This is the accepted manuscript made available via CHORUS. The article has been
published as:

Universal Scaling Bounds on a Quantum Heat Current
Shunsuke Kamimura, Kyo Yoshida, Yasuhiro Tokura, and Yuichiro Matsuzaki
Phys. Rev. Lett. 131, 090401 — Published 31 August 2023
DOI: 10.1103/PhysRevLett.131.090401

Universal Scaling Bounds on a Quantum Heat Current
Shunsuke Kamimura,1, 2, ∗ Kyo Yoshida,1 Yasuhiro Tokura,1, † and Yuichiro Matsuzaki2, ‡1

Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan
2
Research Center for Emerging Computing Technologies,
National Institute of Advanced Industrial Science and Technology (AIST),
1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan.

In this Letter, we derive new bounds on a heat current flowing into a quantum L-particle system
coupled with a Markovian environment. By assuming that a system Hamiltonian and a systemenvironment interaction Hamiltonian are extensive in L, we prove that the absolute value of theheat current scales at most as Θ(L3 ) in a limit of large L. Furthermore, we present an example of noninteracting particles globally coupled with a thermal bath, for which this bound is saturated in termsof scaling. However, the construction of such a system requires many-body interactions induced by
the environment, which may be difficult to realize with the existing technology. To consider morefeasible cases, we consider a class of the system where any non-diagonal elements of the noise operator(derived from the system-environment interaction Hamiltonian) become zero in the system energy
basis, if the energy difference exceeds a certain value ∆E. Then, for ∆E = Θ(L0 ), we derive anotherscaling bound Θ(L2 ) on the absolute value of the heat current, and the so-called superradiancebelongs to a class for which this bound is saturated. Our results are useful for evaluating thebest achievable performance of quantum-enhanced thermodynamic devices, including far-reaching
applications such as quantum heat engines, quantum refrigerators and quantum batteries.

Quantum mechanics successfully describes thecounter-intuitive behaviors of microscopic objects,
which cannot be explained by classical theory. Recently,
considerable effort has been devoted toward engineeringlarge quantum systems while sustaining quantum effects
such as entanglement and coherence. Quantum technology is a new field to seek novel industrial applications
utilizing such quantum properties.
To quantify the performance of quantum devices, wetypically consider its scaling in a limit of a large numberof qubits, and we compare quantum and classical methods with equal resources (e.g., using the same amount oftime). Shor demonstrated that a fault-tolerant quantum
computer can solve factorization problems exponentially faster than the best-known classical algorithm [1].
Quantum metrology exploits the non-classical properties of probe qubits to measure external fields withhigher sensitivity. The estimation uncertainty is known
to decrease by L−1/2 with L classical probes; in principle, the uncertainty can decrease by L−1 with L entangled qubits [2–4].
Meanwhile, since the Industrial Revolution, thermodynamics has been conventionally adopted to describe
the macroscopic behaviors of classical systems. Its extension to fluctuating non-equilibrium classical small
systems, referred to as stochastic thermodynamics, hasbeen studied extensively in recent decades [5]. Specifically, stochastic thermodynamics can provide a richer
set of limitations on heat engine performances, as exemplified by fluctuation theorems [6–9] and trade-off relations [10, 11].
Quantum thermodynamics refers to the extension ofthermodynamics to quantum systems [12–15]. Quantum versions of heat engines [16–18] and batteries [19]have been proposed, where the device is typically mod-eled as an open quantum system. Several experimental demonstrations of such quantum thermodynamic
devices have been reported [20–24]. As with otherquantum technologies, scaling advantages in the performances of heat engines [25–34] and quantum batteries [35–41] have been demonstrated for specific models.
Pioneered in the seminal work for the discovery of superradiance [42], many previous studies regarding a scalingenhancement (with the system size) of a heat current
have been reported, which plays a critical role in theadvantage of quantum thermodynamic devices. Despite
recent discoveries such as a quadratic heat current ina high-temperature environment [25, 26] or interacting
particles [27, 31] and more theoretical studies behindthese systems [29, 32], universal scaling bounds on aheat current for arbitrary open quantum systems remain unknown.
In this Letter, we derive new bounds on a heat current J(t) flowing into an open quantum system. We
consider an L-particle system whose Hamiltonian is extensive with respect to L. A straightforward approachfor generating the heat current is the parallel use of
L particles, where each particle is individually coupledwith the environment. We refer to this approach as aparallel scheme. For this scheme, the heat current scales
as |J(t)| = Θ(L). Throughout this Letter, a functionf (L) is written as f (L) = Θ (g(L)) if there exist constants k1 , k2 and L0 , such that k1 g(L) ≤ f (L) ≤ k2 g(L)is satisfied for any L ≥ L0 . Meanwhile, in a quantum scheme, the particles are collectively coupled withthe environment (see Fig. 1). For fair comparison ofa quantum scheme and a parallel scheme, we assume
that a system-environment interaction Hamiltonian isextensive as well. According to the convention of quantum thermodynamics, we specifically focus on an open

