Smooth Metric Adjusted Skew Information Rates
Koji Yamaguchi1 and Hiroyasu Tajima2,3
1

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

2

Department of Communication Engineering and Informatics, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu,
Tokyo, 182-8585, Japan

arXiv:2211.12522v3 [quant-ph] 8 May 2023

3

JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

Metric adjusted skew information, induced from quantum Fisher information,
is a well-known family of resource measures in the resource theory of asymmetry. However, its asymptotic rates are notvalid asymmetry monotone since it has an
asymptotic discontinuity. We here introduce a new class of asymmetry measureswith the smoothing technique, which weterm smooth metric adjusted skew information. We prove that its asymptotic supand inf-rates are valid asymptotic measures in the resource theory of asymmetry. Furthermore, it is proven that thesmooth metric adjusted skew information
rates provide a lower bound for the coherence cost and an upper bound for the distillable coherence.

1 Introduction
Symmetry and conservation laws are basic yet
profound concepts in modern physics. As firstpointed out by Wigner [1] and further examined by Araki and Yanase [2], conservation laws
place limitations on the precision of the measurements. Based on this observation, Wigner
and Yanase [3] introduced a measure of information content in the presence of a conservation law, called the Wigner-Yanase skew information. It quantifies how a state is “askew" (i.e.,
non-diagonal, or coherent) to the eigenbasis of
the conserved quantity. This quantity was latershown to be closely related to quantum information geometry [4, 5]. Generalizing this connection, metric adjusted skew information is introduced from a family of quantum Fisher information metrics as quantifiers of information content
Koji Yamaguchi: koji.yamaguchi@uwaterloo.ca

relative to a conserved quantity [6].
A deeper understanding of the above relation
between symmetries and skew information has
been gained through the recent development of
quantum resource theories. Quantum resource
theories [7] provide a powerful framework forstudying the connection between a resource andthe physical restriction posed by our inability to
prepare a state and perform an operation. Due to
its wide applicability, various kinds of resources
have been investigated, such as entanglement [8],coherence [9–11], athermality [12, 13] and asymmetry [14]. The resource theory of asymmetry
is one of the most actively studied quantum resource theories, where the dynamics and states
are restricted by symmetries of the system.
In the resource theory of asymmetry, a class ofcoherence called asymmetry is considered to be aresource to implement operations that break thesymmetry. Asymmetry captures consequencesof symmetry that cannot be captured by the
Noether theorem [15] and has various applications. For example, time-translation asymmetryis mandatory for creating accurate clocks [16–21]
and accelerating quantum operations [22], and is
known to be a resource independent of entropy
in quantum thermodynamics [23]. Furthermore,
asymmetry is shown to be an essential resourcein various problems under conservation laws suchas quantum measurements [1, 2, 24–28], gate
implementation in quantum computing [29–33],
quantum error correction [32–37], and the Hayden–Preskill model for evaporating black holes[32, 33].
As a resource under conservation laws, asymmetry is closely related to metric adjusted skew
information. Indeed, metric adjusted skew information plays a significant role in both one-shotand asymptotic settings. In the one-shot setting,a family of metric adjusted skew information is

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1

known as valid resource measures [38, 39] for U (1)symmetry. It is lifted to a family of skew information matrices [40] for a general Lie groupG, which are valid asymmetry measures as wellas G-asymmetry [41] and the relative entropy ofG-frameness [42]. The convertibility conditionsin the one-shot regime [14, 43–46] are significantly simplified in the asymptotic regime, anda thermodynamic structure based on skew information appears for i.i.d. pure states [14, 47]. Furthermore, the conversion theory in the non-i.i.d.
regime [45] has been established by extending thequantum Fisher information, which is an example
in the family of metric adjusted skew information.
Despite the above progress in the resource theory of asymmetry, little is known about asymmetry measures in the asymptotic regime. In
particular, it is known that the asymptotic rateof metric adjusted skew information is not avalid asymptotic asymmetry measure due to a
discontinuity in the asymptotic regime [42, 47].
Also, the asymptotic rate of the relative entropy of G-frameness becomes trivial for any i.i.d.
states since the regularized relative entropy of Gframeness vanishes in the asymptotic limit [42].
These results show that useful asymptotic asymmetry measures cannot be obtained from a oneshot asymmetry measure just by calculating its
asymptotic rate.
In this paper, we introduce a family of valid
asymptotic asymmetry measures, which we callthe smooth metric adjusted skew information
rates. For this purpose, we first introduce the smooth metric adjusted skew information parameterized by the smoothness parameter  ∈ (0, 1]
with the smoothing technique [48, 49]. Then,
we define the smooth metric adjusted skew information rates as the sup- and inf-rates of the smooth metric adjusted skew information in thelimit of  → 0. We show that they are valid
asymptotic asymmetry measures. For an i.i.d.
sequence of a pure state, we further show thatthey are equal to the metric adjusted skew information of the state. Combining these results, we
relate the smooth metric adjusted skew information rates to the coherence cost and the distillable
coherence, which are central operational quantities in the asymptotic conversion theory. Concretely, by using an argument of Lieb-Yngvason’s
non-equilibrium thermodynamics [50], we provethat the smooth metric adjusted skew informa-tion rates provide a lower bound of the coherence
cost and an upper bound of the distillable coherence.
This paper is organized as follows: In Sec. 2,
we first review the basics of the resource theoryof asymmetry. We also briefly summarize theproperties of metric adjusted skew information,
including its asymptotic discontinuity. In Sec. 3,
we introduce a family of the -smooth metric adjusted skew information and its asymptotic rates.
We state our main theorems on the propertiesof the smooth metric adjusted skew information
rates. Theorem 1 shows that the smooth metricadjusted skew information rates are valid asymptotic asymmetry measures. In Theorem 2, thesmooth metric adjusted skew information rates
for i.i.d. states are explicitly calculated. We alsopresent a general asymptotic behavior of metricadjusted skew information for states near i.i.d.
pure states, which are used to prove Theorem 2.
The details of the proofs are postponed to Appendix. In Sec. 4, we prove inequalities that relate the smooth metric adjusted skew information
rates to the coherence cost and the distillable coherence by using Theorems 1 and 2, based on
Lieb-Yngvason’s non-equilibrium thermodynamics.

2 Resource theory of asymmetry
2.1 Definition
In this subsection, we review the definition of theresource theory of asymmetry. Fundamental elements of any resource theory are free states andfree operations that are prepared and performed
freely. In the resource theory of asymmetry, they
are defined with respect to a symmetry group G.
We say that a state ρ is symmetric if and only if it
is invariant under a group action, i.e., ρ = Ug ρUg†
for any g ∈ G, where Ug denotes the unitary representation of g. A symmetric state can be prepared without access to the reference frame related to the group G [14] and is considered a freestate in the resource theory of asymmetry. Anystate that breaks the symmetry is called asymmetric and is regarded as a resource.
A free operation in the resource theory of asymmetry is the so-called covariant operation E satisfying E(Ugin (ρ)Ugin† ) = Ugout E(ρ)Ugout† for allstates ρ and g ∈ G, where Ugin and Ugout are unitary representations of g in the input and output

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2

systems, respectively. A covariant operation describes a process that can be implemented without access to the reference frame related to thegroup G. By using the covariant Stinespring dilation theorem [43, 51], it can also be interpreted
as the dynamics of a system that couples to an
ancillary system initially in a symmetric state under a unitary evolution with a conservation lawof additive observables associated with G.
Although the resource theory of asymmetry isa general framework for investigating the consequences of symmetries described by any group G,
we here focus on U (1) and (R, +) groups wherea unitary representation is given by Ut = e−iHtwith an observable H on the system. This corresponds to the case where a single additive quantity is conserved. Such an additive quantity canvary depending on the physical situation we areinterested in. To simplify the terminologies, wealways call it the Hamiltonian of a system in thispaper. In this case, a quantum channel E is covariant if and only if it commutes with the timetranslation, that is,00E(e−iHt ρeiHt ) = e−iH t E(ρ)eiH t ,∀ρ, ∀t ∈ R,
(1)

where H and H 0 are the Hamiltonians of the input and output systems of E. Furthermore, asymmetric state can be understood as a state diagonalized by the energy eigenbasis since ρ =e−iHt ρeiHt implies [ρ, H] = 0. Since an asymmetric state ρ satisfies [ρ, H] 6= 0, the resource
theory of asymmetry with time-translation symmetry is a branch of resource theories investigating the properties of energetic coherence.
For later convenience, we here introduce notations for convertibility in the resource theoryof asymmetry. The most basic setup in conversion theory is one-shot conversion without error,
which has been analyzed, e.g., in [14, 43, 45, 46].
Suppose that for a given state ρ of a system with
Hamiltonian H and a given state σ of a systemwith Hamiltonian H 0 , there exists a covariant operation E such that σ = E(ρ). In this case, we
say that ρ is convertible to σ and denotecov(ρ, H)  (σ, H 0 ).

(2)

It should be noted that convertibility depends not
only on the states ρ and σ, but also on the Hamiltonians H and H 0 since the covariance of channels
is defined with respect to the time translations

cov

generated by them. The binary relation  is apreorder since the identity operation is covariant
and any product of covariant operations is covariant.
Since the exact conversion is too restrictivefrom a practical point of view, conversion with anerror is often analyzed. In this paper, we focus
on the setup in which the error vanishes in theasymptotic limit. Consider sequences of statesb = {σm }m of systems withρb = {ρm }m and σc0 = {H 0 }m ,b
Hamiltonians H = {Hm }m and H
mrespectively. We say that ρb is asymptoticallyb by covariant operations if andconvertible to σ
only if for any  ∈ (0, 1], there exists a sequence of covariant operations {Em }m such thatlim supm→∞ D(Em (ρm ), σm ) ≤ , where D denotes the trace distance defined by D(ρ, σ) :=1
2 kρ − σk1 . Here, the covariance of Em is defined
0 for each m. In thiswith respect to Hm and Hmcase, we denote
cov

c0 ).
b  a (σb, H(ρb, H)(3)

cov

The subscript in  a indicates that the binary
relation is defined for the asymptotic conversion.
The asymptotic conversion theory has been analyzed in [14, 20, 45, 47]. We will briefly review
their results in Sec. 4, where we relate the smooth
metric adjusted skew information rates to the coherence cost and the distillable coherence. Simicov
larly to the one-shot case, the binary relation  a
is a preorder.

