Universal complementarity between coherence and intrinsic concurrence for two-qubit states

Entanglement and coherence are two essential quantum resources for quantum information processing. A natural question arises of whether there is a direct link between them. In this work, we propose a definition of intrinsic concurrence for two-qubit states. Although the intrinsic concurrence is not a measure of entanglement, it embodies the concurrence of four pure states which are members of a special pure state ensemble for an arbitrary two-qubit state. And we show that intrinsic concurrence is always complementary to first-order coherence. In fact, this relation is an extension of the complementary relation satisfied by two-qubit pure states. Interestingly, we apply the complementary relation in some composite systems composed by a single-qubit state coupling with four typical noise channels respectively, and discover their mutual transformation relation between concurrence and first-order coherence. This universal complementarity provides reliable theoretical basis for the interconversion of the two important quantum resources.


Introduction
Entanglement and coherence represent two crucial natural properties which are widely applied to quantum information processing and computation [1]. For a physical system, commonly used entanglement measures mainly consider correlations between their subsystems, whereas we usually think the physical system as a whole in the research of coherence omitting its structure [2]. Entanglement as one of earlier resources is a crucial ingredient for various quantum information processing protocols [3], such as remote state preparation [4,5], quantum teleportation [6], super-dense coding [7] and so on. With the development of the entanglement measures, entanglement of formation [8], concurrence [9], relative entropy of entanglement [10] and negativity [11] have been proposed. Although entanglement can be measured by a variety of methods, there exist intrinsic relations between them. For instance, a functional relation between the entanglement of formation and concurrence has been put forward by [9].
On the other hand, coherence is a consequence of the superposition of quantum states, which can be used to characterize the interference capability of interaction fields. But in quantum physics, coherence can be known as the entanglement or correlation when it is further broadened to that between two or more subsystems [3]. Svozilík et al [12] investigated relation between first-order coherence and Clauser-Horne-Shimony-Holt Bell'slike inequality in 2015. They showed that the classical coherence of a given subsystem can be converted to the quantum correlations between subsystems for multipartite quantum systems. Černoch et al experimentally verified the conservation of the maximally accessible first-order coherence while it migrated between classical coherence and quantum correlations [13].
Now that both entanglement and coherence are characterized by the resource theory, the understanding of common evolution of coherence and entanglement will be crucial. In particular, the researches of the intrinsic relations hidden in these resources have been made in recent years [14,15]. Since the chosen types of resources and measure approaches are various, there exist distinct differences for these intrinsic relations among the quantum states. Thus, the main goal of our research is how to obtain a universal intrinsic relation. In this paper, Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
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we accomplish two main tasks: we put forward a definition of intrinsic concurrence for a general two-qubit state; its intrinsic concurrence can be complementary to its coherence.
The remainder of this paper is organized as follows. In section 2, we review the quantification of first-order coherence and concurrence. In section 3, we provide detailed proofs of the existence about a special pure state decomposition and put forward a definition of intrinsic concurrence. In section 4, we give detailed proofs of the complementarity and describe the mutual transformation of coherence and intrinsic concurrence. In section 5, we give out the unified complementary relation of single-qubit state in open systems. In final, we end up our paper with a brief conclusion.

Preliminaries
Concurrence is usually used as a measure for entanglement of two-qubit states [3,16]. For a two-qubit pure state yñ | , its spin-flipped state is defined as , where * y ñ | is the complex conjugate of yñ | and σ 2 is . The concurrence is defined as [9] y yy For a general two-qubit state ρ, its spin-flipped density matrix r can be expressed as The concurrence is defined by the convex-roof [17,18] as follows The minimization is taken over all possible decompositions ρ into pure states. An analytic solution of concurrence can be calculated [9] r l l where l n (n ä {1, 2, 3, 4}) are the eigenvalues, in decreasing order, of the non-Hermitian matrix rr . The definition of concurrence is based on the convex-roof construction, and it is suitable for use in both pure states and mixed states [19][20][21].
A widely used measure of coherence in optical systems is the first-order coherence [22], which is similar with the degree of polarization coherence [23]. We consider a two-qubit state r y y . This quantum state ρ can be obtained by applying a unitary operation V to the non-entangled state ρ Λ . Here, the unitary operation V contains the corresponding eigenvectors y ñ | n and the state ρ Λ is a diagonal matrix with the eigenvalues p n . In other words, the state ρ can be generally written (spectral decomposition) as [1] Each subsystem of the state ρ is characterized by the reduced density matrix r r = ( ) Tr . The degree of first-order coherence of each subsystem can be given by [22] Therefore, a measure of coherence for both subsystems, when they are considered independently, has the following form When both subsystems are coherent, that is, 0<D1, while only if both subsystems show no coherence, D=0.

