# Thermodynamics of majority-logic decoding in information erasure

Shiqi Sheng1, Tim Herpich2, Giovanni Diana3, and Massimiliano Esposito2
1Division of Interfacial Water and Key Laboratory of Interfacial Physics and
Technology, Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai 201800, China  2Complex Systems and Statistical Mechanics,
Physics and Materials Science Research Unit, University of Luxembourg, L-1511
Luxembourg, Luxembourg  3Centre for Developmental Neurobiology and MRC Center
for Neurodevelopmental Disorders, King’s College London, Guy’s Hospital Campus
London SE1 1UL, United Kingdom  Electronic Mail<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We investigate the performance of majority-logic decoding in both reversible
and finite-time information erasure processes performed on macroscopic bits
that contain $N$ microscopic binary units. While we show that for reversible
erasure protocols single-unit transformations are more efficient than
majority-logic decoding, the latter is found to offer several benefits for
finite-time erasure processes: Both the minimal erasure duration for a given
erasure and the minimal erasure error for a given erasure duration are
reduced, if compared to a single unit. Remarkably, the majority-logic decoding
is also more efficient in both the small erasure error and fast erasure
region. These benefits are also preserved under the optimal erasure protocol
that minimizes the dissipated heat. Our work therefore shows that majority-
logic decoding can lift the precision-speed-efficiency trade-off in
information erasure processes.

##  I Introduction

For modern society that is characterized by ubiquitous digitalization,
information storage and processing are of utmost importance. Since it becomes
increasingly challenging to keep the capacity of the ever-smaller storage
media constant, the speed and efficiency of information processing and thus
the underlying thermodynamics of the storage medium are becoming more crucial.
Moreover, the exceedingly large thermodynamic costs of performing computation
poses a tremendous technological challenge. Researchers have addressed these
technological challenges by studying the precision-speed-efficiency trade-off
inherent to bit erasure—arguably, the simplest type of computation—using
recent developments from nonequilibrium statistical mechanics.

The relationship between thermodynamics and information theory traces back to
Maxwell who proposed the notorious Maxwell demon Leff and Rex (1990). Later,
Szilárd derived a relationship between the information consumed by the demon
and the extractable work from a thermal reservoir Szilard (1929). Shannon
established the foundations of information theory by his seminal works Shannon
(1948a, b), in which information is interpreted as the uncertainty of an
outcome of an event and the so-called Shannon entropy is introduced to
quantify the amount of information. Based on these works, a milestone in
understanding the connection between thermodynamics and information processing
was achieved by the so-called Landauer’s principle which was established by
the work from Landauer Landauer (1961), Bennett Bennett (1973, 1982) and
Penrose Penrose (2005). Landauer’s principle provides an explicit lower bound
for the heat dissipated by the system when one bit of information is erased.
The validity of this lower bound has been verified experimentally Berut  _et
al._ (2012); Bérut  _et al._ (2015); Jun  _et al._ (2014) and reproduced in
more general theoretical works such as Piechocinska (2000); Shizume (1995);
Esposito and Van den Broeck (2011). The lower bound provided by Landauer’s
principle is only achieved in quasi-static processes, i.e., for information
erasing processes of infinite duration.

For practical purposes one is naturally interested in fast-erasure processes,
hence limiting the applicability of Landauer’s principle and prompting
multiple works on finite-time information erasure and other elementary logical
operations Owen  _et al._ (2017); Wolpert  _et al._ (2017); Kolchinsky and
Wolpert (2018); Wolpert and Kolchinsky (2018); Sagawa (2019); Parrondo  _et
al._ (2015). A novel theory referred to as stochastic thermodynamics Seifert
(2012); Van den Broeck and Esposito (2015), allows to systematically address
the thermodynamic properties in out-of-equilibrium systems. Owing to the
development of stochastic thermodynamics and to the aforementioned work of
Shannon, novel methods to study finite-time erasure processes became available
and prompted several works, e.g., Esposito  _et al._ (2010); Diana  _et al._
(2013); Zulkowski and DeWeese (2014); Gavrilov  _et al._ (2017). This
fundamental understanding of the thermodynamic costs in finite-time bit
erasure may provide new computing strategies for the information and
technology industry.

As a universal result, it was, for instance, proven that in the low-
dissipation limit the optimal, i.e., the least work-intense, transformation
protocol between two sets of probabilities will lead to an irreversible
entropy production, i.e., dissipation, that is inversely proportional to the
transformation duration Aurell  _et al._ (2012); Schmiedl and Seifert (2008);
Zulkowski and DeWeese (2015). This result has also been applied beyond the
theory of information processing, e.g., for the study of efficiencies of
finite-time heat engines Esposito  _et al._ (2010a); Schmiedl and Seifert
(2007); Sheng and Tu (2014, 2015a, 2015b); Esposito  _et al._ (2010b).
However, away from the low-dissipation regime, the relationship between
irreversible entropy production and operating duration is model-dependent.
While two-state systems with Fermi transition rates as a representation of a
bit have been investigated Esposito  _et al._ (2010); Diana  _et al._ (2013),
their applicability to real systems is limited. In general, the correspondence
between stored information and a two-state unit is not a one-to-one, but one
bit of information is normally stored in a macroscopic bit that, in turn, is
composed of an array of non-interacting microscopic binary units. As an
example, in magnetic storage media one bit of information is stored in a
macroscopic bit that contains tens to hundreds of microscopic magnetic grains.
How the collective physical information stored in the array of bistable
microscopic units is translated into macroscopic logical information stored by
the bit is determined by specific decoding rules. The decoding procedure can
formally be understood as a coarse-graining of microscopic physical
information. Employing decoding procedures, the safety of the information
processing is enhanced since the signal-to-noise ratio is proportional to
$\sqrt{N}$, where $N$ is the number of microscopic grains contained in a
macroscopic bit Richter (1999).

It is the aim of this paper to extend the existing studies of the
thermodynamics of finite-time information erasure processes in single
microscopic two-state systems by considering non-interacting ensembles of them
under a specific decoding procedure. To the best of our knowledge, there is no
systematic discussions on the energetics of information erasure processes
under coarse-grained decoding in the literature. Hence, in this work we
explore the costs and benefits of employing the majority-logic decoding, which
is the simplest and most used coarse-graining method in information processing
Dholakia (1991). We show that for reversible erasure protocols, information
erasure in single units is more efficient than symmetric majority-logic
decoding. Conversely, we find that for finite-time erasure protocols the
majority-logic decoding can accelerate the process of erasing a fixed amount
of information or compress the minimal erasure error of a fixed-time erasure
process. While these benefits in terms of speed and precision for most erasure
processes come at the expense of a lower erasure efficiency, we show that,
remarkably, the majority-logic decoding will however also be more efficient
than a single-unit process when the erasing is fast, or the erasure error is
small. When imposing the optimal erasure protocol that minimizes the
dissipated heat, we find that for the two unit models investigated in this
work (Fermi- and Arrhenius-rates units), these advantages are preserved.
Hence, we conclude that the majority-logic decoding lifts the trade-off
between erasure speed, precision, and efficiency when compared to a single
unit.

The plan of the paper is as follows: In Section II we introduce the
microscopic storage unit and define two types of macroscopic bits, the single-
unit bit and the majority-logic decoding bit. The latter is mathematically
formulated. Section III is devoted to studying the energetics of information
erasure processes with majority-logic decoding for different protocols: First,
a constant state-energy protocol of the microscopic unit is considered. Next,
two microscopic unit models, the Fermi-rates and the Arrhenius-rates model,
are introduced and studied if the optimal protocol for the erasure process is
imposed on the system. In the high- and low-dissipation limit analytic results
are derived. Finally, we conclude with a summary and an outlook to potential
future projects in Section IV.

##  II Majority-Logic Decoding

###  II.1 Two Setups of Macroscopic Bits

As an elementary storage unit, we consider a microscopic binary unit with
states $0$ and $1$ and denote by $p$ the probability to observe it in state
$1$. Based on these microscopic binary units we can construct macroscopic
(logical) bits in two different ways as illustrated in Figure 1. First, we
consider a single-unit bit (SUB) in Figure 1a that consists of only one
microscopic unit and is in contact with a heat bath at inverse temperature
$\beta$. The probability of finding the SUB in state $1$ is represented by
$P$, and is, of course, equal to probability $p$ for the microscopic unit to
be in state $1$. Alternatively, a majority-logic decoding bit (MLB) can be
thought of as an array of $N$ identical and non-interacting microscopic units
that are subjected to the same experimental protocol and connected to a heat
bath at inverse temperature $\beta$ as sketched in Figure 1b. The probability
of the MLB to be in state $1$ is denoted by $P$ and is determined via
majority-logic decoding. This decoding scheme prescribes that the information
encoded in the MLB corresponds to the state that is occupied most at the level
of the microscopic units and is therefore a coarse-graining procedure that is
mathematically formulated in the following.

Figure 1: Schematics of a single-unit bit (SUB) in (a) and a $N$-majority-
logic decoding bit (MLB) in (b).

###  II.2 From Microscopic to Macroscopic Probability

We proceed by formulating the relation between the microscopic probability $p$
associated with each microscopic unit and the macroscopic probability $P$
specifying the occupation probability of the MLB. This relation can be
expressed as

$\displaystyle
P(p,N,\zeta)=\sum_{k=\zeta}^{N}{\binom{N}{k}}p^{k}\,(1-p)^{N-k},$ (1)

where $\zeta$ represents the threshold number of the detector of the same
microscopic state during the measurement process. In Appendix A the equality

$\displaystyle P(p,N,\zeta)=\mathcal{I}_{p}(\zeta,N+1-\zeta).$ (2)

is proven. Here, $\mathcal{I}$ refers to the regularized incomplete beta
function

$\displaystyle\mathcal{I}_{x}(a,b)=\frac{\int_{0}^{x}t^{a-1}(1-t)^{b-1}\mathrm{d}t}{\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t}.$
(3)

The bijectivity of the regularized incomplete beta function $\mathcal{I}$
allows furthermore to determine the microscopic probability $p$ given the
macroscopic probability $P$ of an MLB as follows

$\displaystyle p(P,N,\zeta)=\mathcal{I}_{P}^{-1}(\zeta,N+1-\zeta).$ (4)

The qualitative features of the majority-logic decoding are illustrated in
Figure 2 that compares the macroscopic probability $P(p,N,\zeta)$ of a SUB and
MLB for different array sizes $N$ and threshold values $\zeta$.

Figure 2: The curves of $P$ with respect to $p$ at different values of $N$ and
$\zeta$. The blue line refers to the SUB. While the green ($N=11,\zeta=6$) and
red curves ($N=101,\zeta=51$) refer to symmetric MLBs, the orange
($N=11,\zeta=8$) and the brown dotted curves ($N=100,\zeta=51$) refer to
asymmetric MLBs with even $N$.