2.2 Metric adjusted skew informations asasymmetry measures
Resource theories have the advantage of providing a concrete way to quantify a resource. In general, a resource measure is defined with a conversion relation. We say that R(ρ, H) is an asymmetry measure (in the one-shot setting) if and only
if the following two conditions are satisfied:
1) Monotonicity: If (ρ, H)
R(ρ, H) ≥ R(σ, H 0 ).

cov

(σ, H 0 ), then2) If ρ is symmetric, then R(ρ, H) = 0.
If a measure R further satisfies the converse of
the second condition, i.e., if R(ρ, H) = 0 implies
that ρ is symmetric, we say R is faithful. We do
not impose faithfulness as one of the minimal requirements for an asymmetry measure, as well as
entanglement measures in entanglement theory.

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3

In the one-shot regime, there are several known
asymmetry measures [14, 38, 39, 41, 42]. Our
main focus in this paper is the family of metric adjusted skew information [6]. They are first
introduced as quantifiers of non-commutativityof a state and an observable by extending the
Wigner-Yanase skew information [3], which arelater shown to be asymmetry measures [38, 39].
The metric adjusted skew information is closely
related to the Riemannian geometry of the state
space. In classical information geometry, there isa unique Riemannian metric that monotonicallycontracts under information processing, which iscalled the Fisher information metric. The studyof monotone metrics in quantum information theory is initiated by Morozova and Chentsov in [52].
Its classification is completed by Petz in [53],showing a one-to-one correspondence between amonotone metric and an operator monotone function. We say that a function f is a standard
monotone function if and only if it satisfies thefollowing three conditions:i) 0 ≤ A ≤ B =⇒ f (A) ≤ f (B).
ii) f (x) = xf (1/x)
iii) f (1) = 1
It is known thatfRLD (x) ≤ f (x) ≤ fSLD (x),

x>0

(4)

for any standard monotone functions f [54–56],
where fRLD := 2x/(x + 1) and fSLD := (x + 1)/2.
Here, RLD and SLD are abbreviations for rightlogarithmic derivative and symmetric logarithmic
derivative. For a family of states ρt parameterized by a single real number t ∈ R, the familyof quantum Fisher information is defined by thenorm of ∂t ρt with respect to the monotone metric[53, 57], given by
J f (ρt ) := Tr



∂ρt
∂ρt
cf (Lρt , Rρt )∂t∂tA standard monotone function satisfying
f (0) 6= 0 is called regular. Hansen [6] defined
the metric adjusted skew information for a regular standard monotone f as
I f (ρ, H) :=f (0) fJ (ρt )
2t=0

(7)

where ρt denotes a unitary model defined by ρt :={e−iHt ρeiHt }t∈R . The prefactor f (0)/2 is chosen
so thatI f (ψ, H) = Var(ψ, H)(8)holds for any pure state ψ, where Var(ρ, H) :=Tr(ρH 2 ) − (Tr(ρH))2 denotes the variance. For ageneric state ρ, it holds I f (ρ, X) ≤ Var(ρ, X).
By using the eigenvalue decomposition ρ =Pi λi |ii hi|, the metric adjusted skew informations are written as
I f (ρ, H) =f (0) X (λi − λj )2| hi|H|ji |2 . (9)2 i,j λj f (λi /λj )An example of the metric adjusted skew information is the Wigner-Yanase-Dyson skew information, which is known as a one-parameter extension of the Wigner-Yanase skew information.
The corresponding operator monotone function
is given by fWYD,p (x) = p(1 − p)(x − 1)2 /((xp −1)(x1−p − 1)) [6]. The Wigner-Yanase skew information is obtained as a special case for p = 1/2.
Another important example is skew information
for fSLD , which corresponds to the SLD Fisher information. It is known that the SLD skew information is equal to the convex roof of variance [58,P59]: I fSLD (ρ, H) = min{pi ,φi } i pi Var(φi , H),
where {pi , φi } runs over the set of all probability
distributions {pi }i and pure states {φi }i satisfyPing ρ = i pi φi . The SLD skew information is
the largest among the family of metric adjusted
skew information. Precisely, it holds



,

(5)

1
I f (ρ, H) ≥ I fSLD (ρ, H) ≥ I f (ρ, H), (10)2f (0)

where Lρ and Rρ denote the left and right multiplication operators by ρ, i.e., Lρ (O) = ρO andRρ (O) = Oρ for an operator O. The function
cf (x, y) is called the Morozova–Chentsov function
associated with a standard monotone function f ,which is defined bywhere f is an arbitrary regular standard monotone function [60].
From the viewpoint of the resource theory ofasymmetry, critical properties of metric adjusted
skew informations are the following two:1,
yf (x/y)1) Monotonicty [38]: If (ρ, H)  (σ, H 0 ), thenI f (ρ, H) ≥ I f (σ, H 0 ).

cf (x, y) =cov

x, y > 0.

(6)

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42) If ρ is symmetric, then I f (ρ, H) = 0.
In other words, metric adjusted skew informations are valid asymmetry measures. Furthermore, since J f (ρt ) = 0 only if ∂t ρt = 0, metricadjusted skew informations are faithful asymmetry measures.
Another property which we shall use later isthe convexity of the metric adjusted skew informations: For any set of density operator {ρk } andprobability distribution {pk }k such that pk ≥ 0
Pand k pk = 1, it holds!
Xf

pk I (ρk , H) ≥ I

f

k

X

pk ρk , H .

(11)

k

Such a convexity is one of the requirementsthat Wigner and Yanase [3] imposed on quantifiers of information content under conservation laws. The convexity of the Wigner-Yanaseskew information is proved in [3]. The convexity of the Wigner-Yanase-Dyson skew information
was called the Wigner-Yanase-Dyson conjecture,
which was later proved by Lieb [61]. The convexity for all metric adjusted skew informations
is proven by Hansen [6].

2.3 Asymptotic discontinuity of metric adjusted skew informations
The metric adjusted skew informations are valid
asymmetry measures that monotonically decrease
under covariant operations.
However, their
asymptotic rates
1 f
I (ρm , Hm ),m→∞ m

lim sup

1 f
I (ρm , Hm )
m→∞ m
(12)

lim inf

are not asymptotic asymmetry measures. Here,b defined for sewe say that a quantity R(ρb, H)
b is anquences of states ρb and Hamiltonians H
asymptotic asymmetry measure if and only if itsatisfies the following two conditions:covc0 ), thenbb, H1) Monotonicity: If (ρb, H)
 a (σcb0b , H ).
R(ρb, H) ≥ R(σ2) If ρb is a sequence of symmetric states, thenb = 0.
R(ρb, H)We usually do not impose faithfulness for asymptotic resource measures as a minimal requirement. This is because an operationally important

asymmetry measure called the distillable coherence is not faithful. A similar argument can also
be found in entanglement theory [62].
Intuitively, we can understand the reason why
the metric adjusted skew information rates in
Eq. (12) are not asymptotic asymmetry measures
as follows: From Eq. (9), we can estimate themaximal change in metric adjusted skew informations as ∼ kHm k2 ×  when the state changes on
the order of  in the trace distance. This implies
b are interconvertible, theirthat even when ρb and σmetric adjusted skew information rates can takedifferent values, violating condition 1) above. We
remark that this was one of the non-trivial issues
in establishing the conversion theory in the i.i.d.
regime [47].
Indeed, such a discontinuity can be seen by
examining the variance. A quantity A is called
asymptotically continuous [42, 63, 64] if
A(ρm ) − A(σm )lim=0m→∞ 1 + log(dim(Hm ))(13)holds for any states ρm and σm of a Hilbert
space Hm such that limm→∞ D(ρm , σm ) = 0.
In a typical setup, including i.i.d.
setup,
1 + log(dim(Hm )) = O(m) for m → ∞.
1
A(ρm ) =Therefore, Eq. (13) implies limm→∞ m
1
limm→∞ m A(σm ) if limm→∞ D(ρm , σm ) = 0. In[42], it is pointed out that the variance is not
asymptotically continuous. Since metric adjusted
skew informations are equal to the variance forpure states, this result proves that the metric adjusted skew information rates cannot be asymptotic asymmetry measures.

