Distribution of coherence in bipartite systems based on incoherent-quantum coherence measures
Kaifeng Bu Lu Li Shao-Ming Fei Junde Wu
quant-ph math-ph math.MP

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 kfbu@fas.harvard.edu
 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China
 Department of Physics, Harvard University, Cambridge, MA 02138, USA

  
 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China
 
  
    
    feishm@cnu.edu.cn
 School of Mathematical Sciences, Capital Normal University, Beijing 100048, PR China
  Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany



 
 School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, PR China
 




The distribution of quantum coherence in multipartite systems is one of the basic problems in the resource theory of coherence. While the usual coherence measures are defined on a single system and  cannot capture the nonlocal correlation
between subsystems, in order to  deal with the distribution of coherence it is crucial to quantify the coherence in bipartite systems properly. Here, we introduce incoherent-quantum (IQ) coherence measures on  bipartite systems, which can characterize the correlations between systems. According to the  IQ
coherence measures on bipartite systems, we find the distribution of coherence of formation and assistance  in bipartite systems: the
 total coherence of formation  is lower  bounded by the sum of
 coherence of formation  in each subsystem and the entanglement of formation between the subsystems,
 while the total coherence of assistance is upper bounded by the sum of
 coherence of assistance in each subsystem and the entanglement of assistance between subsystems.
 Besides, we also obtain the tradeoff relation between
 the coherence cost  and entanglement cost,  distillable coherence and distillable
 entanglement in bipartite systems.
Thus, the IQ coherence measures introduced here
truly capture the nonlocal correlation between subsystems and  reveal the distribution of coherence in bipartite systems.


Distribution of coherence in bipartite systems based on incoherent-quantum coherence measures
    Junde Wu
    
=============================================================================================






§ INTRODUCTION





Quantum coherence, stemming from the superposition rule of quantum mechanics, can capture the feature of quantumness in a single system,
and  play an important role in
 a variety of applications
ranging from   thermodynamics <cit.> to metrology <cit.>.
Recently, following the method in quantum information theory, the resource theory of coherence has been developed <cit.>.
Besides quantum coherence, there are other resource theory including quantum entanglement <cit.>, asymmetry <cit.>, thermodynamics <cit.>, and steering <cit.>, where all these quantum resource are helpful to quantum information processing tasks.

Any resource theory consists of two basic elements:
free states and free operation. The state (operation)
outside the sets of free states (operation) is called
resource. For example, the free states in the resource theory of coherence is called incoherent states
and the corresponding free operations is called incoherent operations <cit.>.
The resource measures are introduced to
quantify the amount of  resource in a given quantum state.
To quantify the coherence in  a single system, several
operational coherence measures has been proposed, namely,
relative entropy of coherence <cit.>, l_1 norm of coherence <cit.>, coherence of formation <cit.>,
robustness of coherence <cit.>, coherence weight <cit.>  and max-relative entropy of coherence <cit.>,  where relative entropy of coherence characterizes the optimal rate to distill maximally coherent state
from a given quantum state <cit.>, coherence of formation is equal to minimal cost of maximal coherent state to prepare
the given state <cit.> and max-relative entropy of coherence can be interpreted as the maximal overlap with the maximally coherent state
under incoherent operations <cit.>.

However, as the observation and characterization of the properties of quantum systems is often affected by the
coupling to the environment,
the effect of environment on quantifying coherence has to be taken into account.  Thus, incoherent-quantum (IQ) coherence measures on bipartite systems are introduced here, which can not only quantify the coherence in local subsystem but also the collective coherence between systems and thus
plays a crucial role in the distribution of coherence in multipartite systems <cit.>.
In view of the significance of
IQ coherence measures, we investigate the properties of IQ coherence measures in details and  the distribution of coherence in bipartite systems in terms of other coherence measures
such as coherence of formation and assistance.


Here, we introduce  incoherent-quantum (IQ) coherence measures defined by
relative entropy, max-relative entropy and l_1 norm on bipartite systems to quantify the
coherence in the system with the access to a quantum memory.
We also introduce the IQ coherence of formation and assistance on bipartite systems, by which we find the distribution of coherence formation and  assistance  in bipartite systems: the total coherence of formation is lower  bounded by the sum of coherence of formation  in each  subsystem and
entanglement of formation  between subsystems,
while the total coherence of assistance is  upper bounded by the sum of coherence of assistance in each  subsystem and
entanglement of  assistance between subsystems.
Besides, we  find the relationship between coherence cost ( distillable coherence ) and entanglement cost ( distillable entanglement) in bipartite systems.
Moreover, we obtain the monogamy relationship for IQ coherence measures ( such as IQ coherence of assistance and formation ) in tripartite systems, which illustrates the distribution of coherence
in multipartite systems.
Furthermore, we discuss the relationship between different IQ coherence measures, such as the equivalence between IQ coherence measures
defined by max-relative entropy and l_1 norm.



§ PRELIMINARIES


Let  be a d-dimensional Hilbert space and () be the set
of density operators acting on . Let us first recall some
basic facts about max- and min- relative entropies and the resource theory of coherence.

Max-and min-relative entropy.—Given two operators ρ and σ
with ρ≥0, ρ≤ 1 and σ≥0, the max-relative entropy of ρ with respect to σ <cit.> is defined as

    D_max(ρ||σ):=minλ∈_+:ρ≤2^λσ,

where D_max(ρ||σ) is well defined if supp[ρ]⊂supp[σ] with supp[ρ] being the
support of ρ.
The min-relative entropy of ρ with respect to σ <cit.> is defined as

    D_min(ρ||σ):=-logΠ_ρσ,

where Π_ρ is the projector on the support of ρ.


Resource theories of quantum coherence.—Given a fixed  reference basis |i⟩^d-1_i=0
for some d-dimensional Hilbert space, any quantum
state which is diagonal in the reference basis is the free state
in the resource theory of coherence and the set of incoherent states is denoted by . However, there is still general
consensus on the set of free operations in the resource theory of coherence. Here, we refer
incoherent operations (IO) <cit.> as the free operations, where
incoherent operations (IO) is the set of all  quantum operations Φ that
admit a set of Kraus operators K_i_i such that Φ(·)=∑_iK_i(·)K^†_i and
K_i K^†_i⊂ for any i <cit.>. Besides,
several operational coherence measures have been proposed, which are listed as follows,

(i) l_1 norm of coherence <cit.>,

    C_l_1(ρ) = ∑_i,j=0
    i≠ j^d-1 |⟨i|ρ|j⟩|,


(ii)  relative entropy of coherence <cit.>,

    C_r(ρ)=S(Δ(ρ))-S(ρ),

where S(ρ)=-ρlogρ is von Neumann entropy,

(iii)  max-relative entropy of coherence <cit.>,

    C_max(ρ)=min_σ∈D_max(ρ||σ),




(iv) coherence of formation <cit.>,

    C_f(ρ)=min_ρ=∑_ip_iψ_i∑_ip_iS(Δ(ψ_i)),

where the minimization is taken over all pure state decomposition of
ρ,


(v) coherence of assistance <cit.>,

    C_a(ρ)=max_ρ=∑_ip_iψ_i∑_ip_iS(Δ(ψ_i)),

where the maximization is taken over all pure state decomposition of ρ,

(vi) coherence weight <cit.>,

    C_w(ρ)   =   min{s≥0:ρ=(1-s)σ+sτ,  σ∈ℐ, .
          .τ∈()}.