First, we observe that the macroscopic probability $P$ is monotonically
increasing with $p$. Next, for any value of $N$ and $\zeta$ it holds for
$x=0,1$ that $P(x,N,\zeta)=x$ since the complete beta function becomes a unity
operator, $\mathcal{I}=\bm{I}$. Physically, this means that perfect
information erasure in an MLB ($P=0,1$) is realized by perfect erasure in each
microscopic unit the MLB consists of. For odd values of $N$ the symmetric
case, $\zeta=(N+1)/2$, corresponds to the _majority_ -logic decoding in the
strict sense, that is

$\displaystyle
P\left(p,N,\tfrac{N+1}{2}\right)=\mathcal{I}_{p}\left(\tfrac{N+1}{2},\tfrac{N+1}{2}\right).$
(5)

We note that for symmetric majority-logic decoding, $\zeta=(N+1)/2$, the curve
is symmetric with respect to $(1/2,1/2)\;\forall N$. This is readily derived
by noting that according to the binomial theorem, one has

$\displaystyle
P\big{(}\tfrac{1}{2},N,0\big{)}=\frac{1}{2^{N}}\sum_{k=0}^{N}{\binom{N}{k}}=\frac{1}{2^{N}}(1+1)^{N}=1.$
(6)

Thus, it immediately follows from the symmetry of the binomial probability
distribution (Equation (1)) that one has
$P\big{(}1/2,N,\zeta=(N+1)/2\big{)}=1/2$. It can furthermore be seen in Figure
2 that for large $N$ the curves converge to a step function centered at
$p=1/2$. For large $N$, Taylor-expanding Equation (1) for the symmetric case
around $p=1/2$ up to linear order in $|p-1/2|$ yields

$\displaystyle\left|\frac{1}{2}-P\right|=\sqrt{\frac{2N}{\pi}}\left|\frac{1}{2}-p\right|+\mathcal{O}\bigg{(}\left|\frac{1}{2}-p\right|^{3}\bigg{)}\,,$
(7)

where we denote by $\mathcal{O}(x^{2})$ all terms that are at the order
$x^{2}$ or higher. According to Equation (7), the slope at the symmetric point
$p$ is increasing with $\sqrt{N}$ in the symmetric decoding case. We therefore
arrive at the first important result that large macroscopic erasure in the MLB
can be achieved at the cost of small microscopic erasure in each microscopic
unit. The good agreement of the curves corresponding to $N=100$ and $N=101$
suggests that for $\zeta\approx(N+1)/2$ the symmetry decoding case is
approached if $N$ becomes large.

###  II.3 Majority-Logic Decoding as a Coarse-Graining Procedure

The equivalent probabilistic descriptions on the microscopic and macroscopic
level prompt the question of how to define the underlying physical processes
at these levels. In information erasure processes, important quantities are
the heat that is generated during the operation, the change in Shannon entropy
that measures the amount of information erased by that process as well as the
erasure efficiency Esposito and Van den Broeck (2011); Diana  _et al._ (2013).

On the level of a single microscopic unit, the heat dissipated during an
erasure process from $p_{i}$ to $p_{f}$ is denoted by $q$ and defined as a
negative quantity $q<0$. With this convention, the first law of thermodynamics
in differential form reads

$\displaystyle\mathrm{d}_{t}\,e=\dot{q}+\dot{w},$ (8)

where $\dot{w}$ denotes the work current and $\dot{e}$ the rate of energy
change. The microscopic Shannon entropy

$\displaystyle s(p)=-p\ln{p}-(1-p)\ln{(1-p)},$ (9)

allows to quantify $\Delta s(p_{i},p_{f})\equiv s(p_{f})-s(p_{i})$ as the
amount of information that is erased during an erasure process from $p_{i}$ to
$p_{f}$. We set the Boltzmann constant $k_{b}\equiv 1$. The second law of
thermodynamics reads

$\displaystyle\mathrm{d}_{t}\,s=\beta\dot{q}+\dot{\sigma},$ (10)

where $\dot{\sigma}\geq 0$ refers to the irreversible entropy production rate
in the microscopic unit.

Throughout this work, we consider the case in which the initial state is the
maximum information state given by $p_{i}=1/2$ and $s(1/2)=\ln 2$. For this
case, the change in entropy is always negative $\Delta s(p_{i},p_{f})<0$ if
$p_{f}\neq 1/2$. Therefore, a suitable definition of the microscopic erasure
efficiency for this process reads

$\displaystyle\eta_{s}(p_{i},p_{f})=\frac{\Delta s(p_{i},p_{f})}{\beta\,q}.$
(11)

The heat $q$ generated by the microscopic unit is naturally dependent on
specific models and operating protocols. The optimal protocols that minimize
the dissipated heat and thus maximize the erasure efficiency are investigated
in Section III.

On the level of macroscopic bits, the heat dissipation refers to the cumulated
heat generated by all the microscopic units the bit consists of. Thus, the
microscopic definitions from above are, of course, also physically significant
for the erasure process at the level of the SUB that is $s(P)=s(p)$ and
$\eta_{s}(1/2,p_{f})=\eta_{s}(1/2,P_{f})$. Conversely, to perform a
macroscopic erasure in the MLB from $P_{i}=1/2$ to $P_{f}$, an amount of
information specified by the majority-logic decoding needs to be erased in
each microscopic unit contained in the MLB. Here, $P_{f}$ is the probability
that the final state of the MLB after the erasure is logically decoded as
state 1. According to Equation (4), this amounts to change in each microscopic
unit from the initial state $p_{i}=\mathcal{I}^{-1}_{1/2}(\zeta,N+1-\zeta)$ to
the final state $p_{f}=\mathcal{I}^{-1}_{P_{f}}(\zeta,N+1-\zeta)$. The heat
dissipated by the MLB, $Q$, is thus determined as follows

$\displaystyle Q(1/2\to
P_{f})=N\,q\left(\mathcal{I}^{-1}_{1/2}(\zeta,N+1-\zeta)\to\mathcal{I}^{-1}_{P_{f}}(\zeta,N+1-\zeta)\right).$
(12)

The first law of thermodynamics at the level of the MLB reads

$\displaystyle\mathrm{d}_{t}E=\dot{Q}+\dot{W},$ (13)

where the thermodynamic quantities of the MLB are naturally given by the sum
of the microscopic ones, i.e., one has $E=Ne,Q=Nq,W=Nw$. It furthermore holds
that $\Sigma=N\sigma$ since the irreversible entropy production of the MLB
must be equal to the sum of the irreversible entropy production of the
microscopic units contained in the MLB. Next, we define the entropy
quantifying the information stored in the MLB as for the microscopic unit,
i.e.,

$\displaystyle S(P)=-P\ln{P}-(1-P)\ln{(1-P)}.$ (14)

From the definition for the entropy associated with an MLB made in Equation
(14) follows that $S\neq Ns$. This however implies that the entropy balance at
the level of the MLB is broken Herpich  _et al._ (2018); Esposito (2012);
Herpich and Esposito (2019)

$\displaystyle\mathrm{d}_{t}S\neq\beta\,\dot{Q}+\dot{\Sigma}.$ (15)

Hence the macroscopic Shannon entropy in Equation (14) should be thought of as
logical but not strictly physical information. Finally, the macroscopic
efficiency associated with the erasure in the MLB is defined as

$\displaystyle\eta_{m}(1/2,P_{f})=\frac{\Delta S(1/2,P_{f})}{\beta\,Q},$ (16)

with the change in macroscopic Shannon entropy $\Delta S(1/2,P_{f})\equiv
S(P_{f})-S(1/2)=S(P_{f})-\ln 2$.

###  II.4 Reversible Erasure Protocols

In the limit of reversible erasure, the irreversible entropy production
vanishes, and one has for the heat in a SUB $\beta\,q=s(p_{f})-\ln 2$. Thus,
from Equation (11) follows that $\eta_{s}^{rev}=1$ for any erasure process in
this limit. Turning to the MLB, the dissipated heat during the erasure process
reads

$\displaystyle\beta\,Q=N\left[s\Big{(}\mathcal{I}^{-1}_{P_{f}}(\zeta,N+1-\zeta)\Big{)}-s\Big{(}\mathcal{I}^{-1}_{\tfrac{1}{2}}(\zeta,N+1-\zeta)\Big{)}\right].$
(17)

Figure 3 depicts in the reversible limit the heat dissipated during an erasure
process from $P_{i}=1/2$ to $P_{f}$ by a SUB and MLB [panel (a)] and the
associated efficiencies for symmetric decoding processes [panel (b)]. We
observe in panel (a) that the SUB always dissipates less heat than the
symmetric MLBs which generate more heat as $N$ increases. For asymmetric
majority-logic decoding, $\zeta\neq(N+1)/2$, the heat generated by a MLB is
reduced with respect to the one of a SUB and even takes positive values for
probabilities in the range
$[1/2,\mathcal{I}_{1-\mathcal{I}^{-1}_{P_{f}}(\zeta,N+1-\zeta)}(\zeta,N+1-\zeta)]$
or
$[\mathcal{I}_{1-\mathcal{I}^{-1}_{P_{f}}(\zeta,N+1-\zeta)}(\zeta,N+1-\zeta),1/2]$,
if the decoding is left-asymmetric or right-asymmetric, respectively. The
positivity of the heat means that one could even extract work during certain
erasure processes by employing asymmetric majority-logic decoding. This
property is similar to the division of logical entropy of the system entropy
and the physical entropy of each subspaces as discussed in Refs. Cover and
Thomas (1991); Sagawa (2014); Gavrilov and Bechhoefer (2016). However, here,
the analogue of physical entropy in each subspace varies according to
Equations (1) and (9) as the macroscopic probability $P$ changes.

Figure 3: (a) Heat dissipated by the macroscopic bits in the reversible limit.
The initial macroscopic state is $P_{i}=1/2$; (b) Efficiencies associated with
reversible erasure processes for the SUB and symmetric MLBs. The dotted lines
from top to bottom represent the values $\eta=2/\pi$, $\eta=1/3$, $\eta=1/11$
and $\eta=1/101$, respectively.

The erasure efficiency of MLBs under symmetric majority-logic decoding,
$\zeta=(N+1)/2$, for different array lengths $N$ is shown in Figure 3b.
Consistently to the observations made in panel (b), the SUB is always more
efficient than the MLB whose erasure efficiency decreases as $N$ increases.
The same applies to the other erasure branch given by $P_{f}>1/2$ that can be
seen as follows: According to Equations (5) and (9), the macroscopic Shannon
entropy of a symmetric MLB is symmetric with respect to $P=1/2$. Therefore,
the energetics of the erasure process from the initial state $P_{i}=1/2$ to
the final state $P_{f}$ is the same as that of the erasure process from
$P_{i}=1/2$ to the final state $1-P_{f}$. Hence, we will restrict to the
symmetric majority-logic decoding characterized by Equation (5) and to the
erasure branch, $P\leq 1/2$, in the following. To ease notation, we omit the
explicit notation $\zeta=(N+1)/2$ and $P_{i}=p_{i}=1/2$.

As already pointed out earlier, the macroscopic entropy associated with an MLB
can be thought of as a coarse-graining of the sum of the physical entropy of
each microscopic unit contained in the MLB. We write the difference between
the physical entropy of all microscopic units contained in the bit and the
logical (Shannon-like) entropy of the macroscopic bit as

$\displaystyle\mathbb{S}(P,N)=Ns\Big{(}\mathcal{I}^{-1}_{P}\Big{(}\tfrac{N+1}{2},\tfrac{N+1}{2}\Big{)}\Big{)}-S(P).$
(18)

In Appendix B, a proof for the monotonic behavior of $\mathbb{S}(P,N)$ with
respect to $P$ in the case of symmetric majority-logic decoding is provided.
In the range of $0\leq P\leq 1/2$, the entropy function $\mathbb{S}(P,N)$ is
monotonically increasing, else it is decreasing monotonically. Evidently,
$\mathbb{S}(P,N)$ takes its maximum value, $(N-1)\ln 2$ if $p=P=1/2$ and its
minimum value, $\mathbb{S}(P,N)=0$, if $p=P=0,1$. Hence, for symmetric
majority-logic decoding the logical information underestimates or is equal to
the physical one $\mathbb{S}(P,N)\geq 0\;\forall N$. As can be seen in Figure
3a, such an inequality does not hold for the more general asymmetric case
where $\zeta\neq(N+1)/2$. In fact, the inequality $\mathbb{S}(P,N)\geq 0$ is
physically equivalent to the well-known universal relationship between fully
microscopic and coarse-grained Shannon entropies Esposito (2012). It is
furthermore important to note that no general statement can be made about the
changes of entropy for a given erasure process, since, according to Equation
(5) the final distribution for the microscopic unit is, in general, different
from the one of the MLB.

From Equation (16) and the monotonicity of $\mathbb{S}(P,N)$ follows that the
reversible erasure efficiency of the symmetric MLB is bounded as follows

$\displaystyle 0<\eta_{m}^{rev}(1/2,P_{f})\leq 1,$ (19)

where the equality only holds when no information is erased, $\Delta
S(1/2,P_{f}=1/2)=0$. Hence we find that a SUB is always more efficient in
reversible information erasure than a symmetric MLB. This can be attributed to
the majority-logic decoding that neglects some microscopic degrees of freedom
and thus associates less logical entropy to the MLB than the cumulated
physical entropy of the microscopic units that constitute the MLB.

We now state two limiting results for the reversible erasure efficiency of an
MLB. First, in case of perfect erasure, $P_{f}=0$, we derive from Equations
(5) and (16) that the erasure efficiency simplifies to

$\displaystyle\eta_{m}^{rev}(1/2,0)=\frac{1}{N}.$ (20)

Secondly, if the amount of erased information is small, $P_{f}\approx
P_{i}=1/2$, and $N$ is large, the erasure efficiency

$\displaystyle\eta_{m}^{rev}(1/2,P_{f})=\frac{2}{\pi}+\mathcal{O}\bigg{(}\left|\frac{1}{2}-P_{f}\right|^{2}\bigg{)},$
(21)

becomes independent of the final macroscopic probability $P_{f}\approx 1/2$ as
can be seen in Figure 3b.

##  III Symmetric Majority-Logic Decoding under Finite-Time Erasure Protocol

Since we have captured the phenomenology of the reversible majority-logic
decoding in the previous section, we now proceed by studying more realistic,
finite-time information erasure processes. In dynamical processes the heat
generation depends on the erasure protocol and the specific model for the
microscopic units. In this section, we formulate finite-time erasure processes
for two commonly employed microscopic models: We consider a two-state system
with either Arrhenius rates or Fermi rates and denote in both cases by $p(t)$
the probability of the unit to be in state $1$ at time $t$.

###  III.1 Preliminaries

####  III.1.1 Master Equation

The transition rate from state $0$ to state $1$ and vice versa is referred to
as $\omega_{10}(t)$ and $\omega_{01}(t)$, respectively, which depends, in
general, on time via the erasure protocol. We assume that the process is
Markovian, such that the dynamics of $p(t)$ is ruled by a master equation:

$\displaystyle\dot{p}(t)=[1-p(t)]\,\omega_{10}(t)-p(t)\,\omega_{01}(t),$ (22)

with the transition rates satisfying the local detailed balance relation

$\displaystyle\frac{\omega_{10}(t)}{\omega_{01}(t)}=\mathrm{e}^{-\beta\,\Delta\epsilon(t)},$
(23)

where $\Delta\epsilon(t)$ is the energy gap from state $0$ to $1$ that is
modulated in time according to the specific erasure protocol. With Equation
(23) the master Equation (22) can be cast into the form

$\displaystyle\dot{p}(t)=\left[\mathrm{e}^{-\beta\,\Delta\epsilon(t)}-(1+\mathrm{e}^{-\beta\,\Delta\epsilon(t)})p(t)\right]\omega_{01}(t).$
(24)

####  III.1.2 Finite-Time Erasing

No protocol can achieve a perfect erasure corresponding to
$s(p_{f})=s(P_{f})=0$, since this requires an infinite amount of time Diana
_et al._ (2013). This can be seen as follows: For a given microscopic model,
there are several ways to decrease the microscopic probability $p(t)$ at
different speeds, which corresponds to a protocol that ensues different
amounts of heat and thus a different erasure efficiency. This, in turn,
implies that the time required to transform the initial probability
$p_{i}(0)=1/2$ into the final one $p_{f}(\tau)\equiv p_{f}$ is cannot be
smaller than a minimal time $\tau_{c}$

$\displaystyle\tau\geq\tau_{c}=-\log 2p_{f},\quad p_{f}<\frac{1}{2},$ (25)

As $p_{f}\to 0$, the minimal time diverges, $\tau_{c}\to\infty$, such that
perfect erasure of a finite initial amount of information can only be realized
by infinite-time protocols.

Equation (25) can be rearranged to obtain a lower bound for the final
probability obtained after an erasure with fixed duration $\tau$ as follows

$\displaystyle p_{f}\geq p_{c}=\frac{1}{2}\mathrm{e}^{-\tau_{c}},\quad
p_{f}<\frac{1}{2},$ (26)

Consequently, in finite-time information erasure processes, the Shannon
entropy of the final state $s(p_{f})$ can be seen as the erasure error.
According to Equation (26), the minimal erasure error of the process is
$s(p_{c})$ and the erasable information within $\tau_{c}$ is $\Delta
s(p_{f},1/2)=\log 2-s(p_{c})$. According to Equations (25) and (26), the
bounds $\tau_{c}$ and $p_{c}$ are rate- and thus model-independent.

These results hold at the level of a microscopic unit. We now proceed by
discussing the bounds on the level of macroscopic bits. First, since each SUB
consists of only one microscopic unit, Equations (25) and (26) are also
applicable to the macroscopic quantities, that is

$\displaystyle\tau\geq\tau_{c}^{s}=\tau_{c}=-\log 2P_{f},\quad P_{f}\geq
P_{c}^{s}=p_{c}=\frac{1}{2}\mathrm{e}^{-\tau},\quad P_{f}<\frac{1}{2}.$ (27)

To calculate the lower bound on the macroscopic erasure time
$\tau_{c}^{m}(1/2,P_{f},N)$ for a symmetric MLB, we recall that a macroscopic
erasure from $P_{i}=1/2$ to $P_{f}$ is achieved by the corresponding erasure
from $p_{i}=1/2$ to $p_{f}$ in each microscopic unit contained in the MLB.
Therefore, one has for the minimal erasure time in an MLB

$\displaystyle\tau_{c}^{m}(1/2,P_{f},N)=\tau_{c}(1/2,p_{f}),$ (28)

where $p_{f}$ and $P_{f}$ are related to each other via the symmetric
majority-logic decoding in Equation (5). Hence, to compute $\tau_{c}^{m}$, the
final microscopic probability $p_{f}$ needs to be determined via inversion of
Equation (5) and plugged into Equation (25). Next, using Equations (5) and
(26) , one straightforwardly obtains the minimal final probability
$P_{c}^{m}(\tau,N)$ after erasure time $\tau$. From Equation (5) follows the
inequalities

$\displaystyle\tau_{c}^{m}(P_{f},N)\leq\tau_{c}^{s}(P_{f}),\quad
P_{c}^{m}(\tau,N)\leq P^{s}_{c}(\tau),$ (29)

which suggest that majority-logic decoding can both accelerate the erasing
process and additionally reduce the minimal erasure error. Finally, for
finite-time erasure processes we define the erasure power for a SUB and MLB as
follows

$\displaystyle\mathcal{P}_{s}=\frac{\Delta
s(1/2,p_{f})}{\tau},\quad\mathcal{P}_{m}=\frac{\Delta S(1/2,P_{f})}{\tau},$
(30)

respectively.

####  III.1.3 Arrhenius-Rates Unit Model

The Arrhenius model consists of two potential well, which are regarded as
states 0 and 1, respectively, that are separated by a barrier. The energy gap
between state 0 and 1 is defined as $\Delta\epsilon$. To be consistent with
the assumption of Arrhenius transition rates, the energy barrier height of the
potential well associated with state 1 is assumed constant, $\epsilon$,
throughout the erasure process. This setup is different from the one used in
Ref. Jun  _et al._ (2014), where the two states are merged together and then
separated during the erasure. Here, the transition rates read

$\displaystyle\omega^{A}_{01}(t)=r_{0}\,\mathrm{e}^{-\beta\epsilon},\quad\mathrm{and}\quad\omega_{10}^{A}(t)=r_{0}\,\mathrm{e}^{-\beta[\epsilon+\Delta\epsilon(t)]},$
(31)

where $r_{0}$ is a constant setting the time scale of the process. Then, the
master Equation (24) can be written as follows

$\displaystyle\dot{p}(t)=\mathrm{e}^{-\beta\Delta\epsilon(t)}-\big{[}1+\mathrm{e}^{-\beta\Delta\epsilon(t)}\big{]}\,p(t),$
(32)

where the constant $r_{0}\,\mathrm{e}^{-\beta\epsilon}$ is absorbed into the
time scale.

####  III.1.4 Fermi-Rates Unit Model

The Fermi-rates unit model can be experimentally realized via a single quantum
dot with a single energy level $E$ that is in contact with a moving metallic
lead corresponding to a time-dependent chemical potential $\mu(t)$ and a heat
bath at inverse temperature $\beta$ Esposito and Van den Broeck (2011). If the
dot is filled by an electron we consider the unit to be in state $1$, else
$0$. The transition rate for an electron leaving or entering the dot reads

$\displaystyle\omega_{01}^{F}(t)=\frac{r_{0}}{\mathrm{e}^{-\beta\,\Delta\epsilon(t)}+1},\quad\omega_{10}^{F}(t)=\frac{r_{0}}{\mathrm{e}^{\beta\,\Delta\epsilon(t)}+1},$
(33)

respectively, where $\Delta\epsilon(t)\equiv E-\mu(t)$ represents the energy
barrier to enter the dot and $r_{0}$ constant setting the time scale of the
process. Then, the master Equation (24) can be written as follows

$\displaystyle\dot{p}(t)=-p(t)+\frac{1}{\mathrm{e}^{\beta\,\Delta\epsilon(t)}+1},$
(34)

where the constant $r_{0}$ is absorbed into the time scale. Since the second
term in Equation (34) is bounded between 0 and 1, the fastest way to decrease
the microscopic probability $p(t)$ is realized by $\dot{p}(t)=-p(t)$, which
corresponds to a protocol that ensues a divergent heat generation and thus a
vanishing erasure efficiency.

###  III.2 Constant State-Energy Protocol

####  III.2.1 Variable Erasure Duration

With the tools to address finite-time information processing at hand, we want
to start with the simplest erasure protocol given by an instantaneous
switching of the energy gap to the same value $\Delta\epsilon$ at time $t=0$.
For generic transition rates with constant energy gaps,
$\omega_{01}(\Delta\epsilon)$, the master equation of a microscopic unit from
Equation (24) is solved by

$\displaystyle
p(t)=\frac{1}{2}\tanh\left(\frac{\beta\Delta\epsilon}{2}\right)\mathrm{e}^{-\left(1+\mathrm{e}^{-\beta\Delta\epsilon}\right)\omega_{01}(\Delta\epsilon)\,t}+\frac{1}{1+\mathrm{e}^{\beta\Delta\epsilon}},$
(35)

where we used the initial condition $p(0)=1/2$. In the infinite-time limit the
probability converges to the lower bound

$\displaystyle p(\infty)=\frac{1}{1+\mathrm{e}^{\beta\Delta\epsilon}}.$ (36)

We emphasize that for fixed $\Delta\epsilon$ the definitions (25) and (26) no
longer apply. However, Equation (36) represents also a bound on the minimal
erasure error and thus plays a similar role as $p_{c}$. From Equation (35) we
obtain the following expression for the erasure duration

$\displaystyle\tau=-\frac{1}{(1+\mathrm{e}^{-\beta\Delta\epsilon})\,\omega_{01}(\Delta\epsilon)}\ln{\left(2\frac{(1+\mathrm{e}^{\beta\Delta\epsilon})p_{f}-1}{\mathrm{e}^{\beta\Delta\epsilon}-1}\right)}.$
(37)

The heat dissipated by the microscopic unit reads

$\displaystyle
q=\int_{0}^{\tau}\Delta\epsilon\dot{p}(t)\,\mathrm{d}t=\Delta\epsilon\int_{p_{i}}^{p_{f}}\\!\mathrm{d}p=\bigg{(}p_{f}-\frac{1}{2}\bigg{)}\Delta\epsilon,$
(38)

which, with Equation (12), results in the total heat generated by the SUB and
MLB

$\displaystyle q$
$\displaystyle=\bigg{(}p_{f}-\frac{1}{2}\bigg{)}\Delta\epsilon,\quad
Q=N\left[\mathcal{I}^{-1}_{P_{f}}\bigg{(}\frac{N+1}{2},\frac{N+1}{2}\bigg{)}-\frac{1}{2}\right]\Delta\epsilon,$
(39)

respectively. With the definitions in Equations (11), (16) and (30) one has
for the macroscopic erasure power

$\displaystyle\mathcal{P}^{s}\bigg{(}\frac{1}{2},p_{f}\bigg{)}=\frac{\Delta
s(1/2,p_{f})}{\tau},\quad\mathcal{P}^{m}\bigg{(}\frac{1}{2},P_{f}\bigg{)}=\frac{\Delta
S(1/2,P_{f})}{\tau},$ (40)

and the macroscopic efficiencies

$\displaystyle\eta_{s}\bigg{(}\frac{1}{2},p_{f}\bigg{)}=\frac{\Delta
s(1/2,p_{f})}{\beta\,q},\quad\eta_{m}\bigg{(}\frac{1}{2},P_{f}\bigg{)}=\frac{\Delta
S(1/2,P_{f})}{\beta\,Q}.$ (41)

The results of the finite-time erasure process for a fixed energy gap are
depicted in Figure 4.

As can be seen in panel (a), to perform an erasure with the same erasure error
the SUB dissipates less heat and thus has a higher efficiency than the
symmetric MLBs, for which the heat production increases and the efficiency
decreases with growing $N$. Hence the SUB is more efficient than the symmetric
majority-logic decoding, as already observed in the reversible case, cf.
Figure 3. However, Figure 4b shows that the erasure duration is reduced, and
the erasure power thus enhanced by employing the majority-logic decoding for
large ensembles of microscopic units. The minimal erasure error characterized
by Equation (36) is indicated by the dotted vertical lines that correspond to
the minimal final probability $P_{c}^{m}(N)$. Therefore, as already derived in
Equation (29), majority-logic decoding reduces the minimal erasure error that
goes to zero as $N$ becomes large corresponding to perfect erasure.

---
(a)
(b)
Figure 4: (a) Comparison of both the modulus of heat dissipation, $-\beta Q$,
and the associated erasure efficiency between a SUB and a symmetric MLB for
different $N$. (b) Comparison of both the erasure time and the modulus of the
power, $|\mathcal{P}|$, between a SUB and a symmetric MLB for different $N$.
The data was generated by using the generic solution of the master Equation
(35) and setting $\omega_{01},\beta\,\epsilon\equiv 1$. The dotted vertical
lines correspond to the minimal erasure error $P_{c}^{s}$ and $P_{c}^{m}(N)$
for the systems under consideration.

####  III.2.2 Fixed Erasure Duration

We now assume that the explicit form of the transition rates of the
microscopic unit model is known and compare the performance of the two types
of macroscopic bits under the protocol with fixed erasing time and
instantaneous switching of the energy gap. For a specific erasure process, the
erasure time and thus the erasure power are equal for the two macroscopic
bits, hence we restrict the discussion to the macroscopic erasure
efficiencies. The microscopic dynamics of a Fermi-rates unit reads

$\displaystyle
p(t)=\frac{1}{2}\tanh\left(\frac{\beta\Delta\epsilon}{2}\right)\mathrm{e}^{-t}+\frac{1}{1+\mathrm{e}^{\beta\Delta\epsilon}}.$
(42)

Plugging the erasure time $\tau$ into Equation (42), yields the final
microscopic probability and the erasure error. The calculation of the
generated heat, erasure power and efficiency both on the microscopic and
macroscopic level is analogous to the one in the last section. As the specific
dynamics of the unit is known and $\tau$ is fixed, the definition of minimal
final probability in Equation (26) is valid under this protocol.

Figure 5 shows that the symmetric majority-logic decoding has additional
advantages for finite-time erasing: As already observed earlier, the minimal
erasure error of the symmetric MLB is smaller than that of the SUB and
approaches $0$ with increasing $N$ as illustrated by the vertical dotted lines
corresponding to the minimal final probability $P_{c}^{m}(N)$ after the
erasure time $\tau$. More importantly, the symmetric MLBs are more efficient
in the region of small-erasure error region, $P_{f}\approx P_{c}^{s}$, as
opposed to the region of large erasure error where the SUB is more efficient.
We find that for Arrhenius rates the results are qualitatively similar and
thus omitted.

Figure 5: Comparison between the erasure efficiency of a SUB and symmetric
MLBs for different $N$ and instantaneously switched $\Delta\epsilon$ using the
Fermi-rates unit model and fixed erasure time $\tau=2$. The vertical dotted
lines correspond to the minimal final macroscopic probabilities $P_{c}^{s}$
and $P_{c}^{m}(N)$, where for $N=11,101$ the probabilities are too close to
zero to be distinguished.

To sum up, Figures 4 and 5 illustrate the important result that the precision-
speed-efficiency trade-off in finite-time information erasure processes is
lifted by symmetric majority-logic decoding.

###  III.3 Optimal Erasure Protocol

In view of applications, the least work-intense erasure processes are of
particular interest. The general method to determine the optimal erasure
protocol that minimizes the generated heat in a Fermi-rates unit model has
been established in Ref. Esposito  _et al._ (2010). The detailed derivation of
the heat-minimizing protocol for Arrhenius rates is deferred to Appendix C.
Given the optimal protocol, both the microscopic and macroscopic heat
dissipation, erasure power and efficiency follow readily from Equations (38),
(40) and (41).

The erasure efficiency of a SUB and a symmetric MLBs for different $N$ are
compared in Figure 6. It is important to note that the advantages in terms of
information erasure inherent to symmetric majority-logic decoding discussed in
the previous section are preserved and enhanced by the optimal erasure
protocol: First, the minimal erasure error is strongly reduced for a symmetric
MLB at the expense of an erasure efficiency that decreases with increasing $N$
in the regime of small erasure, $P_{f}\gg P_{c}^{s}$. Conversely, for small-
erasure error, $P_{f}\approx P_{c}^{s}$, this relation between the macroscopic
efficiencies is inverted. The inset in panel (a) that depicts the relative
erasure efficiency between a SUB and a symmetric MLB,
$(\eta_{s}-\eta_{m})/\eta_{s}$, reveals that the range of small-erasure error
probabilities over which this holds true is increased for the optimal protocol
compared to the fixed energy protocol. Comparing furthermore panels (a) and
(b) also shows that this range of small erasure-error probabilities is
increasing with decreasing erasure time $\tau$ and increasing $N$.

Figure 6: Comparison between the erasure efficiency of a SUB and of a
symmetric MLB based on the Arrhenius model for different $N$ under the optimal
erasure protocol with the erasure time $\tau=2$ [panel (a)] and $\tau=1$
[panel (b)]. The vertical dotted lines correspond to the minimal final
macroscopic $P_{c}^{s}$ and probabilities $P_{c}^{m}(N)$, where for $N=11,101$
the probabilities are too close to zero to be distinguished. The inset in
panel (a) shows the erasure efficiency of a SUB compared to a symmetric MLB
one, $(\eta_{s}-\eta_{m})/\eta_{s}$ for $N=101$ using the optimal and the
fixed energy protocol.

In the limit of low and high dissipation, analytical results can be obtained
for the optimal protocol, where we will focus on the Arrhenius-rates unit
model. The high-dissipation limit corresponds to an erasure duration $\tau$
that approaches the minimal erasure time $\tau_{c}$. Thus, the parameter $K$
in Equation (69), which represents the degree of irreversibility, is diverging
in this limit. Using Equations (71) and (72), the parameter $K$ can be
expressed as

$\displaystyle
K^{high}(p_{f},\tau)=\frac{1}{2}\frac{1-2p_{f}}{\tau+\ln{(2p_{f})}}.$ (43)

Plugging Equation (43) into Equation (73), one has for the heat dissipation

$\displaystyle\beta\,q^{high}(p_{f},\tau)=\left(\frac{1}{2}-p_{f}\right)\ln\left[\frac{1}{2}\frac{1-2p_{f}}{\tau+\ln{(2p^{f})}}\right].$
(44)

We verify that $K,q\to\infty$ as $\tau\to\tau_{c}=-\ln(2p_{f})$ and that
Equation (44) also represents the solution for a Fermi-rates unit. In the low-
dissipation limit, the erasure duration $\tau$ is large compared to
$\tau_{c}$, hence the parameter $K$ is small. Using Equations (71) and (72),
the parameter $K$ can be expressed as

$\displaystyle
K^{low}(p_{f},\tau)=8\left[\frac{1-\sqrt{2p_{f}}}{2\tau+\ln(2p_{f})}\right]^{2}.$
(45)

Plugging Equation (45) into Equation (73), one has for the heat dissipation

$\displaystyle\beta\,q^{low}(p_{f},\tau)=\frac{2\left(1-\sqrt{2p_{f}}\right)^{2}}{\tau}-\Delta
s(1/2,p_{f}).$ (46)

Significantly, the first term in Equation (46), which could be interpreted as
the irreversible dissipation, is consistent with the low-dissipation
assumption made in Refs. Aurell  _et al._ (2012); Schmiedl and Seifert (2008);
Esposito  _et al._ (2010a). Here, the expression for the heat in Equation (46)
differs from that of a Fermi-rates unit at low dissipation. The heat
dissipated by a SUB in the high ($q^{high}$) and low-dissipation limit
($q^{low}$) are the same as those of the microscopic unit and therefore given
by Equations (44) and (46), respectively. The heat dissipated by a symmetric
MLB in the high ($Q^{high}$) and low-dissipation limit ($Q^{low}$) are readily
derived using Equation (5) with results from Equations (44) and (46). For
small-erasure ($p_{f}\to 1/2$), the expression for the heat in the low-
dissipation limit in Equation (46) simplifies to

$\displaystyle q^{low}\approx
2\left(1+\frac{1}{\tau}\right)\left(\frac{1}{2}-p_{f}\right)^{2}.$ (47)

Using the approximation for the majority-logic decoding in the limit of small-
erasure and large-$N$ in Equation (7), the ratio $\kappa$ between the erasure
efficiency of a symmetric MLB and a SUB

$\displaystyle\kappa\equiv\frac{\eta_{m}(P_{f},N,\tau)}{\eta_{s}(p_{f},\tau)}\approx\frac{2}{\pi},$
(48)

is independent of the erasure duration $\tau$ and final erasure error.

Figure 7a compares the dissipated heat of a SUB and a symmetric MLB for
different erasure durations $\log\tau$ using the optimal protocol applied to a
given erasure process from $P_{i}=1/2$ to $P_{f}$.

Figure 7: (a) Comparison between the dissipated heat of a SUB and a symmetric
MLB ($N=101$) based on Arrhenius-rates units as a function of erasure duration
$\log\tau$ for the optimal erasure protocol with $P_{f}=0.1$. In addition to
the full numerical solution of Equation (73), the analytic low- and high-
dissipation solutions in Equations (44) and (46) are displayed. The vertical
dotted line corresponds to the logarithm of the critical time
$\log\tau_{c}^{s}(0.1)$ of a single microscopic unit for this specific erasure
process. (b) Ratio between the erasure efficiencies of a symmetric MLB
($N=101$) and a SUB as a function of the final macroscopic probability $P_{f}$
for the optimal protocol with the erasure times $\tau=3,5$. The red dotted
line corresponds to the small-erasure and large-$N$ approximation given by
Equation (48) and the vertical black dotted lines represent the minimal
erasure error $P_{c}^{s}(\tau)$.

Additionally, the approximate solutions for the low- and high-dissipation
limit in Equations (44) and (46) are overlaid. As can be seen, the symmetric
MLB under the optimal protocol is still more efficient in the fast-erasure
region, where the erasing duration $\tau$ approaches the minimal erasure time
$\tau_{c}^{s}(p_{f})$ of a SUB. Except for extremely fast-erasure processes,
the full numerical solution of the heat dissipated by the symmetric MLB is in
excellent agreement with the low-dissipation approximation in Equation (47).
This suggests that the calculation of the dissipated heat of a symmetric MLB
built upon microscopic Arrhenius-rate units under the optimal erasure protocol
can be simplified by using the more convenient Equation (47) instead of the
numerically more involving procedure elaborated in Appendix C.

In Figure 7b the ratio between the efficiency of a symmetric MLB and a SUB are
compared for the same erasure process. In agreement with the observation made
in panel (a) that the heat dissipation in the fast-erasure region is
significantly reduced for a symmetric MLB, we note in panel (b) that the
erasure efficiency of a symmetric MLB is considerably higher than that of a
SUB ($\kappa>1$) in this region. It is interesting to notice that even though
the approximate expression for the efficiency ratio in Equation (48) is
theoretically only valid in asymptotic limit $P_{f}\rightarrow 1/2$, the full
numerical solutions are in good agreement with the approximate solution
($2/\pi$) for a large range of values for $P_{f}$.

We therefore conclude that with the aid of symmetric majority-logic decoding,
the energetically optimized erasure processes can be accelerated, be performed
more precise at a lower cost. Hence, for large-$N$ symmetric MLBs the speed-
precision-efficiency trade-off is significantly lifted. We remark that the
optimization procedure (Appendix C) of the Fermi-rates unit model was studied
in detail in Esposito  _et al._ (2010). The results of the performance of a
symmetric MLB built upon Fermi-rates units are qualitatively similar to those
for the Arrhenius-rates unit model discussed above and thus omitted. It should
however be emphasized that the advantages of employing the symmetric majority-
logic decoding for information processing are preserved under a change between
these two different microscopic models.

##  IV Conclusions

In this work, we studied the performance of majority-logic decoding in
reversible and finite-time information erasure processes. To this end, we
introduced two macroscopic bits, the single-unit and the majority-logic bit,
that contain one and $N$ microscopic binary units, respectively. The physical
information stored inside the $N$ microscopic units is translated into logical
information stored in the macroscopic bit via the majority-logic decoding
scheme that can be mathematically formulated via an incomplete regularized
beta function.

We found that for reversible erasure protocols the SUB is always more
efficient in erasing a given amount of information than the symmetric
majority-logic bit. Conversely, for finite-time information erasure processes
that are naturally characterized by the constraints of finite erasure time and
erasure error, the majority-logic bit exhibits multiple advantages compared to
a single unit: Both the erasure time can be reduced, and the erasure error can
be compressed, hence accelerating and refining the erasure. Remarkably, in the
region of small erasure or fast erasure, the majority-logic unit also exhibits
a higher efficiency. These advantages are preserved and enhanced as one
changes from a simple (constant energy barrier) to the optimal protocol
(minimal heat dissipation). Hence, our results suggest that majority-logic
decoding considerably lifts the speed-precision-efficiency trade-off in
finite-time information erasure processes. In view of applications, our work
reveals a promising avenue to accelerate the information processing rather
than focusing on the storage media which may become a severe limitation in the
future semiconductor industry. If one is restricted to a few SUBs per
majority-logic bit, it would furthermore be interesting to consider a
hierarchical system of majority-logic bits and to study if the thermodynamic
benefits of majority-logic decoding can be further enhanced by such an
iterative approach.

Finally, there is an important remark to be made, here. As discussed at the
end of Section II.3 the majority-logic decoding can be thought of as a coarse-
graining procedure. Consequently, the logical information stored in the
macroscopic bit is not equal to the sum of microscopic physical information
stored in the units contained in the bit. However, these hidden costs were not
accounted for in the definitions of the macroscopic erasure power and
macroscopic erasure efficiency. A more rigorous approach would possibly be to
implement a physical mechanism that represents the majority-logic decoding,
e.g., an interacting system of microscopic units that exhibits a phase
transition. For instance, a two-dimensional Ising model or a majority-vote
model of the kind found in Ref. Crochik and Tomé (2005) could be used in the
context of information erasure. The need to quantify the costs incurred by
majority-logic decoding to validate the latter as a promising strategy to
enhance information processing might stimulate future works.

###### Acknowledgements.

We thank Christian Van den Broeck for suggesting investigating the majority-
logic decoding and Zhanchun Tu for fruitful discussions and instructive
suggestions. S.S. is grateful for the financial support of the China
Postdoctoral Science Foundation (Grants NO. 2018M642124), the National Nature
Science Foundation of China (Grants No. 11675017) and the Fundamental Research
Funds for the Central Universities (Grants No. 2017EYT24) and China
Scholarship Council. T. H., G. D. and M. E. acknowledge funding by the
National Research Fund, Luxembourg, in the frame of the AFR PhD Grant 2016,
No. 11271777, in the frame of Project No. FNR/A11/02 and by the European
Research Council project NanoThermo (ERC-2015-CoG Agreement No. 681456).

##  Appendix A Equivalence of the Binomial Cumulative Distribution Function
and $\mathcal{I}$

We start from the definition of the regularized incomplete beta function in
(3) and solve the integral in the denominator via partial integration yielding

$\displaystyle\int_{0}^{1}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{(a-1)!(b-1)!}{(a+b-1)!}.$
(49)

The integral in the numerator is determined via consecutive application of
partial integration

$\displaystyle\int_{0}^{x}t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{1}{a}x^{a}(1-x)^{b-1}+\frac{b-1}{a(a+1)}x^{a+1}(1-x)^{b-2}+\ldots+\frac{(b-1)(b-2)\cdot\ldots\cdot
2\cdot 1}{a(a+1)\cdot\ldots\cdot(a+b-1)}x^{a+b-1}.$ (50)

Collecting results, one finds that

$\displaystyle\mathcal{I}_{x}(a,b)=\sum\limits_{k=a}^{a+b-1}{a+b-1\choose
a}x^{a}(1-x)^{b-1}.$ (51)

If we set $a=\zeta,b=N-\zeta+1,x=p$, we immediately arrive at

$\displaystyle P(p,N,\zeta)=\mathcal{I}_{p}(\zeta,N+1-\zeta),$ (52)

which proves Equation (2).

##  Appendix B The Monotonicity of the Entropy Function $\mathbb{S}(P,N)$

The first-order derivative of the function $\mathbb{S}(p,N)$ introduced in
Equation (18) with respect to $p$ can be expressed as

$\displaystyle\frac{\partial\mathbb{S}(p,N)}{\partial
p}=N\,\ln{\frac{1-p}{p}}-\frac{(1-p)^{\tfrac{N-1}{2}}p^{\tfrac{N-1}{2}}}{\mathcal{B}\left(\tfrac{N+1}{2},\tfrac{N+1}{2}\right)}\ln{\frac{1-f(p,N)}{f(p,N)}},$
(53)

where we write the regularized incomplete beta function with symmetric
arguments as
$f(p,N)\equiv\mathcal{I}_{p}\Big{(}\tfrac{N+1}{2},\tfrac{N+1}{2}\Big{)}$ and
introduce the beta function
$\mathcal{B}\left(a,b\right)\equiv\int_{0}^{1}t^{a-1}(1-t)^{b-1}\,\mathrm{d}t$.
We find for the second-order derivative

$\frac{\partial^{2}\mathbb{S}(p,N)}{\partial
p^{2}}=\frac{(N-1)(1-p)^{\frac{N-3}{2}}p^{\frac{N-3}{2}}(2p-1)\ln{\frac{1-f(p,N)}{f(p,N)}}}{2\mathcal{B}(\frac{N+1}{2},\frac{N+1}{2})}-\frac{N}{p(1-p)}\bigg{[}1-\begin{pmatrix}N\\\
\tfrac{N-1}{2}\end{pmatrix}^{2}\frac{N(1-p)^{N}p^{N}}{(1-f)f}\bigg{]}.$ (54)

We easily verify that the first term on the right-hand side of Equation (54)
is non-positive for any $p$.

To determine the sign of the second term on the right-hand side of Equation
(54), we first recast the term $(1-f)f$ into

$\displaystyle[1-f(p,N)]f(p,N)$
$\displaystyle=\sum_{i=\frac{N+1}{2}}^{N}\sum_{j=0}^{\frac{N-1}{2}}{\binom{N}{i}}{\binom{N}{j}}p^{i+j}(1-p)^{2N-(i+j)}$
(55)
$\displaystyle=\sum_{m=\frac{N+1}{2}}^{N-1}\sum_{i=\frac{N+1}{2}}^{m}{\binom{N}{i}}{\binom{N}{m-i}}[p^{m}(1-p)^{2N-m}+p^{2N-m}(1-p)^{m}]\;+$
(56)
$\displaystyle+\sum_{i=\frac{N+1}{2}}^{N}{\binom{N}{i}}{\binom{N}{N-i}}p^{N}(1-p)^{N},$
(57)

where we have introduced the index $m=i+j$. Using the inequality

$\displaystyle p^{s}(1-p)^{4n+2-s}+p^{4n+2-s}(1-p)^{s}\geq
2p^{2n+1}(1-p)^{2n+1},$ (58)

Equation (57) can be transformed into the inequality

$\displaystyle\left[1-f(p,N)\right]f(p,N)\geq\left[2\sum_{m=\frac{N+1}{2}}^{N-1}\sum_{i=\frac{N+1}{2}}^{m}{\binom{N}{i}}{\binom{N}{m-i}}+\sum_{i=\frac{N+1}{2}}^{N}{\binom{N}{i}}{\binom{N}{N-i}}\right]p^{N}(1-p)^{N},$
(59)

where the equal sign holds for $p=1/2$.

Furthermore, using the binomial theorem, we write

$\displaystyle 2^{2N-2}=$
$\displaystyle\left[\sum_{i=\frac{N+1}{2}}^{N}{\binom{N}{i}}\right]\left[\sum_{j=0}^{\frac{N-1}{2}}{\binom{N}{j}}\right]$
(60) $\displaystyle=$
$\displaystyle\sum_{m=\frac{N+1}{2}}^{N-1}\sum_{i=\frac{N+1}{2}}^{m}{\binom{N}{i}}{\binom{N}{m-i}}+\sum_{i=\frac{N+1}{2}}^{N}{\binom{N}{i}}{\binom{N}{N-i}}+\sum_{m=N+1}^{\frac{3N-1}{2}}\sum_{i=m-\frac{N-1}{2}}^{N}{\binom{N}{i}}{\binom{N}{m-\frac{N-1}{2}}}$
(61) $\displaystyle=$ $\displaystyle
2\sum_{m=\frac{N+1}{2}}^{N-1}\sum_{i=\frac{N+1}{2}}^{m}{\binom{N}{m-i}}{\binom{N}{i}}+\sum_{i=\frac{N+1}{2}}^{N}{\binom{N}{i}}{\binom{N}{N-i}},$
(62)

which is exactly equal to the prefactor in Equation (59). Thus, we arrive at
the inequality

$\displaystyle[1-f(p,N)]f(p,N)\geq 2^{2N-2}p^{N}(1-p)^{N}.$ (63)

Substituting Equation (63) into Equation (54), we derive

$\displaystyle\frac{\partial^{2}\mathbb{S}(p,N)}{\partial p^{2}}\leq 0,$ (64)

by using the inequality

$\displaystyle\frac{2}{\pi}<\frac{4N}{2^{2N}}{\binom{N-1}{(N-1)/2}}^{2}\leq
1.$ (65)

Since $\partial\mathbb{S}(p,N)/\partial p$ is a monotonically decreasing
function with respect to $p$, it follows from
$\partial\mathbb{S}(p,N)/\partial p|_{p=1/2}=0$ that
$\partial\mathbb{S}(p,N)/\partial p$ is positive (negative) for $p<1/2$
($p>1/2$). Therefore, $\mathbb{S}(p,N)$ is monotonically increasing
(decreasing) for $p<1/2$ ($p>1/2$). According to Equation (1), $P(p,N)$ is a
monotonically increasing function with respect to $p$ and $P(1/2,N)=1/2$, thus
we prove that $\mathbb{S}(P,N)$ is also monotonically increasing (decreasing)
for $P<1/2$ ($P>1/2$).

##  Appendix C Detailed Derivation of Optimal Erasure Protocol of Arrhenius-
Rates Unit Model

For Arrhenius rates, the master Equation (32) can be recast as follows

$\displaystyle\beta\Delta\epsilon=-\ln\left(\frac{\dot{p}+p}{1-p}\right).$
(66)

The heat dissipated by a microscopic Arrhenius unit is given by the functional

$\displaystyle\beta
q=\beta\int_{0}^{\tau}\Delta\epsilon\dot{p}\mathrm{d}t=\beta\int_{p_{i}}^{p_{f}}\Delta\epsilon\,\mathrm{d}p=\int_{0}^{\tau}\mathcal{L}(p,\dot{p})\mathrm{d}t,$
(67)

with the explicitly time-independent Lagrangian

$\displaystyle\mathcal{L}(p,\dot{p})\equiv-\dot{p}\ln\left(\frac{\dot{p}+p}{1-p}\right).$
(68)

The minimization of the heat functional amounts to solving the Euler-Lagrange
equation

$\displaystyle\mathcal{L}-\dot{p}\frac{\partial\mathcal{L}}{\partial\dot{p}}=K,$
(69)

that admits the solutions

$\displaystyle\dot{p}_{1}=\frac{K-\sqrt{K^{2}+4Kp}}{2},\quad\dot{p}_{2}=\frac{K+\sqrt{K^{2}+4Kp}}{2},$
(70)

where $K$ is constant resulting from the time-integration of the Euler-
Lagrange equation.

Since we consider the erasure branch from the initial state $p_{i}=1/2$ to the
final one $p_{f}\leq p_{i}$, we restrict to the solution $\dot{p}_{1}$. This
ordinary differential equation yields the following explicit expression of the
erasure duration $\tau$

$\displaystyle\tau=\int_{p_{i}}^{p_{f}}\frac{1}{\dot{p}}\mathrm{d}p=F(p_{f})-F(p_{i}),$
(71)

where we defined the function $F(p)$ as

$\displaystyle
F(p)=-\sqrt{1+\frac{4p}{K}}-\ln\left(\sqrt{1+\frac{4p}{K}}-1\right).$ (72)

Substituting Equation (70) into Equation (67), we obtain the following
expression for the dissipated heat

$\displaystyle\beta
q=\beta\int_{p_{i}}^{p_{f}}\Delta\epsilon\,\mathrm{d}p=G(p_{f})-G(p_{i}),$
(73)

where we defined the function $G(p)$ as

$\displaystyle
G(p)=\frac{1}{2}\sqrt{K^{2}+4Kp}-\ln{(1-p)}-p\ln\left(\frac{K+2p-\sqrt{K^{2}+4Kp}}{2(1-p)}\right).$
(74)

## References

  * Leff and Rex (1990) H. S. Leff and A. F. Rex,  _Maxwell’s Demon 2: Entropy, classical and quantum information, computing_ (Princeton University Press, 1990).
  * Szilard (1929) L. Szilard, Zeitschrift fur Physik 53, 840 (1929).
  * Shannon (1948a) C. E. Shannon, The Bell System Technical Journal 27, 379 (1948a).
  * Shannon (1948b) C. E. Shannon, The Bell System Technical Journal 27, 623 (1948b).
  * Landauer (1961) R. Landauer, IBM J. Res. Dev. 5, 183 (1961).
  * Bennett (1973) C. H. Bennett, IBM J. Res. Dev. 17, 525 (1973).
  * Bennett (1982) C. H. Bennett,  _The thermodynamics of computation - a review_ (Kluwer Academic Publishers - Plenum Publishers, 1982).
  * Penrose (2005) O. Penrose,  _Foundations of Statistical Mechanics: A Deductive Treatment_ (Dover Publications, 2005).
  * Berut  _et al._ (2012) A. Berut, A. Arakelyan, A. Petrosyan, S. Ciliberto, R. Dillenschneider, and E. Lutz, Nature 483, 187 (2012).
  * Bérut  _et al._ (2015) A. Bérut, A. Petrosyan, and S. Ciliberto, Journal of Statistical Mechanics: Theory and Experiment 2015, P06015 (2015).
  * Jun  _et al._ (2014) Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. 113, 190601 (2014).
  * Piechocinska (2000) B. Piechocinska, Phys. Rev. A 61, 062314 (2000).
  * Shizume (1995) K. Shizume, Phys. Rev. E 52, 3495 (1995).
  * Esposito and Van den Broeck (2011) M. Esposito and C. Van den Broeck, EPL (Europhysics Letters) 95, 40004 (2011).
  * Owen  _et al._ (2017) J. A. Owen, A. Kolchinsky, and D. H. Wolpert, (2017), arXiv:1709.00765 [cond-mat.stat-mech] .
  * Wolpert  _et al._ (2017) D. H. Wolpert, A. Kolchinsky, and J. A. Owen, ArXiv e-prints (2017), arXiv:1708.08494 [cond-mat.stat-mech] .
  * Kolchinsky and Wolpert (2018) A. Kolchinsky and D. H. Wolpert, Interface Focus 8 (2018), 10.1098/rsfs.2018.0041.
  * Wolpert and Kolchinsky (2018) D. H. Wolpert and A. Kolchinsky, ArXiv e-prints (2018), arXiv:1806.04103 [cond-mat.stat-mech] .
  * Sagawa (2019) T. Sagawa,  _Energy Limits in Computation - A Review of Landauer’s Principle, Theory and Experiments. Chapter: Second law, entropy production, and reversibility in thermodynamics of information_ (Springer, Boston, MA, USA, 2019).
  * Parrondo  _et al._ (2015) J. Parrondo, J. Horowitz, and T. Sagawa, Nature Physics 11, 131–139 (2015).
  * Seifert (2012) U. Seifert, Reports on Progress in Physics 75, 126001 (2012).
  * Van den Broeck and Esposito (2015) C. Van den Broeck and M. Esposito, Physica A: Statistical Mechanics and its Applications 418, 6 (2015).
  * Esposito  _et al._ (2010) M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, EPL (Europhysics Letters) 89, 20003 (2010).
  * Diana  _et al._ (2013) G. Diana, G. B. Bagci, and M. Esposito, Phys. Rev. E 87, 012111 (2013).
  * Zulkowski and DeWeese (2014) P. R. Zulkowski and M. R. DeWeese, Phys. Rev. E 89, 052140 (2014).
  * Gavrilov  _et al._ (2017) M. Gavrilov, R. Chetrite, and J. Bechhoefer, Proceedings of the National Academy of Sciences 114, 11097 (2017).
  * Aurell  _et al._ (2012) E. Aurell, K. Gawedzki, C. Mejia-Monasterio, R. Mohayaee, and P. Muratore-Ginanneschi, J. Stat. Phys. 147, 487 (2012).
  * Schmiedl and Seifert (2008) T. Schmiedl and U. Seifert, EPL (Europhysics Letters) 81, 20003 (2008).
  * Zulkowski and DeWeese (2015) P. R. Zulkowski and M. R. DeWeese, Phys. Rev. E 92, 032117 (2015).
  * Esposito  _et al._ (2010a) M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010a).
  * Schmiedl and Seifert (2007) T. Schmiedl and U. Seifert, Phys. Rev. Lett. 98, 108301 (2007).
  * Sheng and Tu (2014) S. Sheng and Z. C. Tu, Phys. Rev. E 89, 012129 (2014).
  * Sheng and Tu (2015a) S. Q. Sheng and Z. C. Tu, New Journal of Physics 17, 045013 (2015a).
  * Sheng and Tu (2015b) S. Sheng and Z. C. Tu, Phys. Rev. E 91, 022136 (2015b).
  * Esposito  _et al._ (2010b) M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, Phys. Rev. E 81, 041106 (2010b).
  * Richter (1999) H. J. Richter, Journal of Physics D: Applied Physics 32, R147 (1999).
  * Dholakia (1991) A. Dholakia,  _Majority-Logic Decoding. In: Introduction to Convolutional Codes with Applications. The Springer International Series in Engineering and Computer Science (Communications and Information Theory), vol 275._ (Springer, Boston, MA, USA, 1991).
  * Herpich  _et al._ (2018) T. Herpich, J. Thingna, and M. Esposito, Phys. Rev. X 8, 031056 (2018).
  * Esposito (2012) M. Esposito, Phys. Rev. E 85, 041125 (2012).
  * Herpich and Esposito (2019) T. Herpich and M. Esposito, Phys. Rev. E 99, 022135 (2019).
  * Cover and Thomas (1991) T. M. Cover and J. A. Thomas,  _Elements of Information Theory_ (Wiley-Interscience, New York, NY, USA, 1991).
  * Sagawa (2014) T. Sagawa, Journal of Statistical Mechanics: Theory and Experiment 2014, P03025 (2014).
  * Gavrilov and Bechhoefer (2016) M. Gavrilov and J. Bechhoefer, Phys. Rev. Lett. 117, 200601 (2016).
  * Crochik and Tomé (2005) L. Crochik and T. Tomé, Phys. Rev. E 72, 057103 (2005).


[a]Alexander Studenikin

# Electromagnetic neutrino: The theory and reactor and solar bounds on
neutrino magnetic moments

###### Abstract

A brief overview of the electromagnetic properties of neutrinos is presented
with a discussion of the most important fundamental aspects of the problem.
Then using data from the ground-based reactor and solar neutrino-electron
scattering experiments the best upper bounds on neutrino effective magnetic
moments are discussed.

##  1 The theory

It is usually assumed that the entirety of the electromagnetic properties of
neutrinos are embodied by the structure of the amplitude corresponding to the
Feynman diagramme shown in Fig. 1 that describes the interaction of a neutrino
with a real photon.

Figure 1: A neutrino effective one-photon coupling.

Two incoming and outgoing lines represent initial and final neutrino states
and the third line stands for the real photon. These three lines are connected
by an effective vertex, which in general contains the whole set of the
neutrino electromagnetic characteristics. The diagramme in Fig. 1 corresponds
to the one-photon approximation for the electromagnetic interactions of a
neutrino field $\nu(x)$ that can be described by the effective interaction
Hamiltonian

$H_{em}(x)=j_{\mu}(x)A^{\nu}(x)={\bar{\nu}(x)}\Lambda_{\mu}\nu(x)A^{\mu}(x),$
(1)

where $j_{\mu}(x)$ is the neutrino effective electromagnetic current and
$\Lambda_{\mu}$ is a matrix in the spinor space. Considering the neutrinos as
free particles with the Fourier expansion of the free Dirac fields, for the
amplitude corresponding to the diagramme in Fig. 1 it is possible to get (see
[1] and [2] for the detailed derivations)

$\langle\nu(p_{f})|j_{\mu}(0)|\nu(p_{i})\rangle=\bar{u}(p_{f})\Lambda_{\mu}(q)u(p_{i}),$
(2)

where $q=p_{i}-p_{f}$.

In the most general form the neutrino electromagnetic vertex function
$\Lambda_{\mu}^{ij}(q)$ can be expressed [1] in terms of four form factors

$\Lambda_{\mu}^{ij}(q)=\left(\gamma_{\mu}-q_{\mu}\not{q}/q^{2}\right)\left[f_{Q}^{ij}(q^{2})+f_{A}^{ij}(q^{2})q^{2}\gamma_{5}\right]-i\sigma_{\mu\nu}q^{\nu}\left[f_{M}^{ij}(q^{2})+if_{E}^{ij}(q^{2})\gamma_{5}\right],$
(3)

where $\Lambda_{\mu}(q)$ and form factors $f_{Q,A,M,E}(q^{2})$ are $3\times 3$
matrices in the space of massive neutrinos. Note that in the derivation of the
decomposition (3) the demands followed from the Lorentz-invariance and
electromagnetic gauge invariance are taken into account.

In the case of coupling with a real photon ($q^{2}=0$) the form factors
$f(q^{2})$ provide four sets of neutrino electromagnetic characteristics: 1)
the electric millicharges $q_{ij}=f_{Q}^{ij}(0)$, 2) the dipole magnetic
moments $\mu_{ij}=f_{M}^{ij}(0)$, 3) the dipole electric moments
$\epsilon_{ij}=f_{E}^{ij}(0)$ and 4) the anapole moments
$a_{ij}=f_{A}^{ij}(0)$.

The expression (3) for $\Lambda_{\mu}^{ij}(q)$ is applicable for Dirac and
Majorana neutrinos. However, a Majorana neutrino does not have diagonal
electric charge and dipole magnetic and electric form factors, only a diagonal
anapole form factor can be nonzero. At the same time, a Majorana neutrino can
also have nonzero off-diagonal (transition) form factors.

If one considers the case of equal masses for the initial and final neutrinos,
$m_{i}=m_{f}$, then the decomposition of the neutrino electromagnetic vertex
function reduces to

$\Lambda_{\mu}^{ii}(q)=f_{Q}^{ii}(q^{2})\gamma_{\mu}-i\sigma_{\mu\nu}q^{\nu}\left[f_{M}^{ii}(q^{2})+if_{E}^{ii}(q^{2})\gamma_{5}\right]+f_{A}^{ii}(q^{2})\left(\gamma_{\mu}-q_{\mu}\not{q}/q^{2}\right)\gamma_{5}.$

It is also interesting to consider neutrino electromagnetic properties for the
case of massless neutrinos of the Standard Model when neutrinos are described
by two-component left-handed Weyl spinors. In this case neutrinos have only
one form factor which is equal to the difference of the charge $f_{Q}(q^{2})$
and anapole $f_{A}(q^{2})$ form factors, and the electromagnetic vertex
function is given by

$\Lambda_{\mu}(q)=(\gamma_{\mu}-q_{\mu}\not{q}/q^{2})f(q^{2}),\ \ \ \ \
f(q^{2})=f_{Q}(q^{2})-f_{A}(q^{2}).$ (4)

From these expressions one can expect that at least from the phenomenological
point of view it is not possible to treat consequences of neutrino nonzero
electric charge $f_{Q}(q^{2})$ and anapole $f_{A}(q^{2})$ form factors
separately. This case can approximate the relation between these two form
factors that arises for an ultrarelativistic massive neutrinos.

Taking into account the above-mentioned feature of the anapole form factor
$f_{A}(q^{2})$ (the ambiguity of its allocation against the charge form factor
$f_{Q}(q^{2})$ which manifests itself even in the Standard Model), an
alternative decomposition of the electromagnetic vertex function
$\Lambda_{\mu}(q)$ has been proposed in [3]. In [3] the toroidal dipole form
factor $f_{T}(q^{2})$ is introduced as a characteristic of the neutrino
instead of the anapole form factor $f_{A}(q^{2})$. In this case, the neutrino
electromagnetic vertex can be written in the so-called toroidal
parametrization

$\Lambda_{\mu}^{ij}(q)=f_{Q}^{ij}(q^{2})\gamma_{\mu}-i\sigma_{\mu\nu}q^{\nu}\left[f_{M}^{ij}(q^{2})+if_{E}^{ij}(q^{2})\gamma_{5}\right]+if_{T}^{ij}(q^{2})\epsilon_{\mu\nu\lambda\rho}P^{\nu}q^{\lambda}\gamma^{\rho},$

where $P=p_{i}+p_{f}$ and $\epsilon_{\mu\nu\lambda\rho}$ is the Levi-Civitta
unit antisymmetric tensor. The toroidal parametrisation of the neutrino vertex
has a more clear physical interpretation than the anapole one, because it
provides a one-to-one correspondence between the form factors and the
multipole moments in expansion of electromagnetic fields. The corresponding
toroidal dipole moment $f_{T}(q^{2}=0)$, for the first time introduced in [4],
is more convenient than the anapole moment $f_{A}(q^{2}=0)$ for the
description of T-invariant interactions with nonconcervation of the P and C
symmetries.

From the identity

${\bar{u}}(p_{f})\Big{[}(m_{i}-m_{f})\sigma_{\mu\nu}g^{\mu}+(g^{2}\gamma_{\mu}-{\not{q}}q_{\mu})-\epsilon_{\mu\nu\lambda\rho}P^{\nu}q^{\lambda}\gamma^{\rho}\gamma_{5}\Big{]}\gamma_{5}u_{i}(p_{i})=0$
(5)

it follows [3] that in the static limit the toroidal $f_{T}(q^{2}=0)$ and
anapole $f_{A}(q^{2}=0)$ dipole moments coincide when the masses of the
initial and final neutrino states are equal to each other, $m_{i}=m_{f}$.

##  2 Neutrino electromagnetic properties in scattering experiments

The possible electromagnetic characteristics of neutrinos can manifest
themselves in astrophysical environments, where neutrinos propagate in strong
magnetic fields and dense matter, and also in ground-based laboratory
measurements of neutrinos fluxes from various sources. So far, there are no
any evidenced in favour of neutrino nonzero electromagnetic properties either
from laboratory measurements of neutrinos from ground-based sources or from
observations of neutrinos from astrophysical sources. Only constraints (the
upper bounds on neutrino electromagnetic characteristics) are obtained in
different experiments. The available constraints are discussed in the review
paper [1] (see also [5], [6], [7], [8] and [9] for the latest developments and
progress in this field).

A widely used method to probe the neutrino electromagnetic properties is based
on the direct measurements of low-energy elastic (anti)neutrino-electron
scattering in reactor, accelerator, and solar neutrino experiments. Possible
nonzero electromagnetic characteristics of neutrinos, such as the electric
millicharges, charge radii, dipole magnetic and electric moments, and anapole
(toroidal) moments can provide additional contributions to the scattering of
neutrinos on a target which are measured in the corresponding experiments.

A general strategy of such experiments consists in determining deviations of
the scattering cross section differential with respect to the energy transfer
from the value predicted by the Standard Model of the electroweak interaction.
The desired goal of such experiments may be, firstly, to obtain information
about the magnitude of the contributions of electromagnetic interactions of
neutrinos to the cross sections. Then this information (a modification of the
scattering cross-section confirmed in a specific experiment due to
electromagnetic interactions of neutrinos) can be used to study the patterns
of neutrino propagation in various media or to analyze data from various other
experiments. Within the framework of this approach, we are not interested in
the nature of the occurrence of nontrivial electromagnetic properties of
neutrinos. In this case, only restrictions on the numerical values of specific
electromagnetic characteristics of neutrinos extracted from the data of the
neutrino scattering experiment are used. The question of the origin of the
nonzero electromagnetic properties of neutrinos, that is, about a more
fundamentally theoretical model predicting the nonzero properties of
neutrinos, remains behind the scenes.

The second main purpose of neutrino scattering experiments may be to compare
the obtained experimental data on the cross section (more precisely, from the
obtained upper bounds of the possible electromagnetic contribution to the
cross section) with the predictions of a more general theoretical model that
provides nonzero electromagnetic characteristics of neutrinos from the first
fundamental principles. In this context, we can say that the study of the
electromagnetic properties of neutrinos opens a window into new physics.

The most consistent approach to the theoretical description of the
electromagnetic properties of neutrinos involves the initial introduction of
nonzero electromagnetic characteristics for the mass states of neutrinos. At
the same time, since in neutrino scattering experiments flavour neutrinos are
registered in detectors, it is necessary to translate electromagnetic
properties from the mass into the flavour neutrino basis. Therefore, due to
the neutrino mixing and oscillations along the neutrino path from the source
to the detector the observed (constrained) neutrino electromagnetic
characteristics depends on the neutrino flavour composition in the detector.
The recent and most comprehensive study of neutrino electromagnetic properties
in the neutrino electron scattering with account for neutrino mixing and
oscillations can be found in [10].

Consider the most stringent constraints on the effective neutrino magnetic
moments that are obtained with the reactor antineutrinos: $\mu_{\nu}<2.9\times
10^{-11}\mu_{B}$ (GEMMA Collaboration [11]), and solar neutrinos:
${\mu}_{\nu}\leq 2.8\times 10^{-11}\mu_{B}$ (Borexino Collaboration [12]).
Both these constraints are obtained with investigations of the elastic
scattering of a flavour neutrino $\nu_{l}$ (or an antineutrino
${\bar{\nu}}_{l}$ ) on an electron at rest: $\nu_{l}+e^{-}\to\nu_{l}+e^{-},\ \
l=e,\mu,\tau$. There are two contributions, one from the Standard Model weak
interaction and another one from the neutrino magnetic moment interaction, to
the electron neutrino cross section [13],

$\frac{d\sigma_{\nu_{l}e^{-}}}{dT_{e}}=\left(\frac{d\sigma_{\nu_{l}e^{-}}}{dT_{e}}\right)_{SM}+\left(\frac{d\sigma_{\nu_{l}e^{-}}}{dT_{e}}\right)_{\mu}.$
(6)

The weak-interaction cross section is

$\left(\frac{d\sigma_{\nu_{l}e^{-}}}{dT_{e}}\right)_{SM}=\frac{G^{2}_{F}m_{e}}{2\pi}\bigg{\\{}(g_{V}^{\nu_{l}}+g_{A}^{\nu_{l}})^{2}+(g_{V}^{\nu_{l}}-g_{A}^{\nu_{l}})^{2}\left(1-\frac{T_{e}}{E_{\nu}}\right)^{2}+\left[(g_{A}^{\nu_{l}})^{2}-(g_{V}^{\nu_{l}})^{2}\right]\frac{m_{e}T_{e}}{E_{\nu}^{2}}\bigg{\\}},$
(7)

with the standard coupling constants $g_{V}$ and $g_{A}$ given by:
$g_{V}^{\nu_{e}}=2\sin^{2}\theta_{W}+1/2,g_{A}^{\nu_{e}}=1/2,\ \ \
g_{V}^{\nu_{\mu,\tau}}=2\sin^{2}\theta_{W}-1/2,\ \ \
g_{A}^{\nu_{\mu,\tau}}=-1/2$. The cross section depends on the initial
neutrino energy $E_{\nu}$ and also contains the electron recoil energy
$T_{e}$. For antineutrinos one must substitute $g_{A}\to-g_{A}$. If to account
that the born in the source flavour neutrino $|\nu_{l}\rangle$ arrives to the
detector in the flavour state given by

$|\nu_{l}(L)\rangle=\sum_{k=1}^{3}U^{\ast}_{lk}e^{-\frac{m^{2}_{k}}{2E_{\nu}}L}|\nu_{k}\rangle,$
(8)

then the neutrino magnetic-moment contribution to the cross section is given
by

$\left(\frac{d\sigma_{\nu_{l}e^{-}}}{dT_{e}}\right)_{\mu}=\frac{\pi\alpha^{2}}{m_{e}^{2}}\left(\frac{1}{T_{e}}-\frac{1}{E_{\nu}}\right)\left(\frac{\mu_{\nu_{l}}}{\mu_{B}}\right)^{2},$
(9)

here $\mu_{B}$ is the Bohr magneton. The cross section contains the effective
magnetic moment $\mu_{\nu_{l}}$ [1], [10]

${\mu_{\nu_{l}}}^{2}(L,E_{\nu})=\sum_{j}\left|\sum_{k}U_{lk}^{*}e^{-i\Delta{m}^{2}_{kj}L/2E_{\nu}}\left(\mu_{jk}-i\epsilon_{jk}\right)\right|^{2}$
(10)

that indeed receives equal contributions from the neutrino electric and
magnetic dipole moments, both the diagonal and transition, which are given by
the static values of the corresponding form factors,
$\mu_{jk}=f_{M}^{jk}(q^{2}=0),\ \ \ \ \epsilon_{jk}=f_{E}^{jk}(q^{2}=0)$.

It is just straightforward that scattering experiments based on detection of
neutrinos arriving from different distances probe (or constraint) different
combinations of the fundamental magnetic moments. Now for simplicity we omit
possible contributions from the dipole electric moments. In the reactor short-
baseline experiments (for instance, in the GEMMA experiment) one studies (and
constrains) the effective magnetic moments in the flavor basis. Whereas in the
case of long-baseline experiments (such as the Borexino experiment with the
solar neutrinos) a more convenient interpreting the results is based on
effective magnetic moments in the fundamental mass basis, rather than in the
flavour basis. These should be accounted for when one compares quite similar
values for effective magnetic moments obtained form the GEMMA and Borexino
experiments.

Note that with the inclusion [5] of the effect from possible nonzero neutrino
millicharge to the analysis of the GEMMA collaboration data on the
antineutrino-electron scattering provides the most severe reactor upper bound
on the neutrino millicharge: $q_{\nu}<1.5\times 10^{-12}e_{0}$. For a detailed
and recent discussion on the main processes of the electromagnetic interaction
of neutrinos in astrophysics and the corresponding limitations on millicharges
and effective magnetic moments of neutrinos see [9].

The author is thankful to Konstantin Kouzakov for useful discussions. The work
is supported by the Interdisciplinary Scientific and Educational School of
Moscow University “Fundamental and Applied Space Research” and by the Russian
Foundation for Basic Research under grant No. 20-52-53022-GFEN-a.

## References

  * [1] C. Giunti and A. Studenikin, _Neutrino electromagnetic interactions: A window to new physics, Rev. Mod. Phys._ 87 (2015) 531.
  * [2] M. Nowakowski, E. A. Paschos and J. M. Rodriguez, _All electromagnetic form-factors, Eur. J. Phys._ 26 (2005) 545.
  * [3] V. Dubovik and V. Kuznetsov, _The toroid moment of Majorana neutrino, Int. J. Mod. Phys. A_ 13 (1998) 5257.
  * [4] Ya. Zeldovich, _Parity nonconservation in the first order in the weak-interaction constant in electron scattering and other effects, Sov. Phys. JETP_ 6 (1958) 1184.
  * [5] A. Studenikin, Europhys.Lett. 107 (2014) 21001.
  * [6] A. Parada, Adv. High Energy Phys. 2020 (2020) 5908904.
  * [7] A. Studenikin, _Electromagnetic neutrino properties: new constraints and new effects, PoS_ ICHEP2020 (2021) 180.
  * [8] V. Brdar, A. Greljo, J. Kopp and T. Opferkuch, _The neutrino magnetic moment portal: Cosmology, astrophysics, and direct detection, JCAP_ 01 (2021) 039.
  * [9] A. Studenikin, Electromagnetic neutrino: The basic processes and astrophysical probes, to appear in proc. of TAUP 2021, J. Phys. Conf. Ser. (2022).
  * [10] K. Kouzakov and A. Studenikin, _Electromagnetic properties of massive neutrinos in low-energy elastic neutrino-electron scattering, Phys. Rev. D_ 95 (2017) 055013, [erratum: Phys. Rev. D 96 (2017) 099904].
  * [11] A. Beda, V. Brudanin, V. Egorov et al., Adv. High Energy Phys. 2012 (2012) 350150.
  * [12] M. Agostini et al. [Borexino Collaboration], Phys. Rev. D 96 (2017) 091103.
  * [13] P. Vogel and J. Engel, _Neutrino electromagnetic form-factors, Phys. Rev. D_ 39 (1989) 3378.