Relating quantum coherence and correlations with entropy-based measures

Quantum coherence and quantum correlations are important quantum resources for quantum computation and quantum information. In this paper, using entropy-based measures, we investigate the relationships between quantum correlated coherence, which is the coherence between subsystems, and two main kinds of quantum correlations as defined by quantum discord as well as quantum entanglement. In particular, we show that quantum discord and quantum entanglement can be well characterized by quantum correlated coherence. Moreover, we prove that the entanglement measure formulated by quantum correlated coherence is lower and upper bounded by the relative entropy of entanglement and the entanglement of formation, respectively, and equal to the relative entropy of entanglement for all the maximally correlated states.

Remarkably, any state ρ will generate an incoherent state ρ diag by removing all off-diagonal terms from its density matrix in the reference basis under Π. As a quantifier of coherence, we will use the relative entropy of coherence which is defined as C 2 2 is the quantum relative entropy 36 and the minimization is taken over the set of incoherent states  . It has been shown that ⋅  ( ) re satisfies all the conditions mentioned in ref. 4 . Crucially, this quantity can be evaluated exactly: is the von Neumann entropy 36 . In this paper, unless otherwise stated, we will often refer to a bipartite composite quantum system AB, where A and B are local subsystems. For convenience, we say the subsystems A and B are held by Alice and Bob, respectively. For a given state ρ AB in system AB, the local states of Alice and Bob are denoted as ρ A = Tr B (ρ AB ) and ρ B = Tr A (ρ AB ), respectively, which are obtained by performing a partial trace on ρ AB .
Quantum entanglement [20][21][22][23][24] is a popular kind of quantum correlations which can not be prepared by local operations and classical communication (LOCC). Any state prepared by LOCC is separable. As a quantifier of entanglement, we will employ the relative entropy of entanglement 21 . Quantum discord measures the disturbance induced by local measurements to multipartite states [25][26][27][28][29] . Here, we use the discord measure which is based on the entropy theory. Let {|i〉 A } and {|j〉 B } be some orthonormal bases of subsystems A and B, respectively. If Bob performs the von Neumann measurement Π B = {|j〉 B 〈j|} on his subsystem, the post-measurement state of ρ AB is denoted as where I A is the identity operator on subsystem A. Then, the asymmetric quantum discord with respect to Π B can be written in terms of a difference of relative entropies [25][26][27] , In the classical-quantum dichotomy, to remove the measurement-basis dependence, the asymmetric quantum discord is defined as with the minimization over all local von Neumann measurements Π B . If Alice only performs the von Neumann measurement Π A = {|i〉 A 〈i|} on her subsystem, the similar results are available. Besides, if both Alice and Bob perform local von neumannn measurements Π A and Π B on their respective subsystems, the symmetric quantum discord (global quantum discord in bipartite system 27 ) with respect to A B Π ⊗ Π is defined as: Then, the symmetric quantum discord is defined as , with the minimization over all the local von Neumann measurements of Alice and Bob.
Recently, Yadin et al. 37 studied the asymmetric basis-dependent discord D ( ) , which can be seen as the basis-dependent measure of quantumness of correlation, and the properties of under the strictly incoherent operations were investigated. Here, we will connect the basis-dependent discord and quantum correlated coherence.
Quantum correlated coherence and quantum discord. Let {|i〉 A } and {|j〉 B } be the local reference bases of subsystems A and B, respectively, and we usually use their tensor product {|ij〉 AB } as the reference basis of the composite system AB. For a state ρ AB in system AB, its total coherence is ( )  ρ are known as local coherences. Whenever ρ AB is a product state, the sum of local coherences is equal to the total coherence. Generally, the relative entropy of coherence admits the super-additive property 11 : Thus, the definition of quantum correlated coherence with respect to the relative entropy of coherence is given in the following.  Obviously, quantum correlated coherence is the total coherence between subsystems and non-negative. In fact, quantum correlated coherence is a 'correlation function' , which is similar as quantum mutual information 36 . For a state ρ AB , whatever the reference bases of subsystems are, its quantum correlated coherence is always zero, if and only if ρ AB has no correlations i.e., AB A B ρ ρ ρ = ⊗ . The 'only if ' part is directly derived from the Theorem 1 below. In this sense, quantum correlated coherence can be seen as the basis-dependent measure of quantumness of correlation and accounts for quantum correlations, for example, quantum discord.