3 Smooth metric adjusted skew information rates
So far, we have seen that the asymptotic ratesof the metric adjusted skew informations are
not asymptotic asymmetry measures. Intuitively,
this is because metric adjusted skew informations
change too drastically with a small perturbationin the state space and hence they are not goodquantifiers in the asymptotic conversion theory
with an error.
To find an asymptotic asymmetry measure related to the metric adjusted skew informations,
we here apply the smoothing technique to metric adjusted skew informations. The smoothing

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5technique [48, 49] is commonly used to investigate information-theoretic tasks with an error.
It is closely related to the information-spectrum
method [65], a powerful and universal tool to analyze asymptotic problems in information theory.
The smoothing technique is used in a recent study[45] on the asymptotic conversion theory in the
resource theory of asymmetry in the non-i.i.d.
regime. In [45], the smoothing technique is applied for the max- and min-quantum Fisher information to construct an information-spectrum
approach for quantum Fisher information. However, it has not been applied to metric adjusted
skew informations.
To begin with, let us introduce a family of the-smooth metric adjusted skew informations. Fora smoothing parameter  ∈ (0, 1], we defineIf (ρ, H) :=infσ∈B (ρ)f

I (σ, H),

(14)

where B  (ρ) is the -ball in the state space defined by B  (ρ) := {σ : states | D(ρ, σ) ≤ }. In
Appendix A, we show that If (ρ, H) monotonically decreases through a covariant channel. In
addition, it trivially vanishes for any symmetricstate. Therefore, If (ρ, H) is an asymmetry measure. Note that the -smooth metric adjusted
skew informations If is not a faithful measure
as it vanishes for a state -close to a symmetricstate. In Appendix A, we prove that If (ρ, H)
inherits the convexity of metric adjusted skew informations.
By using the smooth metric adjusted skew
informations, we now define the sup- and infsmooth metric adjusted skew information rates
as follows:fb := lim lim sup 1 I f (ρm , Hm ), (15)I+(ρb, H)
→0+ m→∞ mfb := lim lim inf 1 I f (ρm , Hm ).
(ρb, H)(16)I−
→0+ m→∞ m
We also call them the smooth metric adjusted
skew information rates for short.
We re1 f
mark that since lim supm→∞ m
I (ρm , Hm ) and1 f
lim inf m→∞ m
I (ρm , Hm ) are monotonic funcfbtions of , their right-hand side limits I±(ρb, H)
always exist.
A key property of the smooth metric adjusted
skew information rates is the following:
Theorem 1. For any regular operator monotonefunction f , the smooth metric adjusted skew information rates are asymptotic measures in the

resource theory of asymmetry. That is, the following two conditions are satisfied:covc0 ), thenbb, H1) Monotonicity: If (ρb, H)
 a (σ
f
f
f
c
b
b
0
b , H ) and I− (ρb, H)≥
I+ (ρb, H) ≥ I+ (σfc0b , H ).
I− (σ2) If ρb is a sequence of symmetric states, thenfb = I f (ρb, H)
b = 0.
I+(ρb, H)−This is one of the main results of this paper.
The proofs of Theorem 1 and the convexity ofsmooth metric adjusted skew information rates
are provided in Appendix B.
Note that the monotonicity implies that for anybsequence of Hamiltonians H,
fb = I f (σbI+(ρb, H)+ b , H),

(17)

f
b
b = I f (σ(ρb, H)I−− b , H)(18)

hold if limm→∞ D(ρm , σm ) = 0.
We remark that from Eq. (10), it holds
1fSLD
f
I f (ρ, H) ≥ I+
(ρ, H) ≥ I+
(ρ, H), (19)2f (0) +1fSLD
f
I f (ρ, H) ≥ I−
(ρ, H) ≥ I−
(ρ, H) (20)
2f (0) −for any regular standard monotone function f .
Therefore, smooth skew information rates for
fSLD are the largest among the family of smooth
metric adjusted skew information rates.
Another main result of this paper is the following:
Theorem 2. Let ψ be a pure state having period 2π for a Hamiltonian H. Assume that thethird absolute moment of the Hamiltonian is finite, i.e., hψ||H|3 |ψi < ∞. For a positive parameter R > 0, define ψbiid (R) := {ψ ⊗dRme }mb iid (R) := {Hand H
iid,dRme }m , where Hiid,k :=
Pk
⊗i−1
⊗k−i
⊗H ⊗I
. The smooth metric adi=1 I
justed skew information rates for this i.i.d. sequence are given byf bb iid (R)) = I f (ψbiid (R), H
b iid (R))
I+
(ψiid (R), H−
1 f ⊗dRme
I (ψ, HdRme ) = I f (ψ, H) R.
= lim
m→∞ m
(21)

This theorem is an immediate corollary of thefollowing lemma, proven in Appendix C:

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6

Lemma 3. Let ψ be a pure state with period 2π
with a Hamiltonian H. Assume that the absolute
third moment is finite, i.e., hψ||H|3 |ψi < ∞. Fix
 to be a sufficiently small real parameter. Letρb = {ρm }m be a sequence of states such that ρm ∈B  (ψ ⊗dRme ) for all sufficiently large m with a realparameter R > 0, where B  (ρ) is the -ball inthe state space defined by B  (ρ) := {σ : states |D(ρ, σ) ≤ }. Then there exists a real function
δ f () of  such that lim→0 δ f () = 0 and

I f ρm , Hiid,dmRe≥ I f ψ ⊗dRme , Hiid,dmRe − mδ f () + o(m)(22)

b iid = {Hm }m , where Hm =with Hamiltonians H
Pm ⊗(i−1)⊗ H ⊗ I⊗(m−i) .
i=1 I
The convertibility among i.i.d. pure states isstudied in earlier research [14] and completed in[47]. Suppose that two pure states ψ and φ havethe same period. For i.i.d. sequences ψbiid ={ψ ⊗m } and φbiid (R) = {φ⊗dRme }, it is shown thatψbiid is convertible to φbiid (R) if and only if R ≤F(ψ, H)/F(φ, H 0 ). Here, F(ρ, H) denotes thesymmetric logarithmic derivative (SLD) quantum Fisher information with respect to the oneparameter family of states ρt = {e−iHt ρeiHt }t∈R ,given byF(ρ, H) := 2

X (λi − λj )2i,j

as m → ∞.

λi + λj| hi|H|ji |2 ,(23)

where ρ = i λi |ii hi| is the eigenvalue decomposition. Conventionally, this quantity is simply called the quantum Fisher information. Notethat the quantum Fisher information is related
to the skew information of fSLD = (1 + x)/2 as
4I fSLD (ρ, H) = F(ρ, H).
The above result in the i.i.d. regime shows thatpure states with the same periods are equivalentcoherence resources in the sense that they are interconvertible with a non-vanishing rate as longas the quantum Fisher information is non-zero.
Since the period of a state can be set to 2π by
rescaling the Hamiltonian, we can pick up any
pure state with period 2π as a reference in analyzing convertibility. Here we adopt a coherencebit1
φcoh := |φcoh i hφcoh | , |φcoh i = √ (|0i + |1i)
2
(24)
P

Intuitively,
this
lemma
shows
that


ρm , Hiid,dmReapproximately takes a

1 fmI

local minimum around i.i.d. states ψ ⊗dRme in
the asymptotic regime. As a consequence, itssmooth metric adjusted skew information rates
become equal to the asymptotic rate of metricadjusted skew informations for i.i.d. states,
proving Theorem 2.

4 Smooth metric adjusted skew information rates and the asymptotic conversion theory
In this section, by using Theorems 1 and 2, we
relate smooth metric adjusted skew information
rates to the coherence cost and the distillable coherence, which are the central quantities in theasymptotic conversion theory in the resource theory of asymmetry.

4.1 Coherence cost and distillable coherence
In order to introduce the coherence cost and thedistillable coherence, let us briefly review theasymptotic conversion theory in the resource theory of asymmetry.
Independent and identically distributed (i.i.d.)
setup is one of the most important and fundamental regimes in the asymptotic conversion. In
this setting, many copies of a state are converted
to copies of another state under the assumptionthat the Hamiltonian is given by a sum of copies
of a free Hamiltonian of a subsystem. Precisely,
we consider a sequence of states ρbiid = {ρ⊗m }mwith Hamiltonian Hcoh = |1i h1| as a reference. This state has period 2π and its quantum Fisher information is normalized to unity:F(φcoh , Hcoh ) = 1. For arbitrary sequences ofbstates ρb = {ρm }∞m=1 and Hamiltonians H =∞{Hm }m=1 , the coherence cost and the distillable
coherence are defined byb
Ccost ρb, Hn cov 



b
[
b, H
:= inf R | φdcoh (R), Hcoh (R)  a ρ

o

,

(25)


b
Cdist ρb, H
n




b
:= sup R | ρb, H

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 cov o[
 a φdcoh (R), Hcoh (R)

,

(26)
7

[
where Hcoh (R) := {Hcoh,dRme }m with Hcoh,k :=Pk⊗i−1 ⊗ H⊗k−i and φd (R) :=coh ⊗ I
coh
i=1 I
⊗dRme
{φcoh }m .
The above result [47] for i.i.d. pure states canbe restated as








b iid = F (ψ, H) ,b iid = Cdist ψbiid , H
Ccost ψbiid , H(27)
where the period of ψ is set to be 2π and ψb

iid =
⊗m
{ψ }m . This result for the coherence cost is directly extended to a general i.i.d. states ρbiid =b iid ) = F (ρ, H) [47]. Al{ρ⊗m } as Ccost (ρbiid , Hthough no general formula on the distillable coherence is known for an i.i.d. mixed states, it
is known that i.i.d. mixed states typically havevanishing distillable coherence [20].

