Max- relative entropy of coherence: an operational coherence measure
Kaifeng Bu Uttam Singh Shao-Ming Fei Arun Kumar Pati Junde Wu
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 bkf@zju.edn.cn
 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China
 
   
 
  uttamsingh@hri.res.in
 Harish-Chandra Research Institute, Allahabad, 211019, India
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
     Mumbai 400085, India
 
   
   feishm@cnu.edn.cn
 School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China
  Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
 
  
  akpati@hri.res.in
Harish-Chandra Research Institute, Allahabad, 211019, India

  
   wjd@zju.edn.cn
 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China








The operational characterization of quantum coherence is the corner stone in the development of  resource theory of coherence.
We introduce a new coherence quantifier based on max-relative entropy. We prove that  max-relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of max-relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states.
This advantage of coherent states in subchannel discrimination can be exactly
characterized by the max-relative entropy of coherence.
By introducing suitable smooth max-relative entropy of coherence,
we prove that the smooth max-relative entropy of coherence
provides a lower bound of one-shot coherence cost, and the max-relative entropy of
coherence is equivalent to the relative entropy of coherence in asymptotic limit.
Similar to max-relative entropy of coherence, min-relative entropy of coherence has also been investigated. We show that the min-relative entropy of coherence provides an upper bound of
one-shot coherence distillation, and in asymptotic limit the min-relative
entropy of coherence is equivalent to the relative entropy of coherence.


Max- relative entropy of coherence : an operational coherence measure
    Junde Wu
    December 30, 2023
=====================================================================






§ INTRODUCTION

Quantumness in a single system is characterized by quantum coherence, namely, the superposition of
a state in a given reference basis. The coherence of a state may quantify the capacity of a system
in many quantum manipulations, ranging from  metrology <cit.> to thermodynamics <cit.> .
Recently, various efforts have been made to develop a resource theory of coherence <cit.>.
One of the earlier resource theories is that of quantum entanglement <cit.>, which is a basic resource for various quantum information processing protocols such as
superdense coding <cit.>, remote state preparation <cit.> and quantum teleportation <cit.>. Other notable examples include the resource theories of asymmetry <cit.>, thermodynamics <cit.>, and steering <cit.>. One of the main advantages that a resource theory offers is the lucid quantitative and operational description as well as the manipulation of the relevant resources at ones disposal, thus operational
characterization of quantum coherence is required in the resource theory of coherence.


A resource theory is usually composed of two basic elements: free states and free operations. The
set of allowed states (operations) under the given constraint is what we call the set of free states (operations).
Given a fixed basis, say |i⟩^d-1_i=0 for a d-dimensional system, any quantum state which is diagonal
in the reference basis is called an incoherent state and is a free state in the resource theory of coherence. The set of
incoherent states is denoted by . Any quantum state can be mapped into an incoherent state by a
full dephasing operation Δ, where Δ(ρ):=∑^d-1_i=0iρii.
However, there is no general consensus on the set of free operations in the resource theory of coherence.
We refer the following types of free operations in this work:
maximally incoherent operations (MIO) <cit.>, incoherent operations (IO) <cit.>, dephasing-covariant operations (DIO) <cit.> and strictly incoherent operations (SIO) <cit.>.
By maximally incoherent operation (MIO), we refer to the maximal set of quantum operations
Φ which maps the incoherent states into incoherent states, i.e., Φ()⊂ <cit.>.
Incoherent operations (IO) is the set of all  quantum operations Φ that
admit a set of Kraus operators K_i such that Φ(·)=∑_iK_i(·)K^†_i and
K_i K^†_i⊂ for any i <cit.>.
Dephasing-covariant operations (DIO) are the quantum operations Φ with
[Δ,Φ]=0 <cit.>. Strictly incoherent operations (SIO) is the set of all quantum operations
Φ admitting a set of Kraus operators K_i such that Φ(·)=∑_iK_i(·)K^†_i and
Δ (K_iρ K^†_i)=K_iΔ(ρ) K^†_i for any i and any quantum state ρ.
Both IO and DIO are subsets of MIO , and SIO is a subset of both IO and DIO <cit.>.
However, IO and DIO are two different types of free operations and there is no inclusion relationship between
them (The operational gap between them can be seen in <cit.>).


Several operational coherence quantifiers have been introduced as candidate coherence measures, subjecting to physical requirements such as monotonicity under certain type of free operations in the resource theory
of coherence. One canonical measure to quantify coherence is the relative entropy of coherence, which
is defined as C_r(ρ)=S(Δ(ρ))-S(ρ), where S(ρ)=-ρlogρ is the von Neumann entropy <cit.>.
The relative entropy of coherence plays an important role in the process of coherence distillation, in which it can be interpreted as the optimal rate to distill
maximally coherent state from a given state ρ by IO in the asymptotic limit <cit.>.
Besides, the l_1 norm of coherence <cit.>, which is defined
as C_l_1(ρ)=∑_i≠ j|ρ_ij| with ρ_ij=iρj, has also attracted
lots of discussions about its operational interpretation <cit.>.
Recently, an operationally motivated coherence measure- robustness of coherence (RoC) -
has been introduced, which quantifies the minimal mixing required to erase the coherence in a given quantum state <cit.>.
There is growing concern about the operational characterization of quantum coherence
and further investigations are needed to provide an explicit and rigorous
operational interpretation of coherence.


In this letter, we introduce a new coherence measure based on max-relative entropy
and focus on its operational characterizations.
Max- and min- relative entropies have been introduced and investigated in <cit.>. The well-known (conditional and unconditional) max- and min- entropies  <cit.>
can be obtained from these two quantities. It has been shown that max- and min-entropies are of operational
significance in the applications ranging from data compression <cit.> to state merging <cit.> and security of key <cit.>. Besides, max- and min- relative entropies have been used to define entanglement monotone and their operational significance in the manipulation of entanglement has been provided in <cit.>.
Here, we define max-relative of coherence C_max based on max-relative entropy and investigate the properties
of C_max. We prove that max-relative entropy of coherence for a given state ρ is the maximum achievable
overlap with maximally coherent states under DIO, IO and SIO, which
gives rise to an operational interpretation of C_max and shows the equivalence among DIO, IO and SIO in an operational task.
Besides, we show that max-relative entropy of coherence characterizes the
role of  quantum states in an operational task: subchannel discrimination.
Subchannel discrimination is an important quantum information task which distinguishes the branches of a quantum evolution
for a quantum system to undergo <cit.>. It has been shown that every entangled or steerable
state is a resource in some instance of subchannel discrmination problems <cit.>. Here, we prove that
that every coherent state is useful in the subchannel discrimination of certain instruments, where the usefulness can be quantified by the
max-relative entropy of coherence of the given quantum state.
By smoothing the max-relative entropy of coherence, we introduce ϵ-smoothed max-relative entropy of coherence
C^ϵ_max for any fixed ϵ>0 and show that the smooth max-relative entropy gives an lower bound of  coherence cost in  one-shot version. Moreover,
 we prove that for any quantum state,
max-relative entropy of coherence  is equivalent to
the relative entropy of coherence in asymptotic limit.