2<latexit sha1_base64="orBgV94rnvnOiwm5OjZmpwhUm/U=">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</latexit><latexit sha1_base64="loPTCc9bT4qzG/0qbGj5gKS1spw=">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</latexit>

NB

Ja (t)⋯<latexit sha1_base64="cUqCDs7PSLX3KVMK2MPaFYz7nCU=">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</latexit><latexit sha1_base64="4vSiPUFK4RZg3Kc/sB0khPl6Sas=">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</latexit>

J1 (t)JNB (t)⋯<latexit sha1_base64="6ovOoUtzzlsOmfjHIyTM8HCqGmM=">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</latexit>

⋯

1

a

⋯

<latexit sha1_base64="Wvzm0AdERH7i8XZ8Ctz+vXLbDF0=">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</latexit>⇥L<latexit sha1_base64="of0UkNFxnUqhrS1DWd4kQ6rmKuU=">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</latexit>

quantum scheme

parallel scheme

FIG. 1. Schematic for a quantum scheme for heat current
generation realized in a system composed of L particles surrounded by NB baths. For a parallel scheme, the L particles
are used in parallel.

Further, we assume that these Hamiltonians are time(a)(a)(a)
independent. Let ρ̂B = e−βa ĤB /Tra [e−βa ĤB ] denotea thermal equilibrium state of the a-th bath with an inNNB (a)ρ̂B denote a
verse temperature βa , and let ρ̂B = a=1
product state of these thermal equilibrium states. Here,
Tra is a partial trace over the degrees of freedom of the a(a)th bath. For the interaction Hamiltonian Ĥint , we intro(a)duce Hermitian noise (bath) operators {Âk }k=1,2,...,ca(a)({B̂k }k=1,2,...,ca ) acting on the degrees of freedom of
the system (a-th bath), where the label k denotes achannel through which the system-bath interaction is
induced, and ca denotes the number of channels. Then,the interaction Hamiltonian can be expressed as follows:(a)Ĥint =
quantum system that is weakly coupled with the environment. Under these conditions, we derive a scaling
bound |J(t)| ≤ Θ(L3 ), and we present an example forwhich it is saturated in terms of scaling. However, themodel requires an L-body interaction, which may bedifficult to realize with the existing technology. Therefore, to derive another bound for more feasible models,
we restrict the class of the system such that the interaction with the environment only induces transitions
between system energy eigenstates whose energy difference is smaller than a certain value ∆E. By mathematically imposing this condition, we obtain another scaling
bound |J(t)| ≤ Θ(L2 ) for ∆E = Θ(L0 ). Furthermore,
we find that in the case of superradiance, which is a wellestablished collective energy emission process observedin light-matter systems [42], this bound is saturated interms of scaling. Our derived bounds universally limit
how a heat current can scale with the system-size, regardless of the choice of the system. Our results areuseful for evaluating the best achievable performance ofquantum-enhanced thermodynamic devices. As an example, we can also construct a new quantum heat engine
with a quantum enhancement.
Model.—We consider a total system composed of aquantum system subject to an environment modeled as
NB heat baths. The total Hamiltonian is expressed asfollows:
Ĥtot = ĤS + ĤB + Ĥint ,
ĤB =

NB
X

(a)

ĤB ,

(1)
(2)

a=1

Ĥint =NB
X

(a)

Ĥint ,

(3)a=1

where ĤS (ĤB ) is a system (an environmental) Hamil(a)tonian, ĤB is a free Hamiltonian of the a-th bath, andĤint is a system-environment interaction Hamiltonianthat is defined as a summation of interaction Hamil(a)tonians Ĥint between the system and the a-th bath.

ca
X
k=1

(a)

(a)

Âk ⊗ B̂k .