4.2 Smooth metric adjusted skew information
rates, the coherence cost and the distillable coherence
We are now ready to relate the smooth metricadjusted skew information rates to the coherence cost and the distillable coherence by using
Lieb-Yngvason’s non-equilibrium thermodynamics [50]. Note that although the original aim of
Lieb-Yngvason’s non-equilibrium thermodynamics was to extend the notion of thermodynamic
entropy to non-equilibrium states, some of theirarguments are applicable to a generic preorderrelation.
Concretely, we here prove the following inequalities, proving that the smooth metric adjusted skew information rates yield a lower boundof the coherence cost and an upper bound of the
distillable coherence:

Since




f
b
φd
I+
coh (Rδ ) , Hcoh (Rδ ) =

Rδ
4

(31)

holds from Theorem 2, we getb + δ ≥ 4I f (ρb, H)
b
Ccost (ρb, H)+(32)

for any δ > 0 and henceb ≥ 4I f (ρb, H).
b
Ccost (ρb, H)+(33)

b > +∞, Ccost (ρb, H)b ≥Note that when Ccost (ρb, H)fb is trivially true. By a similar argument,
I+ (ρb, H)
it is shown thatfb ≥ Cdist (ρb, H).
b(ρb, H)
4I−(34)

Among these inequalities, the tightest lower
bound for the coherence cost can be obtainedfSLD
b as can be directly shown fromby I+
(ρb, H)
Eq. (19).
Theorem 4 is of fundamental importance since
it clarifies a relation among smooth metric adjusted skew information rates and quantities with
operational meaning, i.e., the coherence cost andthe distillable coherence. Although the proof provided here is straightforward, the result itself is
non-trivial since it is derived from Theorems 1
and 2. To understand its importance, we shallapply Theorem 4 to explicit examples in Sec. 5.

4.3 Relation to the spectral QFI rates andcomparison to entropic quantities

For pure states with finite periods, asymptotic
conversion theory has been established beyond
the i.i.d. regime [45]. This result shows thatthe coherence cost and the distillable coherenceb it holds
Theorem 4. For any sequences ρb and H,
for any sequence of pure states are given by thespectral sup- and inf-quantum Fisher informationb ≥ 4I f (ρb, H)
b ≥ 4I f (ρb, H)
b ≥ Cdist (ρb, H).
b
Ccost (ρb, H)−+
rates, respectively. By using Theorem 4, it is(28)
straightforward to show that the smooth metricadjusted skew information rates give lower andb < ∞. For any
Proof. Suppose that Ccost (ρb, H)
upper bounds for the spectral quantum Fisherb + δ. Then it holds
δ > 0, define Rδ := Ccost (ρb, H)
information rates. A similar relation is known cov in entropy rates and the spectral entropy rates.
b
b .
b, H
φd
(29)
coh (Rδ ) , Hcoh (Rδ )  a ρThis subsection aims to compare these relations
and explore their similarities and differences.
From the monotonicity of smooth metric adLet us first briefly review the result in [45]. The
justed skew information rates, we get
definitions of quantities reviewed here are somewhat involved and therefore postponed to Apf
f
b
b
b
I+
φd
(R
)
,
H
(R
)
≥
I
ρ
,
H
. (30)
coh
coh
δ
δ
+
pendix D. In [45], two quantities Fmax (ψ, H) and
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8

Fmin (ψ, H), called the max- and min-quantum
Fisher information, are introduced for pure stateswith finite periods. It is proven [45] that they give
upper and lower bounds of the quantum Fisher
information:
Fmax (ψ, H) ≥ F (ψ, H) ≥ Fmin (ψ, H) .

(35)

For a sequence of pure states with a finite period,
the spectral sup- and inf-quantum Fisher information rates are defined by
1 F(ψm , Hm ) , (36)
m max
b H)
b := lim lim inf 1 F  (ψm , Hm ) , (37)F(ψ,min
→0 m→∞ mb H)b := lim lim supF(ψ,→0 m→∞


where Fmax
and Fminare smooth max- and minquantum Fisher information. See Appendix D fortheir definition. It is proven [45] that the coherence cost and the distillable coherence are given
by

b H),
b H)
b
b = F(ψ,Ccost (ψ,

b H)
b H).
b = F(ψ,b
Cdist (ψ,(38)

Therefore, Theorem 4 implies
b H)
b H)
b H)
b H).
b ≥ 4I f (ψ,b ≥ 4I f (ψ,b ≥ F(ψ,
b
F(ψ,
−
+
(39)

Let us now consider a special case with fSLD .
Since 4I fSLD (ρ, H) = F(ρ, H) holds for any state
ρ, we get

1 
1
b H),
b
F (ψm , Hm ) ≥ lim lim inf F  (ψm , Hm ) ≥ F(ψ,
→0 m→∞ m
→0 m→∞ mb H)b ≥ lim lim supF(ψ,where we definedF  (ψ, H) :=(40)

by using the smoothing method [49, 66]. It is
known thatinfρ∈B  (ψ)

F(ρ, H)(41)

These inequalities are interpreted as the asymptotic version of inequalities (35).
A similar inequalities are known for the spectral sup- and inf-information rates. Let us
first introduce the max-entropy Smax (ρ) :=log (rank (ρ)) and the min-entropy Smin (ρ) :=− log (kρk∞ ), where rank (ρ) and kρk∞ denote
the rank of ρ and the maximum eigenvalue ofρ, respectively. These quantities provide upperand lower bounds for the von Neumann entropyS(ρ) := −Tr(ρ log ρ) as
Smax (ρ) ≥ S(ρ) ≥ Smin (ρ).

(42)

Defining the smooth max- and min-entropies (ρ) :=Smax(ρ) := inf σ∈B  (ρ) Smax (σ) and Sminsupσ∈B  (ρ) Smin (σ), the spectral sup- and infentropy rates S(ρb) and S(ρb) of a sequence ofstates ρb = {ρm }m are expressed as
1 Smax (ρm ),→0 m→∞ m
1 
S(ρb) := lim lim inf Smin(ρm )
→0 m→∞ mS(ρb) := lim lim sup

1
1
S(ρm ) ≥ lim inf S(ρm ) ≥ S(ρb).
m→∞
m
m→∞ m
(45)

S(ρb) ≥ lim sup

(43)
(44)

hold [67]. These inequalities can be understoodas an asymptotic version of inequalities (42).
In the resource theory of entanglement, thespectral entropy rates play an essential role inthe conversion theory of non-i.i.d. states. Precisely, for a sequence of bipartite pure states
ψbAB = {ψAB,m }m , the entanglement cost andthe distillable entanglement are given by thespectral sup- and inf-entropy rates for the sequence of reduced states ρb := {ρm }m , whereρm := TrB (ψAB,m ), respectively [68, 69]. Furthermore, the sup- and inf-rates of the von Neumann entropy for ρb are asymptotic entanglement
measures since the entanglement entropy of apure state SEE (ψAB,m ) := S(ρm ) is an entanglement measure that is asymptotically continuous[62, 63, 70–72] as shown by the Fannes inequality[73].
It is worth emphasizing that there is a crucial difference between Eqs. (40) and (45) for
the asymptotic rates of the quantum Fisher information F and the entanglement entropy SEE .

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9

In Eq. (40), smoothing parameter  must be included in since the quantum Fisher information
F is not asymptotically continuous [42]. On theother hand, in Eq. (45), the smoothing technique

is not required for the entanglement entropy SEE
since it is asymptotically continuous [62, 63, 70–72]. The correspondence is summarized in Table 1.

Table 1: Summary of asymptotic resource measures in the resource theory of entanglement and the resource
theory of asymmetry, which are based on entropies and the quantum Fisher information (QFI), respectively.

Entropythe quantum Fisher information (QFI)

sup-rate of smooth max-entropy
sup-rate of smooth max-QFI§
1 b H)
b = lim lim sup 1 F  (ψm , Hm )(ρm ) F(ψ,S(ρb) = lim lim sup Smax
max
→0 m→∞ m
→0 m→∞ m
Spectral rates
inf-rate of smooth min-entropy
inf-rate of smooth min-QFI§
1 b H)
b = lim lim inf 1 F  (ψm , Hm )S(ρb) = lim lim inf Smin
F(ψ,
(ρm )
min
→0 m→∞ m
→0 m→∞ m
sup-rate of entanglement entropy*
sup-rate of smooth QFI
1
1
lim sup S(ρm )lim lim sup F  (ψm , Hm )
→0 m→∞ mm→∞ m
Asymptotic rates*
inf-rate of entanglement entropy
inf-rate of smooth QFI
1
1
lim inf S(ρm )lim lim inf F  (ψm , Hm )
m→∞ m
m→∞
→0
m
* Smoothing technique is not required.
§ Defined for a sequence of pure states with a finite period.

5 Applications
To better understand our results established in
the present paper, we here apply them to concrete
examples.

5.1 Upperbound for distillable cohereceDistillable coherence is particularly relevant for
mixed states since it quantifies the number ofpure resources that can be extracted from noisy
resources. In [20], the following sufficient condition for the distillable coherence to vanish isproven:b iid = 0,[Πρ , H] = 0 =⇒ Cdist ρbiid , H(46)

where Πρ is the projector to the support ofb iid = {Hiid,m }m forρ, ρbiid = {ρ⊗m }m and H
Pm ⊗i−1
Hiid,m := i=1 I⊗ H ⊗ I⊗m−i . However, despite its importance, no general formula is known
for the distillable coherence of mixed states, evenin the i.i.d. regime.
We here derive a simple upper bound for the
distillable coherence applicable to general se-

quence of states. From
Theorem
4, the distillbbable coherence Cdist ρ, H is upper bounded byfb . From the definition of smooth metρb, H
4I−
ric adjusted skew information inf-rate, it holds

1 f
f
b
lim inf m→∞ m I− (ρm , Hm ) ≥ I ρb, H . Therefore,



1 f
b
I (ρm , Hm ) ≥ Cdist ρb, H
m→∞ m

4 lim inf



(47)



b .
holds for any ρb, H
To simplify the argument, let us now consider
the i.i.d. case. Since the metric adjusted skew
informations are additive for product states [6],we have1
lim inf I f (ρ⊗m , Hiid,m ) = I f (ρ, H),
(48)
m→∞ m

implying thatb iid .
4I f (ρ, H) ≥ Cdist ρbiid , H(49)

Note that for fSLD , this bound is equivalent to atrivial inequalityb iid ≥ Cdist ρbiid , H
b iid
Ccost ρbiid , H

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

(50)
10





b iid = 4I fSLD (ρ, H) if ρ has asince Ccost ρbiid , H
period 2π [47]. Our bound in Eq. (49) gives abetter bound in general since I fSLD (ρ, H) is thelargest element in the family of metric adjusted
skew information.
As a concrete example, let us analyze an i.i.d.
copies of qutrit systems with Hamiltonian H =P2
n=0 n |ni hn|. Consider a mixed stateρ := (1 − q)ψ1 + qψ2(51)

where q ∈ (0, 1) and ψi := |ψi i hψi | for orthonormal states {|ψi i}2i=1 . From the definition, themetric adjusted skew information is explicitly calculated as
I f (ρ, H)= (1 − q)Var(ψ1 , H) + qVar(ψ2 , H)

4

3

2

1

0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 1: Comparison of upper bounds of the distillable
coherence for an i.i.d. mixed state. 4I fSLD corresponds
to a trivial bound in Eq. (50), while 4I fWYD,p is a special
case of our new bound in Eq. (49). The plot shows thatthe latter gives a tighter bound.

