Fundamental limits on low-temperature quantum thermom-
etry with finite resolution
Patrick P. Potts, Jonatan Bohr Brask, and Nicolas Brunner
Department of Applied Physics, University of Geneva, Switzerland
July 2, 2019
While the ability to measure low tempera-
tures accurately in quantum systems is impor-
tant in a wide range of experiments, the pos-
sibilities and the fundamental limits of quan-
tum thermometry are not yet fully understood
theoretically. Here we develop a general ap-
proach to low-temperature quantum thermom-
etry, taking into account restrictions arising
not only from the sample but also from the
measurement process. We derive a funda-
mental bound on the minimal uncertainty for
any temperature measurement that has a fi-
nite resolution. A similar bound can be ob-
tained from the third law of thermodynamics.
Moreover, we identify a mechanism enabling
sub-exponential scaling, even in the regime of
finite resolution. We illustrate this effect in
the case of thermometry on a fermionic tight-
binding chain with access to only two lattice
sites, where we find a quadratic divergence of
the uncertainty. We also give illustrative ex-
amples of ideal quantum gases and a square-
lattice Ising model, highlighting the role of
phase transitions.
A sizable part of contemporary physics is focused on
phenomena that occur at low temperatures [1]. While
for some of these phenomena, finite temperatures con-
stitute an undesirable nuisance (usually in the form of
noise), in other scenarios temperature gradients are at
the heart of the physics under investigation [2, 3]. In
both cases, accurately determining temperature is de-
sirable for characterizing the examined processes and
ultimately leads to an increase in control and a better
understanding of the underlying physics [4, 5].
If the precision is quantified by the relative error,
it is not surprising that thermometry becomes more
challenging as temperature is reduced. However, the
state of the matter is even worse as it turns out that
also the absolute error usually increases as temper-
ature is reduced. Sensitively measuring low temper-
atures is therefore challenging. But how hard is it
exactly?
A number of previous studies considered specific
physical models and found that it is exponentially
hard, in the sense that the absolute error diverges ex-
Patrick P. Potts: patrick.potts@teorfys.lu.se, Formerly known as
Patrick P. Hofer
ponentially at low temperatures [6–17]. In fact, this
exponential scaling can be derived from very general
arguments [18–21], based on the energy spectra of the
sample and the probe, and could therefore appear to
represent a fundamental limit on thermometry. How-
ever, it was recently shown that (using a measure-
ment with infinite resolution) a sub-exponential scal-
ing is possible in a system of strongly coupled har-
monic oscillators [22]. This raises the question of what
the fundamental limits to low-temperature quantum
thermometry really are. In particular, when and how
exponential scaling can be overcome. In addition to
fundamental insight, addressing this question is also
highly desirable given the importance of thermometry
for quantum experiments [23].
With this goal in mind, we present a general ap-
proach to low-temperature quantum thermometry. In
addition to the limitations imposed by the system it-
self, we consider a constraint on the measurement,
namely that measurements can only have finitely
many outcomes (implying finite resolution). Under
this constraint alone, we derive a fundamental bound
on quantum thermometry. Alternatively the same
bound can in fact be recovered from the third law
of thermodynamics, without making any assumptions
on the measurements. More generally, our approach
clearly identifies regimes featuring an exponentially
diverging error, and those where sub-exponential scal-
ing is possible even with finite measurement resolu-
tion. We show that the latter is possible in a physical
system, namely a fermionic tight-binding chain with
access to only two lattice sites. In this case, the un-
certainty diverges only quadratically. Hence the same
scaling as in the model of Ref. [22] is achieved, but
here requiring only finite resolution. Furthermore, we
apply our formalism to illustrative examples of ideal
quantum gases and a square-lattice Ising model, high-
lighting the role of phase transitions.
1 Context and main results
Before we consider bounds that are imposed by lim-
ited experimental access, it is illustrative to consider
the limitations imposed by the system itself. The
resulting bounds are mostly based on the quantum
Fisher information and provide a minimal variance
for temperature estimation (under certain conditions
Accepted in Quantum 2019-06-29, click title to verify 1
arXiv:1711.09827v4 [quant-ph] 30 Jun 2019
Figure 1: Illustration of different low-temperature scalings of the smallest possible error in thermometry. Given a measurement,
an energy can be associated to each outcome, corresponding to the most likely system energy given that outcome. These
energies define a temperature-dependent spectrum E
m
which in general differs from the spectrum of the system Hamiltonian.
Left panel: as long as the ground-state energy does not exhibit a degeneracy lifting, the best achievable measurement error
scales exponentially. Right panel: a degeneracy lifting of the ground-state as a function of temperature leads to polynomial
scaling of the measurement error at low temperatures. Note that for the explicit scaling shown, we assumed that there is no
term linear in T in the lowest gap for both panels.
that are specified below) [18]. For thermal states in
the canonical ensemble, such a bound is derived by
Paris in Ref. [24] (see also Ref. [25]) and reads
δT
2
≥
1
ν
k
B
T
4
h
ˆ
H
2
i − h
ˆ
Hi
2
=
1
ν
k
B
T
2
C
, (1)
where ν denotes the number of independent measure-
ments involved in the estimation,
ˆ
H is the Hamilto-
nian, and C the heat capacity of the system. This
bound holds for all systems and it is interesting
to note that a classical calculation by Landau and
Lifshitz yields the exact same result [26] (see also
App. B). Even though Eq. (1) depends on the system
of interest, we can use the third law of thermodynam-
ics to obtain a bound that is system independent. To
this end, we note that the heat capacity vanishes as
T → 0 (the relations between vanishing heat capaci-
ties and the Nernst postulate are discussed below and
in Ref. [27]). This directly implies that the relative
error δT
2
/T
2
necessarily diverges as T → 0. We note
that Eq. (1) implies that temperature can be mea-
sured precisely whenever the heat capacity diverges.
This can happen at phase transitions and is illustrated
below.
For any system with a finite energy gap ∆ between
the ground and the first excited state, Eq. (1) implies
that the absolute error in any temperature measure-
ment diverges exponentially as absolute zero is ap-
proached (i.e., k
B
T ∆) [24]
δT
2
≥
1
ν
g
0
g
1
k
B
T
4
∆
2
e
β∆
, (2)
where β = 1/(k
B
T ) denotes the inverse temperature
and g
0
(g
1
) the degeneracy of the ground (first ex-
cited) state. The exponentially diverging error is a
consequence of the fact that for temperatures far be-
low the lowest gap, the system is essentially in the
ground state, irrespective of the exact value of T .
Such an exponentially diverging error was found in
various studies on low-temperature thermometry (see,
e.g., [6–17, 19–22]). In particular, Eq. (2) is the rel-
evant bound if a weakly coupled, small probe is used
to determine the temperature of the system (assuming
that the energy of the probe can be measured projec-
tively). In that case, the probe thermalizes with the
system and, due to the weak coupling, is described
by a thermal state determined by the probe Hamil-
tonian. Note that the degeneracy factors in Eq. (2)
imply that it is beneficial to use a probe with a highly
degenerate excited state as discussed in Ref. [21].
The bound in Eq. (2) seems to imply that mea-
suring cold temperatures is exponentially difficult in
general. However, this is not necessarily the case. To
see this, we discuss the limitations of Eq. (2). The first
limitation is the assumption of a finite gap between
ground and excited state. While such a gap is strictly
speaking always present as long as the system is of fi-
nite size, it can be far below any relevant energy scale
(including the resolution of any energy measurement).
For systems which can effectively be described by a
continuous energy spectrum, the relevant bound thus
cannot be given by Eq. (2), as the limit k
B
T ∆
cannot be achieved. This will be illustrated by sev-
eral examples. In this case, one might be tempted
to take a step back and resort to Eq. (1). For elec-
tronic systems for instance, the heat capacity usually
vanishes linearly in T [28]. Equation (1) then implies
that the error δT
2
also vanishes linearly in T . This
brings us to the second limitation of Eqs. (1) and (2):
the assumption of complete accessibility. The bound
in Eq. (1) can be saturated when performing projec-
tive measurements of the system energy (in the large
ν limit) [24, 29]. This requires the possibility to ex-
perimentally distinguish between all (non-degenerate)
eigenstates of the system Hamiltonian. For systems
Accepted in Quantum 2019-06-29, click title to verify 2
that are effectively described by a continuous spec-
trum, this is a hopeless task as it requires an infinite
resolution in energy.