(4)

After applying the Born and Markov approximations,
we obtain the Redfield equation as follows: (ℏ = 1) [43]:NB
Xdρ̂S (t)= −i[ĤS , ρ̂S (t)] +Da [ρ̂S (t)],
dt
a=1
Z ∞
ca
X
 (a)
(a)
(a)
Ckl (s) Ãl (−s)ρ̂Âk
Da [ρ̂] =
ds
0

(5)

k,l=1


(a) (a)− Âk Ãl (−s)ρ̂ + h.c.,(6)where ρ̂S (t) denotes a reduced density operator of thesystem (in the Schrödinger picture), and Da denotes adissipator due to the a-th bath. Here, we define a corre(a)(a)(a) (a)lation function as Ckl (s) = Tra [B̃k (s)B̂l ρ̂B ], wherean operator is transformed as Ô 7→ Õ(t) = eiĤ0 t Ôe−iĤ0 t
with respect to the free Hamiltonian Ĥ0 = ĤS + ĤB .
According to the convention of quantum thermodynamics, we define an instantaneous net heat current J(t)from the environment to the system as follows:dρ̂S (t)J(t) = Tr ĤS.
(7)dt
For a multi-bathenvironment, we obtain an additivity
PNB
J(t) = a=1
Ja (t), where a heat current Ja (t) from thea-th bath to the system is defined as follows:
Ja (t) = Tr ĤS Da [ρ̂S (t)] .
(8)
Bound 1.—We can derive a general upper bound on
the absolute value of the heat current as follows [44]:|Ja (t)| ≤ 4∥ĤS ∥

ca
X
k,l=1

(a)

(a)

(a)

Ξkl ∥Âk ∥∥Âl ∥,

(9)

R∞
(a)
(a)
where we introduce a coefficient Ξkl = 0 ds|Ckl (s)|and an operator norm for apgiven operator Ô (induced
by a vector norm ∥ |ψ⟩ ∥ = ⟨ψ|ψ⟩), which is expressed3
Ô|ψ⟩∥
as ∥Ô∥ = sup|ψ⟩ ∥∥|ψ⟩∥. This newly derived bound isapplicable to an arbitrary open quantum system, as far
as the Born and Markov approximations are validated.
Application of Bound 1 to L-particle systems.—We
apply the general bound (9) to a system composed of(generally interacting) L identical particles (such as an
L-qubit system) subject to NB heat baths, and we analyze how a heat current scales with L (see Fig. 1).
Suppose that the form of the system-environment interaction is given as Eq. (4). In particular, we focus on
the case in which the system Hamiltonian is extensive
in L, i.e. ∥ĤS ∥ = Θ(L). Moreover, we assume thatthe interaction Hamiltonian Ĥint is extensive as well,i.e. ∥Ĥint ∥ = Θ(L), which is a natural assumption fora fair comparison of the heat current with that for a(a)parallel scheme. Hence, we assume ∥Âk ∥ = Θ(L/ca ),which means that the distribution of the interaction energy over each channel k is homogeneous. Furthermore,(a)(a)
we assume that ∥B̂k ∥ = Θ(L0 ), Ξkl = Θ(L0 ) and0
NB = Θ(L ), because these quantities are solely determined by the properties of the environment.
ConsePNB
quently, we show that |Ja (t)| and a=1|Ja (t)| scale atmost as Θ(L3 ). Remarkably, this scaling exceeds thatof the superradiance [42], which is known as a collective
emission of photons from L qubits.
In the derivation explained above, we assume thatthe norm of the system-bath interaction scales linearlywith the system size. This assumption is valid for example when a spin ensemble collectively interacts witha cavity where the wavelength of the cavity photon ismuch larger than the distance between the spins [45, 46].
However, when we consider the thermodynamic limit,
it might be difficult for the system-coupling strength
to scale linearly with the system size [47, 48]. Thus,
we consider the case in which the system is not large
enough to consider the thermodynamic limit but is large
enough to observe the collective coupling between thesystem and environment. For example, the collective
effect between a cavity and thousands of superconducting qubits was experimentally observed [46]. Therefore,
our method could be realized up to thousands of qubits.

