Quantum coherence is an important feature in quantum physics and is of practical significance in quantum computation and quantum communication1,2. The formulation of the resource theory of coherence was initiated in Ref.3, in which some requirements are proposed for a well defined quantum coherence measure. Later on, coherence measures or monotones such as the \(l_1\) norm of coherence3, the relative entropy of coherence3, intrinsic randomness of coherence4, coherence concurrence5, distillable coherence6, coherence cost6, robustness of coherence7, coherence number8 and so on are proposed theoretically or operationally. Many of these coherence measures were either created from entanglement measures using a standard modification or are closely related to one that was. For example, the robustness of coherence and coherence number are defined in a manner similar to that of robustness of entanglement and the Schmidt number in entanglement theory, respectively9,10. The \(l_1\) norm coherence is exactly the twice negativity for pure states11.

Coherence of assistance is another quantifier which quantifies the coherence that can be extracted assisted by another party under local measurements and classical communication12. Suppose Alice holds a state \(\rho ^A=\sum _k p_k |\psi _k\rangle \langle \psi _k|\) with coherence \(C(\rho ^A)\). Bob holds another part of the purified state of \(\rho ^A\). The joint state between Alice and Bob is \(\sum _k p_k |\psi _k\rangle _A \otimes |k\rangle _B\). Bob performs local measurements \(\{|k\rangle \langle k|\}\) and informs Alice the measurement outcomes by classical communication. Alice’s quantum state will be in a pure state ensemble \(\{ p_k,\ |\psi _k\rangle \langle \psi _k|\}\) with average coherence \(\sum _k p_k C(|\psi _k\rangle \langle \psi _k|)\). The process is called assisted coherence distillation. The maximum average coherence is called the coherence of assistance which quantifies the one-way coherence distillation rate12. The coherence of assistance is always greater than or equal to the coherence measure. But it is still not clear whether one can always obtain more coherence with the help of another party. Our answer in this paper is that one can always obtain more coherence for the full ranked quantum states.

As with other measures of coherence and entanglement, the coherence of assistance and the entanglement of assistance are also closely related. In fact, the relative entropy coherence of assistance corresponds to the entanglement formation of assistance12,13,14,15 and the \(l_1\) norm coherence of assistance corresponds to the convex-roof extended negativity entanglement of assistance16,17,18. As intrinsic characteristics of quantum physics, the inextricable relationship between quantum coherence and quantum entanglement is not limited to specific quantum coherence measures and entanglement measures as well as the coherence of assistance and the entanglement of assistance. Ref.19 shows any coherence can be converted to entanglement via incoherent operations, and each entanglement measure corresponds to a coherence measure. It has been further shown that coherence can be converted to bipartite nonlocality, genuine tripartite entanglement and genuine tripartite nonlocality20. In Refs.21,22 the authors construct an entanglement monotone based on any given coherence measure. More generally, Ref.23 establishes a general operational one-to-one mapping between coherence measures and entanglement measures.

Inspired by these results, we aim to construct a general relation between the coherence of assistance and entanglement of assistance for the coherence and entanglement theory. First we review the construction of entanglement measures and coherence measures using the convex roof extension. Then we define the general coherence of assistance and the one-to-one correspondence between entanglement of assistance and coherence of assistance is established afterwards. Subsequently, two special classes of states called assisted maximally coherent states and assisted maximally entangled states are introduced. These states can be turned into the maximally coherent or maximally entangled states with the help of another party’s local measurement and classical communication. The necessary conditions for states to be the assisted maximally coherent states or assisted maximally entangled states are presented. These states can be thought of as potentially perfect coherence or entanglement resources. Then we show the coherence of convex roof and the coherence of assistance, as well as the entanglement measure in convex roof construction and the entanglement of assistance, are not equal for any full rank density matrix. This demonstrates that this kind of states are all distillable in the assisted coherence distillation. The unified framework between coherence and entanglement is shown in Fig. 1.

