# Noncommuting conserved charges in quantum thermodynamics and beyond

Shayan Majidy<EMAIL_ADDRESS>Institute for Quantum Computing,
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Perimeter Institute
for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada  William F.
Braasch, Jr  Joint Center for Quantum Information and Computer Science, NIST
and University of Maryland, College Park, MD 20742, USA  Aleksander Lasek
Joint Center for Quantum Information and Computer Science, NIST and University
of Maryland, College Park, MD 20742, USA  Twesh Upadhyaya  Joint Center for
Quantum Information and Computer Science, NIST and University of Maryland,
College Park, MD 20742, USA  Department of Physics, University of Maryland,
College Park, MD 20742, USA  Amir Kalev  Information Sciences Institute,
University of Southern California, Arlington, VA 22203, USA  Department of
Physics and Astronomy, University of Southern California, Los Angeles,
California 90089, USA  Nicole Yunger Halpern<EMAIL_ADDRESS>Joint Center
for Quantum Information and Computer Science, NIST and University of Maryland,
College Park, MD 20742, USA  Institute for Physical Science and Technology,
University of Maryland, College Park, MD 20742, USA

###### Abstract

Thermodynamic systems typically conserve quantities (“charges”) such as energy
and particle number. The charges are often assumed implicitly to commute with
each other. Yet quantum phenomena such as uncertainty relations rely on
observables’ failure to commute. How do noncommuting charges affect
thermodynamic phenomena? This question, upon arising at the intersection of
quantum information theory and thermodynamics, spread recently across many-
body physics. Charges’ noncommutation has been found to invalidate derivations
of the thermal state’s form, decrease entropy production, conflict with the
eigenstate thermalization hypothesis, and more. This Perspective surveys key
results in, opportunities for, and work adjacent to the quantum thermodynamics
of noncommuting charges. Open problems include a conceptual puzzle: Evidence
suggests that noncommuting charges may hinder thermalization in some ways
while enhancing thermalization in others.

##  I Introduction

A simple story from undergraduate statistical physics motivates this
Perspective: Throughout thermodynamics, we consider a small system
$\mathcal{S}$ exchanging quantities with a large environment $\mathcal{E}$
(Fig. 1). Suppose that the systems are quantum. If they exchange only heat,
$\mathcal{S}$ may thermalize to the canonical state $\rho_{\rm can}\propto
e^{-\beta H^{(\mathcal{S})}}$. The environment’s inverse temperature is
$\beta$, and $H^{(\mathcal{S})}$ denotes the system-of-interest Hamiltonian.
If $\mathcal{S}$ and $\mathcal{E}$ exchange heat and particles, $\mathcal{S}$
may thermalize to the grand canonical state $\rho_{\rm GC}\propto
e^{-\beta(H^{(\mathcal{S})}-\mu\mathcal{N}^{(\mathcal{S})})}$. The chemical
potential is $\mu$, and $\mathcal{N}^{(\mathcal{S})}$ denotes the system-of-
interest particle-number operator. This pattern extends to many exchanged
quantities (electric charge, magnetization, etc.) and many thermal states.

Figure 1: Common thermodynamic paradigm: A small system and large environment
locally exchange quantities that are conserved globally. Common quantities
include energy, as well as particles of different species.

The exchanged quantities are conserved globally (across
$\mathcal{S}\mathcal{E}$), so we call them _charges_. Hermitian operators
$Q_{a}$ represent the conserved quantities; $\mathcal{S}$ has an operator
$Q_{a}^{(\mathcal{S})}$, $\mathcal{E}$ has $Q_{a}^{(\mathcal{E})}$, and the
global system has $Q_{a}^{\rm tot}\coloneqq
Q_{a}^{(\mathcal{S})}+Q_{a}^{(\mathcal{E})}\equiv
Q_{a}^{(\mathcal{S})}\otimes\mathbbm{1}^{(\mathcal{E})}+\mathbbm{1}^{(\mathcal{S})}\otimes
Q_{a}^{(\mathcal{E})}$. The index $a=0,1,\ldots,c$; and the Hamiltonian
$H=Q_{0}$.

Across thermodynamics, we often assume implicitly that the charges commute
with each other: $[Q_{a},Q_{a^{\prime}}]=0$ $\forall a,a^{\prime}$. This
assumption is almost never mentioned (recall your undergraduate statistical-
physics course). However, the assumption underlies derivations of the form of
thermal states [1, 2], linear-response coefficients [3], and more.
Observables’ ability to fail to commute, though, enables quintessentially
quantum phenomena: the Einstein–Podolsky–Rosen paradox [4], uncertainty
relations [5], measurement disturbance [6], etc. Quantum physics therefore
compels us to ask, _what happens to thermodynamic phenomena under dynamics
that conserve charges $Q_{a}^{\rm tot}$ that fail to commute with each other?_

This question arose recently at the intersection of quantum information theory
and quantum thermodynamics, then seeped into many-body physics. This
Perspective surveys results, opportunities, and adjacent work. In the rest of
the introduction, we first present a simple physical example. Then, we sample
example phenomena transformed by charges’ noncommutation. We later establish
notation, followed by the Perspective’s outline.

A simple physical example was proposed theoretically [7, 8] and realized with
trapped ions [9]: a chain, shown in Fig. 2, formed from qubits (quantum two-
level systems). A few (e.g., two) qubits form $\mathcal{S}$, and the other
qubits form $\mathcal{E}$. The chain constitutes a closed quantum many-body
system, of the sort whose internal thermalization has recently been studied
theoretically and experimentally (e.g., [10, 11, 12, 13]). Denote by
$\sigma_{a}$ the Pauli-$a$ operator, for $a=x,y,z$; by $\sigma_{a}^{(j)}$, a
spin component of qubit $j$; and by $\sigma_{a}^{\rm
tot}\coloneqq\sum_{j}\sigma_{a}^{(j)}$, a total spin component. One can
construct a Hamiltonian that overtly transports quanta of each $\sigma_{a}$
locally while conserving the $\sigma_{a}^{\rm tot}$’s globally [7, 8]: Denote
the $\sigma_{z}$ ladder operators by $\sigma_{\pm
z}\coloneqq\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})$. The operator
$\sigma_{+z}^{(j)}\sigma_{-z}^{(j+1)}+{\rm h.c.}$ transports one $\sigma_{z}$
quantum from qubit $j+1$ to qubit $j$ and vice versa, in superposition. Define
ladder operators and couplings analogously for $\sigma_{x}$ and $\sigma_{y}$.
The Hamiltonian

$\displaystyle H_{\rm Heis}^{\rm
tot}=\sum_{j}\sum_{\alpha=x,y,z}\left(\sigma_{+\alpha}^{(j)}\sigma_{-\alpha}^{(j+1)}+{\rm
h.c.}\right)=\sum_{j}\vec{\sigma}^{(j)}\cdot\vec{\sigma}^{(j+1)}$ (1)

transports the $\sigma_{a}$’s locally, while conserving them globally. $H_{\rm
Heis}^{\rm tot}$ is the Heisenberg model, theoretically well-known and
experimentally realizable [7, 9]. Yet $H_{\rm Heis}^{\rm tot}$ is rarely, if
ever, expressed as in (1) outside the thermodynamics of noncommuting
charges—as locally transporting and globally conserving three noncommuting
charges. Furthermore, we can extend $H_{\rm Heis}^{\rm tot}$ to nonintegrable
models (which promote thermalization), to subsystems beyond qubits, and to
charges beyond spin components [8].111 For more quantum thermodynamics of spin
exchanges, see [14, 15, 16, 17].

Figure 2: Example thermodynamic system that conserves noncommuting charges:
Two qubits form the system $\mathcal{S}$ of interest, and the rest form the
environment $\mathcal{E}$. A qubit’s three spin components,
$\sigma_{a=x,y,z}$, form the local noncommuting charges. The dynamics locally
transport and globally conserve the charges.

Having exhibited a thermodynamic system whose $Q_{a}$’s fail to commute, we
sample the physics altered by charges’ noncommutation. This sample only dips
into the known results (detailed in Sec. III) but demonstrates why
noncommuting thermodynamic charges merit study. If the $Q_{a}$’s fail to
commute, how much $\mathcal{S}$ can thermalize is unclear, for several
reasons. First, noncommuting charges impede two derivations of the thermal
state’s form [1, 2]. Second, noncommuting charges force degeneracies on
$H^{\rm tot}$. The reason is Schur’s lemma, a group-theoretic result (App. A
and [18, 19, 7]). Nondegenerate Hamiltonians underlie arguments supporting
thermalization, mixing, and equilibration (e.g., [20, 21, 22, 23])—arguments
challenged by noncommuting charges.

Third, noncommuting charges conflict with the eigenstate thermalization
hypothesis (ETH) [24]. The ETH is a theoretical and conceptual toolkit for
explaining how quantum many-body systems thermalize internally [20, 21, 25].
Myriad numerical and experimental observations support the ETH, which has been
applied across many-body physics. Yet noncommuting charges preclude the ETH
(Sec. III). Hence noncommuting charges challenge expectations about
thermalization in a third manner.

We will encounter even more manners across this Perspective. Resistance to
thermalization matters fundamentally and for applications: Most systems
thermalize, so resistance is unusual. Few mechanisms for resistance in quantum
many-body systems are known [26, 27]. Additionally, thermalization disperses
information about initial conditions, so thermalization resistance can promote
memory storage.

A major research opportunity is to identify to what extent noncommuting
charges hinder thermalization. Another opportunity is to re-examine every
thermodynamic or chaotic phenomenon, asking to what extent it changes if
charges fail to commute. Phenomena known to change, as reviewed in this
Perspective, include the ETH, microcanonical states, thermodynamic-entropy
production, entanglement entropy, and local interactions’ ability to effect
global evolutions. A third opportunity is to marry recent noncommuting-charge
work, based largely in quantum thermodynamics and information theory, with
adjacent work in other fields, such as high-energy theory. Other fields favor
the terminology _non-Abelian symmetries_ ; fundamental thermodynamics
motivates our noncommuting-charge language. A Hamiltonian conserves
noncommuting charges, $[H^{\rm tot},Q_{a}^{\rm tot}]=0$ $\forall a$, if and
only if it has a continuous non-Abelian symmetry: $U_{a}^{\dagger}H^{\rm
tot}U_{a}=H^{\rm tot}$ $\forall a$. The charges generate the unitary
operators: $U_{a}=e^{-iQ_{a}t}$, for $t\in\mathbb{R}$. The $Q_{a}$’s form a
Lie algebra, and the $U_{a}$’s form the associated Lie group [28]. We flesh
out the three above-mentioned opportunities across this Perspective.

Across the Perspective, we use the notation introduced above, as well as the
following. We call a closed quantum many-body system (e.g., the composite
$\mathcal{S}\mathcal{E}$ in Fig. 1) a _global_ system. $N$ denotes the number
of degrees of freedom (DOFs) in a global system. Often,
$\mathcal{S}\mathcal{E}$ will consist of $N$ copies of $\mathcal{S}$. For
example, $N$ denotes the number of qubits in Fig. 2. Large but finite $N$—the
mesoscale—interests us: As $N\to\infty$, $\mathcal{S}\mathcal{E}$ grows
classical, according to the correspondence principle [2]. Noncommutation
enables nonclassical phenomena, so we should expect charges’ noncommutation to
influence thermodynamic phenomena at finite $N$. Continuing with notation, we
ascribe to the Pauli operators $\sigma_{a}$ eigenstates $\lvert a\pm\rangle$
associated with the eigenvalues $\pm 1$. Subscripts index charges (as in
$\sigma_{a}^{(j)}$), whereas superscripts index sites or other subsystems
(e.g., $\mathcal{S}$ and $\mathcal{E}$). We often denote commuting charges, or
other observables that might commute, by $\tilde{Q}_{a}$’s.

