d}\sum_{k,l,m,n=1}^{M}\delta_{E_{k}+E_{l},E_{m}+E_{n}}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\langle\mathbb{P}_{\mathcal{A}^{\prime}}(\Pi_{k}f_{\gamma}\Pi_{m}),\mathbb{P}_{\mathcal{A}^{\prime}}(\Pi_{n}f_{\gamma}\Pi_{l})\rangle\end{split}$
(27)

where we used that
$\mathbb{P}_{\mathcal{A}^{\prime}}(\cdot)=\sum_{\alpha}e_{\alpha}(\cdot)e_{\alpha}^{\dagger}$
is a projector.

i) The NRC+ condition requires that there are no degenerate gaps, which
implies that $E_{k}+E_{l}=E_{m}+E_{n}\Leftrightarrow(k=m,\;l=n)\text{ or
}(k=n,\;l=m)\Leftrightarrow\delta_{E_{k}+E_{l},E_{m}+E_{n}}=\delta_{k,m}\delta_{l,n}+\delta_{k,n}\delta_{l,m}-\delta_{k,l}\delta_{k,m}\delta_{k,n})$.
So, Eq. 27 becomes

$\begin{split}\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC^{+}}&=1-\frac{1}{d}\sum_{\alpha=1}^{d(\mathcal{A})}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\left(\sum_{k,l=1}^{M}\left(\Tr(e_{\alpha}\Pi_{k}f_{\gamma}\Pi_{k}e_{\alpha}^{\dagger}\Pi_{l}f_{\gamma}^{\dagger}\Pi_{l})+\Tr(e_{\alpha}\Pi_{k}f_{\gamma}\Pi_{l}e_{\alpha}^{\dagger}\Pi_{l}f_{\gamma}^{\dagger}\Pi_{k})\right)-\right.\\\
&\hskip
100.0pt\left.-\sum_{k=1}^{M}\Tr(e_{\alpha}\Pi_{k}f_{\gamma}\Pi_{k}e_{\alpha}^{\dagger}\Pi_{k}f_{\gamma}^{\dagger}\Pi_{k})\right)=\\\
&=1-\frac{1}{d}\left[\sum\limits_{\gamma=1}^{d(\mathcal{A}^{\prime})}\left(\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(D_{H}(f_{\gamma}))\rVert_{2}^{2}-\frac{1}{2}\sum\limits_{k=1}^{M}\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(\Pi_{k}f_{\gamma}\Pi_{k})\rVert_{2}^{2}\right)+\right.\\\
&\hskip
50.0pt\left.+\sum\limits_{\alpha=1}^{d(\mathcal{A})}\left(\lVert\mathbb{P}_{\mathcal{A}}(D_{H}(e_{\alpha}))\rVert_{2}^{2}-\frac{1}{2}\sum\limits_{k=1}^{M}\lVert\mathbb{P}_{\mathcal{A}}(\Pi_{k}e_{\alpha}\Pi_{k})\rVert_{2}^{2}\right)\right]\end{split}$
(28)

ii) The NRC condition requires in addition that the energy levels are non-
degenerate, so the eigenprojectors $\Pi_{k}$ are 1-dimensional, i.e.
$\Pi_{k}=\outerproduct{\phi_{k}}{\phi_{k}}\;\forall\,k=1,\dots,d$. Notice that

$\begin{split}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\left\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(D_{H}(f_{\gamma}))\right\rVert_{2}^{2}&=\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\left\lVert\sum_{k=1}^{d}\matrixelement{\phi_{k}}{f_{\gamma}}{\phi_{k}}\mathbb{P}_{\mathcal{A}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{k}})\right\rVert_{2}^{2}=\\\
&=\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\sum_{k,l=1}^{d}\matrixelement{\phi_{l}}{f_{\gamma}}{\phi_{l}}\matrixelement{\phi_{k}}{f_{\gamma}^{\dagger}}{\phi_{k}}\langle\mathbb{P}_{\mathcal{A}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{k}}),\mathbb{P}_{\mathcal{A}^{\prime}}(\outerproduct{\phi_{l}}{\phi_{l}})\rangle=\\\
&=\sum_{k,l=1}^{d}\langle\outerproduct{\phi_{k}}{\phi_{l}},\mathbb{P}_{\mathcal{A}}(\outerproduct{\phi_{k}}{\phi_{l}})\rangle\,\langle\mathbb{P}_{\mathcal{A}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{k}}),\mathbb{P}_{\mathcal{A}^{\prime}}(\outerproduct{\phi_{l}}{\phi_{l}})\rangle=\Tr(R^{(0),\mathcal{A}}\,R^{(1),\mathcal{A}^{\prime}})\end{split}$
(29)