While Eqs. (1) and (2) give lower bounds on any
obtained error, they might thus be of little practical
use when the temperature estimation is bounded by
the experimental access (see also Ref. [30] for the influ-
ence of a limited detector resolution on the obtainable
Fisher information). It is therefore highly desirable to
have bounds on low-temperature thermometry that
take into account limited access to the system. This
includes the interesting scenario of determining tem-
perature using a small probe that is coupled strongly
to a large sample. In this case, the reduced state of
the probe differs from a thermal state described by the
probe Hamiltonian. As shown in Ref. [22], the bound
in Eq. (2) (with ∆ being the gap of the probe), can
then be surpassed. This is possible because the total
system (sample and probe) is gapless. For any total
system that is not gapless, an exponential divergence
according to Eq. (2) cannot be avoided [31].
The main goal of this work is to provide a detailed
investigation of low-temperature thermometry under
limited access, resulting in a number of novel bounds
on the associated measurement error. To this end, we
consider measurements with finitely many outcomes.
One can then assign an energy as well as a proba-
bility to each measurement outcome. These quanti-
ties replace the energies and occupation probabilities
of the Hamiltonian in Eq. (1) and define a spectrum
which determines the error in the temperature esti-
mation. The assumption of finite resolution implies
that this spectrum cannot be treated as gapless. If
the spectrum can be approximated as temperature-
independent at low T , we immediately recover an ex-
ponentially diverging error analogous to Eq. (2), see
Fig. 1 (a). However, the spectrum that is associ-
ated to the measurement is in principle temperature-
dependent. This allows to overcome the exponential
scaling of δT
2
by the following mechanism: Whenever
the ground state exhibits a degeneracy at T = 0 which
is lifted at finite temperatures, the error δT
2
scales
polynomially as T → 0, see Fig. 1 (b). In this work,
we show that such a sub-exponential scaling can be
observed using measurements with finite resolution.
This occurs in a tight-binding chain with measure-
ment access restricted to only two lattice sites, lead-
ing to the scaling δT
2
∝ 1/T
2
. We note that the same
scaling is also obtained in a system of strongly cou-
pled harmonic oscillators using a measurement with
infinite resolution [22].
The tight-binding chain provides an upper bound
on the best possible scaling that can be physically
achieved. Furthermore, we show that the restric-
tion of finite resolution alone results in a fundamental
lower bound on the precision of quantum thermome-
try. Interestingly, this bound turns out to be equiv-
alent to the one implied by the third law of thermo-
dynamics, and is less restrictive than the scaling we
obtain for the tight-binding chain. While the relative
error of any temperature measurement necessarily di-
verges, our bound in principle allows for the absolute
error to vanish in the limit T → 0. It is thus an open
question whether there exists a physical system where
temperature can be determined perfectly using mea-
surements with finite resolution. We note that while
in principle any measurement has a finite resolution,
there may be scenarios where an infinite resolution
can safely be assumed, i.e., where its associated spec-
trum is approximately gapless.
The rest of this paper is structured as follows: in
Sec. 2, we introduce the relevant quantities and the
applied formalism. The low-temperature bounds are
then derived in Sec. 3, identifying the condition un-
der which sub-exponential scaling can be achieved. In
Sec. 4, we illustrate our results considering ideal quan-
tum gases. In Sec. 4.1, we consider finite systems and
show how they behave as infinite systems at elevated
temperatures, illustrating the inadequacy of Eq. (2)
for systems with small gaps. In Sec. 4.2, we discuss a
tight-binding chain with restricted access, where poly-
nomial scaling is achieved. We illustrate the effect of
phase transitions on low-temperature thermometry in
finite systems in Sec. 5. We conclude in Sec. 6 dis-
cussing a number of open questions.
2 Framework for temperature mea-
surements
Here we consider the scenario where temperature
is determined through a general quantum measure-
ment described by a positive-operator valued measure
(POVM) with M elements
ˆ
Π
m
. We note that POVMs
provide the most general measurements in quantum
theory [32]. They include scenarios where auxiliary
systems interact in an arbitrary way with the system
of interest as well as measurements that exploit quan-
tum coherence [6, 9, 10, 33–39]. Each of the POVM
elements corresponds to an outcome m which will be
observed with probability
p
m
= Tr
n
ˆ
Π
m
ˆρ
T
o
=
X
n
e
−βE
n
Z
hn|
ˆ
Π
m
|ni, (3)
where ˆρ
T
denotes the thermal state ˆρ
T
= e
−β
ˆ
H
/Z
with Z = Tr{e
−β
ˆ
H
} and
ˆ
H|ni = E
n
|ni. For sim-
plicity, we consider the canonical ensemble. Ex-
tensions to the grand-canonical ensemble, includ-
ing a temperature-dependent chemical potential, are
discussed in App. A. Throughout, we consider the
POVM elements to be temperature-independent.
The amount of information on temperature that is
encoded in the probability distribution p
m
is given by
Accepted in Quantum 2019-06-29, click title to verify 3
the Fisher information [18]
F
T
=
M−1
X
m=0
(∂
T
p
m
)
2
p
m
. (4)
We note that the Fisher information explicitly de-
pends on the POVM. Through the Cram´er-Rao
bound, the Fisher information gives a lower bound
on the variance of any unbiased estimator for T [40]
δT
2
≥
1
νF
T
, (5)
where ν denotes the number of independent measure-
ment rounds. This variance will serve as our quantifier
for the uncertainty of the temperature measurement.
We now define an energy for each measurement out-
come
E
m
=
1
p
m
Tr
n
ˆ
Π
m
ˆ
H ˆρ
T
o
=
P
n
E
n
e
−βE
n
hn|
ˆ
Π
m
|ni
P
n
0
e
−βE
n
0
hn
0
|
ˆ
Π
m
|n
0
i
,
(6)
which can be interpreted as the optimal guess of the
system energy (before the measurement) given the
outcome m. Note that if the POVM elements are
projectors onto the eigenstates of the Hamiltonian,
then the energies defined in Eq. (6) coincide with the
eigenenergies of the system, i.e. E
m
= E
m
. The
Fisher information can be written as the variance of
the spectrum defined by Eqs. (3) and (6)
F
T
=
1
k
2
B
T
4
M−1
X
m=0
p
m
E
2
m
−
M−1
X
m=0
p
m
E
m
!
2
, (7)
where we used Eq. (4) with
∂
T
p
m
=
E
m
−
M−1
X
n=0
p
n
E
n
!
p
m
k
B
T
2
. (8)
As we discuss in detail below, defining the spectrum
associated to a given POVM allows us to group the
POVMs into different classes. These classes are de-
fined through the thermal behavior of the low-energy
spectrum and they lead to different bounds on the
Fisher information. We stress that the energies E
m
are only a convenient tool. Their definition does in
no way influence the generality of the results.
A measurement independent quantity, the quan-
tum Fisher information (QFI), can be obtained by
maximizing the Fisher information over all possible
POVMs. In the case of determining temperature (in
the canonical ensemble), the optimal POVM corre-
sponds to a projective measurement of the energy of
the system and the QFI is determined by the variance
of the system energy [24]
F
T
=
h
ˆ
H
2
i − h
ˆ
Hi
2
k
2
B
T
4
=
∂
2
β
ln Z
k
2
B
T
4
=
∂
T
S
k
B
T
=
∂
T
h
ˆ
Hi
k
B
T
2
,
(9)
where we have given the QFI in terms of dif-
ferent thermodynamic quantities for future refer-
ence and we introduced the von Neumann entropy
S = −k
B
Tr{ˆρ
T
ln ˆρ
T
}. We note that for thermal
states, the von Neumann entropy unambiguously cor-
responds to the thermodynamic entropy [2, 41].
Using the definition of the heat capacity
C =
δQ
dT
= T ∂
T
S, (10)
where δQ is the infinitesimal amount of heat required
for a temperature change of dT , we can write the QFI
as
F
T
=
C
k
B
T
2
. (11)
We note that the heat capacity is usually defined at
constant volume or pressure. The above relations
hold no matter which quantity is being held constant
as temperature is varied (i.e., if volume is held con-
stant, C denotes the heat capacity at constant vol-
ume and similarly for any other choice). The last
expression implies that temperature can be measured
precisely when the heat capacity diverges. This can
happen at phase transitions, where the heat capacity
exhibits a singular behavior. Using phase transitions
for thermometry has a long history [4, 23, 42] (think
of the historical definition of the Celsius scale using
the freezing and boiling points of water) and is il-
lustrated below with the examples of Bose-Einstein
condensation (where C exhibits a cusp) and the two-
dimensional Ising model (where C diverges at the crit-
ical temperature).