!

f (0)(1 − 2q)2
− 1−
| hψ1 |H|ψ2 i |2
qf ((1 − q)/q)(52)

for the mixed state in Eq. (51).
To compare the bounds in Eqs. (49) and (50),
we calculate the metric adjusted skew information for
11
1|0i + |1i + √ |2i ,
2
2
2
1
1
|ψ2 i := √ |0i − √ |1i
22|ψ1 i :=

(53)

for q ∈ (0, 1).
The metric adjusted skew information for fSLD
and fWYD,p , and the bound in Eq. (57) are plotted in Fig. 1. Of particular interest are their behaviors near q ≈ 0 and q ≈ 1. When q = 0, i.e.,ρ = ψ1 , the distillable coherence is given byb iid
Cdist ρbiid , H= 4I fSLD (ψ1 , H) =
q=0

(54)

Note that the result in Eq. (46) is not applicablesince [Πρ , H] = [ψ1 + ψ2 , H] 6= 0. For this example, the metric adjusted skew information for
fSLD and fWYD,p are evaluated as
11 − 15q + 8q 2
,
(55)
16
11 − 7q
I fWYD,p (ρ, H) =
16
!!


 

1
1−q p
1 − q 1−p
−
1−q
−1
−1
8
q
q
(56)
I fSLD (ρ, H) =for q ∈ (0, 1). Since I fWYD,p is a monotonically increasing continuous function of p for p ∈ (0, 1/2)
and q ∈ (0, 1), Eq. (49) implies

11
,
4
(58)

while when q = 1, it holds
b iid
Cdist ρbiid , H= 4I fSLD (ψ2 , H) = 1. (59)q=1

From Eqs. (57), (58) and(59), we find that theb iid is discontindistillable coherence Cdist ρbiid , Huous at q = 0, 1.
We stress that the bounds in this subsection
is derived by using Eq. (47), which can be calculated without using any smoothing technique.
Nevertheless, the smoothing technique is essential to prove Eq. (47) as we have argued in thispaper. In the next subsection, we will revisit thispoint by analyzing an explicit example.

5.2 Necessity of smoothing
4 lim I
p→0+=

fWYD,p(ρ, H)Let us now consider a sequence of pure statesψb = {ψm }m for qutrit systems given by

q6=0,1
9 − 7qb iid≥ Cdist ρbiid , H4
q6=0,1(57)|ψm i :=√√
1 − m |φcoh i⊗m + m |2i⊗m

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(60)
11

where |φcoh i is defined in Eq. (24) and m :=
cov
cov
√
b
1/ m. Note that ψb  a φbcoh (1) and φcoh (1)  a
ψb since limm→∞ D ψm , φ⊗m= 0. Therefore,coh
we getb H)
b H)
b = Ccost (ψ,b = 1.
Ccost (ψ,(61)

The left-hand side of Eq. (47) is evaluated as
4 lim

1

m→∞ m

I f (ψm , Hm ) = ∞

(62)

since
I f (ψm , Hm ) = Var(ψm , Hm )√√m
(9m − 8 m − 1).
=
4

(63)

Therefore, Eq. (47) does not provide a meaningful
bound for distillable coherence in this example.
This can be understood from the fact that theasymptotic rate of metric adjusted skew information is not a good asymptotic asymmetry measure
due to the asymptotic discontinuity [20, 42].
In the present paper, we showed that thesmoothing technique is essential to overcome thisissue. From Lemma 3, the smooth metric adjusted skew information rates are explicitly evaluated as
f b b
4I±
(ψ, H) = 1.

(64)

From Theorem 4, this quantity gives a lower
bound for the coherence cost and an upper boundfor the distillable coherence, as is consistent with
Eq. (61).

6 Conclusions
In this paper, we investigated the propertiesof metric adjusted skew information particularly
from the perspective of the resource theory ofasymmetry. A family of metric adjusted skew information is induced from a family of monotone
metrics, i.e., quantum Fisher information metrics in information geometry. They are known
as asymmetry resource measure in the one-shot
regime. However, their asymptotic rates are notvalid asymptotic asymmetry monotone due totheir asymptotic discontinuity. To find asymptotic asymmetry measures, we first introduced afamily of the -smooth metric adjusted skew information by using the smoothing technique. We

then defined the sup- and inf-smooth metric adjusted skew information rates as its asymptotic
rates in the limit of  → 0. We proved that thesefamilies of smooth metric adjusted skew information rates are valid asymptotic asymmetry measures. We further proved that the smooth metricadjusted skew information rates inherit the convexity of skew informations, which may be viewed
as an extension of the Wigner-Yanase-Dyson conjecture [3, 61] to the asymptotic regime.
By analyzing a general asymptotic behaviorof the metric adjusted skew information for sequences of states that are close to i.i.d. purestates, we explicitly calculated the smooth metric adjusted skew information for an arbitrary sequence of i.i.d pure states. Combining this resultwith the asymptotic monotonicity of the smooth
metric adjusted skew information rates, we related them to the key operational quantities in
the asymptotic conversion theory in the resource
theory of asymmetry. Concretely, we showedthat the smooth metric adjusted skew information rates provide a lower bound of the coherence
cost and an upper bound of the distillable coherence. As a corollary, we further proved inequalities relating the asymptotic rates of the quantum Fisher information to the spectral quantum
Fisher information rates based on an informationspectrum approach for the quantum Fisher information [45]. They have a structure similar to theinequalities for the von Neumann entropy rates
and the spectral entropy rates. However, ouranalysis shows that smoothing parameters must
be included in the rates of the quantum Fisher
information since it has an asymptotic discontinuity.
The main focus of this paper is to study the
metric adjusted skew information in the resource
theory of asymmetry. However, our results could
also be useful in quantum thermodynamics andquantum estimation theory. In the resource theory of athermality, thermal operations are regarded as free. Here, thermal operations are operations that can be implemented by couplinga system to a thermal bath through an energyconserving unitary evolution [12, 74, 75]. Since
any thermal operation is covariant with respect tothe time translation, Theorem 1 implies that thesmooth metric adjusted skew information ratesfI±
are valid asymptotic resource measures in the
resource theory of athermality. Although relevant

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12quantities in quantum thermodynamics are typically introduced from entropies, new insights are
obtained with the family of quantum Fisher information [20]. Therefore, it will be quite interesting
to explore the role of the smooth metric adjusted
skew information rates not only as asymmetry
measures but also as athermality measures. On
the other hand, the smooth metric adjusted skew
information rates can also be helpful in studies
on quantum estimation theory in the asymptotic
regime. Indeed, the metric adjusted skew information is proportional to quantum Fisher information, a central quantifier in quantum estimation theory, for a unitary model. Further researchin these directions is left for future work since it
is out of the scope of the present paper.

Acknowledgments
KY acknowledges support from the JSPS Overseas Research Fellowships. HT acknowledges
supports from JSPS Grants-in-Aid for Scientific Research (JP19K14610 and JP22H05250),
JST PRESTO (JPMJPR2014), and JST MOONSHOT (JPMJMS2061).

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[56] P. Gibilisco and T. Isola, Journal of Mathematical Analysis and Applications 375, 270(2011).
[57] D. Petz and C. Ghinea, in Quantum Probability and Related Topics, QP-PQ: Quantum
Probability and White Noise Analysis, Vol.
Volume 27 (WORLD SCIENTIFIC, 2011)
pp. 261–281.
[58] G. Tóth and D. Petz, Physical Review A 87,
032324 (2013).
[59] S. Yu, arXiv:1302.5311 [quant-ph] (2013).
[60] P. Gibilisco, D. Imparato, and T. Isola, Proceedings of the American Mathematical Society 137, 317 (2009).
[61] E. H. Lieb, Advances in Mathematics 11, 267(1973).
[62] M. Horodecki, P. Horodecki,andR. Horodecki, Physical Review Letters84, 2014 (2000).
[63] M. J. Donald, M. Horodecki,andO. Rudolph, Journal of Mathematical
Physics 43, 4252 (2002).
[64] M. Plenio and S. Virmani, Quantum Information and Computation 7, 1 (2007).
[65] T. S. Han, Information-Spectrum Methods inInformation Theory, Stochastic Modellingand Applied Probability, Vol. 50 (Springer
Berlin Heidelberg, Berlin, Heidelberg, 2003).
[66] N. Datta and R. Renner, IEEE Transactions
on Information Theory 55, 2807 (2009).
[67] N. Datta, IEEE Transactions on Information
Theory 55, 2816 (2009).
[68] M. Hayashi, in IEEE International Symposium on Information Theory, 2003. Proceedings. (2003) pp. 431–.
[69] G. Bowen and N. Datta, IEEE Transactions
on Information Theory 54, 3677 (2008).
[70] G. Vidal, Journal of Modern Optics 47, 355(2000).

