#  Maximum entropy quantum state distributions

Alexander Altland  Institut für Theoretische Physik, Universität zu Köln,
Zülpicher Str. 77, 50937 Cologne, Germany  David A. Huse  Department of
Physics, Princeton University, Princeton, NJ 08544, USA  Institute for
Advanced Study, Princeton, NJ 08540, USA  Tobias Micklitz  Centro Brasileiro
de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil

###### Abstract

We propose an approach to the realization of many-body quantum state
distributions inspired by combined principles of thermodynamics and mesoscopic
physics. Its essence is a maximum entropy principle conditioned by
conservation laws. We go beyond traditional thermodynamics and condition on
the full distribution of the conserved quantities. The result are quantum
state distributions whose deviations from ‘thermal states’ get more pronounced
in the limit of wide input distributions. We describe their properties in
terms of entanglement measures and discuss strategies for state engineering by
methods of current date experimentation.

In the early nineteenth century, thermodynamics was introduced to describe the
state of complex systems on the basis of a minimal specification. The provided
information would specify the values of certain macroscopic observables, such
as the energy density. In the language of statistical thermodynamics, the
_minimal_ amount of information provided translates to _maximal_ entropy of
the remaining unspecified microscopic properties. For example, the canonical
distribution $\rho=Z^{-1}\exp(-\beta H)$ defines a distribution for a system’s
micro-states maximizing entropy under the condition of fixed average energy.
The entropy, conditioned by the specified information, is
$S=-\mathrm{tr}(\rho\ln\rho)$.

Two centennials later, the principles of that approach have not lost their
power. However, progress in experimentation now makes it possible to apply
them to quantum systems of _mesoscopic_ proportions: systems large enough to
be efficient information scramblers and ‘thermalizing’, yet small enough to
harbor large quantum fluctuations. Under these conditions, it becomes natural
to generalize the specified information from given average values of conserved
quantities to full statistical distributions. For example, for a spin chain
with conserved total $z$-axis magnetization, $S_{z}$, one may consider initial
pure states $\hat{\rho}_{0}=|\Psi_{0}\rangle\langle\Psi_{0}|$ engineered to
realize a chosen magnetization distribution
$p(M)\equiv\langle\Psi_{0}|\delta_{\hat{S}_{z},M}|\Psi_{0}\rangle$. Under
dynamical evolution by a Hamilton operator $\hat{H}(t)$ — assumed to be many-
body chaotic — $\Psi_{0}$ evolves into a pseudo-random pure state,
$\hat{\rho}=|\Psi\rangle\langle\Psi|$, with the same $p(M)$ (since
$[\hat{H}(t),S_{z}]=0$), and perhaps other constraints, such as a conserved
total energy. (By contrast, basic thermodynamics would be content with fixing
just $\langle M\rangle=\langle\Psi|\hat{S}_{z}|\Psi\rangle$.)

In this letter we fuse concepts of thermodynamics and quantum mesosocopics. We
consider pure states defined to maximize entropy, conditional that they
realize a distribution $p(Q)$ of a scalar ‘charge’ , $Q$, — energy, particle
number, uni-axial magnetization, etc. We describe the physical properties of
such states, in particular their entanglement properties. The limiting case of
$P(Q)=\delta_{Q,Q_{0}}$ models eigenstates of $\hat{Q}$ satisfying an
eigenstate thermalization hypothesis (ETH) Deutsch (1991); Srednicki (1994);
Rigol et al. (2008).

The key input determining the entanglement properties of our states are
deviations between the specified distribution $p(Q)$ and the spectral
distribution, $\Omega(Q)$, of the conserved operator, $\hat{Q}$, where
$D\Omega(Q)$ is the number of $Q$-eigenvalue states in a $D$-dimensional
Hilbert space. The unconstrained case, $p(Q)=\Omega(Q)$, has the highest
entanglement, corresponding to Page’s maximally random pure states Page
(1993). Constraining to eigenstates of $\hat{Q}$ reduces the entanglement, but
much larger reductions can be obtained by specifying distributions broad
compared to $\Omega(Q)$. Within a parameter space spanned by the center and
width of $p(Q)$, we will discuss various reference configurations, among them
Page’s random states Page (1993) (no input information provided), thermal
distributions Sugiura and Shimizu (2012, 2013); Nakagawa et al. (2018),
microcanonical distributions Vidmar and Rigol (2017); Bianchi et al. (2021),
and very broadly distributed $p(Q)$. We will argue that for spin systems of
mesoscopic proportions the engineering of the latter is in experimental reach
and should thus lead to tunable non-thermal signatures in, e.g., Rényi
entropies Islam et al. (2015); Kaufman et al. (2016) or spin-correlation
probes.

Maximum entropy distributions:— Without much loss of generality, we consider a
$N$-qubit system, and in it a conserved operator which is subsystem additive:
a partition of the system into two subsystems $A$ and $B$ of size $N_{A}$ and
$N_{B}=N-N_{A}$ implies a decomposition $\hat{Q}=\hat{Q}_{A}+\hat{Q}_{B}$. On
the same basis, $\hat{Q}$’s eigenstates, $|n\rangle$ are labeled by a
$D=2^{N}=2^{N_{A}+N_{B}}=D_{A}D_{B}$ dimensional index
$n=(a,b)\in\mathbb{Z}_{2}^{N}$ with eigenvalues $Q_{n}$.