From Eq. (10), one infers that the heat capacity
vanishes as T → 0 as long as the derivative ∂
T
S
remains finite. While the Nernst postulate only en-
forces the entropy to be constant at T = 0 (implying
∂
X
S|
T =0
= 0) the assumption of a finite derivative
∂
T
S is often included in the third law of thermody-
namics [27]. This implies the following bound on the
QFI
lim
T →0
T
2
F
T
→ 0. (12)
It follows that the relative error δT
2
/T
2
in any tem-
perature measurement has to diverge as absolute
zero is approached [cf. Eq. (5)]. In addition to the
unattainability principle of reaching absolute zero
[43, 44], the third law thus implies an unattainability
principle for precisely measuring temperatures close
to absolute zero. We note that idealized scenarios
(e.g. a free particle not constrained by any container)
may result in finite heat capacities at T = 0. Equa-
tion (12) constitutes our first and weakest bound on
low temperature thermometry. In the next section,
we will give stronger bounds which are based on re-
strictions on accessibility.
We note that all considerations that are measure-
ment independent, i.e. all above considerations based
Accepted in Quantum 2019-06-29, click title to verify 4
on the QFI, can be reproduced by a classical calcu-
lation following Landau and Lifshitz [26]. For com-
pleteness, this derivation is reproduced in App. B.
3 Bounds on low temperature ther-
mometry
To investigate bounds on low-temperature thermom-
etry, we consider an arbitrary POVM with finitely
many outcomes. This restriction is discussed in more
detail at the end of this section and it implies that
the spectrum E
m
is discrete. To find bounds on the
Fisher information, we write the formal solution to
Eq. (8) as
p
m
=
e
R
dT ∆
m
(T )/(k
B
T
2
)
P
M−1
n=0
e
R
dT ∆
n
(T )/(k
B
T
2
)
, (13)
where ∆
m
= E
m
− E
0
≥ 0. We now show how the
low-temperature behavior of the spectrum E
m
leads
to different bounds on the error in temperature esti-
mation.
3.1 No degeneracy splittings
Let us first consider the case where the spectrum E
m
can be approximated as temperature independent at
low T . In this case, the probabilities reduce to the
Boltzmann form
p
m
=
g
m
e
−β∆
m
P
n
g
n
e
−β∆
n
, (14)
where the factors g
m
are the temperature-
independent integration constants in Eq. (13).
They are determined by Eq. (3). Without loss of
generality, we can absorb all degeneracies in the
factors g
m
. Relabeling the energies then results
in a non-degenerate spectrum where ∆
j
> ∆
i
for
j > i. Inserting Eq. (14) in Eq. (7) results in the low
temperature behavior
F
T
=
g
1
g
0
∆
2
1
k
2
B
T
4
e
−β∆
1
+ O(e
−βmin(2∆
1
,∆
2
)
/T
4
). (15)
The temperature measurement is then necessarily ac-
companied by an exponentially divergent error as
T → 0 in complete analogy to Eq. (2) with the only
difference that the gap ∆
1
might be unrelated to and
much bigger than the lowest gap in the spectrum of
the system Hamiltonian. In case the implementable
POVMs are far from the optimal one, the tempera-
ture measurement might perform considerably worse
than predicted by Eqs. (1) and (2). This is of partic-
ular relevance when the optimal POVM requires an
infinite resolution.
Next, we relax the assumption of temperature in-
dependent energies and we assume that the E
m
can
be Taylor expanded around T = 0
∆
m
= ∆
m,0
+ k
B
∞
X
l=1
∆
m,l
T
l
. (16)
In this case, the probabilities read
p
m
=
g
m
T
∆
m,1
e
−β∆
m,0
e
P
∞
l=1
∆
m,l+1
T
l
/l
P
n
g
n
T
∆
n,1
e
−β∆
n,0
e
P
∞
l=1
∆
n,l+1
T
l
/l
, (17)
where g
m
T
∆
m,1
is a dimensionless quantity. We note
that by virtue of Weierstrass’ approximation theorem
[45], any continuous ∆
m
(T ) can be approximated to
arbitrary precision by Eq. (16). To investigate the
scaling of the Fisher information, we implicitly as-
sumed that the Fisher information is a continuous
function of T , implying that also ∆
m
are continuous
functions, justifying the use of Eq. (16).
We now consider the case where no degeneracies
are lifted as a function of temperature [i.e., ∆
i
(0) =
∆
j
(0) ⇒ ∆
i
(T ) = ∆
j
(T )]. In this case, we can again
absorb all degeneracies in the factors g
m
and consider
a gapped, non-degenerate spectrum where ∆
j
> ∆
i
for j > i. We note that we only consider temperatures
that are low enough such that no level crossings have
to be taken into account. Inserting Eqs. (16) and
(17) in Eq. (4) and only keeping the largest term in
the limit T → 0 then results in
F
T
=
g
1
g
0
∆
2
1,0
k
2
B
T
4−∆
1,1
e
−β∆
1,0
+ O(e
−β∆
1,0
/T
3−∆
1,1
).
(18)
We thus see that the temperature dependence of ∆
1
can slightly modify the low-temperature scaling of the
Fisher information. However, we still find an expo-
nentially diverging error.
3.2 Degeneracy splittings
We now turn to the case where a sub-exponential scal-
ing can be observed. To this end, we consider the case
of a ground-state which exhibits a degeneracy split-
ting as a function of temperature. At low tempera-
tures, it is then sufficient to consider the ground-state
manifold. All additional terms of the Fisher informa-
tion vanish exponentially as T → 0. We thus have
∆
m,0
= 0 for all considered energies. As temperature
increases, the ground-state will split into different lev-
els. Let the largest gap that opens up grow with T
l
.
All gaps are then approximated as
∆
j
= k
B
∆
j,l
T
l
+ O(T
l+1
), (19)
i.e., gaps with ∆
j,l
= 0 are neglected and the corre-
sponding energies are absorbed into g
0
. Let us con-
sider the case l > 1. From Eqs. (17), (19), and (7),
we find a polynomial low temperature behavior
Accepted in Quantum 2019-06-29, click title to verify 5
F
T
= T
2(l−2)
X
m
∆
2
m,l
g
m
P
m
g
m
−
X
m
∆
m,l
g
m
P
m
g
m
!
2
+ O(T
3l−5
). (20)
In contrast to the above discussed gapless systems,
where the QFI corresponds to a POVM with an infi-
nite resolution, the sub-exponential nature of Eq. (20)
is accessible under the restriction of finite resolution.
From the last equation, we find that for l = 2, the
Fisher information remains constant at T = 0 imply-
ing that the absolute error of a low temperature mea-
surement does not necessarily diverge. For all higher
l, we find a polynomial divergence of the measurement
error. Finally, let us consider the case l = 1 for which
we find
F
T
=
g
1
g
0
∆
2
1,1
T
∆
1,1
−2
+ O(T
min(2∆
1,1
,∆
2,1
)−2
). (21)
As ∆
1,1
tends to zero, we find that we can obtain
any scaling which respects the third law of thermo-
dynamics, i.e. which fulfills Eq. (12). It is interesting
to note that imposing finite resolution results in the
exact same ultimate scaling that can not be overcome
as the third law of thermodynamics. Equation (21)
thus provides a rigorous lower bound on the best scal-
ing that can be observed with measurements of finite
resolution.
Finally, we consider the case where only the ex-
cited states exhibit a degeneracy splitting as a func-
tion of temperature. At low temperatures, it is suf-
ficient to consider the first excited state. If the first
excited state is p-fold degenerate at T = 0, we have
∆
j,0
= ∆
1,0
for all 0 < j ≤ p. However, ∆
j,1
can
be different for each j. A similar derivation as for
Eq. (18) shows that in this case, the largest term in
the limit T → 0 is still given by Eq. (18). However,
the terms that are dropped are at least of the order
of O(e
−β∆
1,0
/T
4−∆
2,1
).
To summarize, we consider POVMs with finitely
many outcomes. In this case, the only way to
overcome an exponentially diverging error in low-
temperature thermometry is to use a POVM such
that the ground state splits into multiple levels at
finite temperatures [as illustrated in Fig. 1 (b)]. The
thermal variation of the occupation numbers of the
ground-state manifold are then not exponentially
small which allows for more precise thermometry.
This insight allows for designing experimentally acces-
sible POVMs for precise low-temperature thermome-
try. Below we provide a physical example for a system
where the Fisher information scales as T
2
[i.e., l = 3 in
Eq. (20)]. This example shows that sub-exponential
scaling can be obtained under the restriction of finite
resolution. We note that the same scaling was found
in Ref. [22] for a different system and a measurement
with infinitely many outcomes. It remains an open
question if a better scaling can be achieved in a phys-
ical system.