Accepted in Quantum 2023-05-08, click title to verify. Published under CC-BY 4.0.

14

[71] M. Horodecki, Quantum Information andComputation 1, 3 (2001).
[72] B. Synak-Radtke and M. Horodecki, Journalof Physics A: Mathematical and General 39,
L423 (2006).
[73] M. Fannes, Communications in Mathematical Physics 31, 291 (1973).
[74] D. Janzing, P. Wocjan, R. Zeier, R. Geiss,

and T. Beth, International Journal of Theoretical Physics 39, 2717 (2000).
[75] M. Horodecki and J. Oppenheim, Nature
Communications 4, 2059 (2013).
[76] M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information: 10th Anniversary Edition,” (2010).
[77] A. D. Barbour and V. Ćekanavićius, The Annals of Probability 30, 509 (2002).

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15

A Properties of the -smooth skew informations
Let us first prove the monotonicity of the -smooth skew informations.
Proposition 5 (Monotonicity of the -smooth metric adjusted skew information). Let  be a parametersuch that  ∈ (0, 1]. Let (ρ, H) and (σ, H 0 ) denote sets of states and Hamiltonians. If ρ is convertiblecov
to σ by a covariant operation, i.e., (ρ, H)  (σ, H 0 ), then it holdsIf (ρ, H) ≥ If (σ, H 0 ).

(65)

Proof. The proof directly follows from the monotonicity of the metric adjusted skew informaiton andthe trace distance. By using a covariant channel E such that E(ρ) = σ, we getIf (ρ, H) =≥infIf (ξ, H)

(66)

inf

If (E(ξ), H 0 )(67)ξ∈B  (ρ)ξ∈B  (ρ)

from the monotonicity of metric adjusted skew informations. Since the trace distance is contractive
under a quantum channel, E(ξ) ∈ B  (E(ρ)) holds for any ξ ∈ B  (ρ), implying thatinfξ∈B  (ρ)

If (E(ξ), H 0 ) ≥infχ∈B  (σ)

If (χ, H 0 ).

(68)

Therefore, we getIf (ρ, H) ≥ If (σ, H 0 ).

(69)

We now prove the convexity of the -smooth metric adjusted skew informations.
Proposition 6 (Convexity of the -smooth metric adjusted skew information). Let  be a parameter ∈ (0, 1]. For any states {ρ(k) }k and a probability distribution {pk }k , it holds!
XX

pk If (ρk , H) ≥ Ifkpk ρ k , H .

(70)

k

Proof. By using the convexity of the metric adjusted skew information!
Xf

pk I (ρk , H) ≥ I

X

f

k

pk ρ k , H ,(71)

I f (σk , H)

(72)

pk I f (σk , H)

(73)

k

we getX

pk If (ρk , H) =

X

pk

k

k

=

inf

σk ∈B  (ρk )

inf 

X

∀k, σk ∈B (ρk )

k

!

≥

inf

∀k, σk ∈B  (ρk )

≥

inf
P

σ∈B  (If

X

pk σk , H(74)k

I f (σ, H)

(75)

p ρ )
k k k

!

= If

X

pk ρk , H .

(76)

k

In the last inequality, we have used
!

∀k,



σk ∈ B (ρk ) =⇒Xk

pk σk ∈ BX

pk ρ k ,(77)

kwhich follows from the convexity of the trace distance. See, e.g., [76] for proof of the convexity of the
trace distance.
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16

B Properties of smooth metric adjusted skew information rates
Let us first prove the asymptotic monotonicity of the smooth metric adjusted skew information rates.
b andProposition 7 (Monotonicity of the smooth metric adjusted skew information rates). Let (ρb, H)covb 0 ) be sequences of states and Hamiltonians. If (ρb, H)b  a (σb 0 ), then it holds
b, H
b, H
(σ
f
c0
b ≥ I f (σI±(ρb, H)± b , H ).

(78)

cov

b  a (σb 0 ), for any  > 0, there exists a sequence of covariant channels {Em }mb, H
Proof. Since (ρb, H)
such that

D (Em (ρm ), σm ) < (79)holds for all sufficiently large m. By using the triangle inequality, we have∀χm ∈ B  (Em (ρm )) ,D (χm , σm ) ≤ D (χm , E(ρm )) + D (Em (ρm ), σm ) < 2(80)

for all sufficiently large m. Therefore, we get
0If (ρm , Hm ) ≥ If Em (ρm ), Hm
f
≥ I2



(81)

0 
σm , Hm(82)

and hence
1 f
1 f
0 
σm , Hm,
I (ρm , Hm ) ≥ lim sup I2
m→∞ m
m→∞ m
1
1 f
0 
lim inf If (ρm , Hm ) ≥ lim inf I2
σm , Hm
m→∞ m
m→∞ m

lim sup(83)(84)

holds for any  ∈ (0, 1]. In the limit of  → 0, it is proven
1 f
I (ρm , Hm )
m→∞ m
1 f
0 
σm , Hm≥ lim lim sup I2
+
→0
m→∞ m

f
b = lim lim supI+(ρb, H)
→0+

(85)
(86)

f
c0 ),
b, H
= I+
(σ

(87)

f
c0
b ≥ I f (σI−(ρb, H)− b , H ).

(88)

and similarly

Let us now prove the convexity of the smooth metric adjusted skew information rates.
Proposition 8 (Convexity of the smooth metric adjusted skew information rates). Let {pk }k be a(k) a sequences of states for each k. For a sequence of states
probability distribution. We denote ρd
PN
P
(k)
d
N
(k) given by {k=1 pk ρ
k=1 pk ρm }m , it holdsNXfp k I±



(k)

f
≥ I±

N
X

!
(k)

b
pk ρb , H

(89)

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17

k=1

b
ρb , H

k=1

Proof. By using the convexity of smooth metric adjusted skew informations, we immediately get
N
X





f
b =
pk I+
ρb(k) , H

k=1

N
X


1 f  (k)I ρm , Hm
→0 m→∞ m

(90)

N


1 X
(k)
pk If ρm, Hm
→0 m→∞ m
k=1

(91)

pk lim lim sup

k=1= lim lim sup1≥ lim lim sup If
→0 m→∞ m
N
X

f
= I+

N
X

!
(k)
pk ρm
, Hm

(92)

k=1

!
(k)

b .
pk ρb , H

(93)

k=1

P

It should be noted that this proof is valid only if we can exchange the order of the sum k and thePlimits lim→0 lim supm→∞ . When the summation k is taken over a finite set, this condition is always
satisfied.




P
f
f PN
(k) , H(k) , Hbbbb
In the same way, we also get N
p
I
ρ
≥
I
p
ρ
.
−
k=1 k
k=1 k −
Combining the convexity of the smooth metric adjusted skew information rates and Theorem 4, we
can derive the following simple upper bound for the distillable coherence:Xfp k I±

b
ρbk , H



!
f
≥ I±

X

k

(k)

b
pk ρb , H!
X

≥ Cdist

k

(k)

b .
pk ρb , H

(94)k

As a corollary, we also get upper bounds that can be calculated without using the smoothing technique:


1 X
1
pk I f ρ(k), Hm ≥ lim inf I f
lim inf
m
m→∞ m
m→∞ m
k

!
X

pk ρ(k)m , Hm!

≥ Cdist

k

X

(k)

b .
pk ρb , H(95)k

C Smooth metric adjusted skew information rates in the i.i.d. regime
We here explicitly calculate the smooth metric adjusted skew information rates in the i.i.d. regime.
For this purpose, we here prove Lemma 3:
Lemma (Restatement of Lemma 3). Let ψ be a pure state with period 2π with a Hamiltonian H.
Assume that the absolute third moment is finite, i.e., hψ||H|3 |ψi < ∞. Fix  be a sufficiently small
real parameter. Let ρb = {ρm }m be a sequence of states such that ρm ∈ B  (ψ ⊗dRme ) for all sufficiently
large m with a real parameter R > 0, where B  (ρ) is the -ball in the state space defined by B  (ρ) :={σ : states | D(ρ, σ) ≤ }. Then there exists a real function δ f () of  such that lim→0 δ f () = 0 and








I f ρm , Hiid,dmRe ≥ I f ψ ⊗dRme , Hiid,dmRe − mδ f () + o(m)where Hiid,k :=

Pk

i=1 I

(m → ∞),(96)⊗i−1 ⊗ H ⊗ I⊗k−i .

To prove this lemma, let us first derive a general behavior of metric adjusted skew informations forstates close to a pure state. The following lemma is an extension of claims in Corollary 1 in [20].
Lemma 9. Let Φ denote the eigenstate with the largest eigenvalue of an arbitrary state ρ. For a purestate Ψ, the infidelity of ρ and Ψ is defined by δ := 1 − hΨ | ρ | Ψi. Then it holds| hΦ|Ψi | ≥ 1 − 2δ.

(97)Furthermore, when δ < 1/2, the metric adjusted skew informations I f (ρ, H) are lower bounded asf (0)I f (ρ, H) ≥
f



δ
1−δ

 (1 − 2δ)2 Var(Φ, H).

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(98)

18

Proof. Equation (97) is proven in Corollary 1 in [20]. We here prove Eq. (98). Letρ = p |Φi hΦ| +kX

λi |ii hi| =:

i=1

k
X

λk |ii hi|

(99)

i=0

be the eigenvalue decomposition, where we defined λ0 := p and |0i := |Φi. SincehΨ|ρ|Ψi =kX

λk | hΨ|ii |2 ≤ p

i=0

k
X

| hΨ|ii |2 = p,

(100)

i=0

we have1 − δ ≤ p.