Intrinsic concurrence
For a general two-qubit pure state yñ | , its concurrence is defined as [24] y r where r A and ρ B are the reduced density matrix of the pure state yñ | . Combining the definition of the first-order coherence with equation (8), it is obvious that the complementary relation of the pure state yñ | can be written as However, for a general two-qubit mixed state, the square sum of these two quantities is not any longer a conserved quantity. It is well-known that a general two-qubit state ρ has many pure state ensembles and the minimization of equation (3) is taken over all possible decompositions ρ into pure states. In order to generalize the complementary relation from the pure state to the mixed state, we try to look for a pure state ensemble j ñ { | } q , n n of the state ρ, i.e. r j j = å ñá | | q n n n n , where the concurrence j ñ (| ) C n of pure states j ñ | n and corresponding probabilities q n satisfy the relation that is a conserved quantity. Interestingly, we find that when the pure states j ñ | n are the eigenvectors of the non-Hermitian matrix rr , the quanlity is indeed a conserved quantity. We will give a detailed proof of this conservation relation in section 4.
Similar to equation (2), we consider the spin-flipped operatorF corresponding to a Hermitian operator F, whose order is 2n, defined as where * F is the complex conjugate of F. First of all, we introduce some peculiarities about spin-flipped operators. These properties provide a solid basis for the proof of two theories that we are going to talk about. Obviously, the spin-flipped operatorF satisfies the Hermitian property. If the Hermitian operator F can be written as the form F=F 1 F 2 , then the corresponding spin-flipped operatorF has a similar form For the Pauli operators σ i , one obtains some special properties * s s s s This indicates that the spin-flipped state r A , related to the single qubit state ρ A , is reversed with the state ρ A in the Bloch sphere space.
And then, in order to prove the fact that there exists the pure state ensemble j ñ { | } q , n n for the state ρ, whose pure states j ñ | n are the eigenvectors of the non-Hermitian matrix rr , we put forward two theorems.
Theorem 1. If a two-qubit state r has a pure state decomposition r j j = å ñá = | | q n n n n 1 4 , and these pure states j ñ | n satisfy the tilde orthogonal relation j j d j j á ñ = á ñ |˜|m n mn n n , then the eigenvectors of the non-Hermitian matrix rr will be j ñ | n and the corresponding eigenvalues will be expressed as l j = ñ (| ) q C n n n 2 2 .
Proof of theorem 1. Assuming that the two-qubit state ρ has a pure state decomposition Then, combining equation (2), we obtain that the non-Hermitian matrix rr can be given by Hence, one obtain that the eigenvalue-eigenvector equation of the non-Hermitian matrix rr has the form Therefore, we obtain that the eigenvalue spectral decomposition of the non-Hermitian matrix rr can be expressed as [25]  where q n satisfy the relation The theorems 1 and 2 reveal the fact that there exists the pure state ensemble j ñ { | } q , n n for the state ρ, whose pure states j ñ | n are the eigenvectors of the non-Hermitian matrix rr . And this two theorems provide a special pure state decomposition r j j = å ñá = | | q n n n n 1 4 method for a two-qubit state ρ, which provides a convenience for us to solve the pure state decomposition.
And next, we give the definition of the intrinsic concurrence. For a general two-qubit state ρ, its intrinsic concurrence is defined as where these pure states j ñ | n are the eigenvectors of the non-Hermitian matrix rr and q n are the corresponding probabilities.
Finally, we obtain three properties about the intrinsic concurrence, which indicate the rough relation between the concurrence and the intrinsic concurrence.