The rest of this Perspective is organized as follows. Section II introduces
the earliest work on noncommuting thermodynamic charges. Section III surveys
new physics that arises from charges’ noncommutation. In Sec. IV, we discuss
bridges with related research. Section V crystallizes the richest-looking
avenues for future work.

##  II Early work

Jaynes was the first, to our knowledge, to address anything like noncommuting
thermodynamic charges [29]. He formalized the _principle of maximum entropy_ ,
which pinpoints the state $\rho=\sum_{k}p_{k}\lvert k\rangle\\!\langle
k\rvert$ most reasonably attributable to a system about which one knows
little. Imagine knowing about $\rho$ only the expectation values
$\langle\tilde{Q}_{a}\rangle$ of observables $\tilde{Q}_{a}$. The state obeys
the constraints ${\rm Tr}(\rho\tilde{Q}_{a})=\tilde{Q}_{a}^{\rm
avg}\in\mathbb{R}$, plus the normalization condition ${\rm Tr}(\rho)=1$.
Whichever constraint-obeying state maximizes the von Neumann entropy $S_{\rm
vN}(\rho)\coloneqq-{\rm Tr}\bm{(}\rho\log(\rho)\bm{)}$ is most reasonable,
according to the maximum-entropy principle. (This subsection’s logarithms are
base-$e$.) The entropy maximization encapsulates our ignorance of everything
except the constraints. The function maximized is

$\displaystyle S_{\rm vN}(\rho)-0$ (2) $\displaystyle=S_{\rm
vN}(\rho)-\lambda\left[{\rm
Tr}(\rho)-1\right]-\sum_{a}\tilde{\mu}_{a}\left[{\rm
Tr}(\rho\tilde{Q}_{a})-\tilde{Q}_{a}^{\rm avg}\right]$
$\displaystyle\eqqcolon\mathcal{L}(\rho,\lambda,\\{\tilde{\mu}_{a}\\}).$

Lagrange multipliers are denoted by $\lambda,\tilde{\mu}_{a}\in\mathbb{R}$.
The state’s eigenbasis, $\\{\lvert k\rangle\\!\langle k\rvert\\}$, equals an
eigenbasis of $\sum_{a}\tilde{\mu}_{a}\tilde{Q}_{a}$: If $\\{\lvert
k\rangle\\!\langle k\rvert\\}$ did not, decohering $\rho$ with respect to any
$\sum_{a}\tilde{\mu}_{a}\tilde{Q}_{a}$ eigenbasis would raise $S_{\rm vN}$
monotonically, while preserving the other terms in (2) [30]. Hence the
decohered state would achieve at least as large an $\mathcal{L}$ value as
$\rho$. Maximizing $\mathcal{L}(\rho,\lambda,\\{\tilde{\mu}_{a}\\})$ with
respect to $p_{k}$ yields

$\displaystyle\rho=\frac{1}{Z}\,e^{-\sum_{a}\tilde{\mu}_{a}\tilde{Q}_{a}}\,.$
(3)

Maximizing with respect to $\lambda$ fixes the partition function: $Z={\rm
Tr}(e^{-\sum_{a}\tilde{\mu}_{a}\tilde{Q}_{a}})$. Maximizing with respect to
$\tilde{\mu}_{a}$ yields $\tilde{Q}_{a}^{\rm
avg}=-\frac{\partial}{\partial\tilde{\mu}_{a}}\log(Z)$. This procedure works,
Jaynes noted, even if the $\tilde{Q}_{a}$’s do not commute. He showed how to
measure such observables’ $\tilde{Q}_{a}^{\rm avg}$’s using ancillas. One can
derive his results alternatively via analytic properties of $Z$ [31].

Jaynes’s work, though pioneering, left threads hanging. First, the maximum-
entropy principle invokes only information theory and quantum physics, not
thermodynamics or charge conservation. Connecting $\rho$ to equilibrium
requires more-physical arguments. Second, Jaynes wrote only one paragraph
about noncommuting $\tilde{Q}_{a}$’s. What could they represent physically?
Which systems might have them? When might they impact thermodynamics?

Still, Jaynes’s known-about observables $\tilde{Q}_{a}$ can be noncommuting
charges $Q_{a}$. Hence Jaynes was the first to write what was later termed the
_non-Abelian thermal state_ (NATS) [2],

$\displaystyle\rho_{\rm
NATS}\coloneqq\frac{1}{Z}e^{-\beta(H-\sum_{a}\mu_{a}Q_{a})}.$ (4)

$\beta$ denotes an inverse temperature. The $\mu_{a}$’s denote generalized
chemical potentials. $\rho_{\rm NATS}$ has the form of Eq. (3), which has been
called the generalized Gibbs ensemble (GGE), regardless of whether the charges
commute [32, 33, 34, 35]. We use the term NATS for three reasons. First, the
term GGE was initially introduced for integrable systems, whereas the NATS is
a thermal state. Second, most GGE literature concerns commuting charges (see
Sec. IV for exceptions). Third, justifying the NATS’s form physically (not
merely information-theoretically) is more difficult than justifying the form
of a typical (commuting-charge) GGE, as explained later in this section.

Acolytes of Jaynes’s expanded upon his work [36, 37, 38, 39]. Balian and
Balazs [39] sought a more-physical justification for $\rho_{\rm NATS}$’s form.
They imagined $N$ copies of the system of interest, in the ensemble tradition
of thermodynamics. In thermodynamics, we regard all copies except one
($\mathcal{S}$) as forming an effective environment ($\mathcal{E}$) [40].
Imagine $\mathcal{S}$ exchanging energy and particles with $\mathcal{E}$. How
can we prove that $\mathcal{S}$ is in a canonical state $\rho_{\rm can}$? We
assume that $\mathcal{S}\mathcal{E}$ has a fixed particle number and an energy
in a small window—that $\mathcal{S}\mathcal{E}$ is in a microcanonical
subspace. Tracing out $\mathcal{E}$ from the microcanonical state yields
$\mathcal{S}$’s state, which equals $\rho_{\rm can}$ if $\mathcal{S}$ and
$\mathcal{E}$ couple weakly [41]. Suppose that $\mathcal{S}$ and $\mathcal{E}$
exchange several commuting charges $\tilde{Q}_{a}$. The microcanonical
subspace is an eigenspace shared by the $\tilde{Q}_{a}^{\rm tot}$’s (except
$\tilde{Q}_{0}^{\rm tot}=H^{\rm tot}$, the time-evolution generator). What if
$\mathcal{S}$ and $\mathcal{E}$ exchange noncommuting charges $Q_{a}$? The
$Q_{a}^{\rm tot}$’s share no eigenbasis, so they might share no eigenspaces.
Microcanonical subspaces might not exist.

Balian and Balazs tried to overcome this challenge. They observed that the
charge densities $Q_{a}^{\rm tot}/N$ commute in the infinite-$N$ limit:
$\lim_{N\to\infty}\frac{1}{N^{2}}[Q_{a}^{\rm tot},Q_{a^{\prime}}^{\rm tot}]=0$
$\forall a,a^{\prime}$. But they could not construct a well-justified
generalization of microcanonical subspaces for noncommuting charges. That goal
would wait for the maturation of quantum information theory.

For decades afterward, no literature addressed thermodynamic charges’ ability
to not commute, to our knowledge. The topic gained attention a few years ago
at the intersection of quantum information and quantum thermodynamics. Two
publications showed that noncommuting charges can violate thermodynamic
expectations [42, 1].

Lostaglio demonstrated that noncommuting charges overturn an expectation about
free energy [42]. Consider a system $\mathcal{S}$ of interest in a state
$\rho^{(\mathcal{S})}$. Let $\mathcal{S}$ begin uncorrelated with its
environment $\mathcal{E}$, which consists of $c$ subsystems:
$\rho^{(\mathcal{S})}\otimes\rho^{(\mathcal{E})}$, wherein
$\rho^{(\mathcal{E})}$ has the GGE form (3). $\mathcal{S}$ has $c$ commuting
charges $\tilde{Q}_{a}^{(\mathcal{S})}$, and $\mathcal{E}$ has
$\tilde{Q}_{a}^{(\mathcal{E})}$’s. One can attribute to $\mathcal{S}$ a “free
energy”222 The quotation marks reflect the controversy surrounding free
energies defined information-theoretically, for out-of-equilibrium states.
$F_{a}^{(\mathcal{S})}(\rho)\coloneqq-\frac{1}{\beta\mu_{a}}S(\rho)+{\rm
Tr}(\tilde{Q}_{a}\rho)$. Let any charge-conserving unitary $U$ evolve
$\mathcal{S}\mathcal{E}$ to a state $\rho^{(\mathcal{S}\mathcal{E})}_{\rm f}$:
$[U,\tilde{Q}_{a}^{\rm tot}]=0$ $\forall a$. Subscript ${\rm f}$’s will
distinguish $\mathcal{S}$’s and $\mathcal{E}$’s final states. $\mathcal{E}$’s
$j^{\rm th}$ subsystem ends up in $\rho^{(\mathcal{E},j)}_{\rm f}\coloneqq{\rm
Tr}_{\bar{j}}(\rho^{(\mathcal{E})}_{\rm f})$. Three more quantities change:
$\mathcal{S}$’s von Neumann entropy, by $\Delta S^{(\mathcal{S})}_{\rm
vN}\coloneqq S_{\rm vN}(\rho^{(\mathcal{S})}_{\rm f})-S_{\rm
vN}(\rho^{(\mathcal{S})})$; the $a^{\rm th}$ charge’s environmental
expectation value, by
$\Delta\langle\tilde{Q}_{a}^{(\mathcal{E})}\rangle\coloneqq\mathrm{Tr}(\tilde{Q}_{a}^{(\mathcal{E})}[\rho^{(\mathcal{E})}_{\rm
f}-\rho^{(\mathcal{E})}])$; and the system’s $a^{\rm th}$ “free energy,” by
$\Delta F_{a}^{(\mathcal{S})}$. $U$ redistributes charges and information
contents according to

$\displaystyle-\Delta S^{(\mathcal{S})}_{\rm
vN}+D\left(\rho^{(\mathcal{S}\mathcal{E})}_{\rm
f}\left\lvert\right\rvert\rho^{(\mathcal{S})}_{\rm
f}\otimes\rho^{(\mathcal{E})}_{\rm f}\right)$ (5)
$\displaystyle=\beta\sum_{a}\mu_{a}\left(\Delta\langle\tilde{Q}_{a}^{(\mathcal{E})}\rangle-\Delta
F_{a}^{(\mathcal{S})}\right)-D\left(\rho^{(\mathcal{E})}_{\rm
f}\left\lvert\right\rvert\otimes_{j}\rho^{(\mathcal{E},j)}_{\rm f}\right).$

The relative entropy $D(\sigma_{1}||\sigma_{2})=-{\rm
Tr}(\sigma_{1}[\log\sigma_{1}-\log\sigma_{2}])$ quantifies the distance
between quantum states $\sigma_{1,2}$ [43]. Equation (5) relates that the
growth in $\mathcal{S}$’s information content, plus the global state’s final
nonseparability, depends on the average changes in $\mathcal{E}$’s charges,
$\mathcal{S}$’s “free-energy” change, and the correlations formed in
$\mathcal{E}$. If charges fail to commute, the derivation breaks down. No term
decomposes into distinct charges’ $\Delta F_{a}^{(\mathcal{S})}$. Hence
noncommuting charges undermine an expectation about free energy.