Assuming a given charge distribution, $p(Q)$, we now construct a distribution
$P(\Psi)=P(\\{\Psi_{n}\\})$ defined on the space of pure states satisfying two
conditions: first, in the limit of large $N$, states drawn from $P$ satisfy
the condition $p(Q)=\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle$, where
$\delta_{\hat{Q},Q}$ is a $\delta$-function or a Kronecker $\delta$, depending
on whether $Q$ is continuous or discrete. Second, the information entropy
$S[P]=-\langle\ln(P)\rangle_{P}$ be maximal, where we introduced the shorthand
notation $\langle\ldots\rangle_{P}=\int D\Psi P(\Psi)(\ldots)$. A distribution
satisfying these criteria is found by extremizing the functional

$\displaystyle A[P]\equiv S[P]$ $\displaystyle+\lambda\left(\langle
1\rangle_{P}-1\right)+\lambda_{0}\left(\langle\langle\Psi|\Psi\rangle\rangle_{P}-1\right)$
$\displaystyle+\sum_{Q}\lambda(Q)\left(\langle\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle\rangle_{P}-p(Q)\right).$
(1)

Here, the Lagrange multipliers, $\lambda_{0}$ and $\lambda(Q)$ impose state
normalization, $\langle\langle\Psi|\Psi\rangle\rangle_{P}=1$, and the
distribution property,
$\langle\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle\rangle_{P}=p(Q)$, only on
average over the distribution $P[\Psi]$. In exchange for the typicality
assumption $\langle F[\Psi]\rangle_{\Psi}\approx F[\Psi]$, which we assume
holds true in the limit of large Hilbert space dimension, we obtain an
extremization problem that is easy to solve:

In the supplemental material sup , we show that the straightforward variation
of the action functional yields the solution

$\displaystyle\frac{D}{\lambda_{0}+\lambda(Q)}=\frac{p(Q)}{\Omega(Q)},\qquad\sum_{Q}\frac{D\Omega(Q)}{\lambda_{0}+\lambda(Q)}=1,$
(2)
$\displaystyle\rho_{n}\equiv\langle|\Psi_{n}|^{2}\rangle=\frac{p(Q_{n})}{D\Omega(Q_{n})}.$
(3)

The first two of these equations fix the Lagrange multipliers, and the third
states that the distribution of wave function amplitudes is Gaussian, with a
variance set by the specified charge distribution $p(Q)$. With $\rho_{n}\geq
0$ and $\sum\rho_{n}=1$, we interpret $\rho=\\{\rho_{n}\\}$ as the average
distribution of spectral weight over Fock space (see table 1 for an overview
of the various distributions relevant to our discussion.)

To gain some familiarity with these expressions, consider the limiting case of
unconstrained states, $\lambda(Q)=0$. The above equations are then solved by
$\lambda_{0}=D$. For this value, $\Psi_{n}$ are Gaussian variables with
uniform variance $D^{-1}$; the random state vectors considered by Page. In
this particular case, the charge distribution $p(Q)=\Omega(Q)$ is dictated by
the native spectral distribution. In the following, we consider what happens
if we condition to Gaussian charge distributions

$\displaystyle p(Q)=\frac{1}{\sqrt{2\pi}\Delta
Q}\exp\left(-\frac{(Q-\bar{Q})^{2}}{2(\Delta Q)^{2}}\right),$ (4)

of general center $\bar{Q}$ and width $\Delta Q$. The observation that the
limiting case $p(Q)=\Omega(Q)$ corresponds to uniformly distributed states of
minimal structure suggests to quantify the ‘input information’ provided by
$p(Q)$ in terms of the statistical distance to $\Omega(Q)$, i.e. the Kullback-
Leibler divergence $I_{\mathrm{in}}\equiv-
D_{\mathrm{KL}}(p||\Omega)=\sum_{Q}p(Q)\ln\left(\frac{p(Q)}{\Omega(Q)}\right)$.
Using Eqs.(2), it is straightforward to verify that

$\displaystyle I_{\mathrm{in}}=-S(\rho)+\ln D,$ (5)

where $S(\rho)=-\sum_{n}\rho_{n}\ln(\rho_{n})$ is the von Neumann entropy of
the distribution $\rho$. In the following, we calculate the effects of this
specified charge distribution $p(Q)$ on the entanglement entropies defined by
subsystem partitions.

input | $\Omega(Q)$ | spectral distribution of charge
---|---|---
| $p(Q)$ | engineered distribution of charge
output | $P(\Psi)$ | distribution of pure states conditioned via $p$
| $\rho_{n}$ | spectral weight distribution in Fock space
| $\rho_{A,a}$ | subsystem reduction of $\rho$
| $p_{A}(Q_{A})$ | charge distribution over subsystem
Table 1: Overview over the different statistical distributions defined by a
conserved operator $\hat{Q}$ in a many body system: the eigenvalues of
$\hat{Q}$ are distributed according to $\Omega(Q)$; we are interested in
states redistributing charge into $p(Q)$. Embedding this condition with the
maximum entropy principle, we obtain average pure state distributions
$\rho_{n}$, and their subsystem marginals $\rho_{A,a}$ for $n=(a,b)$. Finally,
$p_{A}(Q_{A})$ describes the average distribution of charge on the subsystem
$A$.

Entanglement entropies:— We cut our system into two pieces, $A$, $B$, with
associated Fock spaces $\mathcal{F}=\mathcal{F}_{A}\otimes\mathcal{F}_{B}$ of
dimensions $D_{A}\leq D_{B}$. For a given pure state
$\hat{\rho}=|\Psi\rangle\langle\Psi|$, we consider the reduced density
matrices $\hat{\rho}_{A}=\mathrm{tr}_{B}(\hat{\rho})$, and entanglement
entropies
$S=-\langle\mathrm{tr}(\hat{\rho}_{A}\ln\hat{\rho}_{A})\rangle_{\Psi}\simeq-\mathrm{tr}(\rho_{A}\ln\rho_{A})\equiv
S^{\mathrm{a}}$, where in the ‘average’ entropy, $S^{\mathrm{a}}$, we assume
self averaging
$\hat{\rho}_{A}\simeq\langle\hat{\rho}_{A}\rangle_{\Psi}\equiv\rho_{A}$. Page
has shown that in the absence of conditioning
$S^{\mathrm{a}}=S_{\mathrm{th}}\equiv N_{A}\ln 2$ is trivially thermal. In the
following we show how this changes due to specifying $p(Q)$. For a basis
decomposition $n=(a,b)$, $\rho_{A}=\\{\rho_{A,a}\\}$ is described by a list of
$D_{A}$ coefficients $\rho_{A,a}$. Using the definition of
$S_{A}^{\mathrm{a}}$ in combination with Eq. (2), we obtain