(101)

By using 1 = p + ki=1 λi , we get λi ≤ 1 − p ≤ δ for all i = 1, 2, · · · , k.
Equation (98) is derived asP

I f (ρ, H) =kf (0) X(λi − λj )2| hi|H|ji |2
2 i,j=0 λj f (λi /λj )

(102)

k
k
(λi − p)2
(λi − λj )2
f (0) Xf (0) X2
=2×
| hi|H|Φi | +| hi|H|ji |2
2 i=1 pf (λi /p)
2 i,j=1 λj f (λi /λj )≥ f (0)kX(λi − p)2

pf (λi /p)
i=1| hi|H|Φi |2 .

(103)

(104)

From λi ≤ δ and 1 − δ ≤ p, we have p − λi ≥ 1 − 2δ. Since δ < 1/2 implies 1 − 2δ > 0, it holds(p − λi )2 ≥ (1 − 2δ)2 . On the other hand since f is an operator monotone and hence a monotonic
function, we get


f

λi
p





≤f

δ
,
1−δ


(105)

where we used λi /p ≤ δ/(1 − δ). From these inequalities and p ≤ 1, we obtain a lower bound of metricadjusted skew informations asf (0)I f (ρ, H) ≥
f



f



δ
1−δ

f (0)

=

δ
1−δ

 (1 − 2δ)2kX| hi|H|Φi |2

(106)

i=1

 (1 − 2δ)2 Var(Φ, H)(107)

Consider a case where a state ρ is close to a pure state Ψ. Equation (98) shows that metric adjusted
skew informations I f (ρ, H) of are lower bounded by the variance of the eigenstate Φ of ρ with largest
eigenvalue. Furthermore, Eq. (97) shows that Φ is also close to Ψ in the trace distance. Therefore,
a lower bound of I f (ρm , HdRme ) in Lemma 3 can be derived by analyzing the asymptotic behavior of
the energy variances of pure states that are close to ψ ⊗dRme .
For this purpose, let us briefly review the asymptotic behavior of the energy distribution of ψ ⊗dRme .
Let us define the energy distribution
D

E

⊗dRme
pψ⊗dRme (E) := ψ ⊗dRme Πm,
E ψ

E ∈ Spec(Hiid,dRme ),Accepted in Quantum 2023-05-08, click title to verify. Published under CC-BY 4.0.

(108)
19

where Spec(A) denotes the set of all eigenvalues of an operator A and Πm
E denotes the projectors to
the eigenspace of Hiid,dRme with eigenvalue E. Since ψ has a period 2π, pψ⊗dRme (E) = 0 if E ∈/ Z.
In the asymptotic limit of m → ∞, it is known that the energy distribution pm := {pψ⊗dRme (E)}E
converges to the Poisson distribution up to a shift. Precisely, under the assumption that the absolute
third moment is finite, i.e., hψ||H|3 |ψi < ∞, it is shown [45, 47, 77] that there exists km ∈ Z for eachm such that

lim dTV pψ⊗dRme , Υkm PmRVar(ψ,H) = 0,

m→∞

(109)

where dTV denotes the total variation distance and Υk for k ∈ Z denotes the shift operation for aprobability distribution defined by (Υk p)(n) := p(n − k).
Now, let us prove an asymptotic behavior of the variance of random variables which approximately
follows the Poisson distribution:
Lemma 10. Fix a positive parameter λ > 0. Let q = {qm }m be a sequence of probability distributions
such that∃M > 0,∀m > M,∃km ∈ Z,dTV (qm , Υkm Pmλ ) ≤ (110)

for a sufficiently small parameter . Then there exists a function γλ () such thatlim γλ () = 0(111)

Var (qm ) ≥ Var (Pmλ ) − γλ ()m

(112)

→0

and

holds for all sufficiently large m.
Proof. For simplicity, we hereafter assume that the probability distributions qm are defined on integers
1 . Since the variance is invariant under the translation operation Υ , we can assume that km = 0k
without loss of generality. Let us introduce the following notations:δm (n) := Pmλ (n) − qm (n),µPmλ :=

X(113)

nPmλ (n) = mλ(114)

nqm (n),(115)

nµqm :=

Xn∆µm := µPmλ − µqm =Xnδm (n).

(116)

nFrom Pmλ (n) ≥ 0 and qm (n) ≥ 0, we havePmλ (n) ≥ δm (n) ≥ −qm (n).

(117)

Defining Am := {n ∈ Z|δm (n) > 0}, we haveXδm (n) = dTV (qm , Pmλ ) .

(118)

n∈Am

The variance of qm is decomposed into
Var (qm ) =X(n − µqm )2 qm (n)(119)(n − µqm )2 (Pmλ (n) − δm (n))

(120)

n

=

X
n

= Var(Pmλ ) + ∆µ2m −X(n − µqm )2 δm (n).

(121)

n
1

The proof is valid for a general case if we replace the equalities in Eqs. (118) and (119) with ≤ and ≥, respectively.

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20

To provide a lower bound of this quantity, let us introduce an
Im := [µPmλ −am , µPmλ +am ]∩Z,
√interval√−1
where am = α m. Here, α > 0 for  > 0 is defined by α λ := g (1 − ), whereβ2β2dβ √ e− 2 = Erfg(x) :=
2π
−x

Z x



x
√
2



x2
2x
− √ e− 2 .
2π

(122)

Note that α is uniquely determined since g(x) is monotonic and takes any values in [0, 1] for x ≥ 0.
Defining I¯m := Z \ Im , we haveX(n − µqm )2 δm (n)

(123)

n

=

X

(n − µqm )2 δm (n) +(n − µqm )2 δm (n)

(124)

X

(125)

n∈I¯m

n∈Im

≤

X

X

(n − µqm )2 δm (n) +(n − µqm )2 Pmλ (n)n∈I¯mn∈Im ∩Am≤ (|∆µm | + am )2  +X(n − µqm )2 Pmλ (n)

(126)

n∈I¯m

= (|∆µm | + am )2  + ∆µ2m

X

X

Pmλ (n) + 2∆µmn∈I¯m(n − µPmλ )Pmλ (n) +X(n − µPmλ )2 Pmλ (n).

n∈I¯m

n∈I¯m

(127)
By using the Stirling formula
n! ∼

√

2πn

 n
n

e

,

(128)

it holds
Pmλ (mλ + k) =(mλ)mλ+k −mλe(mλ + k)!

1∼p2π(mλ + k)
β2
1
∼√
e− 2λ
2πmλ

(129)



mλ
mλ + k

mλ+k

ek

(130)
(131)

√
for k = β m as m → ∞.
With this formula, we have
Z α

β2
β
dβ √
e− 2λ = 0,
2πλ
−α

(132)

√
β 2 − β2e 2λ = λg(α λ) = λ(1 − ).
dβ √
2πλ
−α

(133)

1 X
lim √
(n − µPmλ )Pmλ (n) =
m→∞
m n∈I
m

and
1 X
lim
(n − µPmλ )2 Pmλ (n) =
m→∞ m
n∈I
m

Z α

implying that√(n − µPmλ )Pmλ (n) = o( m)(134)(n − µPmλ )2 Pmλ (n) = λm + o(m).

(135)

X
n∈I¯m

X
n∈I¯m

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21

Therefore, for any positive constants u and v which are independent of m,X√(n − µPmλ )Pmλ (n) ≤ u m

(136)

n∈I¯m

X

(n − µPmλ )2 Pmλ (n) = (λ + v)m

(137)

n∈I¯m

√
hold for all sufficiently large m. For future convenience, we take u = α λ and v = λ, which implies
that
√
X
(138)
(n − µPmλ )Pmλ (n) ≤ α mλ
n∈I¯m

X

(n − µPmλ )2 Pmλ (n) ≤ 2λm

(139)

n∈I¯m

hold for all sufficiently large m.
On the other hand, by using Chebyshev’s Inequality we haveX

Pmλ (n) ≤

n∈I¯m

λ
.
α2

(140)

By using these results, the variance of qm is lower bounded as
Var (qm ) = Var(pm ) + ∆µ2m −X(n − µqm )2 δm (n)

(141)

n

≥ Var(pm ) + ∆µ2m −



√
√ 2
λ
|∆µm | + α m  + ∆µ2m 2 + 2∆µm α mλ + 2λm .
α


(142)

Since the right hand side is quadratic in ∆µm , we get√√ 2
λ|∆µm | + α m  + ∆µ2m 2 + 2∆µm α mλ + 2λm
α




√
λ
≥ 1 −  − 2 |∆µm |2 − 4α  mλ|∆µm | − α2  + 2λ m
α∆µ2m −
2 2 λ


4α

 m
≥ −  α2  + 2λ + 



(143)
(144)
(145)

1 −  − αλ2where we have assumed that
1−−

λ
>0
α2

(146)

−1
in the√ last line. This assumption is true for a sufficiently small  since lim→0 α = lim→0 g (1 −)/ λ = ∞ holds.
We now introduce4α2 2 λ

γλ () := α2  + 2λ + 

1 −  − αλ2

.

(147)



From Fig. 2, it holds√
lim α = lim g −1 (1 − )/ λ = 0

→0

→0



lim α2 = lim  g −1 (1 − )

2

(148)

/λ = 0.

(149)

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22

→0

→0

Therefore, we get lim→0 γλ () = 0, completing the proof.

1.2
1.0
0.8

ϵ g-1 (1-ϵ )

0.6

ϵ (g-1 (1-ϵ ))2

0.4
0.2

0.2

0.4

0.6

0.8

1.0

ϵ

Figure 2: The behaviors of g −1 (1 − ) and (g −1 (1 − ))2 for  ∈ [0, 1].