Figure 1
figure 1

Relations between coherence and entanglement. Here \(f\in {\mathfrak {F}}{\setminus } \{0\}\). \(E_f\) is a function defined on bipartite pure states as in Eq. (1). \(C_f\) is a function defined on pure states as in Eq. (4). \(E_c\) is the entanglement measure called the entanglement of convex roof which is the convex roof extension of \(E_f\) from pure states to mixed states in Eq. (2). \(C_c\) is the coherence measure called the coherence of convex roof which is the convex roof extension of \(C_f\) from pure states to mixed states in Eq. (5). \(E_c\) and \(C_c\) are one-to-one corresponded by the real symmetric concave function f. The maximum points of \(E_c\) and \(C_c\) are maximally entangled states (ME) in Eq. (3) and maximally coherent states (MC) in Eq. (6) respectively. \(E_a\) is the entanglement of assistance which is the least concave majorant extension of \(E_f\) from pure states to mixed states in Eq. (7). \(C_a\) is the coherence of assistance which is the least concave majorant extension of \(C_f\) from pure states to mixed states in Eq. (8). \(E_a\) and \(C_a\) are one-to-one corresponded by the real symmetric concave function f. The maximum points of \(E_a\) and \(C_a\) are assisted maximally entangled states (AME) in definition 2 and assisted maximally coherent states (AMC) in definition 1 respectively. The coherence of assistance \(C_a\) as well as the entanglement of assistance \(E_a\) is shown to be strictly larger than the coherence of convex roof \(C_c\) and entanglement of convex roof \(E_c\) for all full rank quantum states.

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Results

Entanglement of assistance and coherence of assistance

A state is called incoherent if the density matrix \(\rho\) is diagonal in the fixed reference basis \(\{|i\rangle \}\), \(\rho =\sum _i p_i |i\rangle \langle i|\) with \(p_i\) the probabilities. Otherwise the state is called coherent. Obviously, coherence is basis dependent. A completely positive trace preserving map \(\Lambda\) acting as \(\Lambda (\rho )=\sum _l K_l\rho K_l^\dagger\) is said to be an incoherent operation if all the Kraus operators \(K_l\) map incoherent states to incoherent states3. A coherence measure \(C(\rho )\) should satisfy3: (1) \(C(\rho )\ge 0\) with \(C(\rho )=0\) if and only if \(\rho\) is incoherent. (2) \(C(\rho )\) is nonincreasing under incoherent operations \(\Lambda\), \(C(\rho )\ge C(\Lambda (\rho ))\). (3) \(C(\rho )\) is nonincreasing on average under selective incoherent operations, \(C(\rho )\ge \sum _l q_l C(\rho _l)\), with \(q_l=tr(K_l \rho K_l^\dagger )\) and \(\rho _l=K_l \rho K_l^\dagger /q_l\). (4) \(C(\rho )\) is a convex function on the density matrices, \(C(\sum _j p_j \rho _j)\le \sum _j p_j C( \rho _j)\).

Let \({\mathfrak {{F}}}=\{f\}\) be the set of functions on the probability simplex \(\Omega =\{\mathbf {x}=(x_0,x_1,\ldots ,x_{n-1})^T|\sum _{i=0}^{n-1} x_i =1 \ \text{ and } \ x_i\ge 0 \}\) such that (i) f is a real symmetric concave function; (ii) \(f((1,0,\ldots ,0)^T)=0\). We assume \(f\in {\mathfrak {{F}}}{\setminus } \{0\}\) in this paper. Under these conditions f yields an entanglement monotone \(E_f\) for the \(n\otimes n\) pure states. If \(|\psi \rangle\) has the Schmidt form \(|\psi \rangle =\sum _{i=0}^{n-1} \lambda _i |i_A \rangle |i_B\rangle\) with \(\lambda _i\ge 0\), \(E_f\) can be defined as

$$\begin{aligned} E_f(|\psi \rangle )=f(\lambda (|\psi \rangle )), \end{aligned}$$
(1)

where \(\{|i_A \rangle \}_{i=0}^{n-1}\) and \(\{|i_B \rangle \}_{i=0}^{n-1}\) are orthonormal bases of the two subsystems, and \(\lambda (|\psi \rangle )=(\lambda _0^2, \lambda _1^2, \ldots , \lambda _{n-1}^2)^T\). The entanglement monotone \(E_f\) can be extended to mixed states by the convex roof construction23,24. The entanglement of convex roof \(E_{c}\) is given by

$$\begin{aligned} E_{c}(\rho )=\min \sum _k p_k {E_f}(|\psi _k\rangle ), \end{aligned}$$
(2)

where \(E_f\) is defined by (1), the minimization is taken over all pure state decompositions of \(\rho =\sum _k p_k |\psi _k\rangle \langle \psi _k|\).

The entanglement of convex roof \(E_{c}\) is an entanglement measure24. Any quantum states that are local unitary equivalent to

$$\begin{aligned} |\phi ^+\rangle =\frac{1}{\sqrt{n}} \sum _{j=0}^{n-1} |jj\rangle , \end{aligned}$$
(3)

are all maximally entangled according to \(E_{c}\). These states are the only ones such that \(E_{c}\) attains its maximum for \(n \otimes n\) systems.