§  ENTROPIC IQ COHERENCE MEASURE 


Given a bipartite system _A_B with a fixed basis |i⟩_A_i of _A, we can define the
 relative entropy of incoherent-quantum (IQ) coherence for any bipartite state ρ_AB∈(_A_B) as follows  <cit.>,

    C^A|B_r(ρ_AB)=min_σ_A|B∈ QS(ρ_AB||σ_A|B),

where  the set of incoherent-quantum  states Q <cit.> is given by

    Q={σ_A|B∈(_A_B)|σ_A|B=∑_ip_iσ^A_iτ^B_i,
    σ^A_i is incoherent, τ^B_i∈(_B)},

and C^A|B_r gives an upper bound for assistant distillation of coherence <cit.>.
Max- and min-relative entropies of IQ  coherence have also been defined  in  Ref. <cit.> as follows,

    C^A|B_max(ρ_AB)
    =min_σ_A|B∈ QD_max(ρ_AB||σ_A|B),
    
    C^A|B_min(ρ_AB)
    =min_σ_A|B∈ QD_min(ρ_AB||σ_A|B),

where C^A|B_max captures the maximal advantage  of bipartite states in certain subchannel discrimination problems <cit.>.

For IQ coherence measure C^A|B, the following properties are considered:
(i) positivity, C^A|B(ρ_AB)≥ 0 and C^A|B(ρ_AB)≥ 0
iff ρ_A|B∈ Q;
(ii) monotonicity under  incoherent operation  on A side, that is, C^A|B(Λ^A_IO𝕀(ρ_AB))≤ C^A|B(ρ_AB);
(iii) strong monotonicity under incoherent operation on A side, that is, for incoherent operation   Λ^A_IO(·)=∑_i K^A_i(·)K^A†_i
with K^A_i K^A†_i⊂,
∑_i p_iC^A|B(ρ̃_i)≤ C^A|B(ρ),
where p_i=K^A_iρ_AB K^A†_i and ρ̃_i=K^A_iρ_AB K^A†_i/p_i;
(iv) monotonicity under quantum operation  on B side, that is, C^A|B(𝕀Λ^B(ρ_AB))≤ C^A|B(ρ_AB);
(v) convexity, that is,  for ρ_AB=∑^n_ip_iρ_i,
C^A|B(ρ_AB)≤∑_iC^A|B(ρ_i).

Note that,
 C^A|B_r satisfies all these properties, and C^A|B_max satisfies all these properties except (v).
 However, C^A|B_max satisfies the quasi-convexity instead of convexity,
that is,  for ρ_AB=∑^n_ip_iρ_i,
C^A|B_max(ρ_AB)≤max_iC^A|B_max(ρ_i) ( See <cit.>).
For bipartite pure state |ψ⟩_AB, it  can be written as |ψ⟩_AB=∑^d_A_i=1√(p_i)|i⟩_A|u_i⟩_B in the local basis |i⟩_A_i of _A, thus
C^A|B_r(ψ_AB)=-∑_ip_ilog p_i=S(Δ_A(ρ_A))≤log d_A and C^A|B_max(ψ_AB)=2log(∑^d_A_i=1√(p_i))≤log d_A.
Due to the convexity of C^A|B_r and quasi-convexity of C^A|B_max, C^A|B_r(ρ_AB)≤log d_A and C^A|B_max(ρ_AB)≤log d_A for
any bipartite state ρ_AB∈(_A_B), which means that the maximal value for IQ coherence measures C^A|B_r and C^A|B_max
does not depend on the subsystem B.
Here, we only consider the monotonicity of IQ coherence measures under
local incoherent operations on A side and local quantum operations on B side, while
 the (strong) monotonicity of IQ coherence under  the local
incoherent-quantum operations and classical communication (LIQCC) <cit.> is still unknown as the
characterization of LIQCC (such as the Kraus operators of LIQCC) is unclear.


According to the definition, C^A|B_r(ρ_AB)≥ C_r(ρ_A) with
 ρ_A=Bρ_AB being the reduced state.
For any pure bipartite  state |ψ⟩_AB, the following relation holds,

    C^A|B_r(ψ_AB)=C_r(ρ_A)+S(ρ_B),

where S(ρ_B) is the von Neumman entropy of the reduced state ρ_B=Aψ_AB on system B.
This comes directly from the definition of C^A|B_r and the fact that S(ρ_A)=S(ρ_B) for
pure bipartite state.
In general, for any bipartite state ρ_AB, C^A|B_r(ρ_AB)≥ C_r(ρ_A)+δ_A→ B <cit.>, where δ_A→ B is the quantum discord between A and B for state ρ_AB <cit.>.
For pure tripartite  states, we have the following proposition.


Given a pure tripartite state |ψ⟩_ABC∈(_A_B_C), it holds that

    C^A|B_r(ρ_AB)-C^A|C_r(ρ_AC)=S(ρ_B)-S(ρ_C),

where ρ_AB, ρ_AC are the corresponding reduced states of ψ_ABC.



Any tripartite pure state |ψ⟩_ABC can be written as
|ψ⟩_ABC=∑_i√(p_i)|i⟩_A|u_i⟩_BC in the local basis |i⟩_A_i of _A with p_i≥0, ∑_ip_i=1, and thus
the reduced states ρ_AB and ρ_AC can be written as

    ρ_AB=∑_i,j√(p_ip_j)ij_ACu_iu_j_BC,
    ρ_AC=∑_i,j√(p_ip_j)ij_ABu_iu_j_BC.

Due to the definition,

    C^A|B_r(ρ_AB)   =   S(Δ_A(ρ_AB))-S(ρ_AB)
       =   S(Δ_A(ρ_A))+∑_ip_iS( Cu_i_BC)
          -S(ρ_AB),
    
    C^A|C_r(ρ_AC)   =   S(Δ_A(ρ_AC))-S(ρ_AC)
       =   S(Δ_A(ρ_A))+∑_ip_iS( Bu_i_BC)
          -S(ρ_AC).

As for pure states, the von Neumann entropy of the reduced states is equal,
thus S( Cu_i_BC)=S(Bu_i_BC) for any i,
and S(ρ_AB)=S(ρ_C), S(ρ_AC)=S(ρ_B). Therefore, we obtain the
result.


The Proposition <ref> illustrates that  the difference between C^A|B_r(ρ_AB) and C^A|C_r(ρ_AC) for tripartite pure state is equal to
the difference between the amount of information encoded in  ancillary systems B and C.


In tripartite systems, the monogamy relation for  relative  entropy of IQ coherence has been proposed as
C^AB|C_r(ρ_ABC)≥ C^A|C_r(ρ_AC)+C^B|C_r(ρ_BC) <cit.>, where ρ_AB and  ρ_AC are the
corresponding reduced states.
However, the relationship between C^A|BC_r(ρ_ABC) and C^A|B_r(ρ_AB)+C^A|C_r(ρ_AC) is still unknown, that is, whether the following relation
holds for all tripartite states   remains to be verified,

    C^A|BC_r(ρ_ABC)≥ C^A|B_r(ρ_AB)+C^A|C_r(ρ_AC).

We give an upper bound for the quantity C^A|BC_r(ρ_ABC)- C^A|B_r(ρ_AB)-C^A|C_r(ρ_AC) in terms of conditional entropy and find that the relation  (<ref>) may not hold in general.


Given a tripartite state ρ_ABC∈(_A_B_C), then

    C^A|BC_r(ρ_ABC)- C^A|B_r(ρ_AB)-C^A|C_r(ρ_AC)
       ≤    -S(A|BC)+S(A|B)+S(A|C),

where the conditional entropy is defined as S(X|Y)=S(ρ_XY)-S(ρ_Y).



Let us take another system _A'=_A and the local basis |i⟩_A'_i=|i⟩_A_i.
Define an unitary operator on _A_A' such that
U_AA'|i⟩_A|0⟩_A'=|i⟩_A|i⟩_A'.
Hence, for any tripartite state ρ_ABC=∑_i,jij_Aρ^BC_ij,

    σ_ABCA'   =   Δ_AΔ_A'(U_AA'ρ_ABC0_A'U^†_AA')
       =   ∑_ii_Aρ^BC_iii_A'.