3.3 Relevant low temperature limit
Finite sized systems always exhibit a finite gap be-
tween the ground and the first excited states. There-
fore, Eq. (2) always becomes relevant at sufficiently
low temperatures. However, in many systems of ex-
perimental interest, the gaps between the low-lying
states are by far the smallest energy scales of the prob-
lem and we can predict all experimental observables
by formally letting the gaps go to zero. In this case,
any scaling allowed by Eq. (12) can in principle be
achieved.
Just as a large but finite system may be assumed
to have no gap in the spectrum of the Hamiltonian,
there may be POVMs which have corresponding spec-
tra that, for all practical purposes, can be treated as
gapless. This corresponds to scenarios where temper-
ature is always sufficiently big such that the terms
that are dropped in the bounds discussed in this sec-
tion remain relevant. The relevant low-temperature
limit for the provided scalings is thus
∆ k
B
T ω, ∆
1
. (22)
Here ∆ denotes the lowest gap in the spectrum of
the Hamiltonian, ω stands for any energy scale in the
system that does not vanish as ∆ → 0, and ∆
1
cor-
responds to the lowest gap in the spectrum of the
considered POVM.
4 Examples
In this section, we illustrate our results in a number of
systems. Before turning to a system which exhibits a
sub-exponential scaling of the Fisher information due
to ground-state splitting, we investigate the quantum
Fisher information in finite size systems. In all ex-
amples, we consider a system described by a thermal
state (in the canonical or grand-canonical ensemble)
and we aim to determine the corresponding tempera-
ture.
4.1 Finite vs. infinite systems
Here we illustrate the bounds given in Eqs. (1) and (2)
for finite systems. At sufficiently low temperatures,
we find the universality expressed by Eq. (2), which
holds for any gapped system. However, for sufficiently
small gaps (large systems), the discrete nature of the
spectrum can be neglected and the bound in Eq. (2)
becomes irrelevant for all attainable temperatures.
We focus on non-interacting particles described by
a spectrum of single-particle energies ε
k
and a chemi-
cal potential µ (for a detailed discussion on the quan-
Accepted in Quantum 2019-06-29, click title to verify 6
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
10
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
5
10
15
20
Figure 2: Quantum Fisher information for different systems. In all panels blue (solid) is the exact expression, green (dash-
dotted) the exponential low-temperature limit given [cf. Eq. (27)], and red (dotted) the continuum approximation valid in the
thermodynamic limit (where L → ∞ and ∆ → 0). (a) One dimensional gas of photons contained in a space of length L. The
smallest gap is ∆ = π/L and the first excited state is non-degenerate. (b) Two dimensional gas of non-interacting fermions
of mass m contained in an area L
2
. Here the smallest gap is given by ∆ = π
2
/(4mL
2
) and the first excited state is two-fold
degenerate. (c) Fermionic, one-dimensional tight-binding chain with hopping strength t, periodic boundary conditions, and
N = 500 sites. The smallest gap in this system is given by ∆ = 4πt/N and the first excited state is four fold degenerate
(addition/removal of a right-/left-mover). In all three panels, the QFI diverges as 1/T in the continuum approximation.
Overall, these results show that for temperatures above ∆ the system can be effectively described as being gapless. For
sufficiently small ∆, the exponential low-temperature limit becomes thus irrelevant for all attainable temperatures.
tum Fisher matrix with respect to β and µ, see
Ref. [29]). In this case, the partition function in the
grand-canonical ensemble reads for fermions
ln Z
F
µ
=
X
k
ln
h
1 + e
−β(ε
k
−µ)
i
, (23)
and for bosons
ln Z
B
µ
= −
X
k
ln
h
1 − e
−β(ε
k
−µ)
i
. (24)
From Eq. (57), we find for fermions
F
F
T
=
1
4k
2
B
T
4
X
k
[ε
k
− µ + T ∂
T
µ]
2
cosh
2
[(ε
k
− µ)/(2k
B
T )]
, (25)
and bosons
F
B
T
=
1
4k
2
B
T
4
X
k
[ε
k
− µ + T ∂
T
µ]
2
sinh
2
[(ε
k
− µ)/(2k
B
T )]
. (26)
In the low temperature limit, the last expressions both
reduce to (assuming that ∂
T
µ remains finite as T → 0)
F
T →0
=
g∆
2
k
2
B
T
4
e
−β∆
, (27)
in agreement with Eq. (2). Here ∆ = min
k
|ε
k
− µ|
denotes the smallest gap and g its degeneracy. Note
that for bosons ε
k
− µ > 0, which implies that the
ground-state is always given by zero bosons (i.e. non-
degenerate) while the first excited state contains one
boson. For fermions, the first excited state can either
contain one extra fermion or one extra hole (missing
fermion with ε
k
− µ < 0). Note that the presence of
single-particle energies equal to the chemical poten-
tial imply a degenerate ground-state. However, the
same degeneracy is present in all excited states (we
can always remove or add particles with ε
k
− µ = 0
without changing the energy). This degeneracy there-
fore cancels in the QFI which is the reason that there
is no ground-state degeneracy in Eq. (27).
4.1.1 Photons
As a first example, we consider (polarized) photons in
a container with hard walls and a cubic volume L
d
,
where d denotes the spatial dimension. The single-
particle energies are then given by ε
k
= ck (we set
~ = 1 throughout the paper), where c denotes the
speed of light and the wave number k can take on the
values
k =
π
L
v
u
u
t
d
X
i=1
n
2
i
, (28)
with n
i
being positive integers. From Eq. (26) (with
µ = 0), we find that the low temperature behavior
of the QFI is given by Eq. (27) with g = 1 and ∆ =
c
√
dπ/L.
For a macroscopic system, i.e., in the thermody-
namic limit where L → ∞, the gap ∆ becomes van-
ishingly small and we can replace the sum in Eq. (26)
with an integral. Doing so results in the QFI
F
T
=
L
d
4k
2
B
T
2
Z
d
d
k
(2π)
d
c
2
k
2
sinh
2
[ck/(2k
B
T )]
= η
d
(k
B
L/c)
d
T
d−2
,
(29)
where η
1
= π/3, η
2
= 3ζ(3)/π, and η
3
= 2π
2
/15
with ζ(x) denoting the Riemann-zeta function. As
expected, the QFI scales with the system size L
d
[29].
The QFI for photons in one dimension is plotted in
Fig. 2 (a), together with its low-temperature behav-
ior and the thermodynamic limit. We find that for
temperatures as low as k
B
T & 2∆, the QFI is well
captured by the expression for the thermodynamic
limit given in Eq. (29). This expression diverges for
small temperatures, allowing for precise thermometry
in principle. However, this implies having access to
an infinite system and being able to measure its en-
ergy with infinite precision (i.e., a POVM with infinite
Accepted in Quantum 2019-06-29, click title to verify 7
outcomes). In practice, the precision of any temper-
ature measurement will in this case be limited by the
classical Fisher information of the best experimentally
implementable POVM.
4.1.2 Massive particles
As a next example, we consider a gas of massive, non-
interacting particles in a hard wall container. The
single-particle energies then read ε
k
= k
2
/2m, where
the quantum number k can take on the values given in
Eq. (28). The QFI is then given by Eqs. (26) and (25)
for bosons and fermions respectively. The low tem-
perature limit is again of the form given in Eq. (27),
where the gap ∆ and the degeneracy g depend on the
chemical potential and on the statistics of the parti-
cles (bosonic or fermionic).
The thermodynamic limit of massive particles is
discussed in detail in App. C. For fermions with a
positive and fixed chemical potential, we find that
the QFI in the thermodynamic limit scales as 1/T
at low temperatures, implying a heat capacity which
vanishes linearly in T . This is illustrated in Fig. 2 (b),
where we show the QFI for non-interacting fermions
in two-dimensions with a fixed chemical potential. As
for photons in one dimension, we find a diverging QFI
in the limit ∆ → 0 which provides a good description
of the system for temperatures k
B
T & 2∆.
0 100 200 300 400 500
1.5
1.0
0.5
0.0
0.5
1.0
1.5
Figure 3: Spectrum of fermionic tight-binding chain with lin-
ear approximation for µ = ε. Blue (solid): Correct spectrum
as a function of k. Red (dashed): Linear approximation. For
physical observables, only the partially filled state around the
chemical potential are relevant. This is illustrated by the grey
shading for k
B
T = 0.2t. Here, N = 500.
