# Finite-Time Quantum Landauer Principle and Quantum Coherence

Tan Van Vu<EMAIL_ADDRESS>Department of Physics, Keio University,
3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan  Keiji Saito
<EMAIL_ADDRESS>Department of Physics, Keio University, 3-14-1
Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

###### Abstract

The Landauer principle states that any logically irreversible information
processing must be accompanied by dissipation into the environment. In this
study, we investigate the heat dissipation associated with finite-time
information erasure and the effect of quantum coherence in such processes. By
considering a scenario wherein information is encoded in an open quantum
system whose dynamics are described by the Markovian Lindblad equation, we
show that the dissipated heat is lower-bounded by the conventional Landauer
cost, as well as a correction term inversely proportional to the operational
time. To clarify the relation between quantum coherence and dissipation, we
derive a lower bound for heat dissipation in terms of quantum coherence. This
bound quantitatively implies that the creation of quantum coherence in the
energy eigenbasis during the erasure process inevitably leads to additional
heat costs. The obtained bounds hold for arbitrary operational time and
control protocol. By following an optimal control theory, we numerically
present an optimal protocol and illustrate our findings by using a single-
qubit system.

Introduction.—Any irreversible information processing unavoidably incurs a
thermodynamic cost. This fundamental relationship between information and
thermodynamics is embodied in the Landauer principle Landauer (1961). The
principle states that the amount of heat dissipation $Q$ required to erase
information is lower-bounded by the entropy change $\Delta S$ of information-
bearing degrees of freedom, $\beta Q\geq\Delta S$. Here, $\beta$ is the
inverse temperature of the environment. This inequality – referred to as the
Landauer bound or limit – lays a foundation for the thermodynamics of
information Sagawa (2012); Parrondo  _et al._ (2015); Goold  _et al._ (2016)
and computation Bennett (1982); Wolpert (2019) and provides a resolution to
the paradox of Maxwell’s demon Maruyama  _et al._ (2009). The attainability of
the Landauer bound in the slow quasistatic limit has been experimentally
verified in various systems Bérut  _et al._ (2012); Jun  _et al._ (2014); Yan
_et al._ (2018); Hong  _et al._ (2016); Saira  _et al._ (2020); Dago  _et al._
(2021). Nonetheless, modern computing requires fast memory erasure, which
generally comes with a thermodynamic cost far beyond the Landauer limit. Thus,
improving our understanding of heat dissipation in finite-time information
processing is relevant to the development of efficient computing devices.

In recent years, the Landauer principle has been extensively studied in the
framework of stochastic thermodynamics from the classical to the quantum
regime Sagawa and Ueda (2009); Esposito and den Broeck (2011); Hilt  _et al._
(2011); Deffner and Jarzynski (2013); Reeb and Wolf (2014); Browne  _et al._
(2014); Lorenzo  _et al._ (2015); Goold  _et al._ (2015); Alhambra and Woods
(2017); Campbell  _et al._ (2017); Guarnieri  _et al._ (2017); Boyd  _et al._
(2018); Klaers (2019); Shiraishi and Saito (2019); Dechant and Sakurai (2019);
Buscemi  _et al._ (2020); Timpanaro  _et al._ (2020); Wolpert (2020); Van Vu
and Hasegawa (2021a); Riechers and Gu (2021). One central issue is
investigating how much heat needs to be dissipated to erase information in
far-from-equilibrium situations. When the erasure fidelity is predetermined, a
tradeoff between dissipation and operation time apparently occurs, as
indicated in slow driving protocols Zulkowski and DeWeese (2014, 2015a,
2015b). Recently, Proesmans and coworkers have derived a tradeoff relation for
arbitrary driving speed in classical bits modeled by a double-well potential
Proesmans  _et al._ (2020a, b). They showed that the minimum dissipation for
erasing a classical bit is bounded from below by the Landauer cost, as well as
a term inversely proportional to the operational time, which is somewhat
reminiscent of a speed limit Shiraishi  _et al._ (2018); Funo  _et al._
(2019); Vo  _et al._ (2020); Van Vu and Hasegawa (2021b). In the short-time
limit, the finite-time correction term is dominant over the Landauer cost.
From this development, quantum extensions relevant to information erasure in
qubits are strongly desired.