Corresponding to the max-relative entropy of coherence, we also introduce the min-relative entropy of coherence C_min by min-relative entropy, which is not a proper coherence
measure as it may increase on average under IO.
However, it gives an upper bound for the maximum overlap between the given states and the set of incoherent states. This implies that min-relative entropy of coherence
 also provides a lower bound of a well-known coherence measure, geometry of coherence <cit.>.
By smoothing the min-relative entropy of coherence, we introduce ϵ-smoothed min-relative entropy of coherence
C^ϵ_min for any fixed ϵ>0 and show that the smooth max-relative entropy gives an upper bound of  coherence distillation in  one-shot version.
Furthermore, we show that the min-relative of coherence is also
equivalent to distillation of coherence in asymptotic limit. The relationship among C_min, C_max and other
coherence measures has also been investigated.





§ MAIN RESULTS

Let  be a d-dimensional Hilbert space and () be the set
of density operators acting on . Given two operators  ρ and σ with
ρ≥ 0, ρ≤ 1 and σ≥ 0,  the max-relative entropy of ρ with respect to σ is defined by <cit.>,

    D_max(ρ||σ):   =   minλ:ρ≤2^λσ.

We introduce a new coherence quantifier by max-relative  entropy: max-relative entropy of coherence C_max,

    C_max(ρ):=min_σ∈D_max(ρ||σ),

where  is the set of incoherent states in ().

We now show that C_max satisfies the conditions a coherence measure needs to fulfil. First, it is obvious that
C_max(ρ)≥ 0. And since D_max(ρ||σ)=0 iff ρ=σ <cit.>, we have C_max(ρ)=0 if and only if ρ∈.
Besides, as D_max is monotone under CPTP maps <cit.>,  we have
C_max(Φ(ρ))≤ C_max(ρ) for any incoherent operation Φ.
Moreover, C_max is nonincreasing on average under incoherent operations, that is,
for any incoherent operation Φ(·)=∑_i K_i(·)K^†_i
with K_i K^†_i⊂,
∑_i p_iC_max(ρ̃_i)≤ C_max(ρ),
where p_i=K_iρ K^†_i and ρ̃_i=K_iρ K^†_i/p_i, see proof in Supplemental Material <cit.>.

Remark We have proven that the max-relative entropy of coherence C_max is a bona fide measure of coherence.
Since D_max is not jointly convex, we may not expect that C_max has the convexity, which is a desirable (although
not a fundamental) property for a coherence quantifier. However, we can prove that for ρ=∑^n_ip_iρ_i,
C_max(ρ)≤max_iC_max(ρ_i).
Suppose that C_max(ρ_i)=D_max(ρ_i||σ^*_i) for some σ^*_i, then
from the fact that  D_max(∑_ip_iρ_i||∑_ip_iσ_i)≤max_iD_max(ρ_i||σ_i) <cit.>, we have
C_max(ρ)≤ D_max(∑_ip_iρ_i||∑_ip_iσ^*_i) ≤max_iD_max(ρ_i||σ^*_i)=max_iC_max(ρ_i). Besides, although
C_max is not convex, we can obtain a proper coherence measure with
convexity from C_max by the approach of convex roof extension, see Supplemental Material <cit.>.


In the following, we concentrate on the operational characterization of the
max-relative entropy of coherence, and provide
operational interpretations of C_max.

Maximum overlap with maximally coherent states.—At first we show that
2^C_max is equal to the maximum overlap with the maximally coherent state that can be achieved by  DIO, IO and SIO.


 Given a quantum state ρ∈(), we have

    2^C_max(ρ)   =   dmax_ℰ,|Ψ⟩ F(ℰ(ρ),Ψ)^2,

where F(ρ,σ)=|√(ρ)√(σ)| is the fidelity between states ρ and σ <cit.>,
|Ψ⟩∈ and  is the set of maximally coherent states in (), ℰ belongs to either
DIO or IO or SIO.

(See proof in Supplemental Material <cit.>.)

Here although IO, DIO and SIO are different types of free operations in resource theory of coherence <cit.>,
they have the same behavior in the maximum overlap with the maximally coherent states.
From the view of coherence distillation <cit.>, the maximum
overlap with maximally coherent states can be regarded as the distillation of coherence from
given states under IO, DIO and SIO. As fidelity can be used to define certain distance, thus
C_max(ρ)  can also be viewed as the distance between the set of maximally coherent state and the
set of  (ρ)_∈θ, where θ=  DIO, IO  or  SIO.

Besides distillation of coherence, another kind of coherence manipulation is the coherence cost <cit.>.
Now we study the one-shot version of coherence cost under MIO based on smooth max-relative entropy of coherence.
We define the one-shot coherence cost of a quantum state ρ under MIO as

    C^(1),ϵ_C,MIO(ρ):=min_ℰ∈ MIO
    
    M∈ℤlog M: F(ρ,ℰ(Ψ^M_+))^2≥1-ϵ,

where |Ψ^M_+⟩=1/√(M)∑^M_i=1|i⟩, ℤ is the set of integer and
ϵ>0. The ϵ-smoothed max-relative entropy of coherence of a quantum state ρ is defined by,

    C^ϵ_max(ρ):=min_ρ'∈ B_ϵ(ρ)C_max(ρ'),

where B_ϵ(ρ):=ρ'≥0:ρ'-ρ_1≤ϵ, ρ'≤ρ.
We find that the
smooth max-relative entropy of coherence gives a lower bound of one-shot coherence cost.
Given a quantum state ρ∈(), for any ϵ>0,

    C^ϵ'_max(ρ)≤  C^(1),ϵ_C,MIO(ρ),

where ϵ'=2√(ϵ), see proof in Supplemental Material <cit.>.



Besides, in view of smooth max-relative entropy of coherence, we can obtain the
equivalence between max-relative entropy of coherence and relative entropy of coherence in the asymptotic limit.
Since  relative entropy of coherence is the optimal rate to distill
maximally coherent state from a given state under certain free operations in the asymptotic limit <cit.>,
the smooth max-relative entropy of coherence in asymptotic limit is just the distillation of coherence. That is,
given a quantum state ρ∈(), we have

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n)
    =C_r(ρ).

(The proof is presented in Supplemental Material <cit.>.)

Maximum advantage achievable in subchannel discrimination.–
Now, we investigate  another quantum information processing task: subchannel discrimination, which can also provide an operational
interpretation of C_max. Subchannel discrimination is an important quantum information task which is used to identify the branch of a quantum evolution
to undergo. We consider some special instance of subchannel discrimination problem to show the advantage of coherent states.

A linear completely positive and trace non-increasing map  is called a subchannel. If
a subchannel  is trace preserving, then  is called a channel.
An instrument ℑ=_a_a for a channel  is a collection of subchannels _a with
=∑_a_a  and every instrument has its physical realization <cit.>. A dephasing covariant instrument ℑ^D for a DIO 
is a collection of subchannels _a_a such that =∑_a_a. Similarly, we can define
incoherent instrument ℑ^I and strictly incoherent instrument ℑ^S for channel ∈ IO and ∈ SIO respectively.