4.1.3 Tight-binding chain
As a next example, we consider a fermionic, one-
dimensional tight binding chain described by the
Hamiltonian
ˆ
H =
N
X
j=1
εˆc
†
j
ˆc
j
− t
N
X
j=1
ˆc
†
j+1
ˆc
j
+ ˆc
†
j
ˆc
j+1
, (30)
where ˆc
j
annihilates a fermion on site j, ε denotes the
on-site energy (which is the same for all sites), t the
hopping strength, and N the number of sites. We con-
sider periodic boundary conditions (ˆc
j+N
= ˆc
j
), i.e.,
we consider a tight-binding chain that is arranged into
a ring. Below, we will be interested in the thermody-
namic limit N → ∞, where the boundary conditions
have a negligible effect. Diagonalizing Eq. (30) results
in
ˆ
H =
N
X
k=1
ε
k
ˆc
†
k
ˆc
k
, (31)
with the eigenenergies ε
k
= ε−2t cos(k2π/N) and the
eigenmodes
ˆc
k
=
1
√
N
N
X
j=1
e
ijk
2π
N
ˆc
j
. (32)
The QFI can then be evaluated using Eq. (25). For a
fixed chemical potential, the QFI is plotted in Fig. 2
(c).
In the following, we consider a particularly sim-
ple scenario which allows us to make some analyti-
cal progress. We fix the chemical potential at µ = ε
(half-filling) and we focus on the regime k
B
T t. In
this case, all (low-frequency) observables are deter-
mined by the partially filled states which are within
k
B
T around the chemical potential. This allows us to
linearize the single-particle energies ε
k
−µ ' 2tκ with
κ =
(
k
2π
N
−
π
2
, for 1 ≤ k ≤
N
2
3π
2
− k
2π
N
, for
N
2
< k ≤ N.
(33)
The last approximation is illustrated in Fig. 3.
With the help of this approximation, we can write
the QFI in the thermodynamic limit (N → ∞) as
F
T
=
t
2
N
πk
2
B
T
4
Z
∞
−∞
dκ
κ
2
cosh
2
(κt/k
B
T )
=
πk
B
N
6tT
,
(34)
where we extended the integral from the interval
[−π/2, π/2] to [−∞, ∞]. Again, we find that the QFI
is linear in the system size (N) and diverges as 1/T
(just like a gas of massive fermions with a chemical
potential within the spectrum). The QFI in the ther-
modynamic limit is plotted in Fig. 2 (c). As for the
above examples, the QFI in the thermodynamic limit
is only relevant if energy measurements with arbitrary
precision are available.
4.2 Sub-exponential scaling
We consider again the tight-binding chain discussed
in the last section. However, we now focus the case
where one only has access to two sites of the (infinitely
long) chain. In this case, all accessible observables are
encoded in the reduced state that describes the two
accessible sites. For simplicity, we consider the case
where all POVMs that only act on the two accessible
states can be implemented. As discussed in Ref. [46],
Accepted in Quantum 2019-06-29, click title to verify 8
0.000 0.005 0.010 0.015 0.020 0.025
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.2 0.4 0.6 0.8 1.0
1.0
0.5
0.0
0.5
1.0
Figure 4: Comparison between reduced states describing two sites of an infinite, fermionic tight-binding chain. (a) Quantum
Fisher information. Green (dash-dotted): for a weak coupling between the two sites and the rest of the chain, we find the
usual exponential scaling at low temperatures [cf. Eq. (42)]. Blue (solid): if the coupling between the two sites and the rest
of the chain is strong, sub-exponential scaling can be achieved. In this case, the QFI scales as T
2
[cf. Eq. (40)]. (b) Energy
spectrum of the optimal POVM. For weak coupling, the temperature-independent spectrum results in the exponential scaling
of the quantum Fisher information. For strong coupling, the ground-state splits at finite temperatures resulting in gaps that
scale as T
3
. This results in the quadratic scaling of the quantum Fisher information.
the reduced state can be obtained from the covari-
ance matrix. With the help of the linear spectrum
approximation and in the thermodynamic limit, we
find hˆc
†
j
ˆc
j
i = 1/2 (half filling) and
hˆc
†
j
ˆc
j
0
i =
1
π
Z
π/2
−π/2
dκ
cos
κ +
π
2
(j − j
0
)
e
2tκ/k
B
T
+ 1
'
k
B
T
2t
sin
π
2
(j − j
0
)
sinh [πk
B
T (j − j
0
)/(2t)]
,
(35)
where the last equality holds for k
B
T t. Following
Ref. [46], we find the reduced state for two sites (with
j = 1 and j = 2)
ˆρ
2
=
e
−β
P
σ=±
ε
σ
ˆc
†
σ
ˆc
σ
Z
2
, (36)
with the energies
ε
±
= ±k
B
T ln
t sinh[πk
B
T/(2t)] − k
B
T
t sinh[πk
B
T/(2t)] + k
B
T
, (37)
and modes
ˆc
±
=
1
√
2
(ˆc
1
± ˆc
2
) . (38)
Since we consider the scenario where all POVMs that
only act on the reduced state can be implemented, the
relevant bound on the error in a temperature measure-
ment is determined by the QFI of the reduced state
F
T
=
k
2
B
2t
2
sinh
2
[πk
B
T/(2t)]
[πk
B
T cosh[πk
B
T/(2t)] − 2t sinh[πk
B
T/(2t)]]
2
t
2
sinh
2
[πk
B
T/(2t)] − (k
B
T )
2
. (39)
At low temperatures, this reduces to
F
T
=
π
4
k
2
B
18(π
2
− 4)
(k
B
T )
2
t
4
+ O[k
6
B
T
4
/t
6
]. (40)
We thus find that the QFI of the reduced state of an
infinite tight-binding chain vanishes quadratically at
low temperatures.
The optimal POVM for determining temperature
from the state given in Eq. (36) is a measurement
of the occupation numbers of the modes given in
Eq. (38). As discussed in Sec. 2, we can associate
an energy to each measurement outcome. This is
done explicitly in App. D and the resulting energies
are plotted in Fig. 4 (b). At low temperatures, we
find that these energies show a ground state splitting
that scales as T
3
which implies a Fisher information
that is proportional to T
2
[cf. Eq. (20)] in agreement
with Eq. (40). In the present example, the QFI for
the full system diverges implying that temperature
can in principle be measured precisely in the low T
limit. However, the restriction imposed on the mea-
surements (only two sites can be addressed) necessar-
ily implies a diverging error in low-temperature ther-
mometry.
We can interpret the last scenario as having ac-
cess to a probe that consists of two fermionic sites
which are coupled to each other with a hopping term
of strength t. This probe is then attached to a bath
Accepted in Quantum 2019-06-29, click title to verify 9
that consists of N − 2 fermionic sites. The connec-
tion is made at both ends of the probe with hopping
strength t
0
[cf. inset in Fig. 4 (a)]. The above analysis
then corresponds to the case where t
0
= t which cor-
responds to strong coupling between probe and bath
(note that we made the assumption t k
B
T ). It is
illustrative to compare this case to the weak coupling
case t
0
t. Then the reduced state is given by a
thermal state determined by the local Hamiltonian of
the two sites
˜ρ
2
=
e
−β
P
σ=±
˜ε
σ
ˆc
†
σ
ˆc
σ
˜
Z
2
, (41)
with the energies ˜ε
±
= ∓t, and the states given in
Eq. (38). The QFI of this state reads
F
T
=
t
2
2k
2
B
T
4
1
cosh
2
[t/(2k
B
T )]
T →0
−−−→
2t
2
k
2
B
T
4
e
−
t
k
B
T
.
(42)
At low temperatures, it assumes the usual exponen-
tial form [cf. Eq. (27)] with a degeneracy factor of
g = 2. We thus find that strong coupling can out-
perform weak coupling at low temperatures. This is
illustrated in Fig. 4. Interestingly, both for weak and
for strong coupling, we find that it is the same POVM
that is optimal for thermometry. As discussed in de-
tail in App. D and shown in Fig. 4 (b), at weak cou-
pling the energies associated to the measurement out-
comes are temperature independent while at strong
coupling they are temperature dependent. Further-
more, at strong coupling the energies exhibit a split-
ting of the ground state, allowing for the polynomial
scaling of the Fisher information. Since the optimal
POVM only has four outcomes, it can be ruled out
that the observed sub-exponential scaling originates
from an infinite resolution as it does for the QFI of
gapless systems.
We note that a similar quadratic scaling in the
strong coupling case is found in Ref. [22] for a probe
consisting of a harmonic oscillator coupled to a bath of
harmonic oscillators. In this case, the optimal POVM
depends on the coupling strength and has infinitely
many outcomes.