Here, we present an example of an L-qubit system forwhich the scaling bound |Ja (t)| ≤ Θ(L3 ) obtained from
Eq. (9) is saturated. We introduce a system Hamiltonian as follows:ĤS = ωq Jˆz ,
PL
1

(i)

(i)

(10)

where Jˆz = 2 i=1 σ̂z , and σ̂z is the z-component of
the Pauli operator for the i-th qubit with a frequency ωq .
Then, for Jˆz and an operator Jˆ2 that represents the total angular momentum, we consider a subspace spanned
by Dicke states {|L/2, M ⟩}, which is a simultaneous
eigenstate of Jˆ2 and Jˆz , with eigenvalues L2 L2 + 1
and M , respectively (M = L/2, (L/2) − 1, . . . , −L/2).
Throughout this Letter, we omit the label L/2 of the

Dicke states, i.e., |M ⟩ ≡ |L/2, M ⟩. Explicitly, the Dicke
state |M ⟩ is a superposition state of all the computational states having L2 + M excited states and L2 − Mground states with an equal coefficient, for an odd number L.
For an interaction between the L-qubit system and an
environment modeled by a single bath (i.e. NB = 1), we
consider an m-body interaction (m = 1, 2, . . . , L) [49]. In
particular, we introduce the following system operatorÂ for the interaction Hamiltonian Ĥint = Â ⊗ B̂:gL
σ̂x(1) ⊗ · · · ⊗ σ̂x(m) ⊗ Iˆ(m+1) ⊗ · · · ⊗ Iˆ(L)
Â =
L Cm

+ (all possible permutations) ,
(11)
(i)

where g denotes a constant coupling strength, and σ̂x(Iˆ(i) ) denotes the x-component of the Pauli (identity)
operator for the i-th qubit. Although Â contains a totalL!
terms, we have ∥Â∥ = Θ(L) owingof L Cm = (L−m)!m!

to the normalization by the prefactor; thus, ∥Ĥint ∥ =Θ(L).
Under the Born and Markov approximations, we derive a Redfield equation for the L-qubit system coupledwith a white-noise Markovian bath. Then, for an initial
Dicke state ρ̂S (0) = |L/2⟩⟨L/2|, we obtain an instantaneous heat current J(0) as follows:
J(0) = −γwn ωq

L2 m
,
L Cm

(12)

where γwn denotes a constant dissipation coefficientfor the white-noise environment. For m = L, the interaction Hamiltonian induces a direct transition from
|L/2⟩ (all-excited state) to |−L/2⟩ (all-ground state),and the absolute value of the heat current scales as|J(0)| = Θ(L3 ). Thus, the universal scaling boundΘ(L3 ) becomes saturated. More generally, for the caseof L − m = Θ(L0 ), we can evaluate the scaling as|J(0)| = Θ(L3+m−L ) by using an identity L Cm−1 =m
L−m+1 L Cm . Therefore, in our example, the construction of the interaction for the universal bound Θ(L3 ) to
be saturated requires L-body interactions.
Bound 2.—The previous example requires L-body interactions for the scaling bound to be saturated, whichmight be difficult to realize with the existing technology. Therefore, we consider more feasible cases here. In
practical situations, the transfer energy due to an interaction with an environment should be upper-bounded
by a threshold value, because the transitions should
occur between not-too-distant energy levels of the system Hamiltonian. In particular, we focus on a specificclass of the system by setting a constraint that any nondiagonal elements of the noise operators become zero in
the system energy basis, if the energy difference exceeds(a)a certain value ∆Ek for channel k of the a-th bath.