Given an instrument ℑ=_a_a for a quantum channel , let us consider a Positive Operator Valued Measurement (POVM)
M_b_b with ∑_bM_b=𝕀. The probability of successfully discriminating the subchannels in the instrument
ℑ by POVM M_b_b for input state ρ is given by

    p_succ(ℑ,M_b_b, ρ)
    =∑_a_a(ρ)M_a.

The optimal probability of success in subchannel discrimination of ℑ over
all POVMs is given by

    p_succ(ℑ,ρ)=max_M_b_b
    p_succ(ℑ,M_b_b, ρ).

If we restrict the input states to be incoherent ones, then the optimal probability of success among all incoherent states is given by

    p^ICO_succ(ℑ)=max_σ∈p_succ(ℑ,σ).

We have the following theorem.


Given a quantum state ρ, 2^C_max(ρ)
is the maximal advantage achievable by ρ
compared with incoherent states in all subschannel discrimination problems
of dephasing-covariant, incoherent and strictly incoherent instruments,

    2^C_max(ρ)   =   max_ℑp_succ(ℑ,ρ)/p^ICO_succ(ℑ),

where ℑ is either ℑ^D or ℑ^I or ℑ^S, denoting the dephasing-covariant, incoherent and strictly incoherent instrument, respectively.

The proof of Theorem <ref> is presented in Supplemental Material <cit.>. This result shows that the advantage of coherent states in
certain instances of subchannel discrimination problems can be exactly captured by C_max, which provides another operational interpretation
of C_max and also shows the equivalence among DIO, IO and SIO in the information processing task of subchannel discrimination.

Min-relative entropy of coherence C_min(ρ).–Given  two operators  ρ and σ with
ρ≥ 0, ρ≤ 1 and σ≥ 0,  max- and min- relative entropy of ρ relative to σ are defined as

    D_min(ρ||σ):=-logΠ_ρσ

where Π_ρ denotes the projector onto suppρ, the  support of ρ.
Corresponding to C_max(ρ) defined in (<ref>), we can similarly introduce a quantity defined by min-relative entropy,

    C_min(ρ):=min_σ∈D_min(ρ||σ).

Since D_min(ρ||σ)=0 if supp ρ=supp σ <cit.>,
we have
ρ∈ ⇒ C_min(ρ)=0. However, converse direction may not be true, for example,
let ρ=1/20+1/2+ with |+⟩=
1/√(2)(|1⟩+|2⟩), then ρ is coherent but C_min(ρ)=0.
Besides, as D_min is monotone under CPTP maps <cit.>,
we have  C_min(Φ(ρ))≤ C_min(ρ) for any
Φ∈ IO.
However, C_min may increase on average under IO (see Supplemental Material <cit.>). Thus, C_min is not be a proper coherence measure as C_max.

Although C_min is not a good coherence quantifier, it still has some interesting properties in the manipulation of coherence. First, C_min gives upper bound of the maximum overlap with the set of incoherent states for any given quantum state ρ∈(),

    2^-C_min(ρ)≥max_σ∈F(ρ,σ)^2.

Moreover, if ρ is pure state |ψ⟩, then above equality holds, that is,

    2^-C_min(ψ)=max_σ∈F(ψ,σ)^2,

see proof in  Supplemental Material <cit.>.

Moreover, for geometry of coherence defined by C_g(ρ)=1- max_σ∈F(ρ,σ)^2 <cit.>, C_min also provides a lower bound for C_g as follows
 
    C_g(ρ)≥ 1-2^-C_min(ρ).


Now let us consider again the one-shot version of distillable coherence under MIO by
modifying and smoothing the min-relative entropy of coherence C_min. We define the one-shot distillable coherence of a quantum state ρ under MIO as

    C^(1),ϵ_D,MIO(ρ):=max_ℰ∈ MIO
    
    M∈ℤlog M: F(ℰ(ρ),Ψ^M_+)^2≥1-ϵ,

where |Ψ^M_+⟩=1/√(M)∑^M_i=1|i⟩ and ϵ>0.

For any ϵ>0, we define the smooth min-relative entropy of coherence of a quantum state ρ as follows

    C^ϵ_min(ρ):=max_
    0≤ A≤𝕀
    Aρ≥ 1-ϵmin_σ∈-logAσ,

where 𝕀 denotes the
identity.
It can be shown that C^ϵ_min is a upper bound of one-shot distillable coherence,

    C^(1),ϵ_D,MIO(ρ)≤ C^ϵ_min(ρ)

for any ϵ>0, see proof in Supplemental Material <cit.>.

The distillation of coherence in asymptotic limit can be expressed as

    C_D,MIO=lim_ϵ→ 0lim_n→∞1/nC^(1),ϵ_D,MIO(ρ).

It has been proven that C_D,MIO(ρ)=C_r(ρ) <cit.>. Here we show that the equality in inequality (<ref>) holds in the asymptotic limit as the C_min is equivalent to C_r in the asymptotic limit.
Given a quantum state ρ∈(), then

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n)
    =C_r(ρ).

(The proof is presented in Supplemental Material <cit.>.)

We have shown that C_min gives rise to the bounds for maximum overlap with the incoherent states
and for one-shot distillable coherence. Indeed the exact expression of C_min for some special class of quantum states can be calculated.
For pure state |ψ⟩=∑^d_i=1ψ_i|i⟩ with ∑^d_i=1|ψ_i|^2=1, we have
C_min(ψ)=-logmax_i|ψ_i|^2. For maximally coherent state |Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, we have
C_min(Ψ)=log d, which is the maximum value for C_min in d-dimensional space.

Relationship between C_max and other coherence measures.–
First, we investigate the relationship among C_max, C_min and C_r.
Since D_min(ρ||σ)≤ S(ρ||σ)≤ D_max(ρ||σ)  for any quantum states ρ and σ <cit.>,
one has

    C_min(ρ)≤ C_r(ρ)≤ C_max(ρ).

Moreover, as mentioned before,
these quantities are all equal in the asymptotic limit.

Above all, C_max is equal to the logarithm of robustness of coherence, as
RoC(ρ)=min_σ∈s≥0|ρ≤(1+s)σ and C_max(ρ)=min_σ∈minλ:ρ≤2^λσ <cit.>, that is,  2^C_max(ρ)=1+RoC(ρ).
Thus, the operational interpretations of C_max in terms of maximum overlap with maximally coherent states and
subchannel discrimination, can also be viewed as the operational interpretations of robustness of coherence RoC.
It is known that robustness of coherence plays an important role in a phase discrimination task, which provides an
operational interpretation for robustness of coherence <cit.>. This phase discrimination task
 investigated in <cit.> is just
a special case of the subchannel discrimination in depasing-covariant  instruments. Due to the relationship between C_max and RoC, we can obtain the closed form of
C_max for some special class of quantum states.
As an example, let us consider a pure state |ψ⟩=∑^d_i=1ψ_i|i⟩. Then C_max(ψ)=log((∑^d_i=1|ψ_i|)^2))=2log(∑^d_i=1|ψ_i|).
Thus, for maximally coherent state |Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, we have
C_max(Ψ)=log d, which is the maximum value for C_max in d-dimensional space.