Second, Yunger Halpern reasoned about noncommuting charges in thermodynamic
resource theories [1]. _Resource theories_ are information-theoretic models
for contexts in which restrictions constrain the operations performable and
the systems accessible [44]. Using a resource theory, one calculates the
optimal efficiencies with which an agent, subject to the restrictions, can
perform tasks such as extracting work from nonequilibrium quantum systems. In
thermodynamics, the first law constrains agents to conserve energy. Every
unitary $U$ performable on a closed, isolated system conserves the total
Hamiltonian: $[U,H^{\rm tot}]=0$ [45]. Suppose that $U$ must conserve
commuting global charges $\tilde{Q}_{a}^{\rm tot}$: $[U,\tilde{Q}_{a}^{\rm
tot}]=0$ [46]. Which systems, if freely accessible to a thermodynamic agent,
would render the model nontrivial—would not enable the agent to, say, perform
work for free, achieving a _perpetuum mobile_? Systems in the equilibrium
state (3) [46, 47]. The proof fails, however, if the charges fail to commute
[1].

References [42, 1] spurred three groups to apply quantum-information-theoretic
thermodynamics to noncommuting charges [30, 48, 2] (as overviewed in [49]).
Guryanova et al. quantified tradeoffs among work and charges, using a similar
resource theory [30]. Consider a system $\mathcal{S}$ in an out-of-equilibrium
state $\rho^{(\mathcal{S})}$. $\mathcal{S}$ begins uncorrelated with an
environment $\mathcal{E}$ in a GGE $\rho^{(\mathcal{E})}$. Each of
$\mathcal{S}$ and $\mathcal{E}$ has charges $Q_{a}$ that might or might not
commute. (In this section, only in this paragraph do $Q_{a}$’s denote possibly
commuting charges.) The setup includes also _batteries_ , systems in which
work or charges can be reliably stored and from which the resources can be
reliably retrieved. The authors ascribe a _free entropy_ to every state $\rho$
of $\mathcal{S}$:

$\tilde{F}^{(\mathcal{S})}(\rho)\coloneqq\beta\sum_{a}\mu_{a}{\rm
Tr}\left(\rho Q^{(\mathcal{S})}_{a}\right)-S_{\rm
vN}\left(\rho^{(\mathcal{S})}\right).$ (6)

{$\tilde{F}^{(\mathcal{S})}(\rho)$ is the negative of a nonequilibrium
extension of a Massieu function [40].} Let $\mathcal{S}\mathcal{E}$ evolve
under any unitary $U$ that conserves each total (system–environment–battery)
charge. For all $X=\mathcal{S},\mathcal{E}$, the state $\rho^{(X)}$ evolves to
$\rho^{(X)}_{\rm f}$. $X$’s average $a$-charge changes by $\Delta\langle
Q_{a}^{(X)}\rangle\coloneqq{\rm Tr}(Q_{a}^{(X)}[\rho^{(X)}_{\rm
f}-\rho^{(X)}])$. $\mathcal{S}$’s free entropy changes by
$\Delta\tilde{F}^{(\mathcal{S})}$. The batteries perform on
$\mathcal{S}\mathcal{E}$ the average “charge-$a$ work”
$W_{a}=-\big{(}\Delta\langle Q_{a}^{(\mathcal{S})}\rangle+\Delta\langle
Q_{a}^{(\mathcal{E})}\rangle\big{)}$. This charge work obeys a generalized
second law of thermodynamics:

$\displaystyle\beta\sum_{a}\mu_{a}W_{a}\leq-\Delta\tilde{F}^{(\mathcal{S})}.$
(7)

If $\mathcal{S}$ is trivial (corresponds to a 0-dimensional Hilbert space) or
is unchanged by $U$, $\Delta\tilde{F}^{(\mathcal{S})}=0$. Consequently, one
can extract any amount of any charge from $\mathcal{E}$, by paying a price in
the other charges: $\mu_{a}W_{a}\leq-\sum_{a^{\prime}\neq
a}\mu_{a^{\prime}}W_{a^{\prime}}$. Under what conditions Eq. (7) is saturable
depends on whether the charges commute. Lostaglio _et al._ , too, drew
tradeoff conclusions, which they applied to the erasure of qubits [48].

Rounding out the trio [30, 48, 2], Yunger Halpern _et al._ physically
justified $\rho_{\rm NATS}$’s form in multiple ways, beginning with a
microcanonical-like derivation [2]. Independently of Balian and Balazs [39],
the authors realized that noncommuting charges prevent microcanonical
subspaces from existing (in abundance). The authors therefore generalized
microcanonical to _approximate microcanonical_ (AMC) subspaces. In an AMC
subspace $\mathcal{M}$, every $Q_{a}^{\rm tot}$ has a fairly well-defined
value: Measuring any $Q_{a}^{\rm tot}$ has a high probability of yielding a
value near the expectation value $\langle Q_{a}^{\rm tot}\rangle$. The authors
defined $\mathcal{M}$ and proved its existence under certain conditions [2].
Denote by $\Pi_{\mathcal{M}}$ the projector onto $\mathcal{M}$. Consider
ascribing the AMC state $\Pi_{\mathcal{M}}/{\rm Tr}(\Pi_{\mathcal{M}})$ to the
global system, formed from $N$ copies of the system $\mathcal{S}$ of interest.
Trace out all copies except the $\ell^{\rm th}$: $\rho^{(\ell)}={\rm
Tr}_{\bar{\ell}}\bm{(}\Pi_{\mathcal{M}}/{\rm Tr}(\Pi_{\mathcal{M}})\bm{)}$.
Compare $\rho^{(\ell)}$ with $\rho_{\rm NATS}$ using the relative entropy.
Average over $\ell$. This average distance is upper-bounded as

$\displaystyle\big{\langle}D\big{(}\rho^{(\ell)}||\rho_{\rm
NATS}\big{)}\big{\rangle}_{\ell}\leq\frac{\theta}{\sqrt{N}}+\theta^{\prime}.$
(8)

The $N$-independent constants $\theta,\theta^{\prime}$ depend on parameters
(the number $c$ of charges, their expectation values, etc.). As the global
system grows ($N\to\infty$), the $1/{\sqrt{N}}\to 0$, so the distance shrinks.
Hence a physical argument, based on an ensemble in an AMC subspace,
complements Jaynes’s information-theoretic derivation of $\rho_{\rm NATS}$.
Furthermore, the AMC subspace enabled later noncommuting-charge work [7, 9,
24, 8], described later in this section and in Sec. III.

$\rho_{\rm NATS}$’s form was justified also with the principle of complete
passivity [48, 2, 50]. A state $\rho$ is _completely passive_ if no work can
be extracted from $\rho^{\otimes N}$ adiabatically, for any
$N\in\mathbb{Z}_{\geq 0}$ [51, 52]. Complete passivity extends Carnot’s
version of the second law of thermodynamics: No engine can extract work from
one fixed-temperature heat bath. Consider work extraction that evolves a
battery’s state from $\rho^{(\mathcal{B})}$ to $\rho_{\rm f}^{(\mathcal{B})}$.
Let $Q_{a}^{(\mathcal{B})}$ denote the battery’s $a^{\rm th}$ charge. One can
define the (average) work performed on the battery as ${\rm Tr}(\rho_{\rm
f}^{(\mathcal{B})}[\sum_{a}\mu_{a}Q_{a}^{(\mathcal{B})}])-{\rm
Tr}(\rho^{(\mathcal{B})}[\sum_{a}\mu_{a}Q_{a}^{(\mathcal{B})}])$. Using this
definition and a resource theory, one can prove that $\rho_{\rm NATS}$ is
completely passive [2]. This result resolved the problem spotlighted in [1]:
Noncommuting charges invalidate a resource-theoretic derivation of the thermal
state’s form.333 By _invalidate_ , we mean that, if the charges fail to
commute, the derivation becomes unsound. We use _unsound_ in the technical
sense of the study of logic: A deductive argument is sound if, provided that
the input premises are true, the conclusion is true. Yet work is traditionally
a change in the energy charge [48]. Indeed, Mitsuhashi _et al._ proved
$\rho_{\rm NATS}$’s complete passivity while defining work as a change in
energy alone (imposing charge conservation on just the extracted-from system)
[50]. Furthermore, one can define an analog $W_{a}$ of work for each charge
$Q_{a}$ [48], as in [30]. Consider substituting different $W_{a}$’s into the
definition of complete passivity. Different $W_{a}$’s will anoint different
thermal states as passive, because the $Q_{a}$’s fail to commute [48]. Hence
complete passivity is another thermodynamic principle complicated by
noncommuting charges.

The trio [30, 48, 2] galvanized noncommuting-charge activity at the
intersection of quantum information and quantum thermodynamics. Several
publications elaborated on resource theories [53, 54, 55, 56, 57]. For
example, Gour _et al._ proved more-detailed “second laws,” using the
mathematical tools of matrix majorization [53]. These second laws stipulate
necessary and sufficient conditions for any quantum state $\sigma_{1}$ to
transform into any state $\sigma_{2}$. Other literature showed how different
batteries could store different $Q_{a}$’s despite the charges’ noncommutation
[58, 55, 56, 59].444 Batteries are called _reference frames_ when used to
implement otherwise forbidden transformations. Consider transforming a system-
and-environment composite $\mathcal{S}\mathcal{E}$ with unitaries $U$
restricted to conserving each $Q_{a}^{(\mathcal{S})}+Q_{a}^{(\mathcal{E})}$.
What if a desired $U$ breaks the conservation law? A reference frame may
supply the charge needed, or accept the charge forfeited, by
$\mathcal{S}\mathcal{E}$.,555 Reference [58] predates the trio [30, 48, 2],
lacking thermodynamic motivations but having thermodynamic implications.

The above work, steeped in information theory, is theoretical and abstract.
Yet it inspired the first experimental demonstration that real-world systems
exhibit the thermodynamics of noncommuting charges [9]. Kranzl et al. trapped
6–21 ions in a linear Paul chain. Two qubits formed $\mathcal{S}$, and the
rest formed $\mathcal{E}$ (Fig. 1). The global system was prepared in
$\lvert\psi_{0}\rangle=\left(\lvert y+\rangle\lvert x+\rangle\lvert
z+\rangle\right)^{\otimes N/3}$, in an AMC subspace.666 Reference [7] adapted
the AMC subspace’s definition from the information-theoretic Ref. [2] to many-
body physics. In [2], if $\mathcal{S}\mathcal{E}$ occupies an AMC subspace,
measuring any $Q_{a}^{\rm tot}$ has a high probability of yielding a value
near $\langle Q_{a}^{\rm tot}\rangle$. Constant-in-$N$ expressions quantify
what “high” and “near” mean. In [7, 9, 8], the probability distribution has a
variance of $O(N^{r})$, wherein $r\leq 1$. The native interactions were long-
range Ising couplings ($\vec{\sigma}^{(j)}\cdot\vec{\sigma}^{(k)}$). However,
Heisenberg interactions (also long-range) were simulated via Trotterization
with rotations. The dynamics conserved the charges $\sigma_{x,y,z}^{\rm tot}$
globally while transporting $\sigma_{x,y,z}$ locally. Quantum state tomography
revealed the long-time state $\rho_{\rm f}$ of $\mathcal{S}$.