$\displaystyle\Delta
S^{\mathrm{a}}=-\sum_{Q_{A}}p_{A}\ln\left(\frac{p_{A}}{\Omega_{A}}\right)=-D_{\mathrm{KL}}(p_{A}||\Omega_{A}),$
(6) $\displaystyle\qquad
p_{A}(Q_{A})=\Omega_{A}(Q_{A})\sum_{Q_{B}}\frac{\Omega_{B}(Q_{B})p(Q_{A}+Q_{B})}{\Omega(Q_{A}+Q_{B})},$
(7)

for the difference $\Delta S^{\mathrm{a}}=S_{A}^{\rm av}-S_{\mathrm{th}}$.
Conceptually, $p_{A}(Q_{A})$ describes the unit normalized foo (b) charge
distribution imprinted on subsystem $A$ via the input charge distribution
$p(Q)$ in the maximum entropy ensemble. Eq. (6) states that the resulting
reduction of the entanglement entropy of the average density matrix equals the
(Kullback-Leibler) deviation of the induced distribution from the spectral
distribution of the operator.

To make the result Eq. (6) concrete, we need to evaluate the convolutions in
Eq. (6) in more explicit terms. Reflecting the local additivity principle
obeyed by the variable $Q$, we assume that, except in the far tails of the
spectrum, the density of states is Gaussian

$\displaystyle\Omega(Q)=\Omega(Q,N)=\frac{1}{\sqrt{2\pi}\Gamma}e^{-\frac{Q^{2}}{2\Gamma^{2}}}\equiv\frac{1}{\sqrt{2\pi
N}\gamma}e^{-N\frac{q^{2}}{2\gamma^{2}}},$ (8)

where we define the $Q$-variable such that $Q=0$ corresponds to the center of
the distribution $\Omega(Q)$, and the scaled variables $Q=Nq$,
$\Gamma=\sqrt{N}\gamma$. The same function describes the subsystem spectral
densities as $\Omega_{X}(Q_{X})=\Omega(Q_{X},N_{X})$, $X=A,B$.

In the supplemental material sup  we show that for these spectral densities
the reduced charge density assumes the form

$\displaystyle p_{A}(Q_{A})$
$\displaystyle\approx\frac{N\sqrt{N_{B}}}{\Gamma\sqrt{2\pi N_{A}}}\int
dx\,e^{-\frac{N_{A}N_{B}}{2\Gamma^{2}}x^{2}}p(q_{A}N+xN_{B}),$ (9)

i.e. the input charge distribution convoluted against a smoothing function
which knows about the spectral distribution of $\hat{Q}$ and the relative
sizes of the subsystems. In the limit of large system sizes and fixed
$N_{A}/N_{B}$, the variable $x\sim N^{-1/2}$, and so $xN_{B}\sim N^{1/2}$,
which is small compared to the variable $q_{A}N\sim N$. For a broad input
distribution, $p$, the $x$-dependence of $p$ becomes negligible, and the
$x$-integral yields $p_{A}(Q_{A})\approx\frac{N}{N_{A}}p(Q_{A}(N/N_{A}))$.
This result states that the charge distribution on $A$ inherits that of the
system at large. In the opposite limit of a sharply-peaked input distribution,
$p(Q)=\delta(Q-\bar{Q})$, Eq. (9) collapses to
$p_{A}(Q_{A})\sim\exp(-\frac{N_{A}}{2\Gamma^{2}N_{B}}(q_{A}N-\bar{Q})^{2})$.
This broadening of the $\delta$-function occurs due to the subsystem trace,
which reduces the degree of constraint. More generally, for a Gaussian
distribution of width $\Delta Q$ the computation of the entanglement entropy
Eq. (6) (see supplemental material sup ) yields

$\displaystyle\Delta S^{\mathrm{a}}$
$\displaystyle=-\frac{n_{A}}{2}\left(\delta^{2}+\kappa^{2}-1\right)+\frac{1}{2}\ln\left(1+n_{A}\left(\delta^{2}-1\right)\right),$
$\displaystyle n_{A}=\frac{N_{A}}{N},\qquad\delta=\frac{\Delta
Q}{\Gamma},\qquad\kappa=\frac{|\bar{Q}|}{\Gamma}.$ (10)

According to this result, the state information output via the average
entanglement entropy depends on the three parameters: relative system size,
$n_{A}$; and deviation, $\kappa$, and width, $\delta$, of the input
distribution relative to the width of $\hat{Q}$’s spectral density. Figure 1
shows this quantity for a centered distribution, $\kappa=0$, as a function of
the parameter $\delta$ for multiple values of $n_{A}$. Its most striking
feature is the non-monotonic dependence, with a minimum $\Delta
S^{\mathrm{a}}=0$ for $\delta=1$, or $\Delta Q=\Gamma\propto N^{1/2}$ equal to
the width of the native spectral distribution $\Omega(Q)$. The entanglement
entropy may be reduced by sharpening the input distribution (thus increasing
the specified information), with a limit $\Delta
S^{\mathrm{a}}=(n_{A}+\ln{(1-n_{A})})/2$ for $\delta\rightarrow 0$. However,
perhaps unexpectedly, output information also results from _broadening_ the
input. Already for distributions with $\delta\approx\sqrt{2}$, widening
becomes a stronger information booster than sharpening. For extensively wide
distributions, $\delta\sim\sqrt{N}$, or $\Delta Q\sim N$, the anomalous
contribution to the entanglement entropy is likewise extensive, $\Delta
S^{\mathrm{a}}\approx-n_{A}\delta^{2}\sim N_{A}$. (The extreme limit within
this class of distributions is realized for a flat input
$p(Q)=\mathrm{const}$. In this case, a straightforward estimate yields $\Delta
S^{\mathrm{a}}\approx-N_{A}/24$, respecting the positive definiteness of the
total entropy $S=S_{\mathrm{th}}+\Delta S^{\mathrm{a}}$.)