where
$R_{lk}^{(0),\mathcal{A}}\coloneqq\left\lVert\mathbb{P}_{\mathcal{A}}\left(\ket{\phi_{k}}\bra{\phi_{l}}\right)\right\rVert_{2}^{2}$,
$R_{kl}^{(1),\mathcal{A}}\coloneqq\left\langle\mathbb{P}_{\mathcal{A}}\left(\Pi_{k}\right),\mathbb{P}_{\mathcal{A}}\left(\Pi_{l}\right)\right\rangle$.
Similarly,

$\begin{split}&\sum_{\alpha=1}^{d(\mathcal{A})}\lVert\mathbb{P}_{\mathcal{A}}(D_{H}(e_{\alpha}))\rVert_{2}^{2}=\Tr(R^{(0),\mathcal{A}^{\prime}}\,R^{(1),\mathcal{A}})\\\
&\sum_{k=1}^{d}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\lVert\mathbb{P}_{\mathcal{A}^{\prime}}(\Pi_{k}f_{\gamma}\Pi_{k})\rVert_{2}^{2}=\Tr(R_{D}^{(0),\mathcal{A}}\,R_{D}^{(1),\mathcal{A}^{\prime}})\\\
&\sum_{k=1}^{d}\sum_{\alpha=1}^{d(\mathcal{A})}\lVert\mathbb{P}_{\mathcal{A}}(\Pi_{k}e_{\alpha}\Pi_{k})\rVert_{2}^{2}=\Tr(R_{D}^{(0),\mathcal{A}^{\prime}}\,R_{D}^{(1),\mathcal{A}})\end{split}$
(30)

where $R_{D}=\operatorname{diag}(R)$. Plugging Eqs. 29 and 30 in Eq. 28 one
gets Eq. 7.

###  A.2 Proof of Proposition 2

Recall that given the Hilbert space decomposition Eq. 2, we can express the
projectors over $\mathcal{A}$ and $\mathcal{A}^{\prime}$ as:

$\begin{split}&\mathbb{P}_{\mathcal{A}^{\prime}}(\cdot)=\bigoplus_{J=1}^{d_{\mathcal{Z}}}\left(\Tr_{d_{J}}(\cdot)\otimes\frac{\mathds{1}_{d_{J}}}{d_{J}}\right)\\\
&\mathbb{P}_{\mathcal{A}}(\cdot)=\bigoplus_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\mathds{1}_{n_{J}}}{n_{J}}\otimes\Tr_{n_{J}}(\cdot)\right)\end{split}$
(31)

Also, we have a partial-trace trick:

$\Tr_{A}(\Tr_{B}(\ket{\phi_{k}}\bra{\phi_{l}})\,\Tr_{B}(\ket{\phi_{l}}\ket{\phi_{k}}))=\Tr_{B}(\Tr_{A}(\ket{\phi_{k}}\bra{\phi_{k}})\,\Tr_{A}(\ket{\phi_{l}}\ket{\phi_{l}}))$
(32)

Using the above we have

$\begin{split}&\Tr(R^{(0),\mathcal{A}}\,R^{(1),\mathcal{A}^{\prime}})=\\\
&\hskip
30.0pt=\sum_{k,l=1}^{d}\left\lVert\bigoplus_{J=1}^{d_{\mathcal{Z}}}\frac{\mathds{1}_{n_{J}}}{n_{J}}\otimes\Tr_{n_{J}}(\outerproduct{\phi_{k}^{J}}{\phi_{l}^{J}})\right\rVert_{2}^{2}\,\left\langle\bigoplus_{K=1}^{d_{\mathcal{Z}}}\Tr_{d_{K}}(\outerproduct{\phi_{k}^{K}}{\phi_{k}^{K}})\otimes\frac{\mathds{1}_{d_{K}}}{d_{K}},\bigoplus_{L=1}^{d_{\mathcal{Z}}}\Tr_{d_{L}}(\outerproduct{\phi_{l}^{L}}{\phi_{l}^{L}})\otimes\frac{\mathds{1}_{d_{L}}}{d_{L}}\right\rangle=\\\
&\hskip
30.0pt=\sum_{k,l=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\Tr_{n_{J}}(\outerproduct{\phi_{l}^{J}}{\phi_{k}^{J}}),\Tr_{n_{J}}(\outerproduct{\phi_{l}^{J}}{\phi_{k}^{J}})\rangle}{n_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\Tr_{d_{K}}(\outerproduct{\phi_{k}^{K}}{\phi_{k}^{K}}),\Tr_{d_{K}}(\outerproduct{\phi_{l}^{K}}{\phi_{l}^{K}})\rangle}{d_{K}}\right)=\\\
&\hskip
30.0pt=\sum_{k,l=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{J}},\rho_{l}^{n_{J}}\rangle}{n_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{K}},\rho_{l}^{n_{K}}\rangle}{d_{K}}\right)\end{split}$
(33)