Thus, σ_AB=A'Cσ_ABCA'=∑_ii_Aρ^B_i,
σ_A'C=ABσ_ABCA'=∑_ii_A'ρ^C_i, and
S(σ_AB)=S(Δ_A(ρ_AB)), S(σ_A'C)=S(Δ_A(ρ_AC)) where
ρ^B_i=Cρ^BC_ii and ρ^C_i=Bρ^BC_ii. Then, as
S(σ_ABCA')=S(Δ_A(ρ_ABC)), we have

    S(σ_ABCA'||σ_ABσ_A'C)
       =   S(ρ_AB)+S(ρ_A'C)-S(σ_ABCA')
       =   S(Δ_A(ρ_AB))+S(Δ_A(ρ_AC))-S(Δ_A(ρ_ABC)).

Since relative entropy is
monotone under partial trace,
then

    S(σ_ABCA'||σ_ABσ_A'C)
    ≥ S(σ_BC||σ_Bσ_C),

where σ_BC=ρ_BC, σ_B=ρ_B and σ_C=ρ_C,
that is,

    S(Δ_A(ρ_AB))+S(Δ_A(ρ_AC))-S(Δ_A(ρ_ABC))
       ≥    S(ρ_B)+S(ρ_C)-S(ρ_BC).

Therefore,

    C^A|BC_r(ρ_ABC)- C^A|B_r(ρ_AB)-C^A|C_r(ρ_AC)
       =   S(ρ_AB)+S(ρ_AC)-S(ρ_ABC)
          -[S(Δ_A(ρ_AB))+S(Δ_A(ρ_AC))-S(Δ_A(ρ_ABC))]
       ≤   S(ρ_AB)+S(ρ_AC)-S(ρ_ABC)
          -[ S(ρ_B)+S(ρ_C)-S(ρ_BC)]
       =   -S(A|BC)+S(A|B)+S(A|C).










The negative conditional entropy quantifies the amount of entanglement as
S(A|B)<0 indicates the entanglement between A and B <cit.>.
Thus, the following relation

    -S(A|BC)≥-S(A|B)-S(A|C),

can be viewed as a monogamy relation of entanglement, which holds for any pure tripartite state.
Besides, Lemma <ref> illustrates that the violation of the relation
(<ref>) will lead to the violation of the  relation
(<ref>).


There exists some tripartite state ρ_ABC∈(_A_B_C) such that

    C^A|BC_r(ρ_ABC)≤ C^A|B_r(ρ_AB)+C^A|C_r(ρ_AC).



It is easy to verify that the tripartite state with the form
ρ_ABC=ρ_A_1Bρ_A_2B violates the relation (<ref>)
 where _A=_A_1_A_2. Thus, the relation (<ref>)
does not hold in general.


In view of the discussion in Ref. <cit.>, the relation (<ref>) cannot hold in general as the term
C^A|B_r(ρ_AB)+C^A|C_r(ρ_AC) contains two copies of local coherence C_r(ρ_A), whereas the term C^A|BC_r(ρ_ABC) only contains
one copy of C_r(ρ_A). The relation  (<ref>) will be violated  for the tripartite state ρ_ABC
with weak correlation between B and C, e.g., ρ_ABC=ρ_A_1Bρ_A_2B where _A=_A_1_A_2.




By introducing smooth max and min-relative entropies of IQ coherence,
the distribution of coherence quantified by relative entropy in multipartite systems has been obtained in Ref. <cit.>.
Besides relative entropy of coherence, we  find the distribution of coherence of formation C_f and assistance C_a in bipartite
systems by introducing the corresponding IQ  coherence measures. The IQ coherence of formation on bipartite systems
is defined as follows,

    C^A|B_f(ρ_AB):   =   min∑_ip_iC^A|B_r(ψ_i_AB)
       =   min∑_ip_iS(Δ_A(ψ_i_AB)),

where the minimization is taken over all pure state decomposition ρ_AB=∑_ip_iψ_i_AB. Since S(Δ_A(ψ_i_AB))=S(Δ_A(Bψ_i_AB))  ( see Lemma <ref> in Appendix <ref> )
and von Neumann entropy is concave, then we have

    C^A|B_f(ρ_AB)=min∑_ip_iS[Δ_A(Bρ^AB_i)],

where the minimization is taken over all state decomposition of ρ_AB=∑_ip_iρ^AB_i
 without the restriction of ρ^AB_i to be pure state. C^A|B_f satisfy the properties (i)-(v), where
 (i) and (v) are obvious, (iii) and (iv) are presented in Appendix <ref>, and (ii) comes directly from (iii) and (v).




Here, we consider the distribution of coherence of formation in bipartite systems in terms of  the
IQ coherence of formation C^A|B_f, where C^A|B_f contains not only the local coherence in  subsystem but also the entanglement of formation E_f <cit.> between A and B, for which we have
the following relation.

Given a bipartite state ρ_AB∈(_A_B), then

    C^A|B_f(ρ_AB)≥ C_r(ρ_A)+E_f(ρ_AB),

where ρ_A is the reduced state on subsystem A, and E_f(ρ_AB)=min∑_i p_iS(Aψ_i_AB) with the minimization being taken over all
pure state decomposition of ρ_AB=∑_ip_iψ_i_AB.


For any pure state decomposition of ρ_AB=∑_ip_iψ_i_AB with
ρ^A_i=Bψ_i_AB and ρ^B_i=Aψ_i_AB, we have

    ∑_ip_iS(Δ_A(ψ_i_AB))
       =   ∑_ip_i[C_r(ρ^A_i)+S(ρ^B_i)]
       ≥    C_r(ρ_A) +E_f(ρ_AB),

where the first line comes from (<ref>)
and the second line comes from the convexity of C_r and definition of E_f. Thus, we
 get the result.





Given a bipartite state ρ_AB∈(_A_B), then

    C_f(ρ_AB)≥ C^A|B_f(ρ_AB)+C_f(ρ_B),

where ρ_B is the reduced state of ρ_AB on subsystem B.


There exists an optimal pure state decomposition of ρ_AB=∑_ip_iψ_i_AB such that
C_f(ρ_AB)=∑_ip_iS(Δ_AΔ_B(ψ_i_AB)), where |ψ_i⟩_AB=∑_j√(λ_i,j)|j⟩_A|u_i,j⟩_B
with ∑_jλ_i,j=1 for any i
and |j⟩_A_j is the reference basis of subsystem A. Thus,

    C_f(ρ_AB)   =   ∑_ip_iS(Δ_AΔ_B(ψ_i_AB))
       =   ∑_ip_i[S(Δ_A(ρ^A_i))+∑_jλ_i,jS(Δ_B(u_i,j_B))]
       =   ∑_ip_iS(Δ_A(ρ^A_i))+∑_i,jp_iλ_i,jS(Δ_B(u_i,j_B))
       ≥    C^A|B_f(ρ_AB)+C_f(ρ_B),

where the inequality results from the definitions of C^A|B_f and C_f,  and the fact that
ρ_B=∑_i,jp_iλ_i,ju_i,j_B.


Combining the above two lemmas, we can obtain  the distribution
of coherence of formation in bipartite systems, where the total coherence of formation is lower bounded by the sum
of  local coherence of formation in subsystems A and B and  the entanglement of formation between the subsystems.

Given a bipartite state ρ_AB∈(_A_B),
it holds that,

    C_f(ρ_AB)≥      maxC_r(ρ_A)+C_f(ρ_B), C_r(ρ_B)+C_f(ρ_A) 
                      +E_f(ρ_AB),

where ρ_A and ρ_B are the corresponding reduced states of ρ_AB.


Based on Lemmas <ref> and <ref>, we have

    C_f(ρ_AB)   ≥    C^A|B_f(ρ_AB)+C_f(ρ_B)
       ≥    C_r(ρ_A)+E_f(ρ_AB)+C_f(ρ_B).

Similarly, C_f(ρ_AB)≥ C_r(ρ_B)+C_f(ρ_A) +E_f(ρ_AB) can be obtained.