Correspondingly, for the fixed reference basis \(\{|i\rangle \}\) and any \(f\in {\mathfrak {{F}}}{\setminus } \{0\}\), a coherence measure for pure state \(|\psi \rangle =\sum _{i=0}^{n-1} \psi _i |i\rangle\) can be defined as

$$\begin{aligned} C_f(|\psi \rangle )=f(\mu (|\psi \rangle )), \end{aligned}$$
(4)

where \(\mu (|\psi \rangle )=(|\psi _0|^2, |\psi _1|^2, \ldots , |\psi _{n-1}|^2)^T\) is the coherence vector. The coherence measure \(C_f\) can be extended to mixed states by the convex roof construction23,25. The coherence of convex roof \(C_{c}\) is given by

$$\begin{aligned} C_{c}(\rho )=\min \sum _k p_k {C_f}(|\psi _k\rangle ), \end{aligned}$$
(5)

where \(C_f\) is defined by (4), the minimization is taken over all pure state decompositions of \(\rho =\sum _k p_k |\psi _k\rangle \langle \psi _k|\).

The coherence of convex roof \(C_{c}\) is a coherence measure1,2,3,23. According to the coherence measure \(C_{c}\), all maximally coherent states in an n-dimensional system can be transformed into the pure states in the following set by unitary incoherent operations26:

$$\begin{aligned} \left\{ \frac{1}{\sqrt{n}} \sum _{j=0}^{n-1} e^{\mathrm{i}\theta _j} |j\rangle \ |\ \theta _1, \ldots , \theta _{n-1}\in [0,\ 2\pi )\right\} . \end{aligned}$$
(6)

For any function \(f\in {\mathfrak {F}}{\setminus } \{0\}\), the entanglement monotone \(E_f\) can be also extended to mixed states by the least concave majorant extension, giving rise to entanglement of assistance. The entanglement of assistance can be defined by

$$\begin{aligned} E_a(\rho )=\max \sum _k p_k E_f(|\psi _k\rangle ), \end{aligned}$$
(7)

where the maximization is taken over all pure state decompositions of \(\rho =\sum _k p_k |\psi _k\rangle \langle \psi _k|\).

The entanglement of assistance has been introduced with respect to some specific functions13,14,15,17. Definition (7) presents a general notion of entanglement of assistance for arbitrary function \(f\in {\mathfrak {F}}{\setminus } \{0\}\). It is a dual construction to the entanglement of convex roof. Unlike the entanglement of convex roof which is an entanglement measure, the entanglement of assistance is not a measure of entanglement, as it is not monotonic under local operations and classical communications27. But the entanglement of assistance describes the hidden entanglement that can be unlocked with the help of another party’s local measurement and classical communication.

Correspondingly, we can define the coherence of assistance,

$$\begin{aligned} C_a(\rho )=\max \sum _k p_k C_f(|\psi _k\rangle ), \end{aligned}$$
(8)

with \(C_f\) defined in Eq. (4), where the maximization is taken over all pure state decompositions of \(\rho =\sum _k p_k |\psi _k\rangle \langle \psi _k|\).

We observe that \(C_a\) vanishes if the quantum state is incoherent and pure. Additionally, \(C_a\) is not monotonic under incoherent operations. For example, consider \(\rho =|0\rangle \langle 0|\) and an incoherent operation \(\Lambda (\rho )=K_1\rho K_1^\dagger + K_2\rho K_2^\dagger\), where \(K_1=\frac{1}{\sqrt{2}}I\) and \(K_2=\frac{1}{\sqrt{2}}(|0\rangle \langle 1|+|1\rangle \langle 0|)\) satisfying \(K_1^\dagger K_1 + K_2^\dagger K_2=I\). After the incoherent operation, \(\Lambda (\rho )=\frac{1}{2}(|0\rangle \langle 0|+|1\rangle \langle 1|)=\frac{1}{2}(|\psi _1\rangle \langle \psi _1|+|\psi _2\rangle \langle \psi _2|)\), with \(|\psi _1\rangle =\cos \theta |0\rangle + \sin \theta |1\rangle\) and \(|\psi _2\rangle =-\sin \theta |0\rangle + \cos \theta |1\rangle\). By the assumptions of f we know that there exists an angle \(\theta\) such that \(C_f(|\psi _1\rangle )=C_f(|\psi _2\rangle )>0\). Hence, \(0=C_a(\rho )< C_a(\Lambda (\rho ))\), which violates the monotonicity of coherence measures under incoherent operations. Therefore, the coherence of assistance is actually not a coherence measure.