Since RoC(ρ)≤ C_l_1(ρ) <cit.> and 1+RoC(ρ)=2^C_max(ρ), then
C_max(ρ)≤log(1+C_l_1(ρ)). We have the relationship among these coherence measures,

    C_min(ρ)≤ C_r(ρ)≤ C_max(ρ)
        =   log(1+RoC(ρ))
       ≤   log(1+C_l_1(ρ)),

which implies that 2^C_r(ρ)≤ 1+C_l_1(ρ) (See also <cit.>).




§ CONCLUSION

We have investigated the properties of max- and min-relative entropy of coherence, especially the operational interpretation of the max-relative entropy of coherence.
It has been found that the max-relative entropy of coherence characterizes the
maximum overlap with the maximally coherent states under DIO, IO and SIO, as
well as the maximum advantage achievable by coherent states compared with all incoherent states in
subchannel discrimination problems of all dephasing-covariant, incoherent and strictly incoherent instruments, which
also provides new operational interpretations of robustness of coherence and illustrates the equivalence of DIO, IO and SIO in these two
operational taks. The study of C_max and C_min also makes the relationship between the operational
coherence measures (e.g. C_r and C_l_1 ) more clear.
These results may highlight the understanding to the operational resource theory of coherence.

Besides, the relationships among smooth max- and min- relative relative entropy of coherence
and one-shot coherence cost and distillation have been investigated explicitly. As
both smooth max- and min- relative entropy of coherence
are equal to relative entropy of coherence in the asymptotic limit and the significance of relative entropy of coherence in the distillation of coherence, further studies are desired on the one-shot coherence cost
and distillation.


This work is supported by the Natural Science Foundation of China (Grants No. 11171301, No. 10771191, No. 11571307 and No. 11675113) and the Doctoral Programs Foundation of the Ministry of Education of China (Grant No. J20130061).



apsrev4-1

 






§ STRONG MONOTONICITY UNDER IO FOR C_MAX 


We prove this property based on the method in  <cit.> and the basic facts of D_max <cit.>. Due to the definition of C_max, there exists an optimal σ_*∈ such that C_max(ρ)=D_max(ρ||σ_*). Let σ̃_i=K_iσ_* K^†_i/K_iσ_* K^†_i, then we have

    ∑_ip_i D_max(ρ̃_i||σ̃_i)
       ≤   ∑_i D_max(K_iρ K^†_i||K_iσ_* K^†_i)
       ≤   ∑_i D_max(E𝕀iUραU^†𝕀i||
          ×E𝕀iUσ_*αU^†𝕀i)
       ≤   ∑_i D_max(𝕀iUραU^†𝕀i||
          ×𝕀iUσ_*αU^†𝕀i)
       =    D_max(UραU^†||Uσ_*αU^†)
       =   D_max(ρα||σ_*α)
       =   D_max(ρ||σ_*)
       =   C_max(ρ),

where the first inequality comes from the proof of Theorem 1 in <cit.>, the second inequality comes from the fact that there exists an extended Hilbert space _E, a pure |α⟩∈_E and a
global unitary U on _E such that E𝕀iUραU^†𝕀i=K_iρ K^†_i <cit.>, the third
inequality comes from the fact that D_max is monotone under partial trace <cit.>, the last inequality
comes from the fact that for any set of mutually orthogonal projectors P_k,
D_max(∑_kP_kρ_1 P_k||∑_kP_kρ_2 P_k)=∑_kD_max(P_kρ_1 P_k||P_kρ_2 P_k) <cit.>
 and the first equality comes
from the fact that D_max is invariant under unitary operation and
D_max(ρ_1 P||ρ_2 P)=D_max(ρ_1||ρ_2) for any projector P <cit.>.
Besides, since C_max(ρ̃_i)=min_τ∈D_max(ρ̃_i||τ)≤ D_max(ρ̃_i||σ̃_i), we have ∑_i p_iC_max(ρ̃_i)≤ C_max(ρ).





§ COHERENCE MEASURE INDUCED FROM C_MAX

Here we introduce a proper coherence measure from
C_max by the method of convex roof and prove that it satisfies all the
conditions (including convexity) a coherence measure need to fulfil.
We define the convex roof of C_max as
follows

    C̃_max(ρ)
    =min_ρ=∑λ_iψ_i∑_iλ_iC_max(ψ_i),

where the minimum is taken over all the
pure state decompositions of state ρ.
Due to the definition of C̃_max
and the properties of C_max, the
positivity and convexity of C̃_max
are obvious.
We only need to prove that it is nonincreasing on average under IO.



Given a quantum state ρ∈(),
for any incoherent operation Φ(·)=∑_μ K_μ(·)K^†_μ
with K_μ K^†_μ⊂,

    ∑_μ p_μC̃_max(ρ̃_μ)≤C̃_max(ρ),

where p_μ=K_μρ K^†_μ and ρ̃_μ=K_μρ K^†_μ/p_μ.


Due to the definition of C̃_max(ρ), there exists a pure state decomposition of state ρ=∑_jλ_jψ_j
such that C̃_max(ρ)=∑_jλ_jC_max(ψ_j). Then

    ρ̃_μ   =   K_μρ K^†_μ/p_μ
       =   ∑_jλ_j/p_uK_μψ_jK^†_μ
       =   ∑_jλ_jq^(μ)_j/p_μϕ^(μ)_j,

where |ϕ^(μ)_j⟩=K_μ|ψ_j⟩/√(q^(μ)_j) and
 q^(μ)_j=K_μψ_jK^†_μ.
 Thus, C̃_max(ρ̃_μ)
 ≤∑_jλ_jq^(μ)_j/p_μC_max(ϕ^(μ)_j)
 and
 
    ∑_μp_μC̃_max(ρ̃_μ)
        ≤   ∑_j,μλ_jq^(μ)_jC_max(ϕ^(μ)_j)
       =   ∑_jλ_j∑_μq^(μ)_jC_max(ϕ^(μ)_j)
       =   ∑_jλ_j∑_μq^(μ)_jlog(1+C_l_1(ϕ^(μ)_j))
       ≤   ∑_jλ_jlog(1+∑_μq^(μ)_j C_l_1(ϕ^(μ)_j))
       ≤   ∑_jλ_jlog(1+C_l_1(ψ_j))
       =   ∑_jλ_jC_max(ψ_j)
       =   C̃_max(ρ),

where the third line comes from the fact that for pure state ψ, C_max(ψ)=log(1+C_l_1(ψ)), the
forth line comes from the concavity of logarithm and the fifth lines comes from the fact that monotonicity of C_l_1 under
IO as Φ(ψ_j)=∑_μK_μψ_jK^†_μ=∑_μq^(μ)_jϕ^(μ)_j.