With respect to the local reference bases of {|i〉 A } and {|j〉 B } of subsystems A and B, respectively, the local von Neumann measurements of Alice and Bob are denoted as Π A = {|i〉 A 〈i|} and Π B = {|j〉 B 〈j|}, respectively. Π A and Π B are also called completely resource (coherence) destroying maps which play a crucial role in the resource theory of coherence 38 . By direct calculation, we get that the consumption of quantum correlated coherence for any state ρ AB under Π B coincides with the asymmetric basis-dependent discord D ( )  37 . Proposition 1 provides a concise relationship between quantum correlated coherence and quantum-classical states 29 for some local reference bases of subsystems A and B.
Using the very similar arguments as D ( ) , we obtain that quantum correlated coherence is corresponding to the symmetric basis-dependent discord . Moreover, we also find the condition for quantum correlated coherence (the symmetric basis-dependent discord) to vanish.
such that p kl are probabilities, and all the states ρ A k and ρ B l are perfectly distinguishable by the local von Neumann measurements with respect to the local reference bases {|i〉 A } and {|j〉 B }, respectively.
Proof. To prove the sufficiency, we will use the following property of von Neumann entropy 36 , is Shannon entropy and all ρ i have support on orthogonal subspaces. Since all A k ρ and B l ρ are perfectly distinguishable by the local von Neumann measurements in the local reference bases {|i〉 A } and {|j〉 B }, respectively, are sets of states with support on orthogonal subspaces. Direct calculation shows that ( ) 0 where Π A and Π B are the local von Neumann measurements in the local reference bases {|i〉 A } and {|j〉 B }, respectively. To prove the necessity, we will use the condition for the quantum relative entropy which is unchanged under a quantum operation 39,40 and the explicit proof is left to the Methods. Theorem 1 has several implications. First, it implies that a state with vanished quantum correlated coherence is a specially classical-classical state 41 but not necessary to be a bipartite incoherent state 8,13 . Particularly, a qubit-qubit state with vanished quantum correlated coherence is a product states or a bipartite incoherent state. More complex cases only emerge in higher dimension. For example, the following qutrit-qutrit state with vanished quantum correlated coherence, has yet local coherences: and the reference basis of each subsystem is the computable basis. Second, Theorem 1 indicates the structure of bipartite states which satisfy the super-additive property of the relative entropy of coherence with equality. This settles an important question left open in previous literature 11,42 that whether the equality of Eq. (3) holds if and only if ρ AB is product or incoherent. Finally, if we choose the local eigenbases of ρ A and ρ B as the reference bases of subsystems A and B, respectively, Proposition 1 and Theorem 1 reduce to the corresponding results in ref. 13 . In this sense, our results somewhat generalize the previous results.
The above results show that quantum correlated coherence and the basis-dependent discord are closely related. Due to the equality Recall that the symmetric quantum discord based on the pseudo distance of relative entropy is equal to the basis-free quantum coherence 12,28 , which is denoted as Both of the minimization are taken over all the local generic bases {|i〉 A } and {|j〉 B } of subsystems A and B, respectively. However, one may also consider defining a discord measure D POVM (·) via general local positive-operator-valued measurements (POVMs) on each subsystem 25,29 instead of the local von Neumann measurements. Comparing these three quantifiers of quantum discord, we easily POVM AB . Whenever these three quantities are zero, the corresponding states are classical-classical states 26,41 . Similarly, the asymmetric quantum discord D A|B (ρ AB ) can also be represented by quantum correlated coherence.