4

(a)
(a)
|Ei − Ej | > ∆Ek ⇒ ⟨i| Âk |j⟩ = 0| J(t) | ≤ Θ(L p)pMathematically, this reads as follows:∀i, j,(13)

Prohibited by Bound 1d2
ounB
Our proposal
by
ed
ibit
h
o
Pr

3
<latexit sha1_base64="/YfEerUTJrUTSupZPqd9p7HXNi4=">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</latexit>

where we define a spectral
PNdecomposition of the system
Hamiltonian as ĤS = i=1 Ei |i⟩⟨i|, and N is the dimension of the Hilbert space of the system. Note thatthe system is allowed to be degenerate, and a similarassumption was made in Ref. [50]. Under this condition, by using a relation that was also used in Ref. [50],we rigorously derive another new bound on the absolutevalue of the heat current as follows [51]:|Ja (t)| ≤ 2

ca
X
k,l=1

(a)

(a)

(a)

(a)

Ξkl ∆Ek ∥Âk ∥∥Âl ∥.

(14)

(a)

Therefore, if we assume that ∆Ek = Θ(L0 ) (∀a, k),then |Ja (t)| scales at most as Θ(L2 ), and this provides
a different scaling bound from that provided by Eq. (9).
Moreover, owing to the constraint, we can discuss more
realistic cases for the bound to be saturated without
many-body interactions.
Bound 2 for superradiance and superabsorption.—We
discuss how our results can be applied to non-interacting
qubits coupled with a common bath. For L qubits thatare coupled with a single bath having a single channel (i.e., NB = 1 and ca = 1), a system Hamiltonian
ĤSR and an interaction Hamiltonian Ĥint are respectively given as follows:
ĤSR = ωq Jˆz ,Ĥint = 2g Jˆx ⊗ B̂,

(15)
(16)

PL
(i)
where Jˆx = 12 i=1 σ̂x , ωq is a qubit frequency, B̂ isa bath operator, and g is a coupling strength between
the L-qubit system and the environment. For this system, the condition of Eq. (13) is satisfied for ∆E = ωq ;Consequently, ∆E = Θ(L0 ). Then, using Eq. (14), we
obtain |Ja (t)| ≤ Θ(L2 ).
We compare our results with those of previous studies
on superradiance [52]. Suppose that L is an odd number. Under a Redfield equation derived for the system ofsuperradiance with a zero temperature bath, we obtaina heat current for an initial state |1/2⟩ as follows:1
J(0) = − γ0 ωq (L + 1)2 ,4(17)

where γ0 denotes a constant dissipation coefficient. Thismeans that superradiance saturates the scaling boundΘ(L2 ) obtained from Eq. (14). (see Fig. 2 for a summaryof our results).
Furthermore, we investigate the case of superabsorption, which is interpreted as the reverse process of superradiance [53]. The system Hamiltonian is expressed
as follows:
ĤSA = ωq Jˆz + ΩJˆz2 .

(18)

2<latexit sha1_base64="VJompTfBW9Luv2wYEhWg1yY+a0o=">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</latexit>

Superradiance0<latexit sha1_base64="OnM4oA/LSWgikEjjp39Pv5HrYns=">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</latexit>

ΔE = Θ(L q)

1<latexit sha1_base64="Cq8WyT4Q9pGOSTcgBeZWbQGrhbo=">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</latexit>qFIG. 2. Scaling bounds on a heat current J(t) flowing into
an L-particle open system. The light gray (dark gray) region
represents the prohibited scaling by Bound 1 (Bound 2). For
simplicity, we consider the case of a single bath and introduce ∆E = maxk ∆Ek . Superradiance causes Bound 2 to besaturated, whereas our proposal of using an L-body interaction causes Bound 1 to be saturated. Here, we assume theextensivity of the system Hamiltonian, i.e., ∥ĤS ∥ = Θ(L).