Figure 1:  Entropy Eq. (6) plotted as a function of $\delta$ for
$n_{A}=2^{-9},2^{-7},\ldots 2^{-1}$, with larger entropies corresponding to
larger subsystem sizes.

Fluctuation corrections:— The above construction describes the entropy of the
averaged state over a wide range of parameters. Quantitative corrections to
Eq. (6) arise for average values $\bar{Q}$ in the far tails of the
distribution $\Omega(Q)$, where the Gaussian distribution breaks down, or for
input densities $p(Q)$ which cannot be modeled as locally Gaussian around a
single maximum. While we do not consider such effects in the present paper,
there remains one open conceptual point namely that the average entanglement
entropy of pure states, $S\not=S^{\mathrm{av}}$ differs from that of the
state-average. For example, Page has shown Page (1993) that in the absence of
conditioning, $\Delta S^{\mathrm{a}}=0$, while for the full entanglement
entropy, $\Delta S=-\frac{D_{A}}{2D_{B}}$ is exponentially small for
asymmetric cuts, $D_{A}/D_{B}=2^{N_{A}-N_{B}}$, but becomes sizeable for near-
equal subsystem sizes (see also Foong and Kanno (1994); Sánchez-Ruiz (1995)).

In order to compute (quantum) fluctuation corrections to the result Eq. (6),
we need to go back to a first principles representation
$S=-\partial_{r}|_{r=1}M^{r}$ of the entanglement entropy in terms of the
moments $M^{r}=\langle\mathrm{tr}_{A}(\hat{\rho}_{A}^{r})\rangle_{\Psi}$ of
the reduced density matrix,
$\rho_{A,aa^{\prime}}=\sum_{b}\Psi_{(ab)}\bar{\Psi}_{(a^{\prime}b)}$ Penington
et al. (2019); Liu and Vardhan (2020). For the Gaussian distribution Eq. (11)
this expression becomes a sum over all combinatorial pairings of wave function
amplitudes, where so far we considered the nearest neighbor pairings,
$\langle\Psi_{(ab)}\bar{\Psi}_{(a^{\prime}b)}\rangle_{\Psi}$. In the
supplementary material sup  we consider the contribution of other pairings for
the exemplary case of the microcanonical distribution
$p(Q)=\delta_{Q,\bar{Q}}$. (The computation for generic distributions is more
complicated but does not lead to qualitatively different conclusions although
details change.) It turns out that for $N_{A}\ll N_{B}$ and charges $\bar{Q}$
away from the extreme tails, the average entropy $S^{\mathrm{a}}$ approximates
the full one up to corrections in $D_{A}/D_{B}$. (The scaling in Hilbert space
dimensions follows from the observations that for Gaussian contractions
different from the average one one needs to pay in free summations over the
$B$-subspace.) For $N_{A}\approx N_{B}$, we obtain a correction, $\delta S$,
required to establish the symmetry of the ‘entanglement wedge’ (see Fig. 2)
under exchange $A\leftrightarrow B$. Technically, the wedge reflects zero
singular values in the pure states Schmidt decomposition. For Page states
these zeros only occur for $N_{B}>N_{A}$, however, it is more subtle in
systems with conserved charges. In the supplemental material sup  we show that
for microcanonical distributions the ‘wedge’ is smoothened by a correction
$\delta S\propto-\sqrt{N}\bar{q}$ (as first noted in Vidmar and Rigol (2017)).
Higher order corrections in $D_{A}/D_{B}$ also considered by Page, on the
other hand, remain bounded by $-1/2$ (see supplemental material sup  for
explicit expression). The upshot of this discussion is that for asymmetric
cuts the average entropy approximates the full one, and even for equipartition
$N_{A}=N_{B}$ represents a good approximation.

Figure 2:  Schematic structure of the entanglement entropy as a function of
subsystem size, $N_{A}$. Discussion, see text.

_State preparation: —_ The above discussion has shown how maximum entropy
states realizing broad charge distributions differ from thermal states. For
systems in the thermodynamic limit $N\to\infty$, charge states lie with
overwhelming probability inside the $\sim\sqrt{N}$ tolerance window of the
native distribution $\propto D\Omega(Q)$, and the engineering of pure states
of broader distribution may be hard. However, for the mesoscopic values of $N$
realized in current date AMO experiments or numerical analysis, broad
distributions may be (perhaps even inadvertently) realized. To give an
example, we consider a system of $N$ spin $1/2$ with assumed conservation of
the total $z$-axis spin component, $\hat{Q}=\hat{S}_{z}$. Assume the system
divided into $M=N/L$ blocks of $L$-sized cat states Johnson et al. (2017); Gao
et al. (2010); Yao et al. (2012); Wang et al. (2016); Hacker et al. (2019);
Duan (2019); Omran et al. (2019)
$|\Psi_{a}\rangle=\frac{1}{\sqrt{2}}(|\uparrow\rangle_{L}+|\downarrow\rangle_{L})$
(with $|\uparrow,\downarrow\rangle_{L}$ the $\hat{Q}$-eigenstates at values
$S_{z}=\pm L/2$), and consider
$|\Psi\rangle\equiv\otimes_{a=1}^{M}|\Psi_{a}\rangle$. For this state, a
straightforward calculation shows that
$p(Q)=\langle\Psi|\delta_{Q,\hat{S}_{z}}|\Psi\rangle=\frac{1}{2^{M}}\sum_{a=0}^{M}\binom{M}{a}\delta_{Q,(a-\frac{M}{2})L}=\frac{1}{2^{M}}\binom{M}{\frac{M}{2}+\frac{Q}{L}}\simeq\frac{\sqrt{2}}{\sqrt{\pi
M}}\exp(\frac{2Q^{2}}{ML^{2}})$. This particular state has charge distributed
over a range $\sim L\sqrt{M}=N/\sqrt{M}$. This means that for fixed block
number, $M\sim N^{0}$, we have an extensively broad distribution. In this
scaling regime, the above product cat will evolve under chaotic time evolution
into random states different from structureless ETH states. These differences
show in the entanglement properties emphasized in this paper, but more
directly also in magnetization probes revealing the dynamical conservation of
the charge distribution despite the systems many body chaotic dynamics.