Theorem 1

The coherence of assistance \({C_a}\) corresponds one-to-one to the entanglement of assistance \(E_a\).

See “Methods” section for the proof of the Theorem 1.

Under the product reference bases, the entanglement of assistance \(E_a\) is just the coherence of assistance \(C_a\) for all pure states as well as for Schmidt correlated states \(\rho _{mc}=\sum _{ij} \rho _{ij}|ii\rangle \langle jj|\)28, \(E_a(\rho _{mc})=C_a(\rho _{mc})\). Similar results also hold true for the entanglement of convex roof \(E_c\) and the coherence of convex roof \(C_c\). The correspondence not only bridges coherence theory and entanglement theory, but also generalizes many results in entanglement theory to coherence theory.

The entanglement of assistance \(E_a\) and coherence of assistance \(C_a\) depend on the choice of the functions \(f\in {\mathfrak {F}}{\setminus } \{0\}\). If \(f(\mathbf {p})=-\sum _i p_i \log p_i\) for \(\mathbf {p}=(p_1,p_2,\ldots ,p_n)^T\) in the probability simplex, \(E_a\) becomes the entanglement of formation of assistance13,14,15, and \(C_a\) becomes the relative entropy coherence of assistance12. If \(f(\mathbf {p})=\sum _{i\ne j}\sqrt{p_ip_j}\) for \(\mathbf {p}=(p_1,p_2,\ldots ,p_n)^T\) in the probability simplex, then \(E_a\) becomes the half convex-roof extended negativity of assistance17 and \(C_a\) becomes the \(l_1\) norm coherence of assistance16. Analogously, one can also define various other types of entanglement of assistance and coherence of assistance based on other real symmetric concave functions f. For example, let \(f(\mathbf {p})=\sqrt{2(1-\sum _i p_i^2)}\), then \(E_a\) is the entanglement of assistance in terms of concurrence29, in which an upper bound of entanglement of assistance is provided as \(E_a(\rho )\le \sqrt{2(1-tr(\rho _A^2))}\) with \(\rho _A=tr_B (\rho )\). For this function f, we can define the coherence of assistance \(C_a\) in terms of concurrence similarly and one upper bound is \(C_a(\rho )\le \sqrt{2(1-\sum _i \rho _{ii}^2)}\) with \(\rho _{ii}\) the diagonal entries of \(\rho\) in the reference basis.

Assisted maximally coherent states and assisted maximally entangled states

The average of entanglement and coherence depends on the ensembles of a quantum state. Assisted by another party, the entanglement of assistance and coherent of assistance attain the maximum average entanglement and coherence of the quantum state. Here we investigate two classes of states called assisted maximally coherent states and assisted maximally entangled states for which the maximal average coherence and entanglement are the same as the maximally coherent states and maximally entangled states.

Definition 1

We call an n dimensional quantum state \(\rho\) assisted maximally coherent (AMC) if it is a convex combination of maximally coherent pure states.

The AMC states are a class of states that achieve the maximum of coherence of assistance. Therefore they are a potentially perfect coherence resource. For pure states, all the maximally coherent states are AMC and vice versa. For mixed states, all maximally mixed states \(\rho =\frac{1}{n} \sum _{i=0}^{n-1} |i\rangle \langle i|\) are AMC. This follows from the existence of a maximally coherent pure state decomposition \(\{p_k,\ |\psi _k\rangle \}\) of \(\rho\), where \(p_k=\frac{1}{n}\) for all k and \(|\psi _k\rangle = \frac{1}{\sqrt{n}} \sum _{j=0}^{n-1} e^{2\pi \mathrm{i} (k-1)j/n}|j\rangle\) for \(k=1,2,\ldots ,n\), and \(\mathrm i=\sqrt{-1}\) is the imaginary unit. The Fourier matrix F with its k-th column given by the vector \({\sqrt{n}}|\psi _k\rangle\) satisfies \(FF^\dagger =nI\) and \(|F_{kj}|=1\), \(k=1,\ldots , n\); \(j=0,1,2,\ldots ,n-1\). Therefore, \(\{|\psi _k\rangle \}_{k=1}^n\) is an orthonormal basis of the n dimensional system, which means that \(\sum _{i=0}^{n-1} |i\rangle \langle i|= \sum _{k=1}^n |\psi _k\rangle \langle \psi _k|\).

Theorem 2

If an n dimensional quantum state \(\rho =\sum _{ij} \rho _{ij}|i\rangle \langle j|\) is AMC, then \(\rho _{ii}=\frac{1}{n}\) for all i, which becomes both necessary and sufficient for two and three dimensional systems.