Following [2], Kranzl et al. calculated the state’s distance from
$\rho_{\rm{NATS}}$, using the relative entropy $D(\rho_{\rm f}||\rho_{\rm
NATS})$. They averaged $D(\rho_{\rm f}||\rho_{\rm NATS})$ over the qubit pairs
$\mathcal{S}$ that formed the chain, producing $\left\langle D(\rho_{\rm
f}||\rho_{\rm NATS})\right\rangle$. Similarly, the authors calculated
$\left\langle D(\rho_{\rm f}||\rho_{\rm GC})\right\rangle$ and $\left\langle
D(\rho_{\rm f}||\rho_{\rm can})\right\rangle$. The NATS predicted the
experimental results best: $\left\langle D(\rho_{\rm f}||\rho_{\rm
NATS})\right\rangle<\left\langle D(\rho_{\rm f}||\rho_{\rm
GC})\right\rangle<\left\langle D(\rho_{\rm f}||\rho_{\rm can})\right\rangle$.
Despite the plurality of the conservation laws that decoherence could break,
noncommuting charges’ effects were experimentally observable; dynamical-
decoupling pulse sequences mitigated the noise sufficiently. This experiment,
a realization of the proposal [7], opens the door to experimentally testing
the many theoretical results about noncommuting thermodynamic charges,
discussed in the next section. Along with trapped ions, feasible platforms
include superconducting qudits, ultracold atoms, nitrogen-vacancy centers, and
quantum dots [7].

##  III New physics

Noncommuting thermodynamic charges engender new physics: They conflict with
the ETH, allow for multiple stationary states, and violate the dichotomy of
energy-level statistics. Charges’ noncommutation also constrains the global
dynamics implementable with local charge-conserving unitaries. Additionally,
noncommuting charges decrease thermodynamic entropy production, yet increase
average entanglement entropy.

Conflict with the eigenstate thermalization hypothesis: The ETH explains how
reversible dynamics thermalize closed quantum many-body systems internally
[20, 21, 25]. Consider a system with $N$ DOFs governed by a nondegenerate
Hamiltonian $H^{\rm tot}$. Let $\mathcal{O}$ denote any operator, which we
represent as a matrix relative to the energy eigenbasis. The ETH is an ansatz
for the matrix elements’ forms. Suppose that the system begins in a
“microcanonical” state $|\psi(0)\rangle$—in this context, a pure state with a
small $H^{\rm tot}$ variance. The ETH implies thermalization: The time-
averaged expectation value of $\mathcal{O}$ approximately equals the canonical
expectation value:

$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}\,\langle\psi(t^{\prime})|\mathcal{O}|\psi(t^{\prime})\rangle={\rm
Tr}(\mathcal{O}\rho_{\text{can}})+O(N^{-1}).$ (9)

Throughout this subsection, big-$O$ notation means “scales as.” Noncommuting
charges impede three assumptions behind the argument for thermalization: (i)
Noncommuting charges prevent microcanonical states from existing in abundance
(Sec. II). (ii) Noncommuting charges force degeneracies on the Hamiltonian
(Sec. I). (iii) Noncommuting charges lead the Wigner–Eckart theorem to
supersede the ETH [60]. To review the Wigner–Eckart theorem briefly, we
consider $N$ qubits whose global spin components $S_{x,y,z}^{\rm tot}$ are
conserved, as in Sec. I. Denote by $\\{|\alpha,m\rangle\\}$ the eigenbasis
shared by $H^{\text{tot}}$, $(\vec{S}^{\rm tot})^{2}$, and $S_{z}^{\rm tot}$:
If $\hbar=1$, then $H^{\rm tot}|\alpha,m\rangle=E_{\alpha}|\alpha,m\rangle$,
$(\vec{S}^{\rm
tot})^{2}|\alpha,m\rangle=s_{\alpha}(s_{\alpha}+1)|\alpha,m\rangle$, and
$S_{z}^{\rm tot}|\alpha,m\rangle=m|\alpha,m\rangle$. The Wigner–Eckart theorem
governs spherical tensor operators formed from components $T_{q}^{(k)}$ [60].
The $T_{q}^{(k)}$’s form a basis for the space of operators defined on the
system’s Hilbert space. For example, consider an atom absorbing a photon (of
spin $k=1$), gaining $q=1$ quantum of $z$-type angular momentum.
$T^{(k=1)}_{q=1}$ represents the photon’s effect on the atom’s state. Consider
representing $T^{(k)}_{q}$ as a matrix relative to the energy eigenbasis. That
matrix obeys the Wigner–Eckart theorem [60]:

$\langle\alpha,m|T_{q}^{(k)}|\alpha^{\prime},m^{\prime}\rangle=\langle
s_{\alpha},m|s_{\alpha^{\prime}},m^{\prime};k,q\rangle\langle\alpha||T^{(k)}||\alpha^{\prime}\rangle.$
(10)

$\langle s_{\alpha},m|s_{\alpha^{\prime}},m^{\prime};k,q\rangle$ denotes a
Clebsch-Gordan coefficient, a conversion factor between the product state
$|s_{\alpha^{\prime}},m^{\prime};k,q\rangle\equiv|s_{\alpha^{\prime}},m^{\prime}\rangle\lvert
k,q\rangle$ and the total-spin eigenstate $|s_{\alpha},m\rangle$.
$\langle\alpha||T^{(k)}||\alpha^{\prime}\rangle$ is a reduced matrix
element—the part of
$\langle\alpha,m|T^{(k)}_{q}|\alpha^{\prime},m^{\prime}\rangle$ that does not
depend on magnetic spin quantum numbers. The theorem (10) conflicts with the
ETH, an ansatz for the left-hand side. First, the two constraints are
fundamentally different; the Wigner–Eckart theorem concerns SU(2) symmetry,
while the ETH concerns randomness. Second, the ETH states that off-diagonal
elements $\langle\alpha,m|T_{q}^{(k)}|\alpha^{\prime},m^{\prime}\rangle$ are
exponentially small in $N$. The Wigner–Eckart theorem implies that these
elements may be $O(1)$.

Reference [24] therefore posited a _non-Abelian ETH._ This ansatz depends on
the average energy
$\mathscr{E}\coloneqq\frac{1}{2}(E_{\alpha}+E_{\alpha^{\prime}})$, energy
difference $\omega\coloneqq E_{\alpha}-E_{\alpha^{\prime}}$, average spin
quantum number
$\mathscr{S}\coloneqq\frac{1}{2}(s_{\alpha}+s_{\alpha^{\prime}})$, and
difference $\nu\coloneqq s_{\alpha}-s_{\alpha^{\prime}}$. Denote by
$S_{\text{th}}(\mathscr{E},\mathscr{S})$ the thermodynamic entropy at energy
$\mathscr{E}$ and spin $\mathscr{S}$. The observable $T_{q}^{(k)}$ and
Hamiltonian $H^{\text{tot}}$ satisfy the non-Abelian ETH if, for smooth real
functions $\mathcal{T}^{(k)}(\mathscr{E},\mathscr{S})$ and
$f_{\nu}^{(k)}(\mathscr{E},\mathscr{S},\omega)$,

$\begin{split}\langle\alpha||T^{(k)}||\alpha^{\prime}\rangle=&\mathcal{T}^{(k)}(\mathscr{E},\mathscr{S})\,\delta_{\alpha,\alpha^{\prime}}\\\
&+e^{-S_{\text{th}}(\mathscr{E},\mathscr{S})/2}\,f_{\nu}^{(k)}(\mathscr{E},\mathscr{S},\omega)\,R_{\alpha,\alpha^{\prime}}.\end{split}$
(11)

$\mathcal{T}^{(k)}$ parallels a microcanonical average in the conventional
ETH. $R_{\alpha,\alpha^{\prime}}$ resembles a normalized $O(1)$ random
variable [61, 62, 63]. The matrix element (10) deviates from the ordinary ETH
through $\mathscr{S}$-dependent functions and a Clebsch–Gordan coefficient.
Equation (11) has withstood preliminary numerical checks with a Heisenberg
Hamiltonian on a 2-dimensional qubit lattice [64]. More testing is needed,
however.

The non-Abelian ETH predicts thermalization to the usual extent in some, but
not all, contexts. Consider preparing the system in a state $|\psi(0)\rangle$
in an AMC subspace (Sec. II). Suppose that $|\psi(0)\rangle$ has an extensive
magnetization along an axis that we call $\hat{z}$: $\langle\psi(0)|S_{z}^{\rm
tot}|\psi(0)\rangle=O(N)$. According to the non-Abelian ETH,

$\begin{split}&\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}dt^{\prime}\,\langle\psi(t^{\prime})|T_{q}^{(k)}|\psi(t^{\prime})\rangle\\\
&\quad={\rm
Tr}\left(T_{q}^{(k)}\rho_{\text{NATS}}\right)+O\left(N^{-1}\right).\end{split}$
(12)

However, if $\langle\psi(0)|S_{z}^{\rm tot}|\psi(0)\rangle=0$, the correction
can become $O(N^{-1/2})$—polynomially larger. This result relies on an
assumption argued to be physically reasonable: The smooth function
$\mathcal{T}^{(k)}$ in (11) can contain a nonzero term of $O(s_{\alpha}/N)$.
The unusually large, analytically expected correction constitutes further
evidence that charges’ noncommutation can alter thermalization.

The non-Abelian ETH invites further studies—first, numerical and experimental
tests. Second, from the non-Abelian ETH’s final term, one can infer about
fluctuations around the time average (9). Third, the time required for
equilibration merits study: In quantum many-body physics, most thermalization-
blocking mechanisms slow thermalization in time (Sec. IV). In contrast, the
$O(N^{-1/2})$ “slows” the time-averaged expectation value in $N$, in the
approach to a thermal expectation value. How long does
$\langle\psi(t^{\prime})|T_{q}^{(k)}|\psi(t^{\prime})\rangle$ take to reach
its long-time value? Fourth, using the ETH, Strasberg et al. demonstrated how
classicality, Markovianity, and local detailed balance emerge from the pure-
state dynamics of a system with commuting charges [65]. Their results might be
extended to noncommuting-charge systems, with the non-Abelian ETH and AMC
subspaces. Finally, one might store information in a system that
thermalizes—forgets its initial conditions—less than usual. This memory-
storage opportunity resonates with the work that we explain now.

_Stationary-state multiplicity:_ Consider an open quantum system with a
Liouvilian superoperator $\mathscr{L}$. $\rho_{\rm stat}$ is a stationary
state of $\mathscr{L}$ if $\mathscr{L}\rho_{\rm stat}=0$. Consider an
arbitrary basis for the stationary subspace. One might associate the $j^{\rm
th}$ basis element, $\rho_{\rm stat}^{(j)}$, with a classical alphabet’s
$j^{\rm th}$ letter, $L_{j}$: $\rho_{\rm stat}^{(j)}\leftrightarrow L_{j}$. To
encode $L_{j}$ in the system, one would prepare any state that thermalizes to
$\rho_{\rm stat}^{(j)}$. The larger the stationary subspace, the more
classical information the system may store. Denote by $n_{\rm stat}$ the
stationary subspace’s dimensionality. Zhang et al. derive a lower bound
$n_{\rm NC}$ on $n_{\rm stat}$ for a system with noncommuting charges [66]:
$n_{\rm NC}=\sum_{j}\mathcal{D}_{j}^{2}$, wherein $\mathcal{D}_{j}$ denotes
the symmetry group’s $j^{\rm th}$ irreducible representation. For example,
$n_{\rm NC}\sim N^{3}$ for a Heisenberg model (Sec. I) coupled to an
environment that conserves the system-of-interest charges
$\sigma_{x,y,z}^{(\mathcal{S})}$. If the charges commute, the lower bound
$n_{\rm C}$ scales as the number of simultaneous eigenspaces shared by all the
$Q_{a}^{(\mathcal{S})}$’s. Since $n_{\rm NC}$ and $n_{\rm C}$ scale
differently, noncommuting charges could alter the stationary subspace’s
dimensionality and so the amount of information storable.