Discussion:— We introduced an approach to quantum state design which combines
a maximum entropy principle with premeditated distributions of dynamically
conserved observables in meso-sized many body quantum systems. This
distribution survives unmasked by the chaotic dynamics and shows at various
levels of the system description, beginning with the interpretation of the
states themselves: for a given $p(Q)$, we defined a distribution of states
$P(\Psi)$, such that individual states $\hat{\rho}=|\Psi\rangle\langle\Psi|$
drawn from it, generate the input in the sense
$p(Q)=\mathrm{tr}(\hat{\rho}\delta_{\hat{Q},Q})$. (As an alternative to this
pure state interpretation, one may also consider the mixed state
$\hat{\rho}^{\prime}=\sum_{\Psi}P(\Psi)|\Psi\rangle\langle\Psi|$, which has
the same property $p(Q)=\mathrm{tr}(\hat{\rho}\delta_{\hat{Q},Q})$, see
supplemental material sup  for more on the comparison of the two views.) When
a component $B$ of the system is traced out and in this way becomes part of an
environment, an induced charge distribution $\rho_{A}(Q_{A})$ emerges from the
given one by a convolution over a kernel obtained from $\hat{Q}$’s spectral
distribution, Eq. (9). This effective distribution is broader than the
original one, reflecting the loss of information due to the partial trace. The
statistical distance of the induced distribution to the spectral density of
the conserved operator in $A$ determines the entanglement entropy of our
quantum states, Eq.(6). This central result of our paper quantifies the
difference to thermal states, a principal observation being that broad input
distributions lead to the strongest deviations from ETH states. The
straightforward generalization of this finding to multiple but mutually
commuting conserved quantities (such as energy and particle number) shows that
in this case the individually computed entropies $\Delta S^{\mathrm{a}}$ add.
One point left out here concerns the generalization to distributions of strong
non-Gaussianity. (For recent work discussing the influence of Lorentzian
distributions on the statistics of quantum states we refer to Ref. Bogomolny
and Sieber (2018); Monteiro et al. (2021))

Our analysis illustrates how entire distributions (and not just single average
values) may survive ergodic chaotic time evolution. It would be interesting to
see if this principle of state design may be turned into a creative resource.
Conversely, one needs to watch out, especially in small or medium sized
systems, and check if the application of the ETH hypothesis to a many body
state might be incompatible with an (perhaps inadvertently) introduced defined
initial charge distribution.

_Acknowledgments: —_ D. A. H. thanks Shayan Majidy and Nicole Yunger Halpern
for discussions. D. A. H. is supported in part by NSF QLCI grant OMA-2120757.
T. M. acknowledge financial support by Brazilian agencies CNPq and FAPERJ. A.
A. acknowledges partial support from the Deutsche Forschungsgemeinschaft (DFG)
within the CRC network TR 183 (project grant 277101999) as part of projects
A03.

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##  I Supplementary Material

###  I.1 Solution of the variational equations for the state distribution

We here show how Eqs. (2) follow from variation of the functional Eq. (Maximum
entropy quantum state distributions). The straightforward solution of the
variational equation $\delta_{P}A[P]=0$ yields
$P(\Psi)=\frac{1}{Z}e^{-\lambda_{0}\langle\Psi|\Psi\rangle-\sum_{Q}\lambda(Q)\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle}$,
with a normalization factor $Z$. Using the eigenstate property,
$\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle=\sum_{m}|\Psi_{m}|^{2}\delta_{Q_{m},Q}$,
and integrating over all components $\Psi_{m\not=n}$ but one, we find that
individual amplitudes $\Psi_{n}$ are Gaussian distributed with

$\displaystyle\rho_{n}\equiv\langle|\Psi_{n}|^{2}\rangle=\frac{1}{\lambda_{0}+\lambda(Q_{n})}.$
(11)

This result may now now be used to fix the Lagrange multipliers $\lambda_{0}$
and $\lambda(Q)$. Converting sums into integrals as
$\sum_{n}F(Q_{n})=\sum_{Q}\Omega(Q)F(Q)$, the normalization condition yields
$1=\langle\sum_{n}|\Psi_{n}|^{2}\rangle=\sum_{Q}\frac{D\Omega(Q)}{\lambda_{0}+\lambda(Q)}$.
Treating the second condition in Eq. (Maximum entropy quantum state
distributions) in the same manner, we obtain the first line in Eq. (2).
Substitution of this result into Eq. (11) yields the second line.