Now, we give an example such that the equality  in  Theorem <ref>  holds.
For any quantum state ρ_B∈(_B), there exists an optimal pure state decomposition
of ρ_B=∑_i p_iu_i_B such that C_f(ρ_B)=∑_ip_iS(Δ_B(u_i)). Let us take the pure bipartite state
|ψ⟩_AB=∑_i√(p_i)|i⟩_A|u_i⟩_B, then
C_f(ψ_AB)=C_r(ρ_A)+C_f(ρ_B)+E_f(ψ_AB).

Besides, due to the equivalence between coherence of formation C_f and coherence cost C_c <cit.>, we can obtain the
relationship between coherence cost C_c and entanglement cost E_c in bipartite systems from  Theorem <ref>.

Given a bipartite state ρ_AB∈(_A_B),
it holds that,

    C_c(ρ_AB)≥      maxC_r(ρ_A)+C_c(ρ_B), C_r(ρ_B)+C_c(ρ_A) 
                      +E_c(ρ_AB),

where  ρ_A and ρ_B are the corresponding reduced states of ρ_AB, and the entanglement cost E_c <cit.> is defined as

    E_c(ρ)=inft:lim_n→∞ρ^ n-Λ_LOCC(ϕ^ tn_+)_tr=0,

with |ϕ_+⟩=1/√(2)(|00⟩-|11⟩),  Λ_LOCC being the local operation and classical communication (LOCC) and trace norm A_tr=√(A^† A).


In view of Theorem <ref>, we have the following relationship for
the bipartite state ρ^ n_AB,

    C_f(ρ^ n_AB)≥ C_r(ρ^ n_A)+C_f(ρ ^ n_B)+E_f(ρ^ n_AB).

Since both C_r and C_f are additive <cit.> and  E_c is equivalent to the regularized entanglement of formation E_f <cit.>,
we have

    C_f(ρ_AB)≥ C_r(ρ_A)+C_f(ρ_B)+E_c(ρ_AB).

Similarly, we can also obtain the following relation,

    C_f(ρ_AB)≥ C_r(ρ_B)+C_f(ρ_A)+E_c(ρ_AB).

Therefore, we obtain the result.



It has been proved that relative entropy of coherence C_r is equivalent to distillable coherence C_d <cit.>. Thus we can obtain the relationship between the
distillable coherence and distillable entanglement  in bipartite systems as follows.

Given a bipartite state ρ_AB∈(_A_B),
C_d(ρ_AB) and  E_d(ρ_AB) has the following
relationship,

    C_d(ρ_AB)≥ C_d(ρ_A)+C_d(ρ_B)+ E_d(ρ_AB),

whereρ_A and ρ_B are the corresponding reduced states of ρ_AB and the  distillable entanglement E_d <cit.> is defined as

    E_d(ρ)=inft :lim_n→∞Λ_LOCC(ρ^ n)-ϕ^ t n_+_tr=0.



It has been proved in Ref. <cit.> that

    C_r(ρ_AB)≥ C_r(ρ_A)+C_r(ρ_B)+E^∞_r(ρ_AB),

where E^∞_r is the regularized relative entropy of entanglement <cit.>.
Due to the equivalence between C_r and C_d <cit.>
and the fact that E^∞_r≥ E_d <cit.>, we obtain the result.





In tripartite systems, the monogamy relation of coherence has been considered for relative entropy of coherence C_r and it has been shown in Refs. <cit.> that
 it  does not hold in general  for C_r. However,
the monogamy relation for IQ coherence measure C^A|B_r has been established in Ref. <cit.>. Here,
we obtain the monogamy relation for C^A|B_f in tripartite systems as follows.


Given a bipartite state ρ_ABC∈(_A_B_C), then

    C^AB|C_f(ρ_ABC)
    ≥ C^A|BC_f(ρ_ABC)+C^B|C_f(ρ_BC),

which implies the following monogamy relation,

    C^AB|C_f(ρ_ABC)
    ≥ C^A|C_f(ρ_AC)+C^B|C_f(ρ_BC).



For any pure tripartite state |ψ⟩_ABC=∑_i√(p_i)|i⟩_A|u_i⟩_BC,

    C^AB|C_f(ψ_ABC)
       =   C^AB|C_r(ψ_ABC)
       =   S(Δ_AΔ_B(ρ_AB))
       =   S(Δ_A(ρ_A))+∑_ip_iS(Δ_B(ρ^B_i))
       =   S(Δ_A(ρ_A))+∑_ip_iS(Δ_B(Tr_Cu_i_BC))
       ≥    C^A|BC_f(ψ_ABC)+C^B|C_f(ρ_BC),

where the last inequality results from the fact that S(Δ_A(ρ_A))=C^A|BC_f(ψ_ABC) for pure state
ψ_ABC and ∑_ip_iS(Δ_B(Tr_Cu_i_BC))≤ C^B|C_f(ρ_BC) due to the definition of
C^B|C_f. For any tripartite states ρ_ABC, there exists an optimal
pure state decomposition of ρ_ABC=∑_iλ_iψ_i_ABC such that
C^AB|C_f(ρ_ABC)=∑_iλ_iC^AB|C_f(ψ_i_ABC). Thus,

    C^AB|C_f(ρ_ABC)   =   ∑_iλ_iC^AB|C_f(ψ_i_ABC)
       ≥   ∑_iλ_i[C^A|BC_f(ψ_i_ABC)+C^B|C_f(ρ^BC_i)]
       ≥    C^A|BC_f(ρ_ABC)+C^B|C_f(ρ_BC).












Similar to coherence of formation C_f, coherence of assistance C_a is also defined by taking the pure state decompositions of the given state <cit.>. Here, we introduce the IQ  coherence of assistance
C^A|B_a on bipartite systems, which is defined as follows

    C^A|B_a(ρ_AB):   =   max∑_ip_iC^A|B_r(ψ_i_AB)
       =   max∑_ip_iS(Δ_A(ψ_i_AB)),

where the maximization is taken over all pure state decomposition ρ_AB=∑_ip_iψ_i_AB.
Following the similar method, we can obtain the relationship between coherence of assistance C_a and entanglement of assistance
E_a <cit.> in bipartite systems as follows.

Given a bipartite state ρ_AB∈(_A_B), it holds that,

    C^A|B_a(ρ_AB)   ≤    C_a(ρ_A)+E_a(ρ_AB),
    
    C_a(ρ_AB)   ≤    C^A|B_a(ρ_AB)+C_a(ρ_B),
    
    C_a(ρ_AB)   ≤    C_a(ρ_A)+C_a(ρ_B)+E_a(ρ_AB),

where E_a(ρ_AB)=max∑_i p_iS(Aψ_i_AB) with the maximization being taken over all
pure state decomposition of ρ_AB=∑_ip_iψ_i_AB and ρ_A, ρ_B are the reduced states of ρ_AB on subsystems A and B, respectively.

Theorem <ref>  illustrates that the total coherence of assistance
in bipartite systems is upper bounded by the sum of coherence of assistance
in each subsystem and the entanglement of formation between subsystems.
The proof of Theorem <ref> is almost the same as that of Theorem <ref>, thus we omit it here.
The regularized version of coherence of assistance C^∞_a has also been proposed in
Ref. <cit.>, which is defined as C^∞_a(ρ):=lim_n→∞1/nC_a(ρ^ n)=S(Δ(ρ)).
 Moreover, for any state
extension ρ_AB of a given state ρ_A, i.e., Aρ_AB=ρ_A,
C^∞_a(ρ_A) is a upper bound of C^A|B_r(ρ_AB). In fact,
C^∞_a(ρ_A) is the maximum value of C^A|B_r(ρ_AB) for the state extension ρ_AB
of ρ_A.

Given a quantum state ρ_A∈(_A), then

    C^∞_a(ρ_A)=max_ρ_AB,Bρ_AB=ρ_AC^A|B_r(ρ_AB).