In multipartite systems, the global quantum discord (GQD) 27 can even be represented with quantum correlated coherence. It is worth noting that the reference basis of a multipartite system is the tensor product of the local reference bases of all the subsystems. For a N-partite state ρ , where the minimization is taken over all the generic bases of the multipartite system. With respect to some reference basis of the N-partite system, Using the super-additive property of the relative entropy of coherence given in inequality (3), we are easy to get that the GQD of any N-partite state is non-negative and for a multipartite classical state it is equal to zero. This provides a simple proof of the non-negativity of the GQD in ref. 27 . These results mean that quantum discord in multipartite systems can be better understood with the framework of quantum coherence.
Quantum correlated coherence and quantum entanglement. According to the above discussion, we know that if for arbitrary local reference bases of subsystems A and B, the quantum correlated coherence of ρ AB does not vanish, there must exist quantum correlation (quantum discord) between subsystems A and B. Moreover, it is also possible to characterize entanglement with quantum correlated coherence via state extensions, for example, the entanglement of coherence (EOC) 13 . Then, entanglement can be seen as the irreducible residue of quantum correlated coherence. This highlights the non-locality of quantum entanglement.
In the following, we will discuss some properties of the EOC. For a given state ρ AB in system AB, a bipartite state ρ AA′BB′ is an extension of ρ AB if ρ AA′BB′ satisfies Tr A′B′ (ρ AA′BB′ ) = ρ AB , where subsystems AA′ and BB′ are held by Alice and Bob, respectively 13,43 . The following definition of the EOC establish a connection between entanglement and quantum correlated coherence. Definition 2. (K. C. Tan et al. 13 ) For a given state ρ AB , ρ AA′BB′ is its unitarily symmetric extension and let the local eigenbases of ρ AA′ and ρ BB′ be the local reference bases of subsystems AA′ and BB′, respectively. The entanglement of coherence (EOC) of ρ AB is defined as where the minimization is taken over all possible unitarily symmetric extensions ρ AA′BB′ .
In Definition 2, the extension ρ AA′BB′ is unitarily symmetric if it remains invariant up to local unitary operations on AA′ and BB′ under a system swap between Alice and Bob. It has been shown that the EOC has the properties 13 : non-negative, vanished for separated states, invariant under local unitary operations, non-increasing under LOCC operations, and convex. Furthermore, using entropy-based measures, we give the bounds of the EOC.
Theorem 2. For a given state ρ AB , it holds that If ρ AB is a pure state, these three quantities in inequality (8) are equal.
Proof. Taking some unitarily symmetric extension ρ AA′BB′ of ρ AB , we have where the first inequality is due to that the relative entropy of coherence is no less than the relative entropy of entanglement for a state 12 , and the last inequality is due to that entanglement is un-increased under LOCC operations 21,22 . Then, the inequality (9) where the second equality is due to the property of von Neumann entropy given in Eq. (6). The above equality implies If ρ AB is a pure state, its relative entropy of entanglement is equal to its entanglement of formation, and then equal to its EOC. Hence, the desired results of Theorem 2 are obtained.