Property 1.
For an arbitrary two-qubit pure state yñ | , there is an equivalence relation between its concurrence and intrinsic concurrence. The relation can be expressed as For an arbitrary two-qubit pure state yñ | , its pure state decomposition is just itself. Therefore For an arbitrary two-qubit state r, there is a relation between its intrinsic concurrence and the quantity rr (˜) Tr . The formula can be expressed as Proof. For an arbitrary two-qubit state ρ, its quantity rr l = å = (˜) Tr  Property 3. For a general two-qubit state ρ, there is a lower bound of its intrinsic concurrence. And the lower bound is its concurrence. The inequality about its concurrence and intrinsic concurrence can be written as  r r ( ) ( ) ( ) C C . According to the proof of the property 3, one obtain an inference that the necessary and sufficient condition of is the rank of the non-Hermitian matrix rr .

Complementary relation between intrinsic concurrence and first-order coherence
In this section, we introduce the complementary relation between the intrinsic concurrence and the first-order coherence.
Theorem 3. For a general two-qubit state r, there is a complementary relation Proof of theorem 3. In general, a two-qubit state r is denoted as m n mn m n , 1 3 where  stands for identity operator of single qubit, σ n stand for three Pauli operators, 3 are vectors in R 3 and s s s s =  ( ) , , 1 2 3 . Just to make it easy to calculate, we rewrite the state ρ as . According to equation (6), we obtain that the coherence of each subsystem has the following form , . Therefore, combining equations (7) and (31), one obtains that its first-order coherence can be given by the following formula It is obvious that the spin-flipped operator r corresponding to the state ρ can be expressed as . Therefore, for each subsystem, the definition of the first-order coherence can be rewritten as Similarly, for the whole system, the definition of the first-order coherence can be rewritten as Tr  Tr  2 Tr . 35 Therefore, the square sum of its first-order coherence and intrinsic concurrence is a conserved quantity. The derivative process can be described as follows r r r r r r r r r r r r r r r r Tr  Tr  Tr  . 36 Note that, the theorem 3 reveals that for a general two-qubit state ρ, its first-order coherence r ( ) D and intrinsic concurrence r ( ) C I are a pair of complementary quantities. And it shows that if the evolution of the whole system is unitary, there is a mutual transformation relationship between its first-order coherence and intrinsic concurrence. Combining with equation (23), we can rewrite the complementarity as According to the property 3, one obtain an inference: if the state ρ satisfies condition rr Î (˜) { } R 0, 1 , then this complementarity can be rewritten as For a general two-qubit pure state , its coherence and concurrence satisfy the complementarity (q 1 =1, q 2 =q 3 =q 4 =0) for equation (37) , i.e. that of equation (9). If the mixed state ρ satisfies condition rr = (˜) R 1, then its coherence and concurrence satisfy the complementarity that of equation (38). For a general two-qubit mixed state ρ, the coherence of the state ρ and the concurrence of the four pure states j ñ | n satisfy the complementarity.

The complementary relation of quantum state in open systems
In fact, a quantum system is inevitably coupled with the surrounding environment, and quantum resources are constantly exchanged between the quantum system and the environment. In this section, we consider only the interactions between a single qubit state  r s = +   ( · ) A A 1 2 and the environment, and the dimension of the environment is the same as the dimension of the particle A. Here, =  ( ) A a a a , , 1 2 3 and s s s s =  ( ) , , 1 2 3 . In order to obtain the complementary relation between coherence and concurrence, we assume that the initial state of the environment is ñá | | e e 0 0 . Then the state of the whole quantum system at the initial time (IT) can be described as are the Kraus operators of the environment. We will discuss only a few common channels, namely amplitude damped (AD) channel, bit flip (BF) channel, bit-phase flip (BPF) channel, and phase flip (PF