First, for pure bipartite state ψ_AB with Bψ_AB=ρ_A, then
C^A|B_r(ψ_AB)=S(Δ_A(ρ_A))=C^∞_a(ρ_A). Besides, for mixed bipartite state ρ_AB with
Bρ_AB=ρ_A, there exists a purification ψ_ABC of ρ_AB such that
Cψ_ABC=ρ_AB. Since C^A|B_r is monotone under completely positive and trace preserving
(CPTP) maps on B side, then
C^A|B_r(ρ_AB)≤ C^A|BC_r(ψ_ABC)=S(Δ_A(ρ_A))=C^∞_a(ρ_A).












§ L_1 NORM OF IQ COHERENCE


In order to introduce  l_1 norm of IQ coherence on bipartite systems, let us first introduce a new norm ·_l_1tr on (_A_B) with a fixed basis |i⟩_A_i of _A. For any operator Q∈(_A_B) written as Q=∑^d_A_i,j=1ij_A Q^B_ij,
the norm Q_l_1tr is defined as follows,

    Q_l_1tr:=∑^d_A_i,j=1Q^B_ij_tr,

where A_tr=√(A^† A).
It is easy to show that ·_l_1tr is a norm, that is, it satisfies the following properties:
(i) Positivity, Q_l_1tr≥ 0 and Q_l_1tr=0⇔ Q=0;
(ii) α Q_l_1tr=|α|Q_l_1tr for any α∈;
(iii) Triangle inequality, Q+P_l_1tr≤Q_l_1tr+P_l_1tr for any operators Q,P∈(_A_B).


Based on this new norm, we define  l_1 norm of IQ coherence on bipartite systems as follows,

    C^A|B_l_1(ρ_AB)
       :=   min_σ_A|B∈ Qρ_AB-σ_A|B_l_1tr
       =   ∑_i≠ jρ^B_ij_tr,

where ρ^B_ij=iρ_ABj_A.
Note that, C^A|B_l_1 satisfies the
properties (i)-(v), where
 the positivity of C^A|B_l_1 ( i.e., property (i) ) comes from the positivity
of the norm ·_l_1tr, (iv) results from the contractivity  of ·_tr under
CPTP maps, (v) comes from the
triangle inequality of the norm ·_l_1tr, (iii) and (v) lead to the property (ii). Thus, we only need to prove (iii), which is presented
in the Appendix <ref>.


Due to the definition,  C^A|B_l_1(ρ_AB)≥ C_l_1(ρ_A) with ρ_A being the reduced state of ρ_AB, which comes
from the fact that ρ^B_ij_tr≥ρ^B_ij.
If the subsystem B is a trivial system, i.e., dim _B=1, then C^A|B_l_1(ρ_AB) reduces to C_l_1(ρ_A).
Besides, for bipartite pure state |ψ⟩_AB, which can be written as |ψ⟩_AB=∑^d_A_i=1√(p_i)|i⟩_A|u_i⟩_B,
C^A|B_l_1(ψ_AB)=(∑^d_A_i=1√(p_i))^2-1≤ d_A-1.
Thus, the maximum value for C^A|B_l_1 is d_A-1 which does not depend on the subsystem B.



Given an bipartite state ρ_AB∈(_A_B), then

    C_l_1(ρ_AB)≥ C^A|B_l_1(ρ_AB)+C_l_1(ρ_B),

where ρ_B is the reduced state of ρ_AB.


For any bipartite state ρ_AB=∑_i,jij_Aρ^B_ij,
 the reduced state ρ_B can be written as ρ_B=∑_iρ^B_ii. Thus

    C^A|B_l_1(ρ_AB)   =   ∑_i≠ jρ^B_ij_tr,
    
    C_l_1(ρ_B)   =   ∑_iρ^B_ii_l_1-1=∑_j≠ k|∑_ijρ^B_iik_B|,
    
    C_l_1(ρ_AB)   =   ∑_j≠ k∑_i|jρ^B_iik_B|+∑_i≠ jρ^B_ij_l_1.

Since ∑_j≠ k|∑_ijρ^B_iik_B|≤∑_j≠ k∑_i|jρ^B_iik_B|
and ∑_i≠ jρ^B_ij_tr≤∑_i≠ jρ^B_ij_l_1 ( See Lemma <ref> in Appendix <ref>), we get the result.





This relation (<ref>) is stronger than the known result C_l_1(ρ_AB)≥ C_l_1(ρ_A)+C_l_1(ρ_B) as
C^A|B_l_1(ρ_AB)≥ C_l_1(ρ_A). Besides, C^A|B_l_1  contains not only the local coherence in subsystem A but also
the nonlocal correlation between A and B from the following proposition.


Given a bipartite state ρ_AB∈(_A_B), then we have the following relationship,

    C^A|B_l_1(ρ_AB)^2-C_l_1(ρ_A)^2≥ 2(ρ^2_AB-ρ^2_A).





Any bipartite state ρ_AB
can be written as ρ_AB=∑^d_A_i,j=1ij_Aρ^B_ij and thus

    C^A|B_l_1(ρ_AB)=2∑_i<j|ρ^B_ij|,
    
    C_l_1(ρ_A)=2∑_i<j|ρ^B_ij|.


Moreover, the term ρ^2_AB-ρ^2_A
has the following upper bound,

    ρ^2_AB-ρ^2_A
       =   (∑_i|ρ^B_ii|^2+2∑_i<j|ρ^B_ij|^2)
          -(∑_iρ^B_ii^2+2∑_i<j|ρ^B_ij|^2)
       ≤    2∑_i<j(|ρ^B_ij|^2-|ρ^B_ij|^2)
       ≤    2∑_i<j(|ρ^B_ij|^2-|ρ^B_ij|^2),

where the first and the second inequalities come from the fact that |ρ^B_ij|^2≤|ρ^B_ij|^2.
Therefore,

    C^A|B_l_1(ρ_AB)^2-C_l_1(ρ_A)^2
       =   4[(∑_i<j|ρ^B_ij|)^2-(∑_i<j|ρ^B_ij|)^2]
       =   4[∑_i<j(|ρ^B_ij|-|ρ^B_ij|)][∑_i<j(|ρ^B_ij|+|ρ^B_ij|)]
       ≥    4 ∑_i<j [|ρ^B_ij|^2-|ρ^B_ij|^2]
       ≥    2(ρ^2_AB-ρ^2_A),

where the first inequality comes directly from the fact that |ρ^B_ij|≤|ρ^B_ij|  and the
second inequality comes from the upper bound of ρ^2_AB-ρ^2_A.





The term ρ^2_AB-ρ^2_A quantifies the entanglement between A and B as

    ρ^2_AB-ρ^2_A>0

only if ρ_AB is entangled <cit.> and the inequality (<ref>)
provides a powerful tool in the detection of entanglement in  experiments <cit.>.
Thus, the above proposition implies that the total coherence in bipartite system  quantified by l_1 norm consists of the  nonlocal correlation between A and B and
the local coherence C_l_1(ρ_A) and C_l_1(ρ_B).
Furthermore,  we obtain the monogamy relation of C^A|B_l_1 in tripartite systems,
which clarifies the distribution of coherence by l_1 norm in multipartite systems.

Given a tripartite state ρ_ABC∈(_A_B_C), then

    C^AB|C_l_1(ρ_ABC)≥ C^A|BC_l_1(ρ_ABC)+C^B|C_l_1(ρ_BC),

which implies the following monogamy relation,

    C^AB|C_l_1(ρ_ABC)≥ C^A|C_l_1(ρ_AC)+C^B|C_l_1(ρ_BC),

where ρ_AC, ρ_BC are the corresponding reduced states of ρ_ABC.