Hybrid energy-level statistics: Energy-level statistics diagnose quantum chaos
and integrability [67]. Denote by $\omega=E_{j+1}-E_{j}$ the spacing between
consecutive eigenenergies of a many-body Hamiltonian. Any given spacing (near
the spectrum’s center) has a probability density $P(\omega)$ of having the
size $\omega$. A Poissonian777 This $P(\omega)$ is the Poisson distribution
whose average-rate-of-occurrence parameter vanishes. The reason is,
$P(\omega)\,d\omega$ equals the probability that zero eigenenergies lie in a
width-$d\omega$ interval. $P(\omega)$ diagnoses integrability [67, Sec. 2.3];
and a Wigner–Dyson distribution,
$A_{\gamma}\,\omega^{\gamma}\exp(-B_{\gamma}\omega^{2})$, chaos [68, 69]. The
parameter $\gamma\in\\{1,2,4\\}$ depends on the Hamiltonian’s time-reversal
and rotational symmetries. Normalization and the mean $\omega$ determine the
coefficients $A_{\gamma}$ and $B_{\gamma}$. Noncommuting charges defy the
Poisson-vs.-Wigner–Dyson dichotomy as commuting charges cannot: The charges
generate a non-Abelian algebra, which has multidimensional irreducible
representations, which force degeneracies on the Hamiltonian. These
representations induce statistics that interpolate between the two
distributions [70, 64, 71]. Noh observed such statistics numerically, using a
2D Heisenberg model [64].

Constraints on charge-conserving dynamics: Natural interactions are spatially
local, motivating a quantum-computational result: Every $N$-qubit unitary
decomposes into gates on pairs of qubits [72, 73, 74]. Can charge-conserving
local unitaries effect every charge-conserving global unitary $U$? Perhaps
surprisingly, the answer is no [75]. Using Lie theory, Marvian proved that
locality-constrained charge-conserving unitaries cannot even approximate
$U$’s. The reason is, the two types of unitaries form Lie groups of different
dimensions [75, 76, 77, 78]. Moreover, noncommuting charges impose four types
of constraints on the implementable global unitaries, whereas commuting
charges impose only two [78]. The extra constraints may restrict chaos, which
enables thermalization. Hence Marvian’s result suggests that noncommuting
charges may restrict thermalization.

Decreased thermodynamic-entropy production: Noncommuting charges reduce
entropy production, which quantifies irreversibility [3]. Throughout this
paragraph and the next, _entropy_ refers to thermodynamic entropy, not
entanglement entropy. Let systems $X=A,B$ have charges $Q_{a}^{(X)}$ that
might or might not commute. (In no other subsection of this section do
$Q_{a}^{(X)}$’s denote possibly commuting charges.) Each system begins in a
GGE
$\rho^{(X)}_{\vec{\mu}^{(X)}}\propto\exp\left(\sum_{a}\mu_{a}^{(X)}Q_{a}^{(X)}\right)$,
wherein $\vec{\mu}^{(X)}=(\mu_{0}^{(X)},\mu_{1}^{(X)},...)$ [79] (Fig. 3).
Hence $AB$ begins in
$\rho(0)\coloneqq\rho^{(A)}_{\vec{\mu}^{(A)}}\otimes\rho^{(B)}_{\vec{\mu}^{(B)}}$.
Let us specialize to the linear-response regime:
$\vec{\mu}^{(A)}\approx\vec{\mu}^{(B)}$. A charge-conserving unitary $U$ can
shuttle charges between the systems, producing entropy. Noncommuting charges
decrease the entropy production [3]: Denote by
$\delta\mu_{a}\coloneqq\mu_{a}^{(A)}-\mu_{a}^{(B)}$ the difference between the
systems’ $a$-type chemical potentials. System $A$’s $a$-type charge changes,
in the Heisenberg picture, by $U^{\dagger}Q_{a}^{(A)}U-Q_{a}^{(A)}$.
(Tensored-on $\mathbbm{1}$’s are implicit where necessary to make operators
act on the appropriate Hilbert spaces.) We combine the foregoing two
quantities into
$\tilde{\Sigma}\coloneqq\sum_{a}\delta\mu_{a}\big{(}U^{\dagger}Q_{a}^{(A)}U-Q_{a}^{(A)}\big{)}$.
Taking the expectation value of $\tilde{\Sigma}$ in the initial state, we
obtain the net entropy production, $\Sigma={\rm
Tr}\bm{(}\tilde{\Sigma}\,\rho(0)\bm{)}$. The initial state lies near the fixed
point
$\pi\coloneqq\rho^{(A)}_{\vec{\mu}^{(A)}}\otimes\rho^{(B)}_{\vec{\mu}^{(A)}}$
of U, by the linear-response assumption.888 In $\pi$, unlike in $\rho(0)$, $A$
and $B$ have the same lists of chemical potentials. $\pi$ and $\tilde{\Sigma}$
have a Wigner–Yanase–Dyson (WYD) skew information
$I_{y}(\pi,\tilde{\Sigma})\coloneqq-\frac{1}{2}{\rm
Tr}\big{(}[\pi^{y},\tilde{\Sigma}][\pi^{1-y},\tilde{\Sigma}]\big{)}$,
parameterized by $y\in(0,1)$. $I_{y}(\pi,\tilde{\Sigma})$ quantifies the
coherence that $\tilde{\Sigma}$ has relative to $\pi$’s eigenbasis. The WYD
skew information contributes to the entropy production:
$\Sigma=\frac{1}{2}{\rm
var}(\tilde{\Sigma})-\frac{1}{2}\int_{0}^{1}\text{d}y\,I_{y}(\pi,\tilde{\Sigma})$
[3]. $I_{y}$ is always $\geq 0$ and is positive if and only if the charges
fail to commute. Therefore, noncommuting charges lower $\Sigma$. Since entropy
production accompanies thermalization, noncommutation may inhibit
thermalization.

Two extensions support [3]. First, Shahidani numerically simulated an
optomechanical system interacting with a squeezed thermal bath [80]. Second,
Upadhyaya _et al._ progressed beyond the linear-response regime [81]. Even
there, charges’ noncommutation can decrease entropy production.

How [3] impacts engine efficiencies remains an open question. Efficiencies
suffer from waste and so may benefit from noncommuting charges’ lowering of
$\Sigma$. However, charges’ noncommutation can decrease the amount $W_{a}$ of
$a$-charge (analogous to work) extractable by an engine [82]. If using finite-
size NATS baths, an engine can extract $W_{a}\lesssim
W_{a}\rvert_{\text{infinite-size bath}}-O(C/\xi)$. $C$ captures the
correlations between the charges in $\rho_{\rm NATS}$; if the charges commute,
$C=0$. $\xi$, encoding the baths’ sizes, diverges in the infinite-bath limit.
Extractable work, here lowered by charges’ noncommutation, tends to trade off
with efficiency. A precise relationship between the two figures of merit, in
the presence of noncommuting charges, merits calculation.

Figure 3: Two thermal reservoirs exchange charges, producing entropy: Blue
spheres represent charges of one type, and red cubes represent charges of
another.

Increased average entanglement entropy: Quantum many-body thermalization
entails entanglement, which charges’ noncommutation can enhance [8]. Consider
initializing an isolated $N$-qubit system in a pure state. Divide the system
into subsystems $A$ and $B$ with sizes $N_{A}$ and $N_{B}$. $A$ and $B$ can
share entanglement quantified with the _entanglement entropy_ , the von
Neumann entropy of the reduced state $\rho_{A}$ of $A$: $S_{AB}\coloneqq
S_{\text{vN}}(\rho_{A})$. Plotting $S_{AB}$ against $N_{A}$ yields the _Page
curve_ [83].

Majidy et al. computed Page curves for two comparable $N$-qubit models, one
with commuting charges only and the other with noncommuting charges [8].
Figure 4 describes the models. They involve no dynamics, so the observables of
interest are not technically charges. However, the authors modeled states
resulting from chaotic dynamics: Haar-random states were sampled from
(approximate, when necessary [2, 7, 9]) microcanonical subspaces. Page curves
were estimated numerically via exact diagonalization and analytically via
large-$N$ approximations of combinatorics. The noncommuting-charge Haar-
averaged Page curves lay above the commuting-charge analogs. The difference
was of $O\big{(}\frac{N_{A}^{2}}{N^{2}N_{B}}\big{)}$, in the simplest
comparison. A possible reason centers on the least entangled basis for each
model’s subspace. If the local charges $\tilde{Q}_{a}$ commute, they share an
eigenbasis. Hence the global commuting $\tilde{Q}_{a}^{\rm tot}$’s share a
tensor-product eigenbasis for the subspace. If the local charges fail to
commute, this argument breaks, and the subspace’s least entangled basis is
entangled. One might therefore expect more entanglement of the noncommuting-
charge model on average across the subspace.

Figure 4:  Analogous noncommuting-charge and commuting-charge models: Each
model consists of $N$ sites, each formed from $\mathcal{A}$ and $\mathcal{B}$
qubits. Local observables distinguish the models. The noncommuting observables
are $Q_{1}=X_{\mathcal{A}}\otimes\mathbbm{1}_{\mathcal{B}}$,
$Q_{2}=Y_{\mathcal{A}}\otimes\mathbbm{1}_{\mathcal{B}}$, and
$Q_{3}=Z_{\mathcal{A}}\otimes\mathbbm{1}_{\mathcal{B}}$. The commuting
observables are $\tilde{Q}_{1}=X_{\mathcal{A}}\otimes X_{\mathcal{B}}$,
$\tilde{Q}_{2}=Y_{\mathcal{A}}\otimes Y_{\mathcal{B}}$, and
$\tilde{Q}_{3}=Z_{\mathcal{A}}\otimes Z_{\mathcal{B}}$. The local charges,
summed across the chain, form global observables that define the (approximate)
microcanonical subspaces. (Adapted from [8].)

Other works concern non-Abelian symmetries’ effects on entanglement entropy
but focus less on what charges’ noncommutation changes. For example, a non-
Abelian symmetry raises the entanglement in Wess–Zumino–Witten models,
$(1+1)$-dimensional conformal field theories with Lie-group symmetries [84,
85]. Second, holographic calculations highlight another correction that non-
Abelian symmetries introduce into entanglement entropy [86, 87].999 This
correction appears to be negative. However, [86, 87] concern symmetry-resolved
Page curves, in contrast with the conventional Page curves of [8]. A symmetry-
resolved Page curve models the entanglement, averaged over time, of a system
whose charges move only within $A$ and within $B$, not between the subsystems.
Conventional Page curves model less-restricted thermalization. Third,
algebraic-quantum-field-theory calculations agree that non-Abelian symmetries
raise Page curves [88].

These works suggest several research opportunities. The entanglement entropy’s
increase merits checking with more comparable models à la Fig. 4.
Additionally, one might adjust the Page-curve calculations following Marvian’s
revelation that local charge-conserving unitaries constrain global $U$’s
tightly [75]. Under locality constraints, the Haar distribution may model
chaotic dynamics inaccurately. Finally, the entanglement entropy of [8, 84,
85, 86, 87, 88] differs from the thermodynamic entropy of [3]. Still,
noncommuting charges’ increase of the former conceptually conflicts, somewhat,
with their decrease of the latter.

##  IV Adjacent work in other fields

Noncommuting charges have arisen in other thermodynamics-related contexts,
usually under the guise of non-Abelian symmetries. We discuss four contexts:
First, integrable systems have more charges, which might fail to commute, than
the thermodynamic systems of Sections I–III. Second, noncommuting charges
destabilize a thermalization-avoidant phase of quantum matter: many-body
localization (MBL). Third, gauge theories naturally have non-Abelian
symmetries, albeit local ones. Fourth, hydrodynamics describes the flow of
some non-Abelian charges, including in heavy-ion collisions.