§ THE OPERATIONAL INTERPRETATION OF C_MAX

To prove the results, we need some preparation.
First of all,
Semidefinite programming (SDP) is a powerful tool in this work—which is a generalization of linear programming problems <cit.>. A SDP over 𝒳=ℂ^N and 𝒴=ℂ^M is a triple (Φ, C, D), where Φ is a Hermiticity-preserving map from ℒ(𝒳) (linear operators on 𝒳) to ℒ(𝒴) (linear operators on 𝒴), C∈ Herm(𝒳) (Hermitian operators over 𝒳), and D∈ Herm(𝒴) (Hermitian operators over 𝒴).
There is a pair of optimization problems associated with every SDP (Φ, C, D), known as the primal and the dual problems. The standard form of an SDP (that is typically followed for general conic programming) is <cit.>

    Primal problem   Dual problem
    
    minimize: ⟨ C,X ⟩,    maximize: ⟨ D,Y ⟩, 
    subject to: Φ(X) ≥ D,    subject to: Φ^*(Y) ≥ C, 
    
    X ∈ Pos(𝒳).     Y ∈ Pos(𝒴).

SDP forms have interesting and ubiquitous applications in quantum information theory. For example, it was recently shown by Brandao et. al <cit.> that there exists a quantum algorithm for solving SDPs that gives an unconditional square-root speedup over any existing classical method.




Given a quantum state ρ∈(),

    min_σ≥0
    Δ(σ)≥ρσ=max_τ≥0
    Δ(τ)=𝕀ρτ.



First, we prove that

    max_τ≥0
    Δ(τ)=𝕀ρτ
    =max_τ≥0
    Δ(τ)≤𝕀ρτ.

For any positive operator τ≥0 with Δ(τ)≤𝕀, define τ'=τ+𝕀-Δ(τ)≥0, then  Δ(τ)=𝕀 and
ρτ'≥ρτ. Thus we obtain the above equation.

Now, we prove that

    min_σ≥0
    Δ(σ)≥ρσ=max_τ≥0
    Δ(τ)≤𝕀ρτ.

The left side of equation (<ref>) can be expressed as the following semidefinite
programming (SDP)

    minBσ,
    s.t.  Λ(σ)≥ C,
    σ≥ 0,

where B=𝕀, C=ρ and Λ=Δ. Then the dual SDP is given by

    maxCτ,
    s.t.  Λ^†(τ)≤ B,
    τ≥ 0.

That is,

    maxρτ,
    s.t.  Δ(τ)≤𝕀,
    τ≥ 0.

Note that the dual is strictly feasible as we only need to choose σ=2λ_max(ρ) 𝕀, where λ_max(ρ) is the
 maximum eigenvalue of ρ. Thus, strong duality holds, and the equation (<ref>) is proved.




For maximally coherent state |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩, we have the following facts,

(i) For any ℰ∈ DIO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.

(ii) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ DIO such that τ=dℰ^†(Ψ_+).


(iii) For any ℰ∈ IO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.

(iv) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ IO such that τ=dℰ^†(Ψ_+).


(v) For any ℰ∈ SIO, τ=dℰ^†(Ψ_+) satisfies τ≥ 0 and Δ(τ)=𝕀.

(vi) For any operator τ≥ 0 with Δ(τ)=𝕀, there exists a quantum operation ℰ∈ SIO such that τ=dℰ^†(Ψ_+).



(i) Since ℰ is a CPTP map, ℰ^† is unital. Besides, as ℰ∈ DIO, [ℰ,Δ]=0 implies that [ℰ^†, Δ]=0. Thus Δ(τ)=dℰ^†(Δ(Ψ_+))=ℰ^†(𝕀)=𝕀.


(ii) For any positive operator τ≥ 0 with Δ(τ)=𝕀,
τ=d, thus τ=dτ̂ with τ̂∈() and Δ(τ̂)=1/d𝕀.
Consider the spectral decomposition of  τ̂=∑^d_i=1λ_iψ_i with ∑^d_i=1λ_i=1, λ_i≥ 0 for any i∈1,.., d.
Besides,  for any i∈1,..., d, |ψ_i⟩ can be written as
|ψ_i⟩=∑^d_j=1c^(i)_j|j⟩ with ∑^d_j=1|c^(i)_j|^2=1.
Let us define K^(i)_n=∑^d_j=1c^(i)_jj for any n∈1,...,d, then K^(i)_n|Ψ_+⟩=1/√(d)|ψ_i⟩ and
∑^d_n=1K^(i)_nΨ_+K^(i)†_n=ψ_i.
Let M_i,n=√(λ_i)K^(i)†_n,
then

    ∑_i,nM^†_i,nM_i,n   =   ∑_i,nλ_iK^(i)_nK^(i)†_n
       =   d∑^d_i=1λ_iK^(i)_1K^(i)†_1
       =   d∑^d_i=1λ_i∑^d_j=1|c^(i)_j|^2j
       =   d∑^d_j=1∑^d_i=1λ_i|c^(i)_j|^2j
       =   d∑^d_j=11/dj=𝕀,

 where ∑^d_i=1λ_i|c^(i)_j|^2=∑_iλ_i|ψ_ij|^2
 =jτ̂j=1/d.
Then ℰ(·)=∑_i,nM_i,n(·) M^†_i,n is a CPTP map. Since M_i,n is diagonal,
the quantum operation ℰ(·)=∑_i,nM_i,n(·) M^†_i,n is a DIO.
Moreover, ℰ^†(Ψ_+)=∑_i,nM^†_i,nΨ_+M_i,n
=∑_i,nλ_iK^(i)_nΨ_+K^(i)†_n=∑_iλ_iψ_i=τ̂.

(iii) If  is an incoherent operation, then there exists a set of Kraus operators K_μ such that
(·)=∑_μK_μ(·) K^†_μ and K_μ K^†_μ∈. Thus

    di^†(Ψ_+)i   =   d∑_μi K^†_μΨ_+K_μi
       =   ∑_μ∑_m,niK^†_μmnK_μi
       =   ∑_μ∑_miK^†_μmmK_μi
       =   ∑_μiK^†_μK_μi
    =1,

where the third line comes from the fact that
for any K_μ, there exists at most one nonzero term in each  column which implies
that ⟨i|K^†_μ|m⟩⟨n|K_μ|i⟩≠ 0 only if m=n, and the forth line
comes from the fact that ∑_μK^†_μK_μ=𝕀. Therefore,
Δ(d^†(Ψ_+))=𝕀.

(iv) This is obvious, as the DIO  given in (ii) is also an incoherent operation.

(v) This is obvious as SIO⊂ DIO.

(vi) This is obvious as the DIO  given in (ii) also belongs to SIO.






Given a quantum state ρ∈(), one has

    max_ℰ∈ DIO
    F(ℰ(ρ),Ψ_+)^2
    =
    max_ℰ∈ DIO
    |Ψ⟩∈F(ℰ(ρ),Ψ)^2.

where |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩ and
 is the set of maximally coherent states.