See “Methods” section for the proof of the Theorem 2.

There exist n-dimensional quantum states \(\rho\) with all diagonal entries \(\frac{1}{n}\) which do not allow for pure state decomposition \(\{p_k,\ |\psi _k\rangle \}\) such that all diagonal entries of \(|\psi _k\rangle \langle \psi _k|\) are \(\frac{1}{n}\) for all k and \(n\ge 4\). Some specific examples are shown in Refs.30,31. We now give an explicit pure state decomposition for three dimensional AMC states. In a three dimensional system, the quantum state \(\rho =\sum _{i,j} \rho _{ij} |i\rangle \langle j|\), with \(\rho _{11}=\rho _{22}=\rho _{33}=\frac{1}{3}\) and real nonzero off diagonal entries, is an example of mixed AMC state that is not a maximally mixed state. Let \(p_1=\frac{1}{4}(1+\rho _{12} +\rho _{13} +\rho _{23} )\), \(p_2=\frac{1}{4}(1-\rho _{12} -\rho _{13} +\rho _{23} )\), \(p_3=\frac{1}{4}(1-\rho _{12} +\rho _{13} -\rho _{23} )\), \(p_4=\frac{1}{4}(1+\rho _{12} -\rho _{13} -\rho _{23} )\), and \(|\psi _1\rangle =\frac{1}{\sqrt{3}}(|1\rangle +|2\rangle +|3\rangle )\), \(|\psi _2\rangle =\frac{1}{\sqrt{3}}(-|1\rangle +|2\rangle +|3\rangle )\), \(|\psi _3\rangle =\frac{1}{\sqrt{3}}(|1\rangle -|2\rangle +|3\rangle )\), \(|\psi _4\rangle =\frac{1}{\sqrt{3}}(|1\rangle +|2\rangle -|3\rangle )\), then \(\{p_k,\ |\psi _k\rangle \}\) is a pure state decomposition of \(\rho\) with components all maximally coherent.

Similar to AMC states, we can define the assisted maximally entangled (AME) states in bipartite systems.

Definition 2

An \(n\otimes n\) bipartite quantum state \(\rho\) is called assisted maximally entangled (AME) if it is a convex combination of maximally entangled pure states.

Theorem 3

The \(n\otimes n\) Schmidt correlated state \(\rho _{mc}=\sum _{ij} \rho _{ij}|ii\rangle \langle jj|\) is AME if and only if the n dimensional state \(\rho =\sum _{ij} \rho _{ij}|i\rangle \langle j|\) is AMC.

See “Methods” section for the proof of the Theorem 3. Combining Theorems 2 and 3 , we get the following necessary condition for Schmidt correlated states to be AME.

Corollary 1

If an \(n\otimes n\) Schmidt correlated state \(\rho _{mc}=\sum _{ij} \rho _{ij}|ii\rangle \langle jj|\) is AME, then \(\rho _{ii}=\frac{1}{n}\) for all i, which is both necessary and sufficient for the cases of \(n=2\) and \(n=3\) systems.

For pure states, all maximally entangled states are AME and vice versa. For mixed states, all maximally correlated states \(\rho =\frac{1}{n} \sum _{i=0}^{n-1} |ii\rangle \langle ii|\) are AME due to Corollary 1. Besides the Schmidt correlated states, there are also other AME states. As examples, consider two-qubit system. Let \(\rho =p|\psi _1\rangle \langle \psi _1|+(1-p)|\psi _2\rangle \langle \psi _2|\) with \(0<p<1\), \(|\psi _1\rangle =\frac{1}{\sqrt{2}}(|00\rangle +|11\rangle )\) and \(|\psi _2\rangle =\frac{1}{\sqrt{2}}(|01\rangle +|10\rangle )\). Clearly, \(\rho\) is AME but not Schmidt correlated. The maximally mixed states \(\rho =\frac{1}{n^2} \sum _{i,j=0}^{n-1} |ij\rangle \langle ij|\) are AME, since they can be written as the average of generalized Bell states \(|\phi _{st}\rangle =I\otimes U_{st}^* |\phi ^+\rangle\), where \(U_{st}=h^tg^s\), \(h|j\rangle =|j+1\mod n\rangle\), \(g|j\rangle =\omega ^j |j\rangle\), \(\omega =\exp (-2\pi \mathrm{i}/n)\), and superscript \(*\) stands for the conjugate32.