Three more connections, we mention briefly: Noncommuting charges may impact
quantum information scrambling [89]. Also, $\rho_{\rm NATS}$ helps explain
phase transitions [90, 91, 92]. Finally, the resource theory of asymmetry
quantifies a state’s noninvariance under operation by a symmetry group’s
elements [19]. The group may be non-Abelian, so the theory’s mathematical
tools merit application to noncommuting thermodynamic charges.

_Conventional integrable systems:_ Garden-variety integrable systems fail to
thermalize. Each has extensively many nontrivial charges, which constrain the
dynamics significantly. In contrast, our thermodynamic setup (Sec. I) entails
a number of charges, $c\ll N$, much less than the number of DOFs. Integrable
and near-integrable systems with noncommuting charges have been studied.
Models include the GGE equal to $\rho_{\rm NATS}$ (4). Several works assess
how the GGE’s accuracy grows with the number of $Q_{a}$’s included in the
state [93, 94, 95, 92]. Outside of GGE studies, an integrable system can
exhibit anomalous transport—diffusion with a diffusion constant
$D\sim\sqrt{N}$ [96, 97]. The system studied in [96] is the one-dimensional
nearest-neighbor Heisenberg model (1). The anomalous transport can be
explained with Bethe-ansatz calculations and the $z$-axis magnetization [98,
99, 100]. Can all three noncommuting charges ($\sigma_{x,y,z}$) predict
anomalous transport alternatively? Another open question follows from
increasingly breaking a noncommuting-charge system’s integrability [101]. How
do the system’s behaviors transmute into thermodynamic behaviors described in
this Perspective?

_Many-body localization:_ Disordered, interacting quantum systems exhibit MBL.
Examples include a qubit chain subject to the disordered Heisenberg
Hamiltonian $H^{\rm tot}_{\rm
MBL}=J\sum_{j=1}^{N}\big{(}\vec{\sigma}^{(j)}\cdot\vec{\sigma}^{(j+1)}+h_{j}\sigma_{z}^{(j)}\big{)}$.
The disorder term, $\sum_{j=1}^{N}h_{j}\sigma_{z}^{(j)}$, acts as an external
field whose magnitude $h_{j}$ varies randomly across sites. Denote by $h$ the
disorder’s standard deviation. If the disorder is much stronger than the
interaction, $h\gg J$, the system localizes: Imagine measuring each qubit’s
$\sigma_{z}$. The qubits approximately maintain the measured configuration
long afterward. This behavior contrasts with how thermalizing systems, such as
classical gases, change configurations quickly. Hence MBL resists
thermalization for long times. The reason is, the Hamiltonian decomposes as a
linear combination of quasilocal DOFs [26].

Noncommuting charges destabilize MBL [102]. Consider forcing a non-Abelian
symmetry on $H^{\rm tot}_{\rm MBL}$. The resulting Hamiltonian, $H^{\rm
tot}_{{\rm MBL}^{\prime}}$, will have degeneracies, by Schur’s lemma (App. A).
So will the quasilocal DOFs, which can therefore become “excited” at no energy
cost. Consider adding to $H^{\rm tot}_{{\rm MBL}^{\prime}}$ an infinitesimal
field that violates the symmetry. The resulting Hamiltonian, $H^{\rm
tot}_{{\rm MBL}^{\prime\prime}}$, can map $H^{\rm tot}_{{\rm MBL}^{\prime}}$
eigenstates $\lvert\psi\rangle$ to same-energy eigenstates
$\lvert\tilde{\psi}\rangle$: $\langle\tilde{\psi}\rvert H^{\rm tot}_{{\rm
MBL}^{\prime\prime}}\lvert\psi\rangle\neq 0$. Two such eigenstates can be
zero-energy “excited” states of neighboring quasilocal DOFs. Hence $H^{\rm
tot}_{{\rm MBL}^{\prime\prime}}$ can transport zero-energy “excitations”
between quasilocal DOFs—across the system. Such transport is inconsistent with
MBL. Therefore, non-Abelian symmetries promote a thermalizing behavior.

_Gauge theories:_ Classical electrodynamics exemplifies a gauge theory, a
model that contains more DOFs than does the physical system modeled [103].
Choosing a gauge eliminates the extra DOFs. The transformations between gauges
form a Lie group $\mathcal{G}$. The elements $U\in\mathcal{G}$ preserve the
theory’s action, $\mathscr{S}$. That is, $U:\mathscr{S}\mapsto\mathscr{S}$.
Gauge theories model elementary-particle physics and condensed matter, both of
whose $\mathcal{G}$’s can be non-Abelian. For example, quantum chromodynamics,
which describes the strong force, has SU(3) symmetry. Hence elementary-
particle physics should realize noncommuting-charge thermodynamics naturally;
$\rho_{\rm NATS}$ might be observable in high-energy and nuclear systems
[104]. Granted, confinement prevents quantum-chromodynamic systems from having
nonzero $\langle Q_{a}^{\rm tot}\rangle$’s. Still, noncommuting charges raise
the average entanglement entropy in spin systems with $\langle Q_{a}^{\rm
tot}\rangle=0$ $\forall a$ [8]. Furthermore, subsystems $j$ can contain
positive and negative charges, $\langle Q_{a}^{(j)}\rangle\neq 0$, that can
undergo dynamics [28]. Gauge symmetries are local, though, unlike the global
symmetries covered in this Perspective. The contrast raises the question of
how much noncommuting-charge quantum thermodynamics ports over into gauge
theories.

_Hydrodynamics and heavy-ion collisions:_ Hydrodynamics models long-wavelength
properties of fluids that are in equilibrium locally [105]. The theory
describes condensed matter, certain stages of heavy-ion collisions [106], and
more. Noncommuting charges can flow similarly to the more-often studied energy
and particles [107]. A few effects of charges’ noncommutation have been
isolated; examples include non-Abelian contributions to conductivity [108] and
entropy currents [109]. Also, non-Abelian symmetries can shorten charge-
neutralization times in heavy-ion collisions [110]. More such effects may be
discoverable. Also, quantum thermodynamics might assist with longstanding
questions about heavy-ion collisions: by what process the system thermalizes,
how the initial state should be modeled, whether the (non-)Abelian ETH
explains the thermalization [111, 112], and why the thermalization time is so
short [113].

##  V Outlook

Noncommuting charges pose a quantum challenge to expectations about
thermodynamics. Charges’ noncommutation impedes derivations of the thermal
state’s form [1, 2], eliminates microcanonical subspaces [2], and decreases
thermodynamic-entropy production [3]. Dynamically, noncommuting charges limit
the global evolutions implementable with local interactions more than
commuting charges do [78]. In many-body physics, noncommuting charges underlie
steady-state degeneracies that might be employed to store information [66].
Noncommuting charges also conflict with the ETH [24] and can increase average
entanglement entropy [8, 84, 85, 88]. These discoveries suggest rich research
opportunities, of which we detail five.

First, the predictions merit experimental testing. The first test of
noncommuting-charge thermodynamics was performed with trapped ions [9]. Other
potential platforms include superconducting qubits, quantum dots, ultracold
atoms, quantum optics, and optomechanics [7, 28, 3].

Second, existing results present a conceptual puzzle. Evidence suggests that
noncommuting charges hinder thermalization to an extent: They invalidate
derivations of the thermal state’s form [1, 2], decrease thermodynamic-entropy
production [3], clash with the ETH [24], and uniquely restrict the global
unitaries implementable via local interactions [78]. Other evidence, however,
suggests that noncommuting charges enhance thermalization: They destabilize
MBL [102] and increase average entanglement entropy [8]. These results do not
conflict with each other technically, stemming from different setups. Yet the
results clash conceptually, suggesting that charges’ noncommutation hinders
thermalization in some ways and enhances it in others. Does the hindrance or
enhancement win out overall? Reconciling these results presents a challenge.

Third, to what extent can classical mechanics reproduce noncommuting-charge
thermodynamics? Observables’ noncommutation underlies quintessentially quantum
phenomena including uncertainty relations, measurement disturbance, and the
Einstein–Podolsky–Rosen paradox. Yet classical mechanics features quantities
that fail to commute with each other—for example, rotations about different
axes. How _nonclassical_ is noncommuting-charge thermodynamics (achievable
only outside of classical physics), beyond being merely _quantum_ (achievable
within quantum physics)?

Fourth, every chaotic or thermodynamic phenomenon merits re-examination. To
what extent does it change under dynamics that conserve noncommuting charges?
Example phenomena include diffusion coefficients, transport relations,
thermalization times, monitored circuits [114], out-of-time-ordered
correlators [115], operator spreading [116], frame potentials [117], and
quantum-complexity growth [118].

Finally, noncommuting-charge thermodynamics merits bridging to similar topics
in neighboring fields. Non-Abelian gauge theories, non-Abelian hydrodynamics,
GGE studies, and dynamical phase transitions overlap with noncommuting
thermodynamic charges. To what extent can these areas inform each other? Do
gauge theories realize noncommuting thermodynamic charges naturally?

For decades, conserved thermodynamic quantities were assumed implicitly to
commute with each other. Noncommutation, however, is a trademark of quantum
theory. The identification and elimination of the assumption, though initiated
where quantum information theory meets quantum thermodynamics, have potential
ramifications across quantum many-body physics.

###### Acknowledgements.

This work received support from the John Templeton Foundation (award no.
62422) and the National Science Foundation (QLCI grant OMA-2120757). The
opinions expressed in this publication are those of the authors and do not
necessarily reflect the views of the John Templeton Foundation or UMD. S.M.
received support from the Vanier C.G.S. T.U. acknowledges the support of the
Joint Center for Quantum Information and Computer Science through the Lanczos
Fellowship, as well as the Natural Sciences and Engineering Research Council
of Canada (NSERC), through the Doctoral Postgraduate Scholarship. N.Y.H.
thanks the Institut Pascal for its hospitality during the formation of this
paper, Maurizio Fagotti for discussions about GGEs, Niklas Mueller for
discussions about lattice gauge theories and hydrodynamics, and Christopher D.
White for MBL discussions.

##  VI Competing interests

The authors declare no competing interests.

##  Appendix A Schur’s lemma implies degeneracy of Hamiltonians that have non-
Abelian symmetries

Consider a quantum system associated with a Hilbert space $\mathcal{H}$. Let
$H^{\rm tot}$ denote the Hamiltonian, which has a symmetry associated with a
Lie group $\mathcal{G}$. Let $k$ label the irreducible representations
(irreps) of $\mathcal{G}$. For example, $\mathcal{G}=$ SU(2) has irreps
labeled by $s=0,1/2,1,\ldots$ A corollary of Schur’s lemma states that [19, p.
9]

$\displaystyle\mathcal{H}$
$\displaystyle=\bigoplus_{k}\mathcal{H}_{k}\otimes\mathcal{M}_{k}.$ (A1)

$\mathcal{H}_{k}$ denotes a representation space. The multiplicity space
$\mathcal{M}_{k}$ has a dimensionality equal to irrep $k$’s dimension.

Correspondingly, $H^{\rm tot}$ decomposes as [119, Sec. 3.2.3]

$\displaystyle H^{\rm tot}$ $\displaystyle=\bigoplus_{k}E_{k}\mathbbm{1}_{k}.$
(A2)

$E_{k}$ denotes an eigenenergy. $\mathbbm{1}_{k}$ denotes the identity
operator acting on the multiplicity space $\mathcal{M}_{k}$. If $\mathcal{G}$
is non-Abelian, then some of the irreps $k$ have dimensionalities $>1$.
Consequently, some $\mathbbm{1}_{k}$’s act on multidimensional subspaces
$\mathcal{M}_{k}$. Those $\mathcal{M}_{k}$’s are degenerate eigenspaces of
$H^{\rm tot}$.