In addition, in the quantum regime, quantum coherence is one of the crucial
aspects. Recently, the role of quantum coherence has been intensively
discussed in the context of finite-time thermodynamics Horodecki and Oppenheim
(2013); Uzdin  _et al._ (2015); Lostaglio  _et al._ (2015); Korzekwa  _et al._
(2016); Francica  _et al._ (2019); Santos  _et al._ (2019); Francica  _et al._
(2020); Tajima and Funo (2021). In the case of conventional heat engines
driven by heat baths with different temperatures, quantum coherence hinders
the thermodynamic performance in the linear response regime with respect to
small driving amplitudes Brandner  _et al._ (2017) or small driving speeds
Brandner and Saito (2020). On the other hand, power outputs can be enhanced by
coherence in several far-from-equilibrium models Scully  _et al._ (2011);
Uzdin  _et al._ (2015); Watanabe  _et al._ (2017); Menczel  _et al._ (2020).
Thus, the role of quantum coherence in nonequilibrium thermodynamics is highly
elusive. Concerning the Landauer principle, Miller and coworkers have examined
the slow-driving case and have found that quantum coherence generates
additional dissipation Miller  _et al._ (2020). This property is consistent
with the aforementioned slow-driving case in heat engines. Hence, as the next
step toward a complete understanding of the quantum Landauer principle,
determining the effect of an arbitrary driving speed on the relation between
quantum coherence and dissipation is clearly important.

From these two backgrounds, in this paper, we address the following questions:
(i) What is the effect of a finite-time protocol with a finite erasure error
on heat dissipation? (ii) What is the role of quantum coherence in heat
dissipation for an arbitrary driving speed? These two are clearly fundamental
for the in-depth understanding of the quantum Landauer principle. To answer
these questions, we consider a dynamical class described by the Markovian
Lindblad equations, which guarantee thermodynamically consistent dynamics. We
then rigorously provide quantitative answers to these questions. The result
for the first question is presented below in Ineq. (3), which is identified as
a quantum extension of the classical version obtained in Ref. Proesmans  _et
al._ (2020a). For the second question, we show the inevitable heat dissipation
caused by quantum coherence as below in Ineq. (10), which explicitly shows
that quantum coherence in the qudit always induces additional heat dissipation
compared to classical protocols. The obtained results hold for an arbitrary
control protocol and the driving speed. By using a single-qubit system, we
present an optimal control protocol obtained numerically by optimizing the
dissipation and erasure fidelity, which supports our findings.

Erasure setup and first main result.—We consider an information erasure
process realized by using a controllable open qudit system with an arbitrary
dimension $d$ and an infinite heat bath at the inverse temperature
$\beta=1/k_{\rm B}T$. The former and latter are referred to as information-
bearing and non-information-bearing degrees of freedom, respectively. The
information content we want to erase is encoded in the density matrix of the
system, which is typically set to a maximally mixed state
$\varrho_{0}=\mathbb{I}/d$. This mixed state is regarded as an ensemble of
many initial pure states that are subject to reset. The maximally mixed state
is sufficient to understand the average dissipated heat of the erasure process
for all initial pure states, and if an erasure protocol can reliably reset the
qudit from the maximally mixed state, then it does so for an arbitrary pure
state (see Supplemental Material (SM) Sup  for details). The information is
erased in a finite time $\tau$ by varying the control Hamiltonian $H_{t}$.
Here, we focus on a dynamical class in which the system is always weakly
attached to the heat bath, and its dynamics are described by the Lindblad
master equation Lindblad (1976); Gorini  _et al._ (1976). Let $\varrho_{t}$ be
the density matrix of the system at time $t$; then, the dynamics are described
as