□ From Theorem 2, we conclude that the EOC is not strictly less than the relative entropy of entanglement for a bipartite state, since for pure states they are equal. Moreover, for a maximally correlated state 44 , it is of the form: where {|s〉 A } and {|t〉 B } are some orthonormal bases of subsystems A and B respectively and ρ st are the matrix elements. Then, its EOC is also equal to its relative entropy of entanglement. We show this result in the following theorem.  re AB . □ Using the proof of Theorem 3, we confirm that for a maximally correlated state ρ AB , its EOC is even equal to its quantum correlated coherence with respect to the local eigenbases of ρ A and ρ B , respectively. Moreover, with Theorem 3, it is easy to find a state for which the EOC is strictly less than the entanglement of formation, for example, the maximally correlated Bell diagonal state in the two-qubit system 20

Theorem 3. For any maximally correlated state ρ AB as given in
However, we do not know whether the EOC is equal to the relative entropy of entanglement for any mixed state. In addition, for any bipartite state ρ AB and τ CD , the EOC satisfies the following sub-additivity, An alternative measure of entanglement formulated by quantum correlated coherence (quantum discord) is defined as AA BB , where the minimization is taken over all possible unitarily symmetric extensions ρ AA′BB′ of ρ AB , and  , its entanglement of coherence is defined as E ( ) min ( ) , where the minimization is taken over all possible unitarily symmetric extensions ρ ′ ′  , and the local fixed bases are the eigenbases of ρ ′ is unitarily symmetric if it remains invariant up to local uni-www.nature.com/scientificreports/ 6 ScIENTIfIc REPORtS | 7: 12122 | DOI:10.1038/s41598-017-09332-9 tary operations on C C i i ′ and ′ C C j j under a system swap between C C i i ′ and C C j j ′ for any i, j = 1, 2, …, N. Referring to the proofs of EOC as an entanglement measure 13 , we can show that ρ has the properties: non-negative and vanished for separated states, invariant under local unitary operations, non-increasing under LOCC operations, and convex. These results show that the entanglement in multipartite systems can also be characterized by quantum correlated coherence via state extensions.

Discussion
In this paper, using entropy-based measures, we have obtained the concise relationships between quantum coherence and quantum discord as well as quantum entanglement. The results mean that quantum discord and entanglement can be well characterized by quantum correlated coherence. In particular, we gave the condition for quantum correlated coherence (symmetric basis-dependent discord)to vanish, and this condition provides the explicit structure of states which satisfy the super-additive property of the relative entropy of coherence with equality. We further proved the lower and upper bounds of the EOC and showed that the EOC is equal to the relative entropy of entanglement in a large number of scenarios including all pure states and maximally correlated states. For pure states, the LOCC monotonicity (monotonicity on average under LOCC operations 24,45 ) of EOC is easily obtained with Theorem 2. However, we do not know whether the EOC of a general mixed state is LOCC monotone 24,45 , and we leave it open for future research. Finally, we also generalized our results to multipartite settings.
Quite remarkably, one-way basis-dependent quantum deficit in the bibapartite quantum system is equal to the amount of the total coherence lost by the von Neumann measurement with respect to the reference basis of one of the subsystems 11 . These results suggest that the quantum properties of correlations originate from the quantum properties of coherence and quantum correlations can be unified understood within the framework of coherence. We hope that this work is helpful for further understanding quantum correlations and developing quantum technologies.

Methods
Proof of Theorem 1 in the main text. Here, we prove that a state ρ AB with vanished quantum correlated coherence has a decomposition given in Eq. (5) in the main text.
For a given state ρ AB with vanished quantum correlated coherence, its symmetric quantum discord is equal to zero, i.e., D(ρ AB ) = 0. Then, ρ AB is a classical-classical state 40 with the form , AB , where {|ψ α 〉} and {|φ β 〉} are orthonormal bases of subsystems A and B, respectively. From the main text, we see that where the von Neumann measurements Π A and Π B are with respect to the local reference bases {|i〉 A } and {|j〉 B } of subsystems A and B, respectively.
Recall that the quantum relative entropy is unchanged under a quantum operation  , i.e., , if and only if there is a recovery operation  satisfying 39 : Moreover, there is an explicit formula for the recovery operation: X X ( ) ( ( ) ( ) ) Applied to the Eq. (12), the recovery condition says that By letting Eqs (11) and (13) be equal, and pre-and post-multiplying by A where all λ ∑ β αβ and |ψ α 〉 are the eigenvalues and eigenvectors of ρ A , and all λ ∑ α αβ and |φ β 〉 are the same requirements of ρ B . Thus in Eq. (14) we exclude terms on either side which are not in the support of ρ A and ρ B .