## References

  * [1] N. Yunger Halpern, “Beyond heat baths ii: Framework for generalized thermodynamic resource theories,” Journal of Physics A: Mathematical and Theoretical, vol. 51, no. 9, p. 094001, 2018.
  * [2] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nature communications, vol. 7, no. 1, pp. 1–7, 2016.
  * [3] G. Manzano, J. M. Parrondo, and G. T. Landi, “Non-abelian quantum transport and thermosqueezing effects,” PRX Quantum, vol. 3, no. 1, p. 010304, 2022\.
  * [4] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev., vol. 47, pp. 777–780, May 1935.
  * [5] P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, “Entropic uncertainty relations and their applications,” Rev. Mod. Phys., vol. 89, p. 015002, Feb 2017.
  * [6] E. Benítez Rodríguez and L. Arévalo Aguilar, “A survey of the concept of disturbance in quantum mechanics,” Entropy, vol. 21, p. 142, Feb 2019.
  * [7] N. Yunger Halpern, M. E. Beverland, and A. Kalev, “Noncommuting conserved charges in quantum many-body thermalization,” Physical Review E, vol. 101, no. 4, p. 042117, 2020.
  * [8] S. Majidy, A. Lasek, D. A. Huse, and N. Yunger Halpern, “Non-abelian symmetry can increase entanglement entropy,” Phys. Rev. B, vol. 107, p. 045102, Jan 2023.
  * [9] F. Kranzl, A. Lasek, M. K. Joshi, A. Kalev, R. Blatt, C. F. Roos, and N. Yunger Halpern, “Experimental observation of thermalization with noncommuting charges,” Physical Review X Quantum, vol. 4, p. 020318, Apr 2023.
  * [10] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, “Quantum thermalization through entanglement in an isolated many-body system,” Science, vol. 353, no. 6301, pp. 794–800, 2016.
  * [11] C. Neill, P. Roushan, M. Fang, Y. Chen, M. Kolodrubetz, Z. Chen, A. Megrant, R. Barends, B. Campbell, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, J. Mutus, P. J. J. O’Malley, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. Polkovnikov, and J. M. Martinis, “Ergodic dynamics and thermalization in an isolated quantum system,” Nature Physics, vol. 12, p. 1037, Jul 2016.
  * [12] G. Clos, D. Porras, U. Warring, and T. Schaetz, “Time-resolved observation of thermalization in an isolated quantum system,” Phys. Rev. Lett., vol. 117, p. 170401, Oct 2016.
  * [13] Z.-Y. Zhou, G.-X. Su, J. C. Halimeh, R. Ott, H. Sun, P. Hauke, B. Yang, Z.-S. Yuan, J. Berges, and J.-W. Pan, “Thermalization dynamics of a gauge theory on a quantum simulator,” Science, vol. 377, no. 6603, pp. 311–314, 2022\.
  * [14] J. A. Vaccaro and S. M. Barnett, “Information erasure without an energy cost,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 467, no. 2130, pp. 1770–1778, 2011.
  * [15] T. Croucher, S. Bedkihal, and J. A. Vaccaro, “Discrete fluctuations in memory erasure without energy cost,” Physical review letters, vol. 118, no. 6, p. 060602, 2017.
  * [16] J. S. Wright, T. Gould, A. R. Carvalho, S. Bedkihal, and J. A. Vaccaro, “Quantum heat engine operating between thermal and spin reservoirs,” Physical Review A, vol. 97, no. 5, p. 052104, 2018.
  * [17] T. Croucher and J. A. Vaccaro, “Memory erasure with finite-sized spin reservoir,” arXiv preprint arXiv:2111.10930, 2021.
  * [18] V. Bouchard, “Ma ph 464 - group theory in physics: Lecture notes.” https://sites.ualberta.ca/ vbouchar/MAPH464/front.html, 2020.  University of Alberta.
  * [19] G. Gour and R. W. Spekkens, “The resource theory of quantum reference frames: manipulations and monotones,” New Journal of Physics, vol. 10, p. 033023, mar 2008.
  * [20] J. M. Deutsch, “Quantum statistical mechanics in a closed system,” Phys. Rev. A, vol. 43, pp. 2046–2049, Feb 1991.
  * [21] M. Srednicki, “Chaos and quantum thermalization,” Phys. Rev. E, vol. 50, pp. 888–901, Aug 1994.
  * [22] T. R. de Oliveira, C. Charalambous, D. Jonathan, M. Lewenstein, and A. Riera, “Equilibration time scales in closed many-body quantum systems,” New Journal of Physics, vol. 20, p. 033032, mar 2018.
  * [23] S. Chakraborty, K. Luh, and J. Roland, “How fast do quantum walks mix?,” Phys. Rev. Lett., vol. 124, p. 050501, Feb 2020.
  * [24] C. Murthy, A. Babakhani, F. Iniguez, M. Srednicki, and N. Yunger Halpern, “Non-abelian eigenstate thermalization hypothesis,” Phys. Rev. Lett., vol. 130, p. 140402, Apr 2023.
  * [25] M. Rigol, V. Dunjko, and M. Olshanii, “Thermalization and its mechanism for generic isolated quantum systems,” Nature, vol. 452, pp. 854–858, Apr 2008\.
  * [26] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, “Colloquium: Many-body localization, thermalization, and entanglement,” Rev. Mod. Phys., vol. 91, p. 021001, May 2019.
  * [27] N. Regnault, S. Moudgalya, and B. A. Bernevig, “Quantum many-body scars and hilbert space fragmentation: A review of exact results,” Reports on Progress in Physics, 2022.
  * [28] N. Yunger Halpern and S. Majidy, “How to build hamiltonians that transport noncommuting charges in quantum thermodynamics,” npj Quantum Information, vol. 8, no. 1, pp. 1–7, 2022.
  * [29] E. T. Jaynes, “Information theory and statistical mechanics. ii,” Physical Review, vol. 108, no. 2, p. 171, 1957.
  * [30] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, “Thermodynamics of quantum systems with multiple conserved quantities,” Nature communications, vol. 7, no. 1, pp. 1–9, 2016.
  * [31] Y.-K. Liu, “Gibbs states and the consistency of local density matrices,” arXiv e-prints, pp. quant–ph/0603012, Mar. 2006.
  * [32] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, “Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1d lattice hard-core bosons,” Physical review letters, vol. 98, no. 5, p. 050405, 2007.
  * [33] M. Rigol, “Breakdown of thermalization in finite one-dimensional systems,” Physical review letters, vol. 103, no. 10, p. 100403, 2009.
  * [34] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer, “Experimental observation of a generalized gibbs ensemble,” Science, vol. 348, no. 6231, pp. 207–211, 2015.
  * [35] L. Vidmar and M. Rigol, “Generalized gibbs ensemble in integrable lattice models,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2016, no. 6, p. 064007, 2016.
  * [36] R. Balian, Y. Alhassid, and H. Reinhardt, “Dissipation in many-body systems: a geometric approach based on information theory,” Physics Reports, vol. 131, no. 1-2, pp. 1–146, 1986.
  * [37] H.-T. Elze and W. Greiner, “Quantum statistics with internal symmetry,” Physical Review A, vol. 33, no. 3, p. 1879, 1986.
  * [38] Y. Alhassid and R. Levine, “Connection between the maximal entropy and the scattering theoretic analyses of collision processes,” Physical Review A, vol. 18, no. 1, p. 89, 1978.
  * [39] R. Balian and N. Balazs, “Equiprobability, inference, and entropy in quantum theory,” Annals of Physics, vol. 179, no. 1, pp. 97–144, 1987.
  * [40] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics.  New York: John Wiley & Sons, 2 ed., 1985.
  * [41] L. D. Landau and E. M. Lifshitz, Statistical Physics: Part 1.  Butterworth-Heinemann, 1980.
  * [42] M. Lostaglio, “Thermodynamics at the quantum scale generalized landauer principle,” Master’s thesis, Imperial College London, 2014.
  * [43] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information.  Cambridge University Press, 2010.
  * [44] E. Chitambar and G. Gour, “Quantum resource theories,” Rev. Mod. Phys., vol. 91, p. 025001, Apr 2019.
  * [45] M. Lostaglio, “An introductory review of the resource theory approach to thermodynamics,” Reports on Progress in Physics, vol. 82, p. 114001, oct 2019.
  * [46] N. Yunger Halpern and J. M. Renes, “Beyond heat baths: Generalized resource theories for small-scale thermodynamics,” Phys. Rev. E, vol. 93, p. 022126, Feb 2016.
  * [47] F. Brandão, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner, “The second laws of quantum thermodynamics,” Proceedings of the National Academy of Science, vol. 112, no. 11, pp. 3275–3279, 2015.
  * [48] M. Lostaglio, D. Jennings, and T. Rudolph, “Thermodynamic resource theories, non-commutativity and maximum entropy principles,” New Journal of Physics, vol. 19, no. 4, p. 043008, 2017.
  * [49] E. Hinds Mingo, Y. Guryanova, P. Faist, and D. Jennings, “Quantum thermodynamics with multiple conserved quantities,” in Thermodynamics in the Quantum Regime, pp. 751–771, Springer, 2018.
  * [50] Y. Mitsuhashi, K. Kaneko, and T. Sagawa, “Characterizing symmetry-protected thermal equilibrium by work extraction,” Physical Review X, vol. 12, no. 2, p. 021013, 2022.
  * [51] W. Pusz and S. L. Woronowicz, “Passive states and KMS states for general quantum systems,” Comm. Math. Phys., vol. 58, pp. 273–290, October 1978\.
  * [52] A. Lenard, “Thermodynamical proof of the Gibbs formula for elementary quantum systems,” J. Stat. Phys., vol. 19, p. 575, December 1978.
  * [53] G. Gour, D. Jennings, F. Buscemi, R. Duan, and I. Marvian, “Quantum majorization and a complete set of entropic conditions for quantum thermodynamics,” Nature communications, vol. 9, no. 1, pp. 1–9, 2018.
  * [54] C. Sparaciari, L. Del Rio, C. M. Scandolo, P. Faist, and J. Oppenheim, “The first law of general quantum resource theories,” Quantum, vol. 4, p. 259, 2020.
  * [55] S. Popescu, A. B. Sainz, A. J. Short, and A. Winter, “Reference frames which separately store noncommuting conserved quantities,” Physical Review Letters, vol. 125, no. 9, p. 090601, 2020.
  * [56] Z. B. Khanian, M. N. Bera, A. Riera, M. Lewenstein, and A. Winter, “Resource theory of heat and work with non-commuting charges,” Annales Henri Poincaré, vol. 24, no. 5, pp. 1725–1777, 2023.
  * [57] Z. B. Khanian, “From quantum source compression to quantum thermodynamics,” arXiv preprint arXiv:2012.14143, 2020.
  * [58] I. Marvian and R. Mann, “Building all time evolutions with rotationally invariant hamiltonians,” Physical Review A, vol. 78, no. 2, p. 022304, 2008\.
  * [59] M. L. Bera and M. N. Bera, “Quantum thermodynamics allows quantum measurement almost without collapse,” arXiv preprint arXiv:1910.13224, 2019.
  * [60] R. Shankar, Principles of Quantum Mechanics.  New York, NY: Springer, 2 ed., 1994.