$\displaystyle\dot{\varrho}_{t}$
$\displaystyle=\mathcal{L}_{t}(\varrho_{t})\,,$ (1)
$\displaystyle\mathcal{L}_{t}(\varrho)$
$\displaystyle\coloneqq-i\hbar^{-1}[H_{t},\varrho]+\sum_{k}\mathcal{D}[L_{k}(t)]\varrho,$

with the dissipator given by $\mathcal{D}[L]\varrho\coloneqq L\varrho
L^{\dagger}-\left\\{L^{\dagger}L,\varrho\right\\}/2$. Here, $L_{k}(t)$ are
time-dependent jump operators that account for transitions between different
energy eigenstates. The dot indicates the time derivative, and $[\circ,\star]$
and $\\{\circ,\star\\}$ denote the commutator and anticommutator of the two
operators, respectively. Assume that the generator $\mathcal{L}_{t}$ obeys the
quantum detailed balance with respect to the Hamiltonian $H_{t}$ at all times
Alicki (1976). This condition ensures that the thermal state
$\pi_{t}:=e^{-\beta H_{t}}/{\rm tr}\left\\{e^{-\beta H_{t}}\right\\}$ is the
instantaneous stationary state of the Lindblad equation,
$\mathcal{L}_{t}(\pi_{t})=0$. Hereinafter, both the Planck constant and the
Boltzmann constant are set to unity: $\hbar=k_{\rm B}=1$.

Resetting the system state to a specific state results in a change in the
system entropy $\Delta S$, which is quantified by the von Neumann entropy. The
entropy decrease in the information-bearing degrees of freedom is compensated
by the amount of heat transferred to the environment $Q$. The change in the
system entropy and the average dissipated heat are, respectively, written as
follows Alicki (1979):

$\displaystyle\Delta S$ $\displaystyle\coloneqq-{\rm
tr}\left\\{\varrho_{0}\ln\varrho_{0}\right\\}+{\rm
tr}\left\\{\varrho_{\tau}\ln\varrho_{\tau}\right\\},$ (2) $\displaystyle Q$
$\displaystyle\coloneqq-\int_{0}^{\tau}{\rm
tr}\left\\{H_{t}\dot{\varrho}_{t}\right\\}dt.$

The Landauer bound can be immediately derived from the second law of
thermodynamics, which states that the irreversible entropy production
$\Sigma_{\tau}$ during period $\tau$ is always nonnegative Spohn (1978),
$\Sigma_{\tau}=-\Delta S+\beta Q\geq 0$.

Under the given setup, we now show the first main result, leaving the details
of the proof in the SM Sup :

$\beta Q\geq\Delta
S+\frac{\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}}{2\tau\overline{\gamma}_{\tau}}\geq\Delta
S+\frac{[2(1-1/d)-\epsilon]^{2}}{2\tau\overline{\gamma}_{\tau}}.$ (3)

Here, $\|...\|_{1}$ is the trace norm,
$\epsilon\coloneqq\|\varrho_{\tau}-|0\rangle\langle 0|\|_{1}$ quantifies the
distance error to the ground state, and $\overline{\gamma}_{\tau}$ is the
time-averaged dynamical activity that characterizes the timescale of the
thermal relaxation, defined as
$\overline{\gamma}_{\tau}\coloneqq\tau^{-1}\int_{0}^{\tau}\sum_{k}{\rm
tr}\left\\{L_{k}(t)\varrho_{t}L_{k}(t)^{\dagger}\right\\}dt$ Shiraishi  _et
al._ (2018); Maes (2020). The obtained bound implies that the cost of erasing
the information within a finite time is at least the Landauer cost plus a
distance term proportional to $\tau^{-1}$. The result holds for arbitrary
operational time, driving speed, and the final state, which can be far from
the ground state.