A critical component of the superabsorption is the additionally introduced term ΩJˆz2 . Combined with an engineering of a spectral structure of the environment,
the system exhibits an energy absorption process from|−1/2⟩ to |1/2⟩ with a rate of Θ(L2 ). This is superabsorption. However, the superabsorption does not cause
the bound in Eq. (14) to be saturated in terms of scaling about L. This is because the interaction induces atransition from |−L/2⟩ to |−(L/2) + 1⟩, and the energy
difference between these two eigenstates is ωq +(L−1)Ω.
To use the constraints (13), we must set ∆E = Θ(L)for this model. Thus, Eq. (14) provides an upper boundthat scales as Θ(L3 ), which cannot be saturated by superabsorption.
Bound on a steady-state heat current.—Although, sofar, we have discussed the upper bounds on a heat current for a given initial state, we can also derive an
upper bound on the heat current in a steady state.
However, it is difficult to define a steady state for the
Redfield equation due to the counter-rotating terms.
Thus, we adopt a rotating-wave approximation (RWA)
for the Redfield equation, and we obtain the following Gorini–Kossakowski–Sudarshan–Lindblad (GKSL)
S (t)= L[ρ̂S (t)] [43]:master equation dρ̂dt

L[ρ̂] = −i[ĤS + ĤLS , ρ̂] +

NB
X
a=1

Da(G) [ρ̂].

(19)Here, the Lamb shift term ĤLS satisfies [ĤS , ĤLS ] = 0,

5(G)

environment spectrum, both of which are expected to improve the performance of quantum thermodynamicca
X X
1devices such as quantum heat engines, quantum refrig(a)†
(a)
(a)
(a)
(a)†
Da(G) [ρ̂] =γkl,ω Âl,ω ρ̂Âk,ω − {Âk,ω Âl,ω , ρ̂} ,erators, and quantum batteries.
2
ω k,l=1
We thank A. Yoshinaga for fruitful discussions. This
(20)
work was supported by MEXT’s Leading Initiative for
Excellent Young Researchers, JST COI-NEXT program
where {Â, B̂} = ÂB̂+ B̂ Â is an anti-commutator. Then,(No. JPMJPF2014) and JST PRESTO (No. JPfor the GKSL equation, we define a steady state ρ̂ss for
MJPR1919), Japan. Y. T. acknowledges support froman initial state ρ̂S (0) as follows:
JSPS KAKENHI (No. 20H01827) and JST’s Moonshot
Lt
R&D (No. JPMJMS2061).
ρ̂ss = lim e [ρ̂S (0)].
(21)
and the dissipator Da