where $\ket{\phi_{k}^{J}}$ is the projection of $\ket{\phi_{k}}$ on the
$J^{th}$ sector and
$\rho_{k}^{n_{J}}\coloneqq\Tr_{d_{J}}(\outerproduct{\phi_{k}^{J}}{\phi_{k}^{J}})$.
Similarly,

$\begin{split}&\Tr(R^{(0),\mathcal{A}^{\prime}}\,R^{(1),\mathcal{A}})=\sum_{k,l=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{J}},\rho_{l}^{d_{J}}\rangle}{d_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{K}},\rho_{l}^{d_{K}}\rangle}{n_{K}}\right)\\\
&\Tr(R_{D}^{(0),\mathcal{A}}\,R_{D}^{(1),\mathcal{A}^{\prime}})=\sum_{k=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{J}},\rho_{k}^{n_{J}}\rangle}{n_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{K}},\rho_{k}^{n_{K}}\rangle}{d_{K}}\right)\\\
&\Tr(R_{D}^{(0),\mathcal{A}^{\prime}}\,R_{D}^{(1),\mathcal{A}})=\sum_{k=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{J}},\rho_{k}^{d_{J}}\rangle}{d_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{K}},\rho_{k}^{d_{K}}\rangle}{n_{K}}\right)\end{split}$
(34)

Using Eqs. 33 and 34 in Eq. 7 we have

$\begin{split}&d\left(1-\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}\right)=\\\
&\hskip
49.0pt=\sum_{k,l=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{J}},\rho_{l}^{d_{J}}\rangle}{n_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{K}},\rho_{l}^{d_{K}}\rangle}{d_{K}}\right)-\frac{1}{2}\sum_{k=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{J}},\rho_{k}^{d_{J}}\rangle}{n_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{d_{K}},\rho_{k}^{d_{K}}\rangle}{d_{K}}\right)+\\\
&\hskip
58.0pt+\sum_{k,l=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{J}},\rho_{l}^{n_{J}}\rangle}{d_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{K}},\rho_{l}^{n_{K}}\rangle}{n_{K}}\right)-\frac{1}{2}\sum_{k=1}^{d}\sum_{J=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{J}},\rho_{k}^{n_{J}}\rangle}{d_{J}}\right)\,\sum_{K=1}^{d_{\mathcal{Z}}}\left(\frac{\langle\rho_{k}^{n_{K}},\rho_{k}^{n_{K}}\rangle}{n_{K}}\right)\end{split}$
(35)

Let us calculate the $\mathcal{A}$-OTOC NRC LTA when the eigenstates are of
the form $\ket{\phi_{k}}=\ket{J,p}\otimes\ket{J,m}$ where
$\ket{J,p}\in\mathbb{C}^{n_{J}}$ and $\ket{J,m}\in\mathbb{C}^{d_{J}}$. Using
the multi-index notation $k=(R,p,m)$:

$\begin{split}&\rho_{R,p,m}^{n_{J}}=\delta_{J,R}\,\ket{R,p}\bra{R,p}\\\
&\langle\rho_{R,p,m}^{n_{J}},\rho_{W,s,t}^{n_{J}}\rangle=\delta_{J,R}\,\delta_{R,W}\,\delta{p,s}\\\
&\langle\rho_{R,p,m}^{d_{J}},\rho_{W,s,t}^{d_{J}}\rangle=\delta_{J,R}\,\delta_{R,W}\,\delta{m,t}\end{split}$
(36)

we then have:

$\begin{split}d\left(1-\overline{G_{\mathcal{A}}(U_{t})}^{NRC}\right)&=\sum_{(R,p,m),(W,s,t)}\delta_{R,W}\frac{\delta_{p,s}}{n_{R}}\,\frac{\delta_{p,s}}{d_{R}}+\sum_{(R,p,m),(W,s,t)}\delta_{R,W}\frac{\delta_{m,t}}{n_{R}}\,\frac{\delta_{m,t}}{d_{R}}-\sum_{(R,p,m)}\frac{1}{n_{R}}\,\frac{1}{d_{R}}\\\
&=\sum_{R}n_{R}\,d_{R}^{2}\,\frac{1}{n_{R}d_{R}}+\sum_{R}n_{R}^{2}\,d_{R}\,\frac{1}{n_{R}d_{R}}-\sum_{R}n_{R}d_{R}\,\frac{1}{n_{R}d_{R}}\\\
&=\sum_{J}(d_{J}+n_{J}-1)=\sum_{J}d_{J}+\sum_{J}n_{J}-d_{\mathcal{Z}}\end{split}$
(37)

Now, consider the family of Hamiltonians that satisfy NRC and are of the form
$H=\oplus_{J}H_{J}$. This implies that each $J$-subspace is spanned by exactly
$n_{J}\,d_{J}$ orthonormal energy eigenstates, i.e.
$\ket{\phi_{k}}=\ket{\phi_{J,a}}$ with $J=1,\dots,\mathcal{Z}$,
$a=1,\dots,n_{J}\,d_{J}$. Eq. 35 takes the form:

$\begin{split}d\left(1-\overline{G_{\mathcal{A}}(U_{t})}^{NRC}\right)=\sum_{J,a,b}\frac{\langle\rho_{J,a}^{n_{J}},\rho_{J,b}^{n_{J}}\rangle^{2}}{n_{J}\,d_{J}}+\sum_{J,a,b}\frac{\langle\rho_{J,a}^{d_{J}},\rho_{J,b}^{d_{J}}\rangle^{2}}{n_{J}\,d_{J}}-\frac{1}{2}\sum_{J,a}\frac{\langle\rho_{J,a}^{n_{J}},\rho_{J,a}^{n_{J}}\rangle^{2}+\,\langle\rho_{J,a}^{d_{J}},\rho_{J,a}^{d_{J}}\rangle^{2}}{n_{J}\,d_{J}}\end{split}$
(38)

Notice that $\rho_{J,a}^{n_{J}}$ are reduced density matrices, so
$\lVert\rho_{J,a}^{n_{J}}\rVert_{1}=1$ and
$\lVert\rho_{J,a}^{n_{J}}\rVert_{2}^{2}=\lVert\rho_{J,a}^{d_{J}}\rVert_{2}^{2}=:P_{J}^{a}\leq
1$. So:

$\begin{split}&\langle\rho_{J,a}^{n_{J}},\rho_{J,b}^{n_{J}}\rangle\leq\lVert\rho_{J,a}^{n_{J}}\rVert_{1}\,\lVert\rho_{J,b}^{n_{J}}\rVert_{\infty}\leq\lVert\rho_{J,b}^{n_{J}}\rVert_{2}=\sqrt{P_{J}^{b}}\\\
&\langle\rho_{J,a}^{d_{J}},\rho_{J,b}^{d_{J}}\rangle\leq\sqrt{P_{J}^{b}}\end{split}$
(39)

Also,

$\begin{split}&\sum_{a}\rho_{J,a}^{n_{J}}=\sum_{a}\Tr_{d_{J}}(\ket{\phi_{J,a}}\bra{\phi_{J,a}})=\Tr_{d_{J}}(\mathds{1}_{n_{J}\,d_{J}})=d_{J}\,\mathds{1}_{n_{J}}\\\
&\sum_{a}\rho_{J,a}^{d_{J}}=n_{J}\,\mathds{1}_{d_{J}}\end{split}$ (40)

So:

$\begin{split}d\left(1-\overline{G_{\mathcal{A}}(U_{t})}^{NRC}\right)&\leq\sum_{J,a,b}\frac{\langle\rho_{J,a}^{n_{J}},\rho_{J,b}^{n_{J}}\rangle\sqrt{P_{J}^{b}}}{n_{J}\,d_{J}}+\sum_{J,a,b}\frac{\langle\rho_{J,a}^{d_{J}},\rho_{J,b}^{d_{J}}\rangle\sqrt{P_{J}^{b}}}{n_{J}\,d_{J}}-\sum_{J,a}\frac{{P_{J}^{a}}^{2}}{n_{J}\,d_{J}}\\\
&=\sum_{J,b}\frac{\langle\mathds{1}_{n_{J}},\rho_{J,b}^{n_{J}}\rangle\sqrt{P_{J}^{b}}}{n_{J}}+\sum_{J,b}\frac{\langle\mathds{1}_{d_{J}},\rho_{J,b}^{d_{J}}\rangle\sqrt{P_{J}^{b}}}{d_{J}}-\sum_{J,a}\frac{{P_{J}^{a}}^{2}}{n_{J}\,d_{J}}\\\
&=\sum_{J,a}\left(\sqrt{P_{J}^{a}}\left(\frac{1}{n_{J}}+\frac{1}{d_{J}}\right)-\frac{{P_{J}^{a}}^{2}}{n_{J}\,d_{J}}\right)=:\sum_{J,a}f(P_{J}^{a})\end{split}$
(41)

Notice that

$\frac{\partial f}{\partial
P_{J}^{a}}=\frac{n_{J}+d_{J}-4{P_{J}^{a}}^{3/2}}{2n_{J}d_{J}\sqrt{P_{J}^{a}}}$
(42)

Now, for all $J$’s for which $n_{J},d_{J}\geq 2$, the partial derivative Eq.
42 is non-negative $\forall$ $P_{J}^{a}\in(1/\min\\{n_{J},d_{J}\\},1)$, so $f$
is maximized for $P_{J}^{a}=1$. Meanwhile, if either $n_{J}=1$ or $d_{J}=1$,
the purity $P_{J}^{a}=1$. So:

$d\left(1-\overline{G_{\mathcal{A}}(U_{t})}^{NRC}\right)\leq\sum_{J,a}f(\\{P_{J}^{a}=1\\})=\sum_{J,a}\left(\frac{1}{n_{J}}+\frac{1}{d_{J}}-\frac{1}{n_{J}d_{J}}\right)=\sum_{J}\left(n_{J}+d_{J}-1\right)$
(43)

Thus, the value Eq. 37 is indeed the minimum.

###  A.3 Calculations for Example 2

  * $\bullet$

For the algebra
$\mathcal{A}^{1}=\langle\mathds{1}_{2S},\mathds{1}_{2}\otimes\sigma_{S}^{z},\mathds{1}_{2}\otimes\sigma_{S}^{x},\mathds{1}_{2}\otimes\sigma_{S}^{y}\rangle$
we have $d_{\mathcal{Z}}=1,d_{1}=n_{1}=2$. It is convenient to work with the
orthogonal basis
$\\{e_{\alpha}\\}_{\alpha=1}^{4}=\\{\frac{\mathds{1}_{2S}}{2},\frac{\mathds{1}_{2}\otimes\sigma_{S}^{z}}{2},\frac{\mathds{1}_{2}\otimes\sigma_{S}^{x}}{2},\frac{\mathds{1}_{2}\otimes\sigma_{S}^{y}}{2}\\}$
of $\mathcal{A}^{1}$. Note that this is not the same with the basis in Eq. 24,
but is unitarily related. We can, then, express the projectors
$\mathbb{P}_{\mathcal{A}^{1}},\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}$ as

$\begin{split}&\mathbb{P}_{\mathcal{A}^{1}}(\cdot)=\sum_{\alpha=1}^{4}\langle
e_{\alpha},(\cdot)\rangle
e_{\alpha}=\Tr(\cdot)\frac{\mathds{1}_{2S}}{4}+\Tr(\Tr_{2}(\cdot)\vec{\sigma}_{S})\frac{\mathds{1}_{2}\otimes\vec{\sigma}_{S}}{4}\\\
&\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}(\cdot)=\sum_{\alpha=1}^{4}e_{\alpha}(\cdot)e_{\alpha}^{\dagger}=\frac{(\cdot)}{4}+\frac{\mathds{1}_{2}\otimes\vec{\sigma}_{S}\,(\cdot)\,\mathds{1}_{2}\otimes\vec{\sigma}_{S}}{4}\end{split}$
(44)