AMC states and AME states are potential maximally coherent states and maximally entangled states, since they can be decomposed as the convex combinations of maximally coherent and maximally entangled pure states, respectively. Furthermore, they can be collapsed to maximally coherent states and maximally entangled states with the help of another party’s local measurements and classical communication operationally, if only one knows the optimal pure state decompositions. As applications, one can transform the AMC states to maximally coherent pure states with the help of another party’s local measurements and classical communication for the purpose of quantum information processing such as the Deutsch-Jozsa algorithm to speedup the computation33. In this sense, the AMC states are potentially perfect quantum resources. In fact, the experimental realization in linear optical systems for obtaining the coherence of assistance with respect to the relative entropy coherence in two dimensional systems has already been presented34.

Relation between the convex roof extension and the least concave majorant extension

The strict relation between the coherence of convex roof and the coherence of assistance, that is, whether \(C_c(\rho )< C_a(\rho )\) holds for all mixed quantum states is an interesting topic. The physical motivation is from the coherence distillation, which is to extract pure coherence from a mixed state by incoherent operations6. All coherent states can be distilled by the coherence distillation process. The assisted coherence distillation is then introduced to generate the maximal possible coherence with the help of another party’s local measurements and classical communication12. The relative entropy coherence of assistance in form of Eq. (8) with a specific function f is proposed first there to quantify the one way coherence distillation rate in the assisted coherence distillation. Generally we can get more coherence in the assisted coherence distillation. But a natural question is whether we can extract more coherence from all mixed states in the assisted coherence distillation. This question is factually equivalent to whether the coherence of assistance is strictly larger than the coherence of convex roof for all mixed quantum states. If it is true, all mixed quantum states are distillable in the assisted coherence distillation process. In order to answer this question, we consider a much more general case as follows.

We now investigate the general relations between the convex roof extension and the least concave majorant extension of an arbitrary nonnegative function. Let \({\mathcal {H}}\) be a finite-dimensional Hilbert space and F a nonnegative function defined on the pure states of \({\mathcal {H}}\). Define \(F_a(\rho )=\max \sum _k p_k F(|\psi _k\rangle )\) to be the least concave majorant extension from F, and \(F_c(\rho )=\min \sum _k p_k F(|\psi _k\rangle )\) the convex roof extension from F, where the maximization and minimization are both taken over all pure state decompositions of \(\rho =\sum _k p_k |\psi _k\rangle \langle \psi _k|\), respectively. The convex roof extension \(F_c\) is the largest convex function which is equal to F on the pure states while the least concave majorant extension \(F_a\) is the smallest concave function. The definitions \(F_c\) and \(F_a\) are more general than \(E_c\), \(C_c\) and \(E_a\), \(C_a\).

Theorem 4

Let \({\mathcal {H}}\) be a finite-dimensional Hilbert space and F a nonnegative function defined on the pure states in \({\mathcal {H}}\). Let \(\rho _o\) be a density matrix on \({\mathcal {H}}\) and \(R(\rho _o)\) the range of \(\rho _o\). If \(F_a(\rho _o)=F_c(\rho _o)\), then there exists a positive semidefinite operator Q on \(R(\rho _o)\) such that \(F(|\psi \rangle )=\langle \psi |Q| \psi \rangle\) for all pure states \(|\psi \rangle\) in \(R(\rho _o)\).

See “Methods” section for the proof of the Theorem 4. Theorem 4 transforms the equation \(F_a(\rho )=F_c(\rho )\) into the existence of a positive semidefinite operator Q on \(R(\rho )\). So in order to check the coincidence of \(F_a(\rho )=F_c(\rho )\), one only needs to check the existence of Q for all pure states in the support of \(R(\rho )\). We apply Theorem 4 to coherence theory and entanglement theory.

Corollary 2

For full rank quantum states \(\rho\), the coherence of assistance is strictly larger than the coherence of convex roof, \(C_c(\rho )< C_a(\rho )\).

Corollary 3

For full rank bipartite quantum states \(\rho\), the entanglement of assistance is strictly larger than the entanglement of convex roof, \(E_c(\rho )< E_a(\rho )\).

See “Methods” section for the proof of the Corollary 2. The proof of Corollary 3 is similar to that of Corollary 2. Combined with the physical explanation of coherence (entanglement) of assistance, Corollaries 2 and 3 demonstrate that for full rank density matrices, their coherence (entanglement) can be strictly increased with the help of another party’s local measurements and classical communication. Hence, this kind of states are distillable in the assisted coherence (entanglement) distillation.