###  I.2 The subsystem charge distribution, $p_{A}(Q_{A})$

In order to derive Eq. (9), we represent the subsystem charge distribution as
an integral over Gaussian spectral densities,

$\displaystyle
p_{A}(Q_{A})=\frac{\mathcal{N}_{A}\mathcal{N}_{B}}{\mathcal{N}_{A\cup
B}}N_{B}\int dq_{B}\,$ $\displaystyle\quad
e^{-\frac{1}{2\gamma^{2}}\left(N_{A}q^{2}_{A}+N_{B}q^{2}_{B}-\frac{1}{N}(q_{A}N_{A}+q_{B}N_{B})^{2}\right)}p(q_{A}N_{A}+q_{B}N_{B}),$

where we introduced scaled, and effectively continuous variables
$Q_{X}=q_{X}N_{X}$, $X=A,B$. Considered as a function of $q_{A,B}$, the
exponential function under the integral possesses the saddle points,
$q_{B}=q_{A}$, indicating that the system favors charge equilibration over its
subsystems. With $q_{B}=q_{A}+q$, the second order expansion of the integral
around this saddle assumes the form Eq.(9).

An explicit representation, not relying on the above assumption, may be
obtained for a Gaussian input distribution $p(Q)=\frac{\Delta
Q}{\sqrt{2\pi}}\exp(-\frac{(Q-\bar{Q})^{2}}{2\Delta Q^{2}})$. In this case,
the integral over $q$ yields

$\displaystyle p_{A}(Q_{A})$
$\displaystyle=\frac{N}{N_{A}}\frac{1}{\sqrt{2\pi}\Lambda}\exp\left(-\frac{(Q_{A}\frac{N}{N_{A}}-\bar{Q})^{2}}{2\Lambda^{2}}\right),$
$\displaystyle\qquad\Lambda^{2}=\Gamma^{2}\frac{N_{B}}{N_{A}}+\Delta Q^{2}.$

In the limit of a wide charge distribution $\Delta
Q^{2}\gg\Gamma^{2}(N_{B}/N_{A})$, this effective Gaussian distribution reduces
to $p_{A}(Q_{A})\approx\frac{N}{N_{A}}p(Q_{A}(N/N_{A}))$. More generally,
however, we obtain a Gaussian distribution of enhanced width.

The computation of the KL-divergence Eq. (6) is now reduced to a
straightforward Gaussian integral over the distributions $p_{A},\Omega_{A}$,
and we obtain

$\displaystyle S^{\mathrm{a}}$
$\displaystyle=S_{\mathrm{th}}-\frac{N_{A}}{2N}\left(\frac{\Delta
Q^{2}}{\Gamma^{2}}+\frac{\bar{Q}^{2}}{\Gamma^{2}}-1\right)+$
$\displaystyle\qquad+\frac{1}{2}\ln\left(1+\frac{N_{A}}{N}\left(\frac{\Delta
Q^{2}}{\Gamma^{2}}-1\right)\right)$

###  I.3 Derivation of the fluctuation entropy

We here compute the dominant fluctuation contributions to the entanglement
entropy for the microcanonical charge distribution from the moments
$M^{r}=\sum\langle\Psi_{a_{1}b_{1}}\bar{\Psi}_{a_{2}b_{1}}\Psi_{a_{2}b_{2}}\bar{\Psi}_{a_{3}b_{2}}\ldots\Psi_{a_{r}b_{r}}\bar{\Psi}_{a_{1}b_{r}}\rangle$.
It will be instructive to first review the computation for the case considered
by Page, where the Gaussian contraction over random state vectors yields a
factor
$\langle\Psi_{ab}\bar{\Psi}_{a^{\prime}b^{\prime}}\rangle=D^{-1}\delta_{a,a^{\prime}}\delta_{b,b^{\prime}}$.
A contribution of maximal number, $r$, of $b$-index summations is obtained by
Gaussian pairing
$\langle\Psi_{a_{l}b_{l}}\bar{\Psi}_{a_{l+1}b_{l}}\rangle=D^{-1}\delta_{a_{l},a_{l+1}}$
for all $l$. This single pairing amounts to taking the average over the
distribution prior to computing the moments, $\langle M^{r}\rangle\to\langle
M\rangle^{r}$, as discussed in the first part of the paper. Stepping down in
the index order, $N(r,2)=\binom{r}{2}$ terms with $r-1$ summations over $b$
and two over $a$ are obtained by one pairing outside the above order. The
summation over indices for these two terms yields the estimate $M_{r}\approx
D^{-r}\left(D_{B}^{r}D_{A}+\binom{r}{2}D_{B}^{r-1}D_{A}^{2}\right)$, and the
straightforward differentiation with respect to $r$ gets us to the Page
result, $S\approx N_{A}\ln 2+\frac{D_{A}}{2D_{B}}$ Page (1993). This
observation conveys a number of messages: first, pairings of generic number of
pairing permutations $1<l<(r-2)$ do not contribute. (Technically, they vanish
in the replica limit $r\rightarrow 1$, up to corrections exponentially small
in $D$.) Second, the single transposition fluctuation contribution, $l=1$, is
exponentially small, except for $N_{A}\approx N_{B}$. Finally, the above
results becomes wrong for $N_{A}>N_{B}$. The resolution to this problem lies
in the inclusion of the opposite limit where just one $b$-summation results
from the likewise unique pairing
$\langle\bar{\Psi}_{a_{l}b_{l-1}}\Psi_{a_{l}b_{l}}\rangle=D^{-1}\delta_{b_{l-1},b_{l}}$.
Finally, symmetry is restored by considering $\binom{r}{2}$ terms with one
pairing outside this scheme, two summations over $b$, $r-1$ over $a$. The
differentiation of these two terms gives $S\approx N_{B}\ln
2+\frac{D_{B}}{2D_{A}}$, i.e. Page with $A\leftrightarrow B$.