where we also used that the $e_{\alpha}$’s are already normalized, $\lVert
e_{\alpha}\rVert_{2}=1$. In $R_{1}$ frame the dynamics
$\mathcal{U}_{t}(\cdot)=\exp(it\,H)(\cdot)\exp(-it\,H)$ are given by the
Hamiltonian
$H_{\overline{1}}=J_{z}\,\sigma_{2}^{z}\otimes\sigma_{S}^{z}+J_{x}\,\sigma_{2}^{x}\otimes\sigma_{S}^{x}+J_{y}\,\sigma_{2}^{y}\otimes\sigma_{S}^{y}$
with the eigenstates being the Bell states
$\\{\phi_{k}\\}_{k=1}^{4}=\\{\ket{\phi^{+}},\ket{\phi^{-}},\ket{\psi^{+}},\ket{\psi^{-}}\\}$.
Writing out Eq. 7 explicitly we have

$\begin{split}\overline{G_{\mathcal{A}^{1}}(\mathcal{U}_{t})}^{NRC}=1-\frac{1}{4}&\left(2\sum_{\begin{subarray}{c}k,l=1\\\
k<l\end{subarray}}^{4}\left\lVert\mathbb{P}_{\mathcal{A}^{1}}(\outerproduct{\phi_{k}}{\phi_{l}})\right\rVert_{2}^{2}\,\left\langle\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{k}}),\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}(\outerproduct{\phi_{l}}{\phi_{l}})\right\rangle\right.+\\\
&\left.\hskip 20.0pt+\,2\sum_{\begin{subarray}{c}k,l=1\\\
k<l\end{subarray}}^{4}\left\lVert\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{l}})\right\rVert_{2}^{2}\,\left\langle\mathbb{P}_{\mathcal{A}^{1}}(\outerproduct{\phi_{k}}{\phi_{k}}),\mathbb{P}_{\mathcal{A}^{1}}(\outerproduct{\phi_{l}}{\phi_{l}})\right\rangle\,+\right.\\\
&\left.\hskip
20.0pt+\sum_{k=1}^{4}\left\lVert\mathbb{P}_{{\mathcal{A}^{1}}}(\outerproduct{\phi_{k}}{\phi_{k}})\right\rVert_{2}^{2}\,\left\lVert\mathbb{P}_{{\mathcal{A}^{1}}^{\prime}}(\outerproduct{\phi_{k}}{\phi_{k}})\right\rVert_{2}^{2}\right)\end{split}$
(45)

where we used the fact that the terms with $k>l$ are equal to those with
$k<l$. Substituting Eq. 44 in Eq. 45 one obtains, in a rather tedious but
straightforward manner, that
$\overline{G_{\mathcal{A}^{1}}(\mathcal{U}_{t})}^{NRC}=3/4$.

  * $\bullet$

The algebra
$\mathcal{A}_{i}^{\vec{\eta}}=\langle\mathds{1}_{iS},\mathds{1}_{i}\otimes\sigma_{S}^{\vec{\eta}}\rangle$
is Abelian and we have $d_{\mathcal{Z}}=2,d_{1}=d_{2}=1,n_{1}=n_{2}=2$. We
work with the orthogonal basis
$\\{e_{\alpha}\\}_{\alpha=1}^{2}=\\{\frac{\mathds{1}_{iS}}{\sqrt{2}},\frac{\mathds{1}_{i}\otimes\sigma_{S}^{\vec{\eta}}}{\sqrt{2}}\\}$.
Then

$\begin{split}&\mathbb{P}_{\mathcal{A}_{i}^{\vec{\eta}}}(\cdot)=\sum_{\alpha=1}^{2}\left\langle\frac{e_{\alpha}}{\lVert
e_{\alpha}\rVert_{2}},(\cdot)\right\rangle\frac{e_{\alpha}}{\lVert
e_{\alpha}\rVert_{2}}=\Tr(\cdot)\frac{\mathds{1}_{iS}}{4}+\Tr(\Tr_{i}(\cdot)\vec{\sigma}_{S})\frac{\mathds{1}_{i}\otimes\sigma_{S}^{\vec{\eta}}}{4}\\\
&\mathbb{P}_{{\mathcal{A}_{i}^{\vec{\eta}}}^{\prime}}(\cdot)=\sum_{\alpha=1}^{2}e_{\alpha}(\cdot)e_{\alpha}^{\dagger}=\frac{(\cdot)}{2}+\frac{\mathds{1}_{i}\otimes\sigma_{S}^{\vec{\eta}}\,(\cdot)\,\mathds{1}_{i}\otimes\sigma_{S}^{\vec{\eta}}}{2}\end{split}$
(46)