  * [61] L. Foini and J. Kurchan, “Eigenstate thermalization hypothesis and out of time order correlators,” Phys. Rev. E, vol. 99, no. 4, p. 042139, 2019.
  * [62] S. Pappalardi, L. Foini, and J. Kurchan, “Eigenstate thermalization hypothesis and free probability,” Physical Review Letters, vol. 129, no. 17, p. 170603, 2022.
  * [63] J. Wang, M. H. Lamann, J. Richter, R. Steinigeweg, A. Dymarsky, and J. Gemmer, “Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time,” Phys. Rev. Lett., vol. 128, p. 180601, May 2022.
  * [64] J. D. Noh, “Eigenstate thermalization hypothesis in two-dimensional x x z model with or without su (2) symmetry,” Physical Review E, vol. 107, no. 1, p. 014130, 2023.
  * [65] P. Strasberg, A. Winter, J. Gemmer, and J. Wang, “Classicality, markovianity, and local detailed balance from pure-state dynamics,” Physical Review A, vol. 108, no. 1, p. 012225, 2023.
  * [66] Z. Zhang, J. Tindall, J. Mur-Petit, D. Jaksch, and B. Buča, “Stationary state degeneracy of open quantum systems with non-abelian symmetries,” Journal of Physics A: Mathematical and Theoretical, vol. 53, no. 21, p. 215304, 2020.
  * [67] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, “From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics,” Advances in Physics, vol. 65, no. 3, pp. 239–362, 2016.
  * [68] E. P. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Annals of Mathematics, vol. 62, no. 3, pp. 548–564, 1955\.
  * [69] F. J. Dyson, “Statistical theory of the energy levels of complex systems. ii,” Journal of Mathematical Physics, vol. 3, pp. 157–165, Jan. 1962.
  * [70] N. Rosenzweig and C. E. Porter, “”repulsion of energy levels” in complex atomic spectra,” Phys. Rev., vol. 120, pp. 1698–1714, Dec 1960.
  * [71] O. Giraud, N. Macé, É. Vernier, and F. Alet, “Probing symmetries of quantum many-body systems through gap ratio statistics,” Physical Review X, vol. 12, no. 1, p. 011006, 2022.
  * [72] D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Physical Review A, vol. 51, pp. 1015–1022, Feb. 1995.  Publisher: American Physical Society.
  * [73] S. Lloyd, “Almost Any Quantum Logic Gate is Universal,” Physical Review Letters, vol. 75, pp. 346–349, July 1995.  Publisher: American Physical Society.
  * [74] D. E. Deutsch, A. Barenco, and A. Ekert, “Universality in quantum computation,” Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, vol. 449, pp. 669–677, Jan. 1997.  Publisher: Royal Society.
  * [75] I. Marvian, “Restrictions on realizable unitary operations imposed by symmetry and locality,” Nature Physics, vol. 18, no. 3, pp. 283–289, 2022.
  * [76] I. Marvian, H. Liu, and A. Hulse, “Qudit circuits with su(d) symmetry: Locality imposes additional conservation laws,” arXiv preprint arXiv:2105.12877, 2021.
  * [77] I. Marvian, H. Liu, and A. Hulse, “Rotationally-invariant circuits: Universality with the exchange interaction and two ancilla qubits,” arXiv preprint arXiv:2202.01963, 2022.
  * [78] I. Marvian, “(Non-)Universality in symmetric quantum circuits: Why Abelian symmetries are special,” arXiv e-prints, p. arXiv:2302.12466, Feb. 2023\.
  * [79] G. Manzano, “Squeezed thermal reservoir as a generalized equilibrium reservoir,” Physical Review E, vol. 98, no. 4, p. 042123, 2018.
  * [80] S. Shahidani, “Thermodynamic forces and flows between a thermal bath and a squeezed thermal bath: Application to optomechanical systems,” Physical Review A, vol. 105, no. 6, p. 063516, 2022.
  * [81] T. Upadhyaya, J. Braasch, William F., G. T. Landi, and N. Yunger Halpern, “What happens to entropy production when conserved quantities fail to commute with each other,” arXiv e-prints, p. arXiv:2305.15480, 2023.
  * [82] K. Ito and M. Hayashi, “Optimal performance of generalized heat engines with finite-size baths of arbitrary multiple conserved quantities beyond independent-and-identical-distribution scaling,” Physical Review E, vol. 97, no. 1, p. 012129, 2018.
  * [83] D. N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett., vol. 71, pp. 1291–1294, Aug 1993.
  * [84] P. Calabrese, J. Dubail, and S. Murciano, “Symmetry-resolved entanglement entropy in wess-zumino-witten models,” Journal of High Energy Physics, vol. 2021, no. 10, pp. 1–32, 2021.
  * [85] M. Goldstein and E. Sela, “Symmetry-resolved entanglement in many-body systems,” Physical review letters, vol. 120, no. 20, p. 200602, 2018.
  * [86] S. Zhao, C. Northe, and R. Meyer, “Symmetry-resolved entanglement in ads3/cft2 coupled to u(1) chern-simons theory,” Journal of High Energy Physics, vol. 2021, no. 7, p. 30, 2021.
  * [87] S. Zhao, C. Northe, K. Weisenberger, and R. Meyer, “Charged moments in w3 higher spin holography,” Journal of High Energy Physics, vol. 2022, no. 5, p. 166, 2022.
  * [88] E. Bianchi and P. Dona, “Typical entanglement entropy in the presence of a center: Page curve and its variance,” Physical Review D, vol. 100, no. 10, p. 105010, 2019.
  * [89] L. Agarwal and S. Xu, “Emergent symmetry in brownian syk models and charge dependent scrambling,” Journal of High Energy Physics, vol. 2022, no. 2, pp. 1–57, 2022.
  * [90] Á. L. Corps and A. Relaño, “Dynamical and excited-state quantum phase transitions in collective systems,” Physical Review B, vol. 106, no. 2, p. 024311, 2022.
  * [91] Á. L. Corps and A. Relaño, “Theory of dynamical phase transitions in quantum systems with symmetry-breaking eigenstates,” Physical Review Letters, vol. 130, no. 10, p. 100402, 2023.
  * [92] Á. L. Corps and A. Relaño, “General theory for discrete symmetry-breaking equilibrium states,” arXiv e-prints, p. arXiv:2303.18020, Mar. 2023.
  * [93] M. Fagotti, “On conservation laws, relaxation and pre-relaxation after a quantum quench,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2014, no. 3, p. P03016, 2014.
  * [94] M. Fagotti, “Charges and currents in quantum spin chains: late-time dynamics and spontaneous currents,” Journal of Physics A: Mathematical and Theoretical, vol. 50, p. 034005, dec 2016.
  * [95] K. Fukai, Y. Nozawa, K. Kawahara, and T. N. Ikeda, “Noncommutative generalized gibbs ensemble in isolated integrable quantum systems,” Physical Review Research, vol. 2, no. 3, p. 033403, 2020.
  * [96] M. Žnidarič, “Spin transport in a one-dimensional anisotropic heisenberg model,” Physical Review Letters, vol. 106, no. 22, p. 220601, 2011.
  * [97] M. Ljubotina, M. Žnidarič, and T. Prosen, “Spin diffusion from an inhomogeneous quench in an integrable system,” Nature communications, vol. 8, no. 1, pp. 1–6, 2017.
  * [98] S. Gopalakrishnan and R. Vasseur, “Kinetic theory of spin diffusion and superdiffusion in x x z spin chains,” Physical review letters, vol. 122, no. 12, p. 127202, 2019.
  * [99] J. De Nardis, S. Gopalakrishnan, R. Vasseur, and B. Ware, “Stability of superdiffusion in nearly integrable spin chains,” Physical review letters, vol. 127, no. 5, p. 057201, 2021.
  * [100] E. Ilievski, J. De Nardis, S. Gopalakrishnan, R. Vasseur, and B. Ware, “Superuniversality of superdiffusion,” Physical Review X, vol. 11, no. 3, p. 031023, 2021.
  * [101] M. Mierzejewski, J. Pawłowski, P. Prelovšek, and J. Herbrych, “Multiple relaxation times in perturbed xxz chain,” SciPost Physics, vol. 13, no. 2, p. 013, 2022.
  * [102] A. C. Potter and R. Vasseur, “Symmetry constraints on many-body localization,” Physical Review B, vol. 94, no. 22, p. 224206, 2016.
  * [103] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory.  Nashville, Tennessee: Westview Press, 1995.
  * [104] N. Mueller, T. V. Zache, and R. Ott, “Thermalization of gauge theories from their entanglement spectrum,” Phys. Rev. Lett., vol. 129, p. 011601, Jun 2022.
  * [105] H. Liu and P. Glorioso, “Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,” in Theoretical Advanced Study Institute Summer School 2017” Physics at the Fundamental Frontier”, vol. 305, p. 008, Sissa Medialab, 2018.
  * [106] R. Baier, P. Romatschke, and U. A. Wiedemann, “Dissipative hydrodynamics and heavy-ion collisions,” Phys. Rev. C, vol. 73, p. 064903, Jun 2006.
  * [107] P. Glorioso, L. Delacrétaz, X. Chen, R. Nandkishore, and A. Lucas, “Hydrodynamics in lattice models with continuous non-abelian symmetries,” SciPost Physics, vol. 10, no. 1, p. 015, 2021.
  * [108] M. Torabian and H.-U. Yee, “Holographic nonlinear hydrodynamics from ads/cft with multiple/non-abelian symmetries,” Journal of High Energy Physics, vol. 2009, no. 08, p. 020, 2009.
  * [109] C. Hoyos, B. S. Kim, and Y. Oz, “Odd parity transport in non-abelian superfluids from symmetry locking,” Journal of High Energy Physics, vol. 2014, no. 10, pp. 1–36, 2014.
  * [110] H.-T. Elze and U. Heinz, “Quark-gluon transport theory,” Physics Reports, vol. 183, no. 3, pp. 81–135, 1989.
  * [111] X. Yao, “Su(2) gauge theory in $2+1$ dimensions on a plaquette chain obeys the eigenstate thermalization hypothesis,” Phys. Rev. D, vol. 108, p. L031504, Aug 2023.
  * [112] L. Ebner, B. Müller, A. Schäfer, C. Seidl, and X. Yao, “Eigenstate thermalization in 2+1 dimensional SU(2) lattice gauge theory,” arXiv e-prints, p. arXiv:2308.16202, Aug. 2023.
  * [113] J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, “Qcd thermalization: Ab initio approaches and interdisciplinary connections,” Rev. Mod. Phys., vol. 93, p. 035003, Aug 2021.
  * [114] S. Majidy, U. Agrawal, S. Gopalakrishnan, A. C. Potter, R. Vasseur, and N. Y. Halpern, “Critical phase and spin sharpening in su(2)-symmetric monitored quantum circuits,” Phys. Rev. B, vol. 108, p. 054307, Aug 2023.
  * [115] B. Swingle, “Unscrambling the physics of out-of-time-order correlators,” Nature Physics, vol. 14, pp. 988–990, Oct 2018.
  * [116] V. Khemani, A. Vishwanath, and D. A. Huse, “Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws,” Phys. Rev. X, vol. 8, p. 031057, Sep 2018.
  * [117] D. A. Roberts and B. Yoshida, “Chaos and complexity by design,” Journal of High Energy Physics, vol. 2017, p. 121, Apr 2017.
  * [118] A. R. Brown and L. Susskind, “Second law of quantum complexity,” Phys. Rev. D, vol. 97, p. 086015, Apr 2018.
  * [119] W. Fulton and J. Harris, Representation Theory: A First Course.  New York, New York: Springer, 2004.