Due to <cit.>, every maximally coherent can be expressed as
|Ψ⟩=1/√(d)∑^d_j=1e^iθ_j|j⟩, that is,
|Ψ⟩=U_Ψ|Ψ_+⟩ where U_Ψ=∑^d_j=1e^iθ_jj.
Obviously, [U_Ψ, Δ]=0, thus U_Ψ∈ DIO and

    F(ℰ(ρ),Ψ)^2
       =   F(ℰ(ρ), U_ΨΨ_+U^†_Ψ)^2
       =   F(U^†_Ψℰ(ρ)U_Ψ,Ψ)^2
       =   F(ℰ'(ρ),Ψ_+)^2,

where ℰ'(·)=U^†_Ψℰ(·)U_Ψ∈ DIO as
ℰ, U_Ψ∈ DIO.




After these preparation, we begin to prove Theorem 1.


[Proof of Theorem 1]
If ℰ belongs to
DIO, that is, we need to prove

    2^C_max(ρ)=dmax_ℰ∈ DIO
    |Ψ⟩∈F(ℰ(ρ),Ψ)^2,

where
 is the set of maximally coherent states.
In view of Lemma <ref>,  we only need to prove

    2^C_max(ρ)=dmax_ℰ∈ DIO
    F(ℰ(ρ),Ψ_+)^2,

where |Ψ_+⟩=1/√(d)∑^d_i=1|i⟩.

First of all,

    2^C_max(ρ)   =   min_σ∈minλ|ρ≤λσ
       =   min_σ≥0σ|ρ≤Δ(σ)
       =   min_σ≥0
    Δ(σ)≥ρσ.


Second,

    dF(ℰ(ρ),Ψ_+)^2
       =   dℰ(ρ)Ψ_+
       =   dρℰ^†(Ψ_+)
       =   ρτ,

where τ=dℰ^†(Ψ_+. According to Lemma <ref>, there is
one to one correspondence between DIO and the set τ≥0|Δ(τ)=𝕀. Thus we have


    dmax_ℰ∈ DIO
    F(ℰ(ρ),Ψ_+)^2
    =max_τ≥0
    Δ(τ)=𝕀ρτ.

Finally, according to Lemma <ref>, we get the desired result (<ref>).
Similarly, we can prove
the case where ℰ belongs to
either IO or SIO based on Lemma <ref>.








§ SUBCHANNEL DISCRIMINATION IN DEPHASING COVARIANT INSTRUMENT



[Proof of Theorem 2]
First, we consider the case where instrument ℑ is dephasing-covariant instrument  ℑ^D. Due to the definition of C_max(ρ), there exists an incoherent state σ
such that ρ≤ 2^C_max(ρ)σ. Thus, for any
dephasing-covariant instrument ℑ^D and
POVM M_b_b,

    p_succ(ℑ^D,M_b_b,ρ)≤ 2^C_max(ρ) p_succ(ℑ^D, M_b_b, σ),

which implies that

    p_succ(ℑ^D,ρ)≤ 2^C_max(ρ) p^ICO_succ(ℑ^D).



Next, we prove that there exists a dephasing-covariant instrument
ℑ^D such that the equality in (<ref>) holds.
In view of Theorem 1, there exists a DIO 
such that

    2^C_max(ρ)=d(ρ)Ψ_+,

where |Ψ_+⟩=1/√(d)∑^d_j=1|j⟩.
Let us consider the following diagonal unitaries

    U_k=∑^d_j=1e^ijk/d2πj,k∈1,..,d.

The set U_k|Ψ_+⟩^d_k=1  forms a basis of the Hilbert space
and ∑^d_k=1U_kΨ_+U^†_k=𝕀.
Let us define subchannels _k_k as _k(ρ)=1/dU_k(ρ) U^†_k. Then
the channel =∑^d_k=1_k is a DIO. That is, the instrument ℑ^D=_k_k is a
dephasing-covariant instrument.

For any POVM M_k_k  and any incoherent state σ, the probability of success is

    p_succ(ℑ^D, M_k_k,σ)
       =   ∑_k_k(σ)M_k
       =   1/d(σ)∑_kU^†_kM_kU_k.

Since M_k_k is a POVM, then
∑_kM_k=𝕀. As U_k_k
are all diagonal unitaries , we have

    Δ(∑_kU^†_kM_kU_k)
       =   ∑_k U^†_kΔ(M_k)U_k 
       =   ∑_kΔ(M_k)
       =   Δ(∑_k M_k)
       =   Δ(𝕀)=𝕀.

Thus,

    p_succ(ℑ^D, M_k_k,σ)
       =   1/d(σ)∑_kU^†_kM_kU_k
       =   1/dΔ((σ))∑_kU^†_kM_kU_k
       =   1/d(σ)Δ(∑_kU^†_kM_kU_k)
       =   1/d(σ)
    =1/d,

where the second equality comes from the fact that (σ)∈ for any incoherent state σ,  and the second last equality
comes from that fact that
Δ(∑_kU^†_kM_kU_k)=𝕀.
That is,

    p^ICO_succ(ℑ^D)=1/d.


Besides, taking the POVM  N_k_k with N_k=U_kΨ_+U^†_k,
one has _k(ρ)N_k=1/d(ρ)Ψ_+ and

    p_succ(ℑ^D, N_k_k,ρ)
       =   ∑_k_k(ρ)N_k
       =   ∑_k1/d(ρ)Ψ_+
       =   (ρ)Ψ_+
       =   2^C_max(ρ)/d
       =   2^C_max(ρ)p^ICO_succ(ℑ^D).

Thus, for this depasing-covariant instrument ℑ^D=_k_k,

    p_succ(ℑ^D,ρ)/p^ICO_succ(ℑ^D)≥2^C_max(ρ).


Finally it is easy to see that the above proof is also true for ℑ  is
ℑ^I or ℑ^S.




Note that the phasing discrimination game studied in <cit.> is just a special
case of the subchannel discrimination in the dephasing-covariant instruments.
In the phasing discrimination game, the phase ϕ_k
is encoded into a diagonal unitary U_ϕ_k=∑_j e^ijϕ_kj.
Thus the discrimination of a collection of phase ϕ_k with
a prior probability distribution p_k is equivalent to the
discrimination of
the set of subchannel _k_k, where _k=p_k𝐔_k and
𝐔_k(·)=U_ϕ_k(·) U^†_ϕ_k.





§ C^Ε_MAX AS A LOWER BOUND OF ONE-SHOT COHERENCE COST 


 The ϵ-smoothed max-relative entropy of coherence of a quantum state ρ is defined by,

    C^ϵ_max(ρ):=min_ρ'∈ B_ϵ(ρ)C_max(ρ'),

where B_ϵ(ρ):=ρ'≥0:ρ'-ρ_1≤ϵ, ρ'≤ρ.
Then

    C^ϵ_max(ρ)   =   min_ρ'∈ B_ϵ(ρ)min_σ∈D_max(ρ'||σ)
       =   min_σ∈ D^ϵ_max(ρ||σ),

where D^ϵ_max(ρ||σ)  is the smooth max-relative entropy <cit.> and defined as

    D^ϵ_max(ρ||σ)=inf_ρ'∈ B_ϵ(ρ)D_min(ρ'||σ).
 