In $R_{1}$ frame the Hamiltonian is
$H_{\overline{1}}=J_{z}\,\sigma_{2}^{z}\otimes\sigma_{S}^{z}+J_{x}\,\sigma_{2}^{x}\otimes\sigma_{S}^{x}+J_{y}\,\sigma_{2}^{y}\otimes\sigma_{S}^{y}$
with the eigenstates being the Bell states
$\\{\phi_{k}\\}_{k=1}^{4}=\\{\ket{\phi^{+}},\ket{\phi^{-}},\ket{\psi^{+}},\ket{\psi^{-}}\\}$,
while in $R_{2}$ frame the Hamiltonian is
$H_{\overline{2}}=J_{z}\,\mathds{1}_{1}\otimes\sigma_{S}^{z}+J_{x}\,\sigma_{1}^{x}\otimes\mathds{1}_{S}-J_{y}\,\sigma_{1}^{x}\otimes\sigma_{S}^{z}$
with the eigenstates being $\\{\ket{+0},\ket{+1},\ket{-0},\ket{-1}\\}$, where
$\ket{\pm}\equiv\frac{1}{\sqrt{2}}(\ket{0}\pm\ket{1})$ are the eigenstates of
$\sigma_{x}$. Using the analogous expression of Eq. 45 and substituting Eq. 46
for the two frames one obtains Eqs. 18 and 19.

###  A.4 Gradient Descent for Test of Conjecture

The distinguished basis corresponding to $\mathcal{A}$ consists of vectors
$\mathbb{B}=\left\\{\ket{p_{J}}\otimes\ket{a_{J}}\right\\}_{p_{J}=1,\
a_{J}=1}^{n_{J},\ d_{J}}$ and is unique up to local unitaries on the virtual
bipartitions in each $J$-block of Eq. 2. In this basis, the eigenstates of the
conjectured Hamiltonian (see Conjecture) are the "canonical" basis vectors of
the full Hilbert space. For an arbitrary NRC Hamiltonian $H$, the eigenbasis
can be expressed as a rotation of $\mathbb{B}$ by a unitary $U$, so that the
one-dimensional eigenprojectors are given as $U\Pi_{k}U^{\dagger}$, where
$\Pi_{k}\equiv\outerproduct{p_{J},a_{J}}{p_{J},a_{J}}$ are the projectors
corresponding to $\mathbb{B}$.

In Section IV.1 we utilize a gradient descent algorithm by Abrudan et al. [59]
in order to numerically minimize Eq. 7 over all possible eigenbases for a
given algebra $\mathcal{A}$. For a given $\mathcal{A}$, the $\mathcal{A}$-OTOC
NRC LTA is a function
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U):U(d)\rightarrow\mathbb{R}$
over the manifold $U(d)$ of $d\times d$ unitaries, which we can embed in a
Euclidean space $\mathbb{C}^{d\times d}$ with inner product $\langle
A,B\rangle_{\mathbb{C}^{d\times d}}\coloneqq\mathscr{R}(\Tr(AB^{\dagger}))$,
where $\mathscr{R}$ denotes the real part. Note that
$\mathcal{U}_{t}(\cdot)=\exp(it\,H)(\cdot)\exp(-it\,H)$ is the unitary
Hamiltonian evolution in the Heisenberg picture, while $U$ is the unitary that
represents the transformation of the eigenbasis of $H$ to $\mathbb{B}$.