Any tripartite state ρ_ABC can be written  as ρ_ABC=∑_i,j∑_m,nij_Amn_Bρ^C_ij,mn
with the local basis |i⟩_A_i and |m⟩_B_m of _A and _B. Then the reduced state ρ_BC=∑_i∑_m,nmn_Bρ^C_ii,mn. Thus

    C^AB|C_l_1(ρ_ABC)   =   ∑_(i,m)≠ (j,n)ρ^C_ij,mn_tr,
    
    C^A|BC_l_1(ρ_ABC)   =   ∑_i≠ j∑_m,nmn_Bρ^C_ij,mn_tr
       ≤   ∑_i≠ j∑_m,nmn_Bρ^C_ij,mn_tr
       =   ∑_i≠ j∑_m,nρ^C_ij,mn_tr,
    
    C^B|C_l_1(ρ_BC)   =   ∑_m≠ n∑_iρ^C_ii,mn_tr≤∑_i∑_m≠ nρ^C_ii,mn_tr,

where (i,m)≠ (j,n) means i≠ j or m≠ n. Therefore, we get the result.

















Now, let us consider the relationship between C^A|B_l_1 and C^A|B_max, where
we find that C^A|B_l_1 is closely related to C^A|B_max and
they are equal for certain type of bipartite states.

Given a bipartite state ρ_AB∈(_A_B), then

    1+1/d_A-1C^A|B_l_1(ρ_AB)≤ 2^C^A|B_max(ρ_AB)≤ 1+C^A|B_l_1(ρ_AB),

where d_A is the dimension of _A.




Any bipartite state ρ_AB can be written as ρ_AB=∑_i,jij_Aρ^B_ij, then
C^A|B_l_1(ρ_AB)=∑_i≠ jρ^B_ij_tr=∑_i,jρ^B_ij_tr-1.
For ρ^B_ij with i<j, there exists a unitary U^B_ij such that
U^B_ijρ^B_ij=|ρ^B_ij|, and thus
ρ^B_jiU^B†_ij=|ρ^B_ij| as (ρ^B_ij)^†=ρ^B_ji.
Now, let us take the  positive operator M as follows,

    M   =   1/d_A-1∑_i<j[i_A𝕀_B+j_A𝕀_B
                 +ij_A U^B†_ij+ji_A U^B_ij]
       =   𝕀_AB+1/d_A-1[ij_A U^B†_ij+ji_A U^B_ij],

where the positivity of M comes from the fact that
i_A𝕀_B+j_A𝕀_B=|ij_A U^B,†_ij+ji_A U^B_ij| and the
fact that |X|+X≥ 0 for any Hermitian operator X.


Due to the definition of C^A|B_max, there exists an incoherent-quantum state
τ_A|B such that

    ρ_AB≤ 2^C^A|B_max(ρ_AB)τ_A|B.

Thus

    Mρ_AB≤ 2^C^A|B_max(ρ_AB)Mτ_A|B,

which leads to

    1+1/d_A-1C^A|B_l_1(ρ_AB)≤ 2^C^A|B_max(ρ_AB).





Besides, let us take the incoherent-quantum state σ_A|B to be

    σ_A|B   =   1/1+C^A|B_l_1(ρ_AB)[∑_ii_Aρ^B_ii
          +∑_i<ji_A |ρ^B_ji|+j_A|ρ^B_ij|].

Then ρ_AB≤ (1+C^A|B_l_1(ρ_AB))σ_A|B, as

    (1+C^A|B_l_1(ρ_AB))σ_A|B-ρ_AB
       =   ∑_i<j[i_A |ρ^B_ji|+j_A|ρ^B_ij|
           -ij_Aρ^B_ij-ji_Aρ^B_ji]
       ≥   0,

where the inequality comes from the fact that
i_A |ρ^B_ji|+j_A|ρ^B_ij|=ij_Aρ^B_ij+ji_Aρ^B_ji for i≠ j.










Given a bipartite state ρ_AB∈(_A_B), if there exists a unitary operator
U_AB=∑_ii_A U^B_i such that
U^B_iρ^B_ijU^B,†_j=|ρ^B_ij| for any i,j, then

    C^A|B_max(ρ_AB)= log(1+C^A|B_l_1(ρ_AB)),

 where |P| is defined as |P|=√(P^† P).



Set λ=2^C^A|B_max(ρ_AB). Due to the definition of
C^A|B_max, there exists a state σ_A|B∈ Q such that

    ρ_AB≤ 2^λσ_A|B.

Applying the unitary operation
U_AB(·) U^†_AB on both sides of the above equation and the taking the partial trace on part B, one obtains

    ∑_i,jρ^B_ij_trij_A≤ 2^λσ_A,

where σ_A is the reduced state of σ_A|B and thus σ_A∈.
Taking the pure state |+⟩=1/√(d_A)∑_i|i⟩_A, we get

    ⟨+|∑_i,jρ^B_ij_trij_A|+⟩≤ 2^λ⟨+|σ_A|+⟩,

which implies that 2^λ≥∑_i,jρ^B_ij_tr as +σ_λ+=1/d_A and
+ij+=1/d_A for any i,j. That is,
C^A|B_max(ρ_AB)≥log(∑_i,jρ^B_ij_tr)=log(1+C^A|B_l_1(ρ_AB)).
Combining with Proposition <ref>, we obtain the result.




It is easy to see that pure bipartite states satisfy the conditions in Proposition <ref>. Thus
the equation (<ref>) holds for any bipartite pure states. Moreover, the bipartite states ρ_AB, which have the following
form  ρ_AB=∑_kρ^AB_k with ρ^AB_k=kρ^B_kk+d_A-kρ^B_d_A-k,d_A-k+kd_A-kρ^B_k,d_A-k
+d_A-kkρ^B_d_A-k,k, also satisfy the conditions in Proposition <ref>, that is, the equation (<ref>) holds for
such states. For example, for the bipartite state ρ_AB(λ)=λϕ_++(1-λ)ψ_+ with 0≤λ≤1,
|ϕ_+⟩=1/√(2)(|00⟩-|11⟩) and |ψ_+⟩=1/√(2)(|01⟩-|10⟩), the  equation (<ref>) holds.

Note that, other coherence measures defined on a single system, such as coherence weight <cit.>,  can also be
used to define the corresponding IQ coherence measures on bipartite systems in a similar way, which is omitted here.
Although the IQ  coherence measure depends on the local basis in subsystem A, it will become
the measures of classical-quantum correlation if we take the minimization over  all the local
basis on system A <cit.>. For example, let us take the minimization over all the local basis for  l_1 norm of IQ coherence as
follows,

    Q^A|B_l_1(ρ_AB)=min_local basis on AC^A|B_l_1(ρ_AB),

where Q^A|B_l_1 is called one-side negativity of quantumness <cit.>.




§ ADDITIVITY OF IQ COHERENCE MEASURES

The above sections show that IQ coherence measures can capture the nonlocal correlation between subsystems. However, the measure of nonlocal
correlation may not be additive, such as the relative entropy of entanglement. Thus we discuss the additivity of IQ coherence measures
in this section. Let us begin with the simplest case, relative entropy and l_1 norm. In view of the definition, it is easy to see the additivity of
C^A|B_r and C^A|B_l_1: for any two bipartite states ρ_A_1B_1∈(_A_1_B_1) and ρ_A_2B_2∈(_A_2_B_2),
then

    C^A_1A_2|B_1B_2_r(ρ_A_1B_1ρ_A_2B_2)
       =   C^A_1|B_1_r(ρ_A_1B_1)+ C^A_2|B_2_f(ρ_A_2B_2),

and

    1+C^A_1A_2|B_1B_2_l_1(ρ_A_1B_1ρ_A_2B_2)
       =   [1+C^A_1|B_1_l_1(ρ_A_1B_1)]·[1+C^A_2|B_2_l_1(ρ_A_2B_2)].


Now, we consider the additivity of IQ coherence measures C^A|B_max and C^A|B_f, for which we have the following propositions.