Physically important aspects of this relation are now in order. First, suppose
that we use a protocol in which the stored information is fully erased – that
is, $\varrho_{\tau}$ is equal to $|0\rangle\langle 0|$. Then, the bound Ineq.
(3) becomes

$\beta Q\geq\ln d+\frac{2(1-1/d)^{2}}{\tau\overline{\gamma}_{\tau}}.$ (4)

For a fast erasure $\tau\bar{\gamma}_{\tau}\ll 1$, the second term in the
lower bound becomes dominant. On the other hand, the lower bound reduces
exactly to the Landauer cost in the slow-erasure limit
$\tau\bar{\gamma}_{\tau}\gg 1$. The inequality (4) is the quantum extension of
the classical finite-time Landauer bound derived in Ref. Proesmans  _et al._
(2020a) (see Ineq. (9) therein). Next, consider the case of imperfect
information erasure; that is, $\varrho_{\tau}$ is not equal to
$|0\rangle\langle 0|$. In this case, we can discuss how quantum coherence
remaining at the final time is related to dissipation, thus revealing an
intrinsic difference between classical and quantum protocols. To this end, let
$\Lambda(\circ)\coloneqq\sum_{n}\Pi_{n}(\circ)\Pi_{n}$ be the dephasing map in
the energy eigenbasis of the ending Hamiltonian $H_{\tau}$, in which $\Pi_{n}$
is the projection operator onto the $n$th eigenspace. Because the trace norm
is contractive under completely positive trace-preserving maps, we can
decompose $\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}$ into classical and quantum
parts as

$\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}=\|\Lambda(\varrho_{0})-\Lambda(\varrho_{\tau})\|_{1}^{2}+C_{\rm
res},$ (5)

where $C_{\rm res}\geq 0$ is a quantum term quantifying the residual coherence
in the final state with respect to the energy eigenbasis
$\\{|n_{\tau}\rangle\\}$. Analogously, the von Neumann entropy production can
also be decomposed as

$\Delta S=\Delta S_{\rm cl}+C_{\rm rel},$ (6)

where $\Delta S_{\rm cl}\coloneqq
S(\Lambda(\varrho_{0}))-S(\Lambda(\varrho_{\tau}))$ is the classical entropy
change in terms of the population distribution and $C_{\rm rel}\coloneqq
S(\Lambda(\varrho_{\tau}))-S(\varrho_{\tau})\geq 0$ is the relative entropy of
coherence in the final state Baumgratz  _et al._ (2014). Consequently, the
lower bound in Ineq. (3) can be separated into classical and quantum terms as

$\beta Q\geq\underbrace{\Delta S_{\rm
cl}+\frac{\|\Lambda(\varrho_{0})-\Lambda(\varrho_{\tau})\|_{1}^{2}}{2\tau\overline{\gamma}_{\tau}}}_{\text{classical}}+\underbrace{C_{\rm
rel}+\frac{C_{\rm res}}{2\tau\overline{\gamma}_{\tau}}}_{\text{quantum}}.$ (7)

The classical term is a bound that can also be derived by using a classical
probabilistic process. The above expression indicates that the quantum
coherence left in the final state contributes a nonnegative term in the lower
bound.

Second main result.—The relation (7) describes the quantum coherence in the
final state. However, this is not enough to capture the role of coherence
because quantum coherence is generated during the time it takes to reach the
final state. Therefore, in addition to the above argument, we discuss the
general relation between quantum coherence and heat dissipation during a
finite time $\tau$. To achieve a general result, we assume a general initial
density matrix and consider the time evolution obeying the dynamics (1).
Quantum coherence in the energy eigenbasis is typically generated because of
the presence of non-commuting terms in the Hamiltonian. For the present aim,
using the following intuitive $\ell_{1}$ norm of coherence Baumgratz  _et al._
(2014) conveniently helps quantify the amount of coherence contained in the
state $\varrho_{t}$:

$C_{\ell_{1}}(\varrho_{t})\coloneqq\sum_{m\neq n}|\langle
m_{t}|\varrho_{t}|n_{t}\rangle|.$ (8)

Here, $\\{|n_{t}\rangle\\}$ is the instantaneous energy eigenbasis of the
Hamiltonian $H_{t}$ fnt . Mathematically, $C_{\ell_{1}}$ is the sum of the
absolute values of all the off-diagonal elements with respect to the energy
eigenbasis. This quantifier was shown to be a suitable measure of coherence
Baumgratz  _et al._ (2014) and is widely used in literature Streltsov  _et
al._ (2017). The amount of coherence accumulated during period $\tau$ can thus
be defined naturally:

$\mathscr{C}_{\tau}\coloneqq\int_{0}^{\tau}C_{\ell_{1}}(\varrho_{t})dt.$ (9)

The second main result of this study is that the average dissipated heat in a
finite time is lower-bounded by the Landauer cost plus a nonnegative coherence
term (see the SM Sup ):

$\beta Q\geq\Delta S+\frac{\overline{\gamma}_{\tau}^{\rm
R}\mathscr{C}_{\tau}^{2}}{2\tau}.$ (10)

In this equation, $\overline{\gamma}_{\tau}^{\rm R}$ is the time average of
the characteristic relaxation rate, given by $(\overline{\gamma}_{\tau}^{\rm
R})^{-1}=\tau^{-1}\int_{0}^{\tau}\left(\sum_{k}\|L_{k}(t)\|_{\infty}^{2}\right)^{-1}dt$
in the $d=2$ case (see the SM Sup  for the form of
$\overline{\gamma}_{\tau}^{\rm R}$ in the generic case). Here,
$\|...\|_{\infty}$ is the spectral norm. We emphasize that the result holds
for the arbitrary operational time and driving speed. The obtained relation
sets a lower bound on dissipation in terms of information-theoretic entropy
and quantum coherence. The inequality (10) implies that the quantum coherence
produced during information erasure must be accompanied by additional heat.
The greater the generation of coherence, the more heat is dissipated. The
relation (10) is valid for the entire driving speed regime, thus covering the
slow-driving limit Miller  _et al._ (2020). Combining this with the first main
result (3), the Landauer bound can be strengthened as $\beta Q\geq\Delta
S+\max\\{\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}/\overline{\gamma}_{\tau},\overline{\gamma}_{\tau}^{\rm
R}\mathscr{C}_{\tau}^{2}\\}/(2\tau)$, when we use the maximally mixed state
for the initial state.

Numerical demonstration with the optimal control theory.—We exemplify our
findings with a simple model of information erasure by using a single qubit.
Two-level qubit systems are relevant in quantum computation and are commonly
used to store memory in measurement-driven engines Elouard  _et al._ (2017);
Bresque  _et al._ (2021). The qubit can be viewed as a spin-$1/2$ particle
weakly coupled to a large bath of bosonic harmonic oscillators Leggett  _et
al._ (1987), evolving according to the Hamiltonian

$H_{t}=\frac{\epsilon_{t}}{2}\left[\cos(\theta_{t})\sigma_{z}+\sin(\theta_{t})\sigma_{x}\right]$
(11)

and two jump operators
$L_{1}(t)=\sqrt{\alpha\epsilon_{t}(N_{t}+1)}|0_{t}\rangle\langle 1_{t}|$,
$L_{2}(t)=\sqrt{\alpha\epsilon_{t}N_{t}}|1_{t}\rangle\langle 0_{t}|$, in which
$\sigma_{x,y,z}$ are the Pauli matrices, $\alpha$ is the coupling strength,
$N_{t}\coloneqq 1/(e^{\beta\epsilon_{t}}-1)$ is the Planck distribution, and
$\epsilon_{t}$ and $\theta_{t}$ are time-dependent control parameters. The
quantity $\epsilon_{t}$ is the energy gap between the instantaneous energy
eigenstates, whereas $\theta_{t}$ controls the relative strength of coherent
tunneling to energy bias Leggett  _et al._ (1987). If $\theta_{t}$ is fixed at
all times, it corresponds to a classical protocol. Otherwise, quantum
coherence in the energy eigenbasis is generated, implying a genuine quantum
protocol.