5 Phase transitions
Given the connection between the QFI and the heat
capacity of a system [cf. Eq. (11)], one could hope to
make use of phase transitions, where the heat capac-
ity can show a singular behavior, in order to enhance
the precision of thermometry. Indeed the connection
between critical behavior and metrology was investi-
gated before [42]. Here we investigate two phase tran-
sitions. Bose-Einstein condensation (BEC), where the
heat capacity exhibits a maximum at the critical tem-
perature, and the Ising model, where the heat capac-
ity diverges at the phase transition to a magnetically
ordered state. We note that while phase transitions
can enhance thermometry at the critical temperature,
there seems to be no direct influence on the scaling of
the Fisher information as T → 0.
5.1 Bose-Einstein condensation
To investigate BEC, we consider a non-interacting
Bose gas in three dimensions with a fixed density.
The chemical potential then becomes temperature-
dependent and is determined by
N =
X
k
1
e
β[ε
k
−µ(T )]
± 1
, (43)
where N denotes the number of particles. In the ther-
modynamic limit, where N, L → ∞ with the density
N/L
3
fixed, the Bose gas condenses into the ground
state at the critical temperature
k
B
T
c
=
2π
m
N
L
3
ζ(3/2)
2/3
. (44)
The heat capacity exhibits a cusp at the critical tem-
perature and reads
C
k
B
N
=
15
4
ζ(5/2)
ζ(3/2)
T
T
c
3/2
, for T ≤ T
c
, (45)
and
C
k
B
N
'
3
2
"
1 +
ζ(3/2)
2
7/2
T
c
T
3/2
#
, for T > T
c
.
(46)
Note that the last expression is only approximative
(see Ref. [47] for a detailed discussion).
Figure 5 (a) shows the QFI of a finite system to-
gether with the thermodynamic limit obtained from
Eqs. (45) and (46). While the maximum in the heat
capacity leads to a shoulder in the QFI, the system
is much more sensitive to temperatures which are of
the order of the smallest gap. For the finite system,
we numerically evaluated Eq. (26) with the chemical
potential determined by Eq. (43) with N = 100.
5.2 Ising model
Here we consider a square-lattice Ising model in two
dimensions described by the Hamiltonian
ˆ
H = −J
X
hi,ji
ˆσ
i
z
ˆσ
j
z
, (47)
where the sum goes over nearest neighbors. This
model exhibits a phase transition (in the ther-
modynamic limit) to a magnetically ordered state
[(anti)ferro-magnetic for J > 0 (J < 0)] at the critical
temperature
k
B
T
c
=
2J
ln(1 +
√
2)
. (48)
In the thermodynamic limit, the heat capacity reads
[48]
Accepted in Quantum 2019-06-29, click title to verify 10
0 1 2 3 4 5 6 7 8
0.0
0.5
1.0
1.5
2.0
2.5
0
1
2
3
4
5
6
7
8
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25 30
0.0
0.5
1.0
1.5
2.0
0 10 20 30 40 50
0.0
0.5
1.0
1.5
2.0
Figure 5: Quantum Fisher information for systems exhibiting phase transitions. For both panels, blue (solid) shows the exact
expressions for a finite system, green (dash-dotted) the corresponding low temperature limit [cf. Eq. (27)], and red (dashed) the
thermodynamic limit N → ∞. (a) Non-interacting Bose-gas with N = 100 bosons in three dimensions. At low temperatures,
the QFI follows the exponential behavior while at high temperatures it is well described by the thermodynamic limit (where
∆ ∝ 1/L → 0). The phase transition (Bose-Einstein condensation) gives rise to a shoulder in the QFI. The inset shows the
heat capacity which has a cusp at the critical temperature T
c
in the thermodynamic limit [the small discontinuity is due to
the approximation in Eq. (46)]. (b) Square-lattice Ising model with N = 16 spins. In this model, the QFI at low temperatures
is well captured by the expression for the thermodynamic limit. This is due to the fact that the smallest gap does not vanish
in the thermodynamic limit (as J remains constant). In the thermodynamic limit, both the QFI as well as the heat capacity
diverge at the critical temperature, allowing for precise thermometry at least in principle.
C
k
B
N
=
4
π
J
k
B
T
coth(2J/k
B
T )
2
n
K
1
(z) − E
1
(z) −
1 − tanh
2
(2J/k
B
T )
h
π
2
+ [2 tanh
2
(2J/k
B
T ) − 1]K
1
(z)
io
,
(49)
where N denotes the number of spins and we intro-
duced
z = 2
sinh(2J/k
B
T )
cosh
2
(2J/k
B
T )
, (50)
and the complete elliptic integrals of the first and sec-
ond kind
K
1
(z) =
Z
π/2
0
dφ
1
p
1 − z
2
sin
2
(φ)
,
E
1
(z) =
Z
π/2
0
dφ
q
1 − z
2
sin
2
(φ).
(51)
In Fig. 5 (b) we plot the QFI for a finite Ising lat-
tice together with the thermodynamic limit obtained
from Eq. 49. For the finite system, we considered a
square lattice of N = 16 spins with periodic boundary
conditions in both directions (i.e., a lattice arranged
on a torus). We numerically evaluated the variance
of the Hamiltonian in a thermal state to determine
the QFI [cf. Eq. (9)]. We note that already for such a
small system, the QFI is substantially enhanced at T
c
as compared to the low-temperature approximation.
In this system, the effect of the gapped structure (the
gap is equal to 8J, corresponding to flipping a single
spin) and the phase transition add up to a large peak
in the QFI.
6 Conclusions
We investigated fundamental limitations on low-
temperature quantum thermometry with finite res-
olution. We developed an approach where restric-
tions from both the sample and the measurement are
taken into account. Formally, we associate an energy
as well as a probability to each measurement out-
come, which fully determine the Fisher information
of the corresponding POVM. This leads to a classi-
fication of POVMs, corresponding to different low-
temperature scalings of the measurement error. While
most POVMs lead to an exponentially diverging er-
ror, this is not the case in general. The tempera-
ture dependence of the associated energies can result
in a sub-exponential scaling of the error, whenever
the ground state exhibits a degeneracy at zero tem-
perature which is lifted at finite temperatures. Ex-
tending our results to POVMs with associated spec-
tra that are approximatively gapless, such that the
finite-resolution assumption breaks down, provides an
interesting avenue of future research. We note that
our bounds are no longer valid if the POVM elements
are allowed to become temperature-dependent as in
Refs. [24, 49].
As illustrated in Fig. 1, the introduced spectra pro-
vide a visual tool to distinguish between scenarios
where the measurement error scales exponentially and
Accepted in Quantum 2019-06-29, click title to verify 11
scenarios where a polynomial scaling can be obtained.
Our approach allowed us to show that infinite reso-
lution is not required for observing a sub-exponential
scaling in the measurement error, and provides an in-
tuitive understanding of the origin of this advantage.
In particular, for the specific system of a fermionic
tight-binding chain with access to only two sites, we
find an error that scales as δT
2
∝ 1/T
2
. An identical
scaling was found in Ref. [22], considering a bosonic
probe strongly coupled to an infinite bath of harmonic
oscillators, and a measurement with infinite resolu-
tion. It would be interesting to investigate the re-
sults of Ref. [22] in the light of our approach, and see
whether the requirement of infinite resolution can be
relaxed.
Furthermore, we derived a fundamental bound on
the precision of quantum thermometry assuming only
that measurements have finite resolution. The same
bound can be obtained from a different assumption
based on the third law of thermodynamics. Inter-
estingly, this bound is not saturated by any of the
known examples of realistic quantum thermometry,
and in principle allows for better schemes. A central
question is now to determine the best possible scal-
ing which is physically achievable. This scaling must
lie somewhere in between the lower bound and the
quadratic scaling provided by our explicit example.
In the case a scaling better than quadratic is possible,
we believe that our formalism represents a promising
approach to identify explicit schemes. Intriguingly,
our lower bound in principle allows the absolute er-
ror to vanish as T → 0, and it would be interesting
to see if this is actually possible in a physical system.
We finally note that throughout this paper we quanti-
fied thermometry through the obtained information.
A complementary measure is provided by the distur-
bance of the procedure [50]. The generalization of our
approach to include disturbance is left open for future
work.
Note added. After completion of this manuscript,
we became aware of related work [31], which considers
thermometry on gapless systems by studying coupled
harmonic oscillators.
Acknowledgements. We thank Paul Skrzypczyk
and Mart´ı Perarnau-Llobet for discussions. We ac-
knowledge support from the Swiss National Science
Foundation (Grant No. 200021 169002, and QSIT).