Starting from a random unitary $U_{0}$ we move on $U(d)$ along the geodesic
with the steepest descent for
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U)$ via
$U_{1}\equiv\exp(-\mu\,G_{U_{0}}U_{0})$. Here, $\mu\in\mathbb{R}^{+}$ is a
dynamic step size adjusted to improve the rate of convergence [59] and
$G_{U_{0}}\in T_{\mathds{1}}U(d)$ is the Riemannian gradient of
$\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U)$ at $U_{0}$ translated
to the tangent space at the identity. Choosing the inner product $\langle
X,Y\rangle_{U}\coloneqq\frac{1}{2}\mathscr{R}(\Tr(XY^{\dagger})$ for the
tangent space of $U(d)$ at $U$, we have
$G_{U}\equiv\Gamma_{U}U^{\dagger}-U\Gamma_{U}^{\dagger}$ [59], where
$\Gamma_{U}\coloneqq\nabla_{U^{*}}\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}$
is the standard Euclidean gradient on $\mathbb{C}^{n\times n}$. Using the
first expression in Eq. 28 for the one-dimensional eigenprojectors
$U\Pi_{k}U^{\dagger}$, the function
$\overline{G_{\mathcal{A}}(\mathcal{U})}^{NRC}(U)$ is written explicitly as

$\begin{split}\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U)&=1-\frac{1}{d}\sum_{\alpha=1}^{d(\mathcal{A})}\sum_{\gamma=1}^{d(\mathcal{A}^{\prime})}\left(\sum_{k,l=1}^{d}\Tr(e_{\alpha}U\Pi_{k}U^{\dagger}f_{\gamma}U\Pi_{k}U^{\dagger}e_{\alpha}^{\dagger}U\Pi_{l}U^{\dagger}f_{\gamma}^{\dagger}U\Pi_{l}U^{\dagger})+\right.\\\
&\hskip
100.0pt+\sum_{k,l=1}^{d}\Tr(e_{\alpha}U\Pi_{k}U^{\dagger}f_{\gamma}U\Pi_{l}U^{\dagger}e_{\alpha}^{\dagger}U\Pi_{l}U^{\dagger}f_{\gamma}^{\dagger}U\Pi_{k}U^{\dagger})-\\\
&\hskip
100.0pt\left.-\sum_{k=1}^{d}\Tr(e_{\alpha}U\Pi_{k}U^{\dagger}f_{\gamma}U\Pi_{k}U^{\dagger}e_{\alpha}^{\dagger}U\Pi_{k}U^{\dagger}f_{\gamma}^{\dagger}U\Pi_{k}U^{\dagger})\right)\end{split}$
(47)

For the Euclidean gradient of a function $f(U):\mathbb{C}^{d\times
d}\rightarrow\mathbb{R}$, we have $\delta f=\langle\nabla_{U}f,\delta
U\rangle_{\mathbb{C}^{d\times d}}+\langle\nabla_{U^{*}}f,\delta
U^{*}\rangle_{\mathbb{C}^{d\times
d}}=2\mathscr{R}\left(\Tr(\left(\nabla_{U^{*}}f\right)^{T}\delta U)\right)$,
where $A^{T}$ denotes the matrix transpose and we used that
$(\nabla_{U}f)^{*}=\nabla_{U^{*}}f$. Performing the variation in Eq. 47 and
comparing with
$\delta\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}=2\mathscr{R}\left(\Tr(\Gamma_{U}^{T}\delta
U)\right)$ we find, after some algebraic manipulation, that

$\begin{split}\Gamma_{U}&=2\sum_{kl}\bigg{(}(1-\delta_{kl}/2)\\\ &\hskip
45.0pt\left(\left[\sum_{\gamma}\left(\mathcal{P}_{\mathcal{A}^{\prime}}\left(U\Pi_{l}U^{\dagger}f_{\gamma}U\Pi_{l}U^{\dagger}\right)U\Pi_{k}U^{\dagger}f_{\gamma}^{\dagger}\right)+\sum_{\alpha}\mathcal{P}_{\mathcal{A}}\left(U\Pi_{l}U^{\dagger}e_{\alpha}U\Pi_{l}U^{\dagger}\right)U\Pi_{k}U^{\dagger}e_{\alpha}^{\dagger}\right]+\text{h.c.}\right)U\Pi_{k}\bigg{)}.\end{split}$
(48)

where h.c. denotes the hermitian conjugate of the expression inside the
brackets.

We iteratively update the unitary $U_{k+1}=\exp(-\mu_{k}G_{U_{k}}U_{k})$ until
the convergence condition
$\absolutevalue{\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U_{k+1})-\overline{G_{\mathcal{A}}(\mathcal{U}_{t})}^{NRC}(U_{k})}<\epsilon$
is met, where $\epsilon$ is a tolerance set to $\epsilon=10^{-8}$.