Resetting the qubit to the ground state $|0\rangle$ with a probability close
to $1$ can be achieved with various control protocols. For example, we can
either gradually increase the energy gap $\epsilon_{t}$ from an initial value
$\epsilon_{0}\approx 0$ to a final value $\beta\epsilon_{\tau}\gg 1$ while
also changing $\theta_{t}$ Miller  _et al._ (2020), or we can quench the
Hamiltonian at $t=0$ and let the system relax to an equilibrium state close to
$|0\rangle\langle 0|$. Here we particularly consider the Pareto-optimal
protocols Miettinen (1999), which optimize two incompatible objectives: the
success probability and the average dissipated heat. Specifically, we solve
the optimization problem of minimizing the following multi-objective
functional:

$\mathcal{F}[\\{\epsilon_{t},\theta_{t}\\}]\coloneqq\lambda\tau\beta
Q-(1-\lambda)F(\varrho_{\tau},|0\rangle\langle 0|).$ (12)

Here, $\lambda\in[0,1)$ is a weighting factor, and $F(\varrho,\sigma)={\rm
tr}\left\\{\sqrt{\sqrt{\varrho}\sigma\sqrt{\varrho}}\right\\}^{2}$ is the
fidelity of the two quantum states $\varrho$ and $\sigma$ Jozsa (1994). The
first and second terms of the functional correspond to the average dissipated
heat and erasure fidelity, respectively. When $\lambda=0$, the reliability of
the information erasure takes precedence over dissipation, and the final state
is optimized to be as close to the ground state as possible. The $\lambda>0$
case indicates that the dissipation is also minimized under a given allowable
error of the final state. Because of physical limitations, imposing
constraints on the control parameters is reasonable. Hereinafter, we set the
following lower and upper bounds on the parameters
$\beta\epsilon_{t}\in[0.4,10]$ and $\theta_{t}\in[-\pi,\pi]$.

Figure 1: Numerical results obtained with the optimal and non-optimal
protocols, depicted by the red and blue lines, respectively. (a) Time
variations of the control parameters $\epsilon_{t}$ and $\theta_{t}$. (b)
Geometrical representation of the time evolution of the qubit on the Bloch
sphere. (c) Average dissipated heat over time. Other parameters are given by
$\alpha=0.2$, $\beta=1$, $\epsilon_{0}=0.4$, $\epsilon_{\tau}=10$, and
$\tau=10$.

Obtaining the analytical solution for the optimization problem in Eq. (12)
under the given constraints is a daunting task. Hence, we numerically solve
the optimal protocol by discretizing the protocol and minimizing the
functional $\mathcal{F}$ by using a nonlinear programming method Sup . To
demonstrate the effect of quantum coherence on information erasure, we
consider $\varrho_{0}=\mathbb{I}/2$. The initial state thus does not contain
any amount of coherence. We examine two control protocols: the optimal
protocol found through the nonlinear programming method for $\lambda>0$
satisfying $(1-\lambda)/\lambda=10^{4}$ and the non-optimal protocol used in
Ref. Miller  _et al._ (2020), in which
$\epsilon_{t}=\epsilon_{0}+(\epsilon_{\tau}-\epsilon_{0})\sin(\pi
t/2\tau)^{2}$ and $\theta_{t}=\pi(t/\tau-1)$. Both protocols drive the system
to the ground state with the same order of error at the final time. Figure
1(a) shows the time variation of the control parameters $\epsilon_{t}$ and
$\theta_{t}$ for each protocol. The energy gap $\epsilon_{t}$ in each protocol
increases gradually in different ways and eventually reaches the same value.
Interestingly, in the optimal protocol, $\theta_{t}$ is always fixed at $0$;
hence, no coherence is created during the period $\tau$. On the other hand,
the non-optimal protocol constantly changes $\theta_{t}$ and generates quantum
coherence. We express the density matrix in the Bloch representation and plot
the time evolution of the Bloch vector in Fig. 1(b). Notice that the qubit
evolves through completely different paths toward the ground state for each
protocol. Figure 1(c) shows the average dissipated heat for each protocol at
each time. Note also that the optimal protocol clearly dissipates less heat
than the non-optimal protocol, which is consistent with our finding that the
generation of coherence incurs additional heat costs.