is defined as follows:t→∞First, by using the commutation relationship(a)(a)[ĤS , Âl,ω ] = −ω Âl,ω (∀l, ω, a) that is satisfied fora microscopically derived GKSL equation, we showthat [ĤS , ρ̂ss ] = 0 when [ĤS , ρ̂S (0)] = 0. Second, weshow that, if [ĤS , ρ̂] = 0 is satisfied, a heat current(G)(G)
Ja (ρ̂) = Tr(ĤS Da [ρ̂]) calculated by the GKSL
master equation (after the RWA) is the same as that
calculated by the Redfield equation (before the RWA)
Ja (ρ̂) = Tr(ĤS Da [ρ̂]). Therefore, we can use the righthand sides of Eqs. (9) and (14) as the upper bounds on(G)
the steady-state heat current Jss,a = Tr(ĤS Da [ρ̂ss ]),which were originally derived for the Redfield equation.
Moreover, we find a specific system to show a steadystate heat current of |Jss | = Θ(L3 ), which saturates
Bound 1 in terms of scaling. Consequently, we canconstruct new quantum-enhanced thermodynamic
devices such as a quantum heat engine whose power
output P scales as P = Θ(L3 ) while its efficiency being
fixed [54].
Conclusion.—In this Letter, we discussed newly derived bounds on a heat current flowing into an open
quantum system weakly coupled with an environment.
First, we derived a general scaling bound (with the number of particles) on the absolute value of the heat current
by assuming the extensivity of a system Hamiltonian
and a system-environment interaction Hamiltonian. In
particular, we found that the best achievable scaling
for the L-particle system is Θ(L3 ) in a limit of large
L. However, for the scaling bound to be saturated inour example, an L-body interaction is required, which
may be difficult to realize with the existing technology.
Then, we derived another bound based on the constraintthat any non-diagonal elements of the noise operators
become zero with respect to the system energy basis, if
the energy difference of these two bases exceeds a certain value ∆E. Based on this second bound, we showedthat the absolute value of the heat current scales atmost as Θ(L2 ) for ∆E = Θ(L0 ), and this bound is saturated for superradiance. We first revealed the bounds
that universally limit how fast a heat current generated
by an open L-particle quantum system can scale with
the number of particles. Our results are applicable not
only to an open quantum system involving an interaction between particles but also to various types of the∗s2130043@s.tsukuba.ac.jp
tokura.yasuhiro.ft@u.tsukuba.ac.jp‡
matsuzaki.yuichiro@aist.go.jp[1] P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” (Proceedings 35th annual symposium on foundations of computer science,1994) pp. 124–134.
[2] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert,
M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865
(1997).
[3] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011).
[4] C. L. Degen, F. Reinhard, and P. Cappellaro, Rev.
Mod. Phys. 89, 035002 (2017).
[5] U. Seifert, Rep. Prog. Phys. 75, 126001 (2012).
[6] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
[7] G. E. Crooks, Phys. Rev. E 60, 2721 (1999).
[8] T. Hatano and S.-i. Sasa, Phys. Rev. Lett. 86, 3463
(2001).
[9] U. Seifert, Phys. Rev. Lett. 95, 040602 (2005).
[10] N. Shiraishi, K. Saito, and H. Tasaki, Phys. Rev. Lett.
117, 190601 (2016).
[11] P. Pietzonka and U. Seifert, Phys. Rev. Lett. 120,
190602 (2018).
[12] S. Vinjanampathy and J. Anders, Contemp. Phys. 57,
545 (2016).
[13] F. Binder, L. A. Correa, C. Gogolin, J. Anders, and
G. Adesso, Fund. Theor. Phys. 195, 1 (2018).
[14] S. Deffner and S. Campbell, Quantum Thermodynamics: An introduction to the thermodynamics of quantum
information (Morgan & Claypool Publishers, 2019).
[15] A. Auffèves, PRX Quantum 3, 020101 (2022).
[16] R. Alicki, J. Phys. A: Math. Gen. 12, L103 (1979).
[17] H.-T. Quan, Y.-x. Liu, C.-P. Sun, and F. Nori, Phys.
Rev. E 76, 031105 (2007).
[18] P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75,
050102(R) (2007).
[19] R. Alicki and M. Fannes, Phys. Rev. E 87, 042123(2013).
[20] C. Bergenfeldt, P. Samuelsson, B. Sothmann, C. Flindt,and M. Büttiker, Phys. Rev. Lett. 112, 076803 (2014).
[21] K. Zhang, F. Bariani, and P. Meystre, Phys. Rev. Lett.
112, 150602 (2014).
[22] J. P. Pekola, Nat. Phys. 11, 118 (2015).
[23] F. Altintas, A. Ü. C. Hardal, and Ö. E. Müstecaplıoğlu,
Phys. Rev. A 91, 023816 (2015).
[24] J. P. S. Peterson, T. B. Batalhão, M. Herrera, A. M.
Souza, R. S. Sarthour, I. S. Oliveira, and R. M. Serra,†