Discussions

We have introduced the general coherence of assistance in terms of real symmetric concave functions on the probability simplex, the coherence of assistance and the entanglement of assistance are shown to be in one-to-one correspondence as entanglement measures and coherence measures in the convex roof construction. Assisted maximally coherent states and assisted maximally entangled states are proposed as the convex combination of the maximally coherent states and maximally entangled states respectively, which can act potentially as perfect resource in quantum information. A necessary and sufficient condition for two or three-dimensional states to be AMC or AME is presented. Moreover, we have shown that the coherence of convex roof and the coherence of assistance are not equal for any full rank density matrix, together with a similar result for the entanglement with convex roof construction and the entanglement of assistance. These results may help strengthen our understanding of the important resources quantum coherence and entanglement.

Methods

Proof of Theorem 1

The coherence measure \(C_f\) in Eq. (4) corresponds one-to-one to the real symmetric concave function \(f\in {\mathfrak {{F}}}{\setminus } \{0\}\)23,25. The entanglement measure \(E_f\) in Eq. (1) also corresponds one-to-one to the real symmetric concave function \(f\in {\mathfrak {{F}}}{\setminus } \{0\}\)24. Therefore, the coherence measure \(C_f\) in Eq. (4) and the entanglement measure \(E_f\) in Eq. (1) are in one-to-one correspondence. As the least concave majorant extension of the coherence measure \(C_f\) and entanglement measure \(E_f\), the coherence of assistance \({C_a}\) and the entanglement of assistance \(E_a\) are also in one-to-one correspondence. \(\square\)

Proof of Theorem 2

Before we prove the theorem, we first introduce the concept of a correlation matrix35. An \(n\times n\) Hermitian matrix is called a correlation matrix if it is a positive semidefinite matrix with all diagonal entries being 1. The set of correlation matrices is compact and convex. The extreme points of the set are the correlation matrices with rank 1 for \(n=2,3\)36. (For \(n\ge 4\), there are extreme n by n correlation matrices which have rank two). Hence all \(n\times n\) correlation matrices can always be decomposed into the convex combination of rank 1 correlation matrices for \(n=2,3\).

Since the diagonal entries of density matrices of maximally coherent pure states are all equal to \(\frac{1}{n}\), as the convex combination of maximally coherent pure states, the diagonal entries of AMC states are \(\rho _{ii}=\frac{1}{n}\) for all i. Therefore, all n dimensional AMC states are the correlation matrices scaled by a multiplicative factor of \(\frac{1}{n}\). Since the maximally coherent pure states correspond to the rank 1 correlation matrices, and all \(2\times 2\) and \(3\times 3\) correlation matrices can be decomposed into a convex combination of rank 1 correlation matrices, the AMC states correspond exactly to the set of correlation matrices for \(n=2,3\). (For \(n\ge 4\), the AMC states correspond to a proper subset of the \(n\times n\) correlation matrices). This implies that all diagonal entries being equal to \(\frac{1}{n}\) is necessary and sufficient for two and three dimensional AMC states. \(\square\)

Proof of Theorem 3

Note that the pure state decompositions of the Schmidt correlated state \(\rho _{mc}\) are all of the Schmidt form \(|\psi ^\prime \rangle =\sum _i a_i|ii\rangle\)37. Then \(\{ p_k,\ |\psi ^{\prime }_k\rangle \}\) is a pure state decomposition of \(\rho _{mc}\) with \(|\psi _k^\prime \rangle =\sum _i a_i^{(k)}|ii\rangle\) if and only if \(\{ p_k,\ |\psi _k\rangle \}\) is a pure state decomposition of \(\rho\) with \(|\psi _k\rangle =\sum _i a_i^{(k)}|i\rangle\). While \(\sum _i a_i^{(k)}|ii\rangle\) is maximally entangled if and only if \(\sum _i a_i^{(k)}|i\rangle\) is maximally coherent. Therefore, \(\rho _{mc}=\sum _{ij} \rho _{ij}|ii\rangle \langle jj|\) is AME if and only if \(\rho =\sum _{ij} \rho _{ij}|i\rangle \langle j|\) is AMC. \(\square\)

Proof of Theorem 4

For all \(0\le \tau \le \rho _o\), define \({\tilde{F}}(\tau )\) as

$$\begin{aligned} {\tilde{F}}(\tau )=\sum _k q_k F(|\psi _k\rangle ), \end{aligned}$$
(9)