For any two bipartite states ρ_A_1B_1∈(_A_1_B_1) and ρ_A_2B_2∈(_A_2_B_2),

    C^A_1A_2|B_1B_2_max(ρ_A_1B_1ρ_A_2B_2)
       =   C^A_1|B_1_max(ρ_A_1B_1)+ C^A_2|B_2_max(ρ_A_2B_2).





Due to definition of max-relative entropy of IQ coherence measure, there exists optimal Q states σ_A_1|B_1 and σ_A_2|B_2
such that ρ_A_iB_i≤ 2^C^A|B_max(ρ_A_iB_i)σ_A_i|B_i.
Hence, we have the following inequality,

    C^A_1A_2|B_1B_2_max(ρ_A_1B_1ρ_A_2B_2)
       ≤   C^A_1|B_1_max(ρ_A_1B_1)+ C^A_2|B_2_max(ρ_A_2B_2).


Now, we prove the converse.
It has been proved in Ref. <cit.> that

    2^C^A|B_max(ρ_AB)=max_τ_AB≥0
    Δ_A𝕀_B(τ_AB)=𝕀_ABρ_ABτ_AB.

Hence, there exist operators τ_A_iB_i such that
τ_A_iB_i≥0, Δ_A_i𝕀_B_i(τ_A_iB_i)=𝕀_A_iB_i and
2^C^A_i|B_i_max(ρ_AB)=ρ_A_iB_iτ_A_iB_i,
for i=1,2. Then the operator τ_A_1A_2B_1B_2:=τ_A_1B_1τ_A_2B_2 satisfies the conditions
τ_A_1A_2B_1B_2≥ 0 and Δ_A_1Δ_A_2𝕀_B_1B_2(τ_A_1A_2B_1B_2)=𝕀_A_1A_2B_1B_2, which implies that
2^C^A_1A_2|B_1B_2_max(ρ_A_1B_1ρ_A_2B_2)≥(ρ_A_1B_1ρ_A_2B_2)τ_A_1A_2B_1B_2, i.e.,

    C^A_1A_2|B_1B_2_max(ρ_A_1B_1ρ_A_2B_2)
       ≥   C^A_1|B_1_max(ρ_A_1B_1)+ C^A_2|B_2_max(ρ_A_2B_2).

Therefore, we obtain the result.






For any two bipartite states ρ_A_1B_1∈(_A_1_B_1) and ρ_A_2B_2∈(_A_2_B_2),

    C^A_1A_2|B_1B_2_f(ρ_A_1B_1ρ_A_2B_2)
       =   C^A_1|B_1_f(ρ_A_1B_1)+ C^A_2|B_2_f(ρ_A_2B_2).





Due to the definition of C^A|B_f, it is easy to get the inequality

    C^A_1A_2|B_1B_2_f(ρ_A_1B_1ρ_A_2B_2)
       ≤    C^A_1|B_1_f(ρ_A_1B_1)+ C^A_2|B_2_f(ρ_A_2B_2).

Thus, we only need to prove the converse. First, we prove that
for any pure state |ψ⟩_A_1A_2B_1B_2, the following inequality holds,

    C^A_1A_2|B_1B_2_f(ψ_A_1A_2B_1B_2)
       ≥    C^A_1|B_1_f(σ_A_1B_1)+ C^A_2|B_2_f(σ_A_2B_2),

where σ_A_1B_1, σ_A_2B_2 are the corresponding reduced states of ψ_A_1A_2B_1B_2.
Since the pure state |ψ⟩_A_1A_2B_1B_2 can be written as
|ψ⟩_A_1A_2B_1B_2=∑_i√(p_i)|i⟩_A_1|u_i⟩_A_2B_1B_2, then

    C^A_1A_2|B_1B_2_f(ψ_A_1A_2B_1B_2)
       =   S(Δ_A_1Δ_A_2(σ_A_1A_2))
       =   S(Δ_A_1(σ_A_1))+∑_ip_iS(Δ_A_2(σ^A_2_i))
       =   S(Δ_A_1(σ_A_1))+∑_ip_iS(Δ_A_2(Tr_B_2σ^A_2B_2_i))
       ≥   C^A_1|B_1_f(σ_A_1B_1)+C^A_2|B_2_f(σ_A_2B_2),

where
S(Δ_A_1(σ_A_1))≥ C^A_1|B_1_f(σ_A_1B_1)
results from the concavity of von Neumann entropy, and
∑_ip_iS(Δ_A_2(Tr_B_2σ^A_2B_2_i))≥ C^A_2|B_2_f(σ_A_2B_2)
comes from (<ref>).


Moreover, there exists an optimal pure state decomposition of ρ_A_1B_1ρ_A_2B_2=∑_iλ_iψ_i_A_1A_2B_1B_2 such that
C^A_1A_2|B_1B_2_f(ρ_A_1B_1ρ_A_2B_2)=∑_iλ_iC^A_1A_2|B_1B_2_f(ψ_i_A_1A_2B_1B_2). Therefore,

    C^A_1A_2|B_1B_2_f(ρ_A_1B_1ρ_A_2B_2)
       =   ∑_iλ_iC^A_1A_2|B_1B_2_f(ψ_i_A_1A_2B_1B_2)
       ≥   ∑_ip_i[C^A_1|B_1_f(σ^A_1B_1_i)+C^A_2|B_2_f(σ^A_2B_2_i)]
       ≥    C^A_1|B_1_f(ρ_A_1B_1)+C^A_2|B_2_f(ρ_A_2B_2),

where the first inequality comes from (<ref>), the second inequality comes from the convexity
of C^A|B_f with σ^i_A_1B_1=Tr_A_2B_2ψ_i_A_1A_2B_1B_2, σ^A_2B_2_i=Tr_A_1B_1ψ_i_A_1A_2B_1B_2, ρ_A_1B_1=∑_ip_iσ^A_1B_1_i and ρ_A_2B_2=∑_ip_iσ^A_2B_2_i.




Note that the additivity of C^A|B_a is still unclear as the method used in the proof of the additivity of C^A|B_f
does not work for C^A|B_a.
Nevertheless, if the subsystems B_i ( i=1,2 ) are trivial, i.e., the dimension is 1, then
one has the additivity of the coherence measures. For example,   the additivity of IQ coherence measures C^A|B_max will
lead to the additivity of C_max if the subsystems B_i ( i=1,2 ) are trivial.

Given two quantum states ρ_1∈(_A_1) and
ρ_2∈(_A_2), it holds that

    C_max(ρ_1ρ_2)=C_max(ρ_1)+C_max(ρ_2).


Due to the additivity of C_max, we  can obtain the additivity of
robustness of coherence ROC <cit.> as follows,

    1+ROC(ρ_1ρ_2)
    =[1+ROC(ρ_1)]·[1+ROC(ρ_2)],

which comes directly from the fact that
C_max(ρ)=log (1+ROC(ρ)) <cit.>.
Following the same method, it is easy to obtain the additivity
of coherence weight C_w<cit.> as following,

    1-C_w(ρ_1ρ_2)
    =[1-C_w(ρ_1)]·[1-C_w(ρ_2)].

Thus, the additivity of
robustness of coherence and
coherence weight are proved here, which will be useful to
the further study on the distribution of coherence in
multipartite systems quantified by robustness of coherence
and coherence weight.





§ CONCLUSION

In this work, we have investigated the properties of the incoherent-quantum coherence measures defined by
relative entropy, max-relative entropy and l_1 norm on bipartite systems. We also introduce the IQ coherence of formation and assistance
on bipartite systems. And we have found the distribution of coherence of formation C_f and assistance C_a in bipartite systems: the total coherence of formation is lower ( upper ) bounded by the sum of coherence of formation (assistance) in each local subsystem and
entanglement of formation ( assistance ) between subsystems.
Besides, we have obtained the tradeoff relation between
coherence cost and entanglement cost, distillable coherence and distillable entanglement  in bipartite systems.
Moreover, we have  obtained the monogamy relationship of the IQ coherence of formation and assistance in tripartite systems.
Furthermore, the additivity of IQ coherence measures have been discussed.
 These results substantially advance the understanding of the physical laws that governs the distribution of quantum coherence in
bipartite systems and pave the way for the further researches in this direction.