6Phys. Rev. Lett. 123, 240601 (2019).
[25] A. Ü. C. Hardal and Ö. E. Müstecaplıoğlu, Sci. Rep. 5,
1 (2015).
[26] W. Niedenzu and G. Kurizki, New J. Phys. 20, 113038(2018).
[27] M. Kloc, P. Cejnar, and G. Schaller, Phys. Rev. E 100,
042126 (2019).
[28] G. Watanabe, B. P. Venkatesh, P. Talkner, M.-J.
Hwang, and A. del Campo, Phys. Rev. Lett. 124,
210603 (2020).
[29] H. Tajima and K. Funo, Phys. Rev. Lett. 127, 190604(2021).
[30] M. Kloc, K. Meier, K. Hadjikyriakos, and G. Schaller,
Phys. Rev. Applied 16, 044061 (2021).
[31] S. Kamimura, H. Hakoshima, Y. Matsuzaki, K. Yoshida,
and Y. Tokura, Phys. Rev. Lett. 128, 180602 (2022).
[32] B. Yadin,
B. Morris,and K. Brandner,
arXiv:2206.12639 (2022).
[33] L. da Silva Souza, G. Manzano, R. Fazio, and F. Iemini,
Phys. Rev. E 106, 014143 (2022).
[34] N. Jaseem, S. Vinjanampathy, and V. Mukherjee, Phys.
Rev. A 107, L040202 (2023).
[35] F. Campaioli, F. A. Pollock, F. C. Binder, L. Céleri,
J. Goold, S. Vinjanampathy, and K. Modi, Phys. Rev.
Lett. 118, 150601 (2017).
[36] D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini,and M. Polini, Phys. Rev. Lett. 120, 117702 (2018).
[37] F. Tacchino, T. F. Santos, D. Gerace, M. Campisi, andM. F. Santos, Phys. Rev. E 102, 062133 (2020).
[38] K. Ito and G. Watanabe, arXiv preprint
arXiv:2008.07089 (2020).
[39] J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse,
B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo,
D. G. Lidzey, and T. Virgili, Sci. Adv. 8, eabk3160(2022).
[40] F. Mayo and A. J. Roncaglia, Phys. Rev. A 105, 062203(2022).
[41] Y. Ueki, S. Kamimura, Y. Matsuzaki, K. Yoshida, andY. Tokura, J. Phys. Soc. Jpn. 91, 124002 (2022).
[42] R. H. Dicke, Phys. Rev. 93, 99 (1954).
[43] H.-P. Breuer, F. Petruccione, et al., The Theory ofOpen Quantum Systems (Oxford University Press on
Demand, 2002).
[44] See the subsection II A of Supplementary Material,
which includes Refs. [55–62], for a detailed derivationof Bound 1.
[45] A. Imamoğlu, Phys. Rev. Lett. 102, 083602 (2009).
[46] K. Kakuyanagi, Y. Matsuzaki, C. Déprez, H. Toida,
K. Semba, H. Yamaguchi, W. J. Munro, and S. Saito,
Phys. Rev. Lett. 117, 210503 (2016).
[47] K. Hepp and E. H. Lieb, Phys. Rev. A 8, 2517 (1973).
[48] P. Kirton, M. M. Roses, J. Keeling, and E. G.
Dalla Torre, Adv. Quantum Technol. 2, 1800043 (2019).
[49] See the subsection III A of Supplementary Material fora detailed calculation.
[50] J.-Y. Gyhm, D. Šafránek, and D. Rosa, Phys. Rev.
Lett. 128, 140501 (2022).
[51] See the subsection II B of Supplementary Material fora detailed derivation of Bound 2.
[52] See the subsection III B of Supplementary Material fora detailed calculation.
[53] K. Higgins, S. Benjamin, T. Stace, G. Milburn, B. W.
Lovett, and E. Gauger, Nat. Commun. 5, 1 (2014).
[54] See the subsection IV D of Supplementary Material forsome details.
[55] A. Lenard, J. Stat. Phys. 19, 575 (1978).
[56] S. P. Pedersen, K. S. Christensen, and N. T. Zinner,
Phys. Rev. Res. 1, 033123 (2019).
[57] M. C. Butler and D. P. Weitekamp, Phys. Rev. A 84,
063407 (2011).
[58] E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys.
Rev. 69, 37 (1946).
[59] C. J. Wood, T. W. Borneman, and D. G. Cory, Phys.
Rev. Lett. 112, 050501 (2014).
[60] A. Bienfait, J. Pla, Y. Kubo, M. Stern, X. Zhou, C. Lo,
C. Weis, T. Schenkel, M. Thewalt, D. Vion, et al., Nat.
Nanotechnol. 11, 253 (2016).
[61] G. Agarwal, Phys. Rev. A 4, 739 (1971).
[62] A. A. Houck, D. Schuster, J. Gambetta, J. Schreier,
B. Johnson, J. Chow, L. Frunzio, J. Majer, M. Devoret,
S. Girvin, et al., Nature 449, 328 (2007).