[Proof of Equation (5)]
Suppose ℰ is MIO  such that
F(ℰ(Ψ^M_+),ρ)^2≥ 1-ϵ and
C^(1),ϵ_C,MIO(ρ)=log M.
Since F(ρ,σ)^2≤ 1-1/4ρ-σ^2_1 <cit.>, then
ℰ(Ψ^M_+)-ρ_1≤2√(ϵ). Thus
ℰ(Ψ^M_+)∈ B_ϵ'(ρ), where ϵ'=2√(ϵ).
As C_max
 is monotone under MIO, we have
C^ϵ'_max(ρ)
≤ C_max(ℰ(Ψ^M_+))≤ C_max(Ψ^M_+)=log M= C^(1),ϵ_C,MIO(ρ).





§ EQUIVALENCE BETWEEN C_MAX AND C_R IN ASYMPTOTIC CASE


We introduce several lemmas first to prove the
result. For any self-adjoint operator Q on a finite-dimensional
Hilbert space, Q has the spectral decomposition as
Q=∑_iλ_iP_i, where P_i is the orthogonal projector onto
the eigenspace of Q. Then we define the positive operator
Q≥ 0=∑_λ_i≥ 0P_i, and  Q>0, Q≤ 0, Q<0 are defined in a similar way.  Moreover, for any two operators
Q_1 and Q_2, Q_1≥ Q_2 is defined as Q_1-Q_2≥ 0.





<cit.>
Given two quantum states ρ,σ∈(), then

    D^ϵ_max(ρ||σ)≤λ

for any λ∈ and ϵ=√(8ρ>2^λσρ).



Note that in <cit.>, the above lemma is proved for bipartite states. However,
this lemma also holds for any state.


(Fannes-Audenaert Inequality <cit.>)
For any two quantum states ρ and σ with ϵ=1/2ρ-σ_1, the following inequality holds:

    |S(ρ)-S(σ)|≤ϵlog(d-1)+H_2(ϵ),

where d is the dimension of the system and H_2(ϵ)
=-ϵlogϵ-(1-ϵ)log(1-ϵ) is the binary
Shannon entropy.


Based on these lemmas, we can prove the equivalence between C_max and C_r in asymptotic limit.
[Proof of Equation (6)]
First, we prove that

    C_r(ρ)≤lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n).


Since

    C^ϵ_max(ρ^ n)
       =   min_σ_n∈_nD^ϵ_max(ρ^ n||σ_n)
       =   D_max(ρ_n,ϵ||σ̃_n),

where ρ_n,ϵ∈ B_ϵ(ρ^ n),
σ̃_n∈_n and _n is the set of incoherent states of
^ n.

Due to the definition of _max, we have

    ρ_n,ϵ≤ 2^C^ϵ_max(ρ^ n)σ̃_n.

Then

    C_r(ρ_n,ϵ)
       ≤    S(ρ_n,ϵ||σ̃_n)
       =   ρ_n,ϵlogρ_n,ϵ-ρ_n,ϵlogσ̃_n
       ≤   ρ_n,ϵ(C^ϵ_max(ρ^ n)+logσ̃_n)-ρ_n,ϵlogσ̃_n
       ≤    C^ϵ_max(ρ^ n),

where ρ_n,ϵ≤ρ^ n=1. Besides, as ρ_n,ϵ-ρ^ n≤ϵ,
due to the Fannes-Audenaert Inequality (<ref>), we have

    C_r(ρ_n,ϵ)
       ≥    C_r(ρ^ n)-ϵlog(d-1)-H_2(ϵ)
       =   nC_r(ρ)-ϵlog(d-1)-H_2(ϵ).

Thus,

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n)
    ≥ C_r(ρ).


Next, we prove that

    C_r(ρ)≥lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n).

Consider the sequence ρ̂=ρ^ n^∞_n=1 and
σ̂_I=σ^ n_I^∞_n=1, where σ_I∈ such that C_r(ρ)=S(ρ||σ_I)=min_σ∈ S(ρ||σ).
Denote

    D(ρ̂||σ̂)
    :=infγ:lim_n→∞supρ^ n≥ 2^nγσ^ n_Iρ^ n=0   .

Due to the Quantum Stein's Lemma <cit.>,

    D(ρ̂||σ̂)
    =S(ρ||σ_I)=C_r(ρ).


For any δ>0, let λ=D(ρ̂||σ̂)+δ=S(ρ||σ_I)=C_r(ρ)+δ.
Due to the definition of the quantity D(ρ̂||σ̂), we have

    lim_n→∞supρ^ n≥ 2^nλσ^ n_Iρ^ n=0.

Then  for any ϵ>0, there exists an integer N_0 such that for any n≥ N_0, ρ^ n≥ 2^nλσ^ n_Iρ^ n<ϵ^2/8.
According to Lemma <ref>, we have

    D^ϵ_max(ρ^ n||σ^ n_I)≤ nλ =nC_r(ρ)+nδ.

for n≥ N_0. Hence,

    C^ϵ_max(ρ^ n)≤ nC_r(ρ)+nδ.

Therefore

    lim_ϵ→ 0lim_n→∞1/n
    C^ϵ_max(ρ^ n)≤ C_r(ρ)+δ.

Since δ is arbitrary,
lim_ϵ→ 0lim_n→∞1/n
C^ϵ_max(ρ^ n)≤ C_r(ρ).





§ 
C_MIN MAY INCREASE ON AVERAGE UNDER IO



For pure state |ψ⟩=∑^d_i=1ψ_i|i⟩, we have
C_min(ψ)=-logmax_i|ψ_i|^2. According to <cit.>,
if C_min is nonincreasing on average under IO,
it requires that C_min should be a concave function of its diagonal part for
pure state. However, C_min(ψ)=-logmax_i|ψ_i|^2 is convex on the
diagonal part of the pure states, hence C_min may increase on average
under IO.





Besides, according the definition of C_min for pure state |ψ⟩, one has

    2^-C_min(ψ)
    =max_σ∈F(ψ,σ)^2.

However, this equality does not hold for any states. For any quantum state, the inequality (13) in the main context holds.

[Proof of Equation (13)]
There exists a σ_*∈ such that
F(ρ,σ_*)^2=max_σ∈F(ρ,σ)^2. Let us consider the spectrum decomposition
of the quantum state ρ, ρ=∑_iλ_iψ_i with λ_i>0 and ∑_iλ_i=1. Then
the projector Π_ρ onto the support of ρ  can be written as
Π_ρ=∑_iψ_i and 2^-C_min(ρ)=max_σ∈Π_ρσ.

Besides, there exists pure state decomposition of σ_*=∑_iμ_iϕ_i such that
F(ρ,σ_*)=∑_i√(λ_iμ_i)ψ_iϕ_i <cit.>. Thus

    F(ρ,σ_*)^2   =   (∑_i√(λ_iμ_i)ψ_iϕ_i)^2
       ≤   (∑_iλ_i)(∑_iμ_iψ_iϕ_i)
       ≤   ∑_iμ_iΠ_ρϕ_i
       =   Π_ρσ_*
       ≤    2^-C_min(ρ),

where the first inequality is due to the Cauchy-Schwarz inequality and the second inequality comes from the fact that
∑_iλ_i=1 and ψ_i≤Π_ρ for any i.







§ EQUIVALENCE BETWEEN C_MIN AND C_R IN ASYMPTOTIC CASE

For any ϵ>0, the smooth min-relative entropy of coherence of a quantum state ρ is defined as follows

    C^ϵ_min(ρ):=max_
    0≤ A≤𝕀
    Aρ≥ 1-ϵmin_σ∈-logAσ,

where 𝕀 denotes the
identity.
Then

    C^ϵ_min(ρ)
       =   max_
    0≤ A≤𝕀
    Aρ≥ 1-ϵmin_σ∈-logAσ
       =   min_σ∈max_
    0≤ A≤𝕀
    Aρ≥ 1-ϵ
    -logAσ
       =   min_σ∈D^ϵ_min(ρ||σ),

where D^ϵ_min(ρ||σ) is the smooth min-relative entropy <cit.> and
defined as

    D^ϵ_min(ρ||σ)=
    sup_
    0≤ A≤𝕀
    Aρ≥ 1-ϵ
    -logAσ.




Given a quantum state ρ∈(), for any ϵ>0,

    C^ϵ_min(ρ)≤ C^ϵ_max(ρ)-log(1-2ϵ).




Since

    C^ϵ_max(ρ)   =   min_σ∈D^ϵ_max(ρ||σ),
    
    C^ϵ_min(ρ)   =   min_σ∈D^ϵ_min(ρ||σ),

we only need to prove that  for any two states ρ and σ,

    D^ϵ_min(ρ||σ)
    ≤ D^ϵ_max(ρ||σ)-log(1-2ϵ).


First, there exists a ρ_ϵ∈ B_ϵ(ρ) such that
D^ϵ_max(ρ||σ)=D_max(ρ_ϵ||σ)=logλ. Hence λσ-ρ_ϵ≥ 0.

Second, let 0≤ A≤𝕀, Aρ≥ 1-ϵ such that
D^ϵ_min(ρ||σ)=-logAσ.
Since for any two positive operators A and B, AB≥ 0. Therefore
(λσ-ρ_ϵ)A≥0, that is,

    Aρ_ϵ≤λAσ.



Since ρ-ρ_ϵ_1=|ρ-ρ_ϵ|<ϵ and
Aρ≥1-ϵ, one gets

    |Aρ_ϵ-Aρ|
       ≤   A|ρ-ρ_ϵ|
       ≤   |ρ-ρ_ϵ|<ϵ.

Thus,

    Aρ_ϵ≥Aρ-ϵ≥ 1-2ϵ,

which implies that

    1-2ϵ≤λAσ.

Take logarithm on both sides of the above inequality,
we have

    -logAσ≤logλ-log(1-2ϵ).

That is,

    D^ϵ_min(ρ||σ)
    ≤ D^ϵ_max(ρ||σ)-log(1-2ϵ).





The following lemma is a kind of generalization of the Quantum Stein' Lemma <cit.> for the special case of the incoherent state set , as the the set of incoherent states satisfies the requirement in <cit.>.  Note that this lemma can be generalized to any quantum resource theory which satisfies some postulates <cit.>
and it is called the exponential distinguishability property (EDP) (see <cit.>).



Given a quantum state ρ∈(),

(Direct part) For any ϵ>0, there exists a sequence of POVMs A_n, 𝕀-A_n_n such that

    lim_n→∞(𝕀-A_n)ρ^ n=0,

and for every integer n and incoherent state w_n∈_n with _n is the
set of incoherent states on ^ n,

    -logA_nw_n/n+ϵ≥ C_r(ρ).


(Strong converse) If there exists ϵ>0 and a sequence of POVMs A_n, 𝕀-A_n_n such that for every integer n>0 and w_n∈_n,

    -logA_nw_n/n-ϵ≥ C_r(ρ),

then

    lim_n→∞(𝕀-A_n)ρ^ n=1.






Now, we are ready to prove the equivalence between C_min and C_r in asymptotic limit.
[Proof of Equation (18)]
First we prove that

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n)
    ≤ C_r(ρ).

Since C_r(ρ)=lim_ϵ→ 0lim_n→∞1/nC^ϵ_max(ρ^ n) and
C^ϵ_min(ρ)≤ C^ϵ_max(ρ)-log(1-2ϵ), then
we have

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n)
    ≤ C_r(ρ).



Now we prove that

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n)
    ≥ C_r(ρ)

According to Lemma <ref>,
for any ϵ>0, there exists a sequence of POVMs A_n such that for sufficient large integer n,  ρ^ nA_n≥1-ϵ, and thus

    C^ϵ_min(ρ^ n)
    ≥min_σ∈-logA_nσ≥ n(C_r(ρ)-ϵ),

where the last inequality comes from the direct part of Lemma <ref>.
Therefore,

    lim_ϵ→ 0lim_n→∞1/nC^ϵ_min(ρ^ n)
    ≥ C_r(ρ).







§ C^Ε_MIN AS AN UPPER BOUND OF ONE-SHOT DISTILLABLE COHERENCE 




Given a quantum state ρ∈(), then for any ℰ∈ MIO,

    C^ϵ_min(ℰ(ρ))
    ≤ C^ϵ_min(ρ).




Let 0≤ A≤𝕀 and Aℰ(ρ)≥ 1-ϵ such that

    C^ϵ_min(ℰ(ρ))
    =min_σ∈-logAσ
    =-logAσ_*.

Then

    C^ϵ_min(ρ)
       ≥    -logℰ^†(A)σ_*
       =   -logAℰ(σ_*)
       ≥   min_σ∈-logAσ
       =   C^ϵ_min(ℰ(ρ)),

where the first inequality comes from the fact that
ℰ^†(A)ρ=Aℰ(ρ)≥ 1-ϵ and
0≤ℰ^†(A)≤𝕀 as 0≤ A≤𝕀 and
ℰ^† is unital.






[Proof of Equation (17)]
Suppose that ℰ is the optimal MIO such that
F(ℰ(ρ), Ψ^M_+)^2≥1-ϵ with log M=C^(1),ϵ_D,MIO(ρ).
By Lemma <ref>, we have

    C^ϵ_min(ρ)
       ≥    C^ϵ_min(ℰ(ρ))
       =   max_
    0≤ A≤𝕀
    Aℰ(ρ)≥ 1-ϵmin_σ∈-logAσ
       ≥   min_σ∈-logΨ^M_+σ
       =   log M=C^(1),ϵ_D,MIO(ρ),

where the second inequality comes from the fact that
0≤Ψ^M_+≤𝕀 and
Ψ^M_+ℰ(ρ)=F(ℰ(ρ),Ψ^M_+)^2≥1-ϵ.