A closer inspection of the full sum, and of its convergence properties in the
limit $r\rightarrow\infty$, shows that the terms with $(r,r-1)$ or $(2,1)$
summations over $b$ need to be kept, depending on whether $D_{B}$ is larger or
smaller than $D_{A}$. In this way, one obtains the full ‘Page curve’, i.e. the
entropy $S\approx N_{X}\ln 2+\frac{D_{X}}{2D_{\bar{X}}}$ with
$(X,\bar{X})=(A,B)$ or $(B,A)$ depending on which of the systems is larger.

These general structures remain valid in the case of more general
distributions, and specifically the microcanonical one with pairing
$\langle\Psi_{ab}\bar{\Psi}_{a^{\prime}b^{\prime}}\rangle=\delta_{Q_{a}+Q_{b},\bar{Q}}(D\Omega(\bar{Q}))^{-1}\delta_{a,a^{\prime}}\delta_{b,b^{\prime}}$.
Substituting these expressions into $M_{r}$, organizing the sum according to
the number of $b$\- and $a$-index summations, and trading the $a$ and $b$
summations for summations over $Q_{A}$ and $Q_{B}=\bar{Q}-Q_{A}$ weighted by
spectral densities, we obtain the two alternative representations

$\displaystyle M_{r}$
$\displaystyle=\frac{1}{F(\bar{Q})^{r}}\sum_{Q_{A},Q_{B}}\delta_{Q_{A}+Q_{B},\bar{Q}}\sum_{k=1}^{r}N(r,k)\times$
(12)
$\displaystyle\qquad\times\left\\{\begin{array}[]{l}F_{B}(Q_{B})^{r+1}\left(\frac{F_{A}(Q_{A})}{F_{B}(Q_{B})}\right)^{k}\cr
F_{A}(Q_{A})^{r+1}\left(\frac{F_{B}(Q_{B})}{F_{A}(Q_{A})}\right)^{k}\end{array}\right.$
(15)

with the abbreviations $F(\bar{Q})=D\Omega(\bar{Q})$,
$F_{X}(Q_{X})=D_{X}\Omega_{X}(Q_{X})$, $X=A,B$, and combinatorial factors
$N(r,1)=1$, $N(r,2)=\binom{r}{2},\ldots$ known as Narayana numbers.
Convergence in the limit $r\to\infty$ (more precisely, the analytic properties
of the representation of the sum over Narayana numbers in terms of
hypergeometric functions Liu and Vardhan (2020); Penington et al. (2019))
implies that the first (second) of these defines an asymptotic $k$-series in
the case $F_{A}(Q_{A})<F_{B}(Q_{B})$ ($F_{A}(Q_{A})>F_{B}(Q_{B})$). The terms
$k=1,2$ are relevant to the computation of the entanglement entropy, and doing
the $r$-derivative we obtain

$\displaystyle
S=-\sum_{Q_{A},Q_{B}}\delta_{Q_{A}+Q_{B},\bar{Q}}\left[\frac{F_{A}(Q_{A})F_{B}(Q_{B})}{F(\bar{Q})}\ln\left(\frac{F_{B}(Q_{B})}{F(\bar{Q})}\right)+\frac{1}{2}\frac{F^{2}_{A}(Q_{A})}{F(\bar{Q})}\right]\Theta(F_{B}(Q_{B})-F_{A}(Q_{A}))+(A\leftrightarrow
B).$ (16)

If $F_{A}(Q_{A})<F_{B}(Q_{B})$ for all terms in the sum, we are back to the
result discussed in the main part of the paper, with corrections in
$F_{A}/F\propto D_{A}/D_{B}$ from the second term. However, more sizeable
modifications arise for larger subsystems, where swaps
$F_{A}(Q_{A})>F_{B}(Q_{B})$ may occur for at least some terms under the sum.

For the Gaussian spectral densities, Eq. (8), the computation of these sums
for general configurations $(\bar{Q},N_{A})$ is a straightforward if tedious
affair. We here limit ourselves to the discussion of equal partitions,
$N_{A}=N_{B}$, for which the strongest corrections to the previous results
will be obtained. In this case, the case distinction in the above sum reduces
to $\Theta(F_{B}(Q_{B})-F_{A}(Q_{A}))=\Theta(|Q_{A}|-|Q_{B}|)$, and assuming
$\bar{Q}\geq 0$, this equals $\Theta(Q_{A}-\bar{Q}/2)$.

Comparing this expression to $S^{\mathrm{a}}$ studied in the main text — the
first sum without $\Theta$-function constraint and $F^{2}_{A}/F$ correction —
we find the deviation

$\displaystyle\delta
S\simeq\sum_{Q_{A}<\bar{Q}/2}\frac{F_{A}(Q_{A})F_{A}(\bar{Q}-Q_{A})}{F(\bar{Q})}\ln\left(\frac{F_{A}(Q_{A})}{F_{A}(\bar{Q}-Q_{A})}\right),$

where we neglected the non-logarithmic as parametrically subleading (see
below). Processing the $Q_{A}$-sum and the spectral density weights as in
section I.2, we obtain (see also Ref. Vidmar and Rigol (2017))

$\displaystyle\delta
S\simeq-\frac{\sqrt{N}}{\sqrt{2\pi}\gamma}\int_{q_{A}<0}dq_{A}\,e^{-\frac{Nq^{2}_{A}}{2\gamma^{2}}}\left(\frac{N\bar{q}q_{A}}{\gamma^{2}}\right)=-\frac{\sqrt{N}\bar{q}}{\sqrt{2\pi}\gamma}.$
(17)

A similar computation shows that the second term in angular brackets in Eq.
(16) yields the correction

$\displaystyle\delta S$
$\displaystyle=-\frac{1}{2}e^{\frac{N\bar{q}^{2}}{2\gamma^{2}}}{\rm
Erfc}\left(\frac{\sqrt{N}\bar{q}^{2}}{\sqrt{2\gamma^{2}}}\right),$ (18)

with $\delta S\simeq-\gamma/(\sqrt{2\pi N}\bar{q})$ for
$\bar{q}\gg\gamma/\sqrt{N}$ and cut off at $\delta S=-1/2$ for $\bar{q}=0$. We
conclude that for generic values $\bar{q}\gtrsim\gamma/N^{-1/2}$, Eq. (17)
defines the dominant fluctuation contribution to the entanglement entropy.