K.F. Bu thanks Prof. Arthur Jaffe and Dr. Zhengwei Liu for their
hospitality in Harvard University and the support of  a grant from the Templeton
Religion Trust.
J.D. Wu is supported by the Natural Science Foundation of China (Grants No. 11171301, No. 10771191, and No. 11571307) and the Doctoral Programs Foundation of the Ministry
of Education of China (Grant No. J20130061).
S.M. Fei is supported by the Natural Science Foundation of China under No. 11675113.



apsrev4-1
 




§ PROPERTIES OF C^A|B_F

[Proof of strong monotonicity under IO on A side for C^A|B_f]
For any incoherent operation
Λ^A_IO on A side with the set of Kraus operators K^A_μ,
we need to prove ∑_μ q_μC^A|B_f(ρ_AB)(ρ_μ)≤ C^A|B_f(ρ_AB).
Due to the definition of C^A|B_f, there exists an optimal pure state decomposition of ρ_AB=∑_ip_iψ_i_AB such that
C^A|B_f(ρ_AB)=∑ p_iC^A|B_r(ψ_i_AB).
Let |ϕ_i,μ⟩_AB=K^A_μ|ψ_i⟩_AB/√(q_i,μ) with q_i,μ=K^A_μψ_i_ABK^A,†_μ. Hence,

    C^A|B_f(ρ_AB)   =   ∑_ip_iC^A|B_r(ψ_i_AB)
       ≥   ∑_ip_i∑_μ q_i,uC^A|B_r(ϕ_i,μ)
       =   ∑_μq_μ∑_i p_iq_i,μ/q_μC^A|B_r(ϕ_i,μ)
       ≥   ∑_μq_μC^A|B_r(ρ_μ),

where the first inequality comes from the fact that C^A|B_r is strong monotonicity under IO on A side for
|ψ_i⟩_AB and the last inequality comes from the convexity of C^A|B_r.



[Proof of  monotonicity under CPTP maps on B side for C^A|B_f]

For CPTP map
Λ^B on B side with the set of Kraus operators K^B_μ,
we need to prove C^A|B_f(ρ_AB)(Λ^B(ρ_μ))≤ C^A|B_f(ρ_AB).
Due to the definition of C^A|B_f,
there exists an optimal pure state decomposition of ρ_AB=∑_ip_iψ_i_AB such that
C^A|B_f(ρ_AB)=∑ p_iS(Δ_A(ψ_i_AB))=∑_ip_iS(Δ_A(ρ^A_i)) with ρ^A_i=Bψ_i_AB. Hence
Λ^B(ρ_AB)=∑_i,μp_iK^B_μψ_i_ABK^B,†_μ=∑_ip_i∑_μq_i,μϕ_i,μ_AB
with |ϕ_i,μ⟩_AB=K^A_μ|ψ_i⟩_AB/√(q_i,μ) and q_i,μ=K^A_μψ_i_ABK^A,†_μ. Thus

    C^A|B_f(Λ^B(ρ_AB))
       ≤   ∑_i,μp_iq_i,μS(Δ_A(ϕ_i,μ))
       =   ∑_i,μp_iq_i,μS(Δ_ATr_B(ϕ_i,μ))
       ≤   ∑_ip_iS(∑_μq_i,μΔ_ATr_B(ϕ_i,μ))
       =   ∑_ip_iS(Δ_A(ρ^A_i))
       =   C^A|B_f(ρ_AB),

where the first inequality comes from the definition of C^A|B_f and the second inequality comes from the fact the concavity
of von Neumann entropy.




Given a bipartite pure state |ψ⟩_AB, it holds that

    S(Δ_A(ψ_AB))=S(Δ_A(Bψ_AB)).



Since the pure state |ψ⟩_AB can be expressed in the given basis |i⟩_A_i of
_A as follows,

    |ψ⟩_AB=∑_i√(p_i)|i⟩_A|u_i⟩_B,

with p_i≥ 0 and ∑_ip_i=1. Then

    Δ_A(ψ_AB)   =   ∑_ip_ii_Au_i_B,
    Δ_A(Bψ_AB)   =   ∑_ip_ii_A,

which implies that S(Δ_A(ψ_AB))=S(Δ_A(Bψ_AB))=-∑_ip_ilog p_i.






§ STRONG MONOTONICITY UNDER IO ON A SIDE 

The strong monotonicity under IO on A side can be proved
following the similar method used in <cit.>.
For incoherent operation on A side
Λ^A_IO(·)=∑_μ K^A_μ(·) K^A,†_μ,
ρ^AB_μ=K^A_nρ_AB K^A,†_μ/p_μ and
p_μ=K^A_μρ_AB K^A,†_μ. Thus

    ∑_μ p_μ C^A|B_l_1(ρ^AB_μ)
       =   ∑_μ p_μ∑_i≠ jρ^AB_μ_tr
       =   ∑_μ∑_i≠ j iK^A_μρ_ABK^A,†_μj_tr
       =   ∑_μ∑_i≠ j∑_r,s[K^A_μ]_ir[K^A,†_μ]_sjρ^B_rs_tr,

where ρ^B_rs=rρ^AB_μs_A∈(_B).

Since Λ^A is incoherent, then
[K^A_μ]_ir[K^A,†_μ]_rj=δ_ij for any r, where
δ_ij=1 if i=j, otherwise δ_ij=0. Therefore

    ∑_μ p_μ C^A|B_l_1(ρ^AB_μ)
       =   ∑_μ∑_i≠ j∑_r≠ s∑_r,s[K^A_μ]_ir[K^A,†_μ]_sjρ^B_rs_tr
       ≤   ∑_μ∑_i≠ j[K^A_μ]_ir[K^A,†_n]_sjρ^B_rs,μ_tr
       =   ∑_r≠ sρ^B_rs,μ_tr∑_μ∑_i≠ j|[K^A_n]_ir[K^A,†_μ]_sj|
       ≤   ∑_r≠ sρ^B_rs_tr,

where the last inequality comes from the
fact that ∑_μ∑_i≠ j|[K^A_μ]_ir[K^A,†_μ]_sj|≤ 1
given in <cit.>. Thus, we obtain the strong monotonicity
of C^A|B_l_1 under IO on A side.



§ RELATION BETWEEN L_1 NORM AND TRACE NORM

The trace norm ·_tr is closely related to the l_1 norm ·_l_1, for which we have the following relationship,

Given an operator P∈() and a fixed reference basis |i⟩_i of . Then

    P_tr=min_U,VUPV_l_1,

where the minimization is taken over all the unitaries U,V acting on  and ·_l_1 is defined by the
the given basis.


First, let us prove that for operator P∈(), P_tr≤P_l_1.
Due to single value decomposition of P, there exists two orthonormal basis |x_i⟩_i and
|y_i⟩_i such that

    P_tr   =   ∑_ix_iPy_i
       =   ∑_i∑_j,kx_ijjPkky_i
       ≤   ∑_j,k|jPk|∑_i|x_ijky_i|
       ≤   ∑_j,k|jPk|,

where the last inequality comes from the fact that
∑_i|x_ijky_i|≤ (∑_i|x_ij|^2)^1/2(∑_i|ky_i|^2)^1/2=1
with |x_i⟩_i and  |y_i⟩_i being the orthonormal basis.
Thus, P_tr=UPV_tr≤UPV_l_1 for any two unitaries.

Besides, there exist unitaries U and V such that
UPV=∑_is_ii with s_i_i being the single value of
P, and thus UPV_l_1=P_tr.










 