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Accepted in Quantum 2019-06-29, click title to verify 13
A Chemical potential
In this section, we consider systems described by the grand canonical ensemble, i.e, we consider states of the
form
ˆρ
µ
=
e
−β(
ˆ
H−µ
ˆ
N)
Z
µ
, (52)
where [
ˆ
H,
ˆ
N] = 0 and the partition sum reads Z
µ
= Tr{exp [−β(
ˆ
H − µ
ˆ
N)]}. Here we only consider a single
chemical potential. Extensions to multiple particle species, each with their own chemical potential, are straight-
forward. For a temperature-independent chemical potential, we can simply replace
ˆ
H →
ˆ
H − µ
ˆ
N in Sec. 2.
The bounds derived in Sec. 3 are then still valid but the energies E
m
are defined as
E
m
=
1
p
m
Tr
n
ˆ
Π
m
(
ˆ
H − µ
ˆ
N)ˆρ
T
o
. (53)
For the QFI, we find in this case
F
T
=
h(
ˆ
H − µ
ˆ
N)
2
i − h(
ˆ
H − µ
ˆ
N)i
k
2
B
T
4
=
∂
T
h(
ˆ
H − µ
ˆ
N)i
k
B
T
2
=
∂
2
β
ln Z
µ
k
2
B
T
4
=
∂
T
S
k
B
T
=
C
k
B
T
2
. (54)
Therefore, the bound in Eq. (12) still holds. We note that in this case, the optimal POVM is given by a
projective measurement of energy and particle number which leads to temperature-independent E
m
[29].
For a temperature-dependent chemical potential, using the definitions in Eq. (3) and (4), the Fisher informa-
tion can still be written as the variance of the energies associated to the POVM elements [cf. Eq. (7)]. However,
these energies now read
E
m
=
1
p
m
Tr
n
ˆ
Π
m
[
ˆ
H − (µ − T ∂
T
µ)
ˆ
N]ˆρ
T
o
. (55)
Note that they can be temperature dependent even for the optimal POVM which is still given by a projective
measurement of both energy and particle number. This implies that even the QFI for a gapped system can
exhibit a sub-exponential scaling at low temperatures. For this to happen, the lowest energy E
0
needs to be
degenerate at T = 0 and this degeneracy must be lifted at finite temperatures. For the optimal POVM, the
energies are of the form
E
m
= E
m
− (µ − T ∂
T
µ)N
m
, (56)
where E
m
and N
m
are the eigenvalues of
ˆ
H and
ˆ
N. This implies that a sub-exponential scaling can only be
achieved if the states involved in the ground-state degeneracy at T = 0 correspond to different particle numbers.
We therefore find that a temperature-dependent chemical potential together with a particle-number uncertainty
in the ground-state manifold are necessary ingredients for a QFI that scales sub-exponentially. We note that
if the chemical potential can be Taylor-expanded around T = 0, then the ground-state degeneracy is lifted at
least proportionally to T
2
. Thus Eq. (20) applies and the QFI scales at least with T
0
.
A temperature dependent chemical potential also alters Eq. (54) and we find
F
T
=
∂
2
β
ln Z
µ
k
2
B
T
4
−
(∂
T
µ)
2
k
B
T
h
ˆ
Ni =
C
k
B
T
2
+ (∂
T
µ)(∂
T
h
ˆ
Ni). (57)
The QFI is thus directly proportional to the heat capacity if either the chemical potential or the mean particle
number is fixed as a function of temperature.
We close this section with a brief discussion on the scenario where both temperature as well as chemical
potential are fixed but unknown parameters that are to be determined by a measurement. This corresponds to
the problem of multi-parameter estimation where the variances of any measurement are bounded by
δT
2
δT δµ
δT δµ δµ
2
≥
1
ν
F
−1
, (58)
with the QFI matrix given by [29]
F =
F
T
F
T µ
F
T µ
F
µ
, (59)
where
F
T
=
∂
2
β
ln Z
µ
k
2
B
T
4
=
∂
T
S
k
B
T
=
∂
T
h
ˆ
H − µ
ˆ
Ni
k
B
T
2
,
F
µ
= ∂
2
µ
ln Z
µ
=
∂
µ
h
ˆ
Ni
k
B
T
,
F
T µ
=
1
k
B
T
∂
β
1
β
∂
µ
ln Z
µ
=
∂
T
h
ˆ
Ni
k
B
T
.
(60)
Accepted in Quantum 2019-06-29, click title to verify 14
We note that if we have perfect knowledge of µ, then we can set δµ = 0 and we recover the results for a
temperature-independent chemical potential given in Eq. (54). In the next section, we reproduce the above
relations by a classical calculation.
B Classical bound on thermometry
Here we reproduce the calculation on fluctuations of thermodynamic quantities given in Ref. [26]. Interestingly,
the variances found for these fluctuations are exactly the minimal variances given in Eqs. (58)-(60). Intuitively,
the variance of any temperature measurement is bounded from below by the variance in the intrinsic temperature
fluctuations.
As in Ref. [26], we consider a large, closed system with a well defined energy E
t
. The probability of finding
the total system in a given configuration is then determined by the entropy as
w ∝ δ(E − E
t
)e
S
t
/k
B
= δ(E −E
t
)e
(S
0
+∆S
t
)/k
B
, (61)
where S
t
denotes the total entropy and S
0
its maximal value (i.e. ∆S
t
≤ 0). We now consider a small part of
the total system which is still macroscopic in size. We denote this part as the system whereas all the rest of the
total system is denoted as the bath. We make the assumption that the system locally equilibrates on time-scales
that are much faster than all other time-scales of the problem. In this case, we can restrict the analysis to local
equilibrium states which are described by a temperature T , a chemical potential µ, and an entropy S. All
these quantities depend on the energy E and the particle number N in the system. Due to energy and particle
exchange with the bath, the thermodynamic quantities that describe the system will fluctuate. In particular,
the energy deviation from its equilibrium value reads
∆E = −T
B
∆S
B
− µ
B
∆N
B
= −T
B
∆S
t
+ T
B
∆S + µ
B
∆N, (62)
where ∆S
B
and ∆N
B
are the entropy and particle number deviation in the bath. For the last equality, we used
S
t
= S
B
+ S and ∆N
B
= −∆N and all quantities without subscript denote the system. From Eq. (62), we find
∆S
t
= −
1
T
B
(∆E − T
B
∆S − µ
B
∆N) ' −
1
2T
(∆S∆T + ∆N∆µ). (63)
To obtain the second equality, we used T
B
= T + ∆T and µ
B
= µ + ∆µ with
T =
∂E
∂S
N
, µ =
∂E
∂N
S
, (64)
and neglected all higher order terms in ∆X by expanding ∆E (∆T , ∆µ) up to second (first) order in ∆S and
∆N [26].
Taking T and µ as independent variables and using the Maxwell relation
∂
µ
S = ∂
T
N, (65)
where we keep µ constant when taking the derivative with respect to T and vice versa, we find
∆S
t
= −
1
2T
∆T ∆µ
∂
T
S ∂
T
N
∂
T
N ∂
µ
N
∆T
∆µ
. (66)
Identifying N with h
ˆ
Ni, we find that the fluctuations of the total entropy is governed by the QFI given in
Eq. (60). The probability of observing a temperature deviation ∆T and a chemical potential deviation ∆µ from
their equilibrium values is thus given by [cf. Eq. (61)]
w(∆T, ∆µ) ∝ exp
−
1
2
∆T ∆µ
F
∆T
∆µ
. (67)
Thus temperature and chemical potential fluctuate with variances given by the inverse of the QFI matrix. Any
measurement with the aim of determining T and µ must thus result in variances that are strictly bigger as
expressed by the Cram´er-Rao bound in Eq. (58).