###  I.4 A note about distributions

There are at least three different distributions, defined over different
support sets, that are relevant for our discussion. To define them, consider
an operator $\hat{Q}$ with eigenstates $|n\rangle$ and eigenstates $Q_{n}\in
I$ inside a certain interval.

  1. 1.

The first (input distribution) is $p(Q)$ some distribution on $I$ one can
choose at will.

  2. 2.

The second is defined by a density matrix represented in the $\hat{Q}$
eigenbasis $\hat{\rho}=\sum_{n}\rho_{n}|n\rangle\langle n|$. The set of
numbers $\\{\rho_{n}\\}$ defines a distribution on the set $\\{n\\}$ (i.e.
according to Born’s rule, the probability to measure the outcome $Q_{n}$ in a
measurement of $\hat{Q}$ is given by $\rho_{n}$).

  3. 3.

The third is a distribution $P(\Psi)$ defined on the Hilbert space of states
$\\{\Psi\\}$, i.e. a complex $D$-dimensional vector space.

To illustrate how these distributions are related to each other, we first note
that distributions 2 and 3 individually define density matrices as

$\displaystyle\hat{\rho}=\sum_{n}\rho_{n}|n\rangle\langle
n|,\qquad\hat{\rho}^{\prime}=\int D\Psi\,P(\Psi)\,|\Psi\rangle\langle\Psi|,$

where in the second expression $D\Psi=\prod_{n}d\Psi_{n}$. (Notice that we are
here summing over a massively overcomplete set.) We can then establish a
connection between the latter, requiring that the two representations
$\hat{\rho}$ and $\hat{\rho}^{\prime}$ individually generate the input
distribution 1. Conceptually, a distribution on the interval, $I$, is
generated from one on $\\{n\\}$ by considering $Q_{n}$ as a random variable
and $Q$ as a dependent random variable. (Much as e.g. the number parity is a
dependent variable of $1,2,3,4,5,6$ when throwing a dice.) The distribution of
$Q$ is then obtained from that of $Q_{n}$ as

$\displaystyle
p(Q)=\langle\delta_{Q,Q_{n}}\rangle_{\rho}\equiv\sum_{n}\rho_{n}\delta_{Q,Q_{n}}.$
(19)

For example, if $\rho_{n}=Z^{-1}e^{-\lambda Q_{n}}$ is a canonical
distribution, this becomes

$\displaystyle p(Q)=\frac{1}{Z}\sum_{n}e^{-\lambda
Q_{n}}\delta_{Q,Q_{n}}=\frac{D\Omega(Q)}{Z}e^{-\lambda Q},$

with $Z=\sum_{n}e^{-\lambda Q_{n}}=D\sum_{Q}\Omega(Q)e^{-Q}$. In this way
$\rho_{n}$ descends to a distribution $p(Q)$. In the same manner, the
distribution $P(\Psi)$ generates a distribution $p(Q)$ too,

$\displaystyle p(Q)$
$\displaystyle=\langle\delta_{\hat{Q},Q}\rangle_{\Psi}\equiv\int D\Psi
P(\Psi)\langle\Psi|\delta_{\hat{Q},Q}|\Psi\rangle.$ (20)

Assuming Gaussianity $P(\Psi)=\mathcal{N}\prod_{n}e^{-c_{n}|\Psi_{n}|^{2}}$,

$\displaystyle p(Q)$ $\displaystyle=\sum_{n}c^{-1}_{n}\delta_{Q,Q_{n}}\equiv
D\Omega(Q)c^{-1}(Q),$ (21)

where in the final step we made the ansatz $c_{n}=c(Q_{n})$. If we now impose
these induced density distributions to be equal, we get the identification

$\displaystyle c(Q)=Ze^{\lambda Q}.$ (22)

As a sanity check, we notice that the states described by the distribution
$P(\Psi)$ are normalized (on average),

$\displaystyle 1$ $\displaystyle=\langle\langle\Psi|\Psi\rangle\rangle_{\Psi}$
$\displaystyle=\int D\Psi
P(\Psi)\sum_{n}|\Psi_{n}|^{2}=\sum_{n}c_{n}^{-1}=Z^{-1}\sum_{n}e^{-\lambda
Q_{n}}.$

Summarizing, there is two a priori different density matrices $\hat{\rho}$ and
$\hat{\rho}^{\prime}$ descending to the same ‘macroscopic’ charge
distribution, and what remains to elaborate on their relation. To this end, we
compare their matrix elements

$\displaystyle\hat{\rho}_{nm}=\langle
n|\hat{\rho}|m\rangle=\frac{1}{Z}\delta_{nm}e^{-\lambda Q_{n}},$

and

$\displaystyle\hat{\rho}^{\prime}_{nm}$ $\displaystyle=\langle
n|\hat{\rho}^{\prime}|m\rangle=\mathcal{N}\int D\Psi
e^{-c_{n}|\Psi_{n}|^{2}}\Psi_{n}\bar{\Psi}_{m}=\frac{\delta_{nm}}{c_{n}}.$
(23)

Employing that $c_{n}=c(Q_{n})$ given in Eq. (22), we conclude that

$\displaystyle\hat{\rho}_{nm}$ $\displaystyle=\hat{\rho}^{\prime}_{nm}.$ (24)

That is, although defined in very different ways, both density matrices are
actually equal.