Combining these lemmas, we here prove Lemma 3:
Proof of Lemma 3. We denote ψm := ψ ⊗dRme . Let ρb = {ρm }m be an arbitrary sequence of states suchthat ρm ∈ B  (ψm ). From the Fuchs-van de Graaf inequalities, an upper bound of the infidelity of ρmand ψm is derived as
1 − hψm | ρm | ψm i ≤ 1 − (1 − D(ρm , ψm )) ≤ 1 − (1 − )2 =: δ1 .

(150)

Let Φm be the eigenstate of ρm with the largest eigenvalue. From Eq. (97), we get|hΦm | ψm i| ≥ 1 − δ1 .

(151)

For the probability distribution pΦm of Φm , it holdsdTV (pΦm , pψm ) ≤ D (Φm , ψm ) ≤q

1 − (1 − δ1 )2 .

(152)

By using Eq. (109), for any  > 0,dTV (pψm , Υkm Pmλ ) ≤ ∃km ∈ Z,

(153)

holds for all sufficiently large m, where λ := RVar(ψ, H). Therefore, from the triangle inequality, weget∃km ∈ Z,dTV (pΦm , Υkm Pmλ ) ≤q

1 − (1 − δ1 )2 +  =: δ2(154)

for all sufficiently large m. Note that lim→0 δ1 = lim→0 δ2 = 0 holds. From Lemmas 9 and 10, we
get
I(ρm , Hiid,dmRe ) ≥

f (0)
f



δ
1−δ1

 (1 − 2δ1 )2 Var(Φm , Hm )≥ m (λ − γλ (δ2 ))

f (0)
f



δ1
1−δ1

 (1 − 2δ1 )2

(155)
(156)

as m → ∞, where γλ is defined in Lemma 10. Definingδ f () := (λ − γλ (δ2 ))

f (0)
f



δ1
1−δ1

 (1 − 2δ1 )2 − λ,(157)

we have lim→0 δ f () = 0 andI(ρm , Hiid,dmRe ) ≥ mλ − mδ f () = mRVar(ψ, H) − mδ f ()Accepted in Quantum 2023-05-08, click title to verify. Published under CC-BY 4.0.

(158)
23

for all sufficiently large m. Since
1  ⊗dRme
I ψ, Hiid,dmRe = RVar(ψ, H),m→∞ m
this result can also be written as

lim





I(ρm , Hiid,dmRe ) ≥ I ψ ⊗dRme , Hiid,dmRe − mδ f () + o(m)(159)

(m → ∞).

(160)

As an immediate corollary, we get Theorem 2:
Theorem (Restatement of Theorem 2). Let ψ be a pure state having period 2π for a Hamiltonian
H. Assume that the third absolute moment of the Hamiltonian is finite, i.e., hψ||H|3 |ψi < ∞. For ab iid (R) := {Hpositive parameter R > 0, define ψbiid (R) := {ψ ⊗dRme }m and H
iid,dRme }m , where Hiid,k :=
Pk
⊗i−1
⊗k−i
⊗H ⊗I
. The smooth metric adjusted skew information rates for this i.i.d. sequence
i=1 I
are given byf bb iid (R))
b iid (R)) = I f (ψbiid (R), H
I+(ψiid (R), H−
1 f ⊗dRme
= lim
I (ψ
, HdRme ) = I f (ψ, H) R.
m→∞ m
Proof. From Lemma 3, we getf bb iid (R)) ≥ I f (ψbiid (R), H
b iid (R)) ≥ I(ψ, H)R.
I+
(ψiid (R), H
−

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(162)

A straightforward calculation shows1
1 f ⊗dRme
I (ψ, Hiid,dRme ) = lim
dRmeI(ψ, H) = I(ψ, H)R.
m→∞ m
m→∞ m
lim

(163)

Since
lim

1

m→∞ m

f b
b iid (R)) ≥ I f (ψbiid (R), H
b iid (R))
I f (ψ ⊗dRme , Hiid,dRme ) ≥ I+
(ψiid (R), H
−

(164)

f
, we getholds by definition of I±f bb iid (R)) = I f (ψbiid (R), H
b iid (R)) = I(ψ, H)R.
I+
(ψiid (R), H
−

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As another corollary, we obtain an alternative proof of the converse part of the conversion theory
for i.i.d. pure states [47]:Corollary 11. Let ψ and φ be pure states with period 2π with Hamiltonians H and H 0 , respectively.
b Hbbb iid ) cov
b 0 (R)) holds for ψb = {ψ ⊗m }m and φ(R)If (ψ,
 a (φ(R),H= {φdRme }m , then it holds
iid
Var(ψ, H) ≥ Var(φ, H 0 )R.

(166)

Var(ψ, H)F(ψ, H)=.
0
Var(φ, H )F(φ, H 0 )

(167)

i.e.,
R≤

Proof. From the monotonicity of the smooth metric adjusted skew information rates, we get
f b b
f b
b 0 (R)).
I±(ψ, H) ≥ I±
(φ(R), H(168)

From Theorem 2, this inequality implies
I(ψ, H) ≥ I(φ, H 0 )R.

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24

D Review of the spectral quantum Fisher information rates
We here briefly review the results in [45]. For a pure state ψ and a Hamiltonian H, the period isdefined asnoτ := inf t | hψ|e−itH |ψi | = 1 .

(170)

t>0

We assume that H is bounded below and 0 < τ < ∞. Without loss of generality, we can set the periodτ
to be 2π by rescaling the Hamiltonian as H → 2π
H. In the following, we always assume that purestates have period 2π for simplicity. In this case, the pure state ψ has support in the eigenspaces ofthe Hamiltonian with eigenvalues given by n + E0 , where n is a positive integer and E0 is a constant.
Shifting the Hamiltonian by a constant, we can assume that E0 = 0. We define the energy distributionof a pure state ψ bypψ (n) := hψ|Πn |ψi(n ∈ Z≥0 ),(171)

where Πn is a projector to the eigenspace of the Hamiltonian with eigenvalue n. An essential fact isthat the exact convertibility among pure states is fully characterized by the energy distribution [14].
cov
cov
0 ,H
0
This is because (ψ, H)  (ψHOHO ) and (ψHO , HHO )  (ψ, H) holds for a state
0
|ψHO
i :=

∞ q
X

pψ (n) |ni

(172)

n=0

of a harmonic oscillator system with Hamiltonian HHO = ∞n=0 n |ni hn|. This observation can also be
extended to the convertibility with vanishing error in the asymptotic regime. For further detail, seealso [45].
To define the max- and min-quantum Fisher information, we introduce several notations. For a real
number λ ∈ R, we define a generalized Poission distribution Pλ = {Pλ (n)}n∈Z by
P(

Pλ (n) :=ne−λ λn!
0(n ≥ 0).
(n < 0)(173)

For λ ≥ 0, this is an ordinary Poisson distribution. However, for λ < 0, it is not a probability
distribution since some of the element becomes negative. For sequences of numbers a = {a(n)}n∈Z
and b = {b(n)}n∈Z , we define the convolution sequence (a ∗ b) = {(a ∗ b)(n)}n∈Z by (a ∗ b)(n) :=Pe denotes its “inverse" sequence with respect to ∗ in thek∈Z a(n − k)b(k). For a given sequence q, q
sense that (q ∗ qe)(n) = δn,0 , where δn,m is the Kronecker delta. We denote a ≥ 0 for a = {a(n)}n if
and only if a(n) ≥ 0 for all n. The max- and min-quantum Fisher information are defined by [45]Fmax (ψ, H) := inf {4λ | Pλ ∗ pfψ ≥ 0} ,Fmin (ψ, H) := sup {4λ | pψ ∗ P−λ ≥ 0} .

(174)
(175)

To define the smooth max- and min-quantum Fisher information, let us first assume that the system
of interest is a harmonic oscillator. The max-quantum Fisher information is extended to a mixed
state ρ by Fmax (ρ, HHO ) = inf Φρ Fmax (Φρ , HHO + HA ), where the infimum is taken over the set of all
purification Φρ of ρ and HA is a Hamiltonian of an ancillary system with integer eigenvalues. The
-smooth max- and min-quantum Fisher information are then defined asFmax(ψ, HHO ) :=inf (ρ, HHO ),ρ∈B  (ψ)

Fmin(ψ, HHO ) :=supφ∈Bpure(ψ)

Fmin(φ, HHO ),

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
where -balls are defined by B  (ψ) := {ρ: states | D(ρ, ψ) ≤ } and Bpure(ψ) := {φ: pure states |D(φ, ψ) ≤ }.

Accepted in Quantum 2023-05-08, click title to verify. Published under CC-BY 4.0.

25

For a system with a generic Hamiltonian, the -smooth max- and min-quantum Fisher information
are defined by


0
Fmax
(ψ, H) := Fmax
(ψHO
, HHO ),



0
Fmin
(ψ, H) := Fmin
(ψHO
, HHO ),

(177)

0where ψHO
is defined in Eq. (172).
bb
Let (ψ, H) = ({ψm }m , {Hm }m ) be any sequences of pure states and Hamiltonians with period 2π.
The spectral sup- and inf- quantum Fisher information rates are defined as [45]1 Fmax (ψm , Hm ) ,→0 m→∞ mb H)b := lim lim inf 1 F  (ψm , Hm ) .
F(ψ,min→0 m→∞ mb H)b := lim lim sup
F(ψ,

(178)
(179)

These quantities are shown to be equal to the coherence cost and the distillable coherence [45], i.e.,
b H)
b H),
b = F(ψ,b
Ccost (ψ,

b H)
b H).
b = F(ψ,b
Cdist (ψ,Accepted in Quantum 2023-05-08, click title to verify. Published under CC-BY 4.0.

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