Accepted in Quantum 2019-06-29, click title to verify 15
C Thermodynamic limit for massive, non-interacting particles
In this section, we consider massive, non-interacting particles enclosed in a cubic container with volume L
d
,
where d denotes the spatial dimension. The single-particle energies are given by ε
k
= k
2
/(2m), where the
quantum number k can take on the values
k =
π
L
v
u
u
t
d
X
i=1
n
2
i
, (68)
with n
i
being positive integers. Here we are interested in the thermodynamic limit, i.e., the limit L → ∞
(keeping the density finite), where k becomes a continuous variable. The QFI for such a system is given in
Eqs. (26) and (25) for bosons and fermions respectively. In the thermodynamic limit, we can replace the sum
over k with an integral resulting in
F
B
T
=
ν
d
T
2−d/2
√
mk
B
L
π
d
Z
∞
−µ/(2k
B
T )
dx [x + ∂
T
µ(T )/(2k
B
)]
2
(x + µ/(2k
B
T ))
d/2−1
sinh
2
(x)
, (69)
and
F
F
T
=
ν
d
T
2−d/2
√
mk
B
L
π
d
Z
∞
−µ/(2k
B
T )
dx [x + ∂
T
µ(T )/(2k
B
)]
2
(x + µ/(2k
B
T ))
d/2−1
cosh
2
(x)
, (70)
where ν
1
= 1, ν
2
= π, and ν
3
= 2π. As expected, the QFI scales with the volume L
d
[29]. In two dimensions,
the above integrals can be solved analytically yielding
F
B
T
=
mk
B
πT
L
2
Li
2
(e
βµ
) +
µ
k
B
T
ln
1 − e
βµ
−
(µ − T ∂
T
µ)
2
4k
2
B
T
2
[1 + coth (µ/(2k
B
T ))]
−
µ∂
T
µ
2k
2
B
T
4
−
∂
T
µ
k
B
ln [2 sinh(−µ/(2k
B
T ))] ,
(71)
and
F
F
T
=
mk
B
πT
L
2
−Li
2
(−e
βµ
) −
µ
k
B
T
ln
1 + e
βµ
+
(µ − T ∂
T
µ)
2
4k
2
B
T
2
[1 + tanh (µ/(2k
B
T ))]
+
µ∂
T
µ
2k
2
B
T
4
+
∂
T
µ
k
B
[ln [cosh(µ/(2k
B
T ))] + ln(2)] ,
(72)
where Li
2
(x) denotes the polylogarithm of order two. Equation (72) is plotted in Fig. 2 (b).
For a temperature-independent chemical potential µ, we can find the low temperature behavior of Eqs. (71)
and (72) in all dimensions. To this end, we distinguish three regimes.
C.1 µ = 0
In this case, we use
Z
∞
0
dx
x
d
2
+1
sinh
2
(x)
=
1
2
d
2
Γ
d
2
+ 2
ζ
d
2
+ 1
, (73)
and
Z
∞
0
dx
x
d
2
+1
cosh
2
(x)
=
2
d
2
− 1
2
d
Γ
d
2
+ 2
ζ
d
2
+ 1
, (74)
where Γ(x) denotes the gamma function and ζ(x) the Riemann zeta function. With the help of the above
identities, we find
F
B
T
=
ν
d
2
d
2
T
2−d/2
√
mk
B
L
π
d
Γ
d
2
+ 2
ζ
d
2
+ 1
, (75)
and
F
F
T
=
(2
d
2
− 1)ν
d
2
d
T
2−d/2
√
mk
B
L
π
d
Γ
d
2
+ 2
ζ
d
2
+ 1
. (76)
We find that the QFI is proportional to T
d
2
−2
in contrast to photons [cf. Eq. (29)], where we find the scaling
T
d−2
. We note that these expressions for the QFI are valid at all temperatures (but they are restricted to
µ = 0). We further note that for a Bose gas with fixed density, the chemical potential is pinned to µ = 0 below
the critical temperature. Indeed, multiplying Eq. (75) with T
2
/N and setting d = 3, we recover Eq. (45).
Accepted in Quantum 2019-06-29, click title to verify 16
C.2 µ < 0
For negative chemical potentials, we perform a shift x → x − µ/(2k
B
T ) in the integrals of Eqs. (69) and (70).
We then make the approximation
sinh[x + |µ|/(2k
B
T )] ' cosh[x + |µ|/(2k
B
T )] '
1
2
e
x
e
|µ|/(2k
B
T )
, (77)
which is valid in the limit k
B
T |µ|. We further use the identity
Γ
d
2
= 2
d
2
Z
∞
0
dxx
d
2
−1
e
−2x
, (78)
resulting in the QFI
F
T
= ν
d
Γ
d
2
2
d
2
√
mk
B
L
π
d
µ
2
e
−|µ|/(k
B
T )
k
2
B
T
4−
d
2
, (79)
which holds both for bosons and for fermions in the limit k
B
T |µ|. Due to the negative chemical potential,
there is a gap of magnitude |µ| between the ground-state (no particles), and the first excited state (presence
of a single particle). This results in an exponential scaling of the QFI similar to what we found for a discrete
spectrum [cf. Eq. (27)]. The fact that there is a continuum of states above the gap, modifies the scaling in
the denominator [T
4
in Eq. (27) vs. T
4−
d
2
in Eq. (79)] similar to what we found if the excited states exhibit a
temperature dependence [cf. Eq. (18)].
C.3 µ > 0
This regime is exclusively of relevance for fermions since µ ≤ 0 for bosons (in order to have finite occupation
numbers). In the limit k
B
T µ, we assume that the integrand in Eq. (70) is only non-zero for x µ/k
B
T . We
can then extend the integral to go from minus infinity to infinity and approximate x + µ/(2k
B
T ) ' µ/(2k
B
T ).
With the identity
Z
∞
−∞
dx
x
2
cosh
2
(x)
=
π
2
6
, (80)
we then find
F
F
T
=
ν
d
2
d
2
π
2
3
r
m
µ
L
π
d
k
B
µT
. (81)
We thus find a QFI that diverges as 1/T for all dimensions as long as the chemical potential is within the
energy spectrum. This scaling implies a heat capacity that vanishes linearly in temperature as expected for free
fermions [28].
D Thermometry with two fermionic sites: optimal POVM
In this section, we consider the optimal POVM for thermometry given the states in Eqs. (36) and (41). The
first state is obtained by tracing out all but two sites of a tight-binding chain described by the Hamiltonian
in Eq. (30). The second state is obtained by weakly coupling two fermionic sites (coupled to each other with
tunnel coupling t) to a thermal bath. In both cases we focus on half-filling, i.e. µ = ε. Both states are diagonal
in the basis given in Eq. (38). The optimal POVM is thus the same for both states and is given by measuring
the occupation numbers in the diagonal basis. The corresponding POVM elements read
ˆ
Π
0
=
1 − ˆc
†
+
ˆc
+
1 − ˆc
†
−
ˆc
−
,
ˆ
Π
±
= ˆc
†
±
ˆc
±
1 − ˆc
†
∓
ˆc
∓
,
ˆ
Π
2
= ˆc
†
+
ˆc
+
ˆc
†
−
ˆc
−
.
(82)
As discussed in Sec. 2, we can associate probabilities [cf. Eq. (3)] and energies [cf. Eq. (6)] to the POVM
elements. For the reduced state of the tight-binding chain, we find
p
0
= p
2
=
1
4
−
(k
B
T )
2
4t
2
1
sinh
2
[πk
B
T/(2t)]
,
p
±
=
1
2
±
k
B
T
2t
1
sinh[πk
B
T/(2t)]
2
,
(83)
Accepted in Quantum 2019-06-29, click title to verify 17
and
E
0
= E
2
=
(k
B
T )
3
t sinh[πk
B
T/(2t)]
πk
B
T cosh[πk
B
T/(2t)] − 2t sinh[πk
B
T/(2t)]
t
2
sinh
2
[πk
B
T/(2t)] − (k
B
T )
2
,
E
±
= ∓
(k
B
T )
2
t sinh[πk
B
T/(2t)]
πk
B
T cosh[πk
B
T/(2t)] − 2t sinh[πk
B
T/(2t)]
t sinh[πk
B
T/(2t)] ± k
B
T
.
(84)
With the help of Eq. (7), we recover the QFI in Eq. (39) which confirms that the POVM in Eq. (82) is indeed
optimal for determining temperature if one only has access to the reduced state. At low temperatures, the
probabilities and energies reduce to
p
0
= p
2
=
1
4
−
1
π
2
+ O[(k
B
T/t)
2
],
p
±
=
1
2
±
1
π
2
+ O[(k
B
T/t)
2
],
(85)
and
E
0
= E
2
=
2π
2
3(π
2
− 4)
(k
B
T )
3
t
2
+ O[(k
B
T )
5
/t
4
],
E
±
= ∓
π
2
3(π ± 2)
(k
B
T )
3
t
2
+ O[(k
B
T )
5
/t
4
].
(86)
We thus find that the energies are degenerate at T = 0. At finite temperatures, gaps open up that scale as
T
3
, giving rise to the polynomial low temperature behavior of the quantum Fisher information discussed in the
main text and illustrated in Fig. 4.
For the weakly coupled system, we find the probabilities
p
0
= p
2
= (2 + 2 cosh[t/(k
B
T )])
−1
,
p
±
=
h
e
∓t/(k
B
T )
+ 1
i
−2
,
(87)
and the energies
E
0
= E
2
= 0, E
±
= ε
±
= ∓t. (88)
With the help of Eq. (7), we recover the QFI in Eq. (42). Since the above energies are temperature independent,
we find an exponential scaling of the Fisher information at low temperatures in accordance with Eq. (15).
Accepted in Quantum 2019-06-29, click title to verify 18