where \(\sum _k q_k |\psi _k\rangle \langle \psi _k|\) is any pure state decomposition of \(\tau\) into a weighted sum of pure states, i.e. \(q_k\ge 0\) for all k and \(\sum _k q_k\le 1\). We claim that \({\tilde{F}}(\tau )\) does not depend on the pure state decomposition at hand. Indeed, let \(\sum _{k^\prime } q_{k^\prime } |\psi ^\prime _{k^\prime }\rangle \langle \psi ^\prime _{k^\prime }|\) be another pure state decomposition of \(\tau\), and \(\sum _h r_h |\phi _h\rangle \langle \phi _h|\) be a fixed pure state decomposition of \(\rho _o-\tau \ge 0\). Then, \(\sum _k q_k |\psi _k\rangle \langle \psi _k|+ \sum _h r_h |\phi _h\rangle \langle \phi _h|\) and \(\sum _{k^\prime } q_{k^\prime } |\psi ^\prime _{k^\prime }\rangle \langle \psi ^\prime _{k^\prime }|+\sum _h r_h |\phi _h\rangle \langle \phi _h|\) are two pure state decompositions of \(\rho _o\), hence the equality \(F_c(\rho _o)=F_a(\rho _o)\) implies \(\sum _k q_k F(|\psi _k\rangle )=\sum _{k^\prime } q_{k^\prime } F(|\psi ^\prime _{k^\prime }\rangle )\) by definition of \(F_a\) and \(F_c\). Clearly, the maps \({\tilde{F}}\) and F coincide on pure states, and moreover \({\tilde{F}}(0)=0\). Further, we claim that

$$\begin{aligned} {\tilde{F}}(t_1\tau _1+t_2\tau _2)=t_1 {\tilde{F}}(\tau _1) + t_2 {\tilde{F}}(\tau _2) \end{aligned}$$
(10)

for all \(0\le \tau _i \le \rho _o\) and \(t_i\ge 0\) with \(t_1+t_2=1\). Indeed, this follows by taking any pure state decompositions of \(\tau _1\) and \(\tau _2\) into weighted sums of pure states and applying (9) to both sides of the equation.

We can now define a functional on the space \(S(\rho _o)\) of all self-adjoint operators with range in \(R(\rho _o)\) as follows: \(p(H)=\inf (k_+{\tilde{F}}(\tau _+)-k_-{\tilde{F}}(\tau _-))\) where the infimum is taken over all nonnegative real numbers \(k_+\) and \(k_-\) and all density matrices \(\tau _+\) and \(\tau _-\) whose range is contained in \(R(\rho _o)\) and for which \(k_+\tau _+-k_-\tau _-=H\). It is easy to verify that \(p(H_1+H_2)\le p(H_1)+p(H_2)\) and \(p(kH)=kp(H)\) for all nonnegative k. Hence p is a sublinear functional on the space of all self-adjoint operators with range in \(R(\rho _o)\). Note also that if \(\rho\) is any density matrix with range in \(R(\rho _o)\), we can see that \(p(\rho )\le {\tilde{F}}(\rho )\) by choosing \(k_+=1\), \(k_-=0\), \(\tau _{+}=\rho\) and \(\tau _{-}\) to be any density matrix. By choosing \(k_+=0\), \(k_-=1\), \(\tau _{+}\) to be any density matrix and \(\tau _{-}=\rho\), we can see that \(p(-\rho )\le -{\tilde{F}}(\rho )\).

By the classical Hahn–Banach theorem, there exists a linear functional L(H) on \(S(\rho _o)\) such that \(L(H)\le p(H)\) for all \(H\in S(\rho _o)\). Now if \(\rho\) is any density matrix with range in \(R(\rho _o)\), we get \(L(\rho )\le p(\rho )\le {\tilde{F}}(\rho )\). We also get \(-L(\rho )=L(-\rho )\le p(-\rho )\le -{\tilde{F}}(\rho )\) which after driving by minus one gives us \(L(\rho )\ge {\tilde{F}}(\rho )\). Combining our inequalities we get \({\tilde{F}}(\rho )=L(\rho )\). Thus, there exists a nonnegative linear operator \(Q:\ R(\rho _o)\rightarrow R(\rho _o)\) such that \({\tilde{F}}(\rho )=tr(Q\rho )\) for all states \(\rho\) with \(R(\rho )\subseteq R(\rho _o)\), which concludes the proof. \(\square\)

Proof of Corollary 2

For full rank quantum states \(\rho\), if \(C_c(\rho )=C_a(\rho )\), then there is a nonnegative linear operator Q such that \(C_f(|\psi \rangle )=\langle \psi |Q| \psi \rangle\) for all pure states in \(R(\rho )={\mathcal {H}}\). Since \(C_f(|i\rangle )=\langle i|Q|i\rangle =0\) for all incoherent pure states \(\{|i\rangle \langle i|\}_{i=0}^{n-1}\) in \({\mathcal {H}}\), Q is a zero operator, which contradicts to \(f \ne 0\). \(\square\)