From an energetic point of view, quantum coherence has been shown to be
detrimental to the erasure of information. To optimize dissipation, the qubit
should behave as a classical bit. Further, we numerically find that this is
the case even when dissipation is not minimized – that is, when $\lambda=0$.
In other words, quenching the Hamiltonian to
$H_{0}=\epsilon_{\tau}\sigma_{z}/2$ at time $t=0$ and relaxing the system to
equilibrium is the best protocol to bring the qubit as close as possible to
the ground state. We conjecture that the quench protocol, namely, thermal
relaxation, is the optimal protocol in terms of erasure reliability. Note that
the conjecture is restricted to the $\varrho_{0}=\mathbb{I}/2$ case because
the shortcut-to-equilibration protocol Dann  _et al._ (2019) or an optimal
protocol Sup  may outperform the quench protocol for the
$\varrho_{0}\neq\mathbb{I}/2$ case. When the initial state is not described by
the maximally mixed state and contains some coherence, we also find that the
optimal protocol hardly produces quantum coherence as compared to the non-
optimal protocol (see the SM Sup  for details). From the analytical and
numerical evidence, we can conclude that the creation of quantum coherence
should be avoided when erasing information.

Figure 2: Numerical illustration of the bound Ineq. (3) with the (a) optimal
and (b) non-optimal protocols. The upper panels show the average dissipated
heat $\beta Q$, the derived bound $\Delta
S+\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}/(2\tau\overline{\gamma}_{\tau})$, and
the Landauer cost $\Delta S$ for each operational time. The lower panels show
the amount of quantum coherence generated in the energy eigenbasis. The
operational time $\tau$ is varied, while the other parameters are set to
$\alpha=0.2$, $\beta=1$, $\epsilon_{0}=0.4$, and $\epsilon_{\tau}=10$.

Next, we investigate the performance of the bound Ineq. (3) with optimal and
non-optimal protocols. We vary the operational time and plot the dissipated
heat $\beta Q$, the derived bound $\Delta
S+\|\varrho_{0}-\varrho_{\tau}\|_{1}^{2}/(2\tau\overline{\gamma}_{\tau})$, the
Landauer cost $\Delta S$, and the coherence term
$\overline{\gamma}_{\tau}^{\rm R}\mathscr{C}_{\tau}^{2}/(2\tau)$ as functions
of $\tau$ in Fig. 2. Notice that the dissipated heat is always bounded from
below by the bound Ineq. (3) and is far beyond the Landauer cost.
Particularly, in the case of the optimal protocol, the derived bound is tight
and asymptotically saturated as $\tau$ increases. The optimal protocol is also
less dissipative than the non-optimal protocol for all operational times.
Simultaneously, the optimal protocol does not create coherence, whereas a
positive amount of coherence is generated in the non-optimal protocol.
Regarding the tightness of the derived bounds, Ineq. (3) can be tighter or
looser than Ineq. (10). If little or no coherence is produced, the former is
generally tighter than the latter. Conversely, when a large amount of
coherence is generated – that is, when $\mathscr{C}_{\tau}\gg 1$ – the latter
is stronger than the former (see the SM Sup  for numerical illustrations).

Summary.—We derived the lower bound on the thermodynamic cost associated with
finite-time information erasure for Markovian open quantum dynamics. The bound
is far beyond the Landauer cost for fast control protocols. We also revealed
the relation between quantum coherence and heat dissipation for the entire
driving speed regime, stating that the creation of quantum coherence
inevitably causes additional heat costs. In the context of the Landauer
principle, this relation implies that quantum coherence is detrimental to
erasing information from an energetic viewpoint. We confirmed the results with
both optimal and non-optimal protocols. Our findings are not only
fundamentally critical but also helpful in establishing a design principle for
efficient memory erasure. The generalization of the results obtained here to
other cases, such as the finite-size environments and the non-Markovian regime
Rivas (2020), is a future study.

###### Acknowledgements.

Acknowledgments.—We are grateful to K. Funo and H. Tajima for the fruitful
discussion. We also thank K. Brandner for telling us about his study on
quantum heat engines. This work was supported by Grants-in-Aid for Scientific
Research (JP19H05603 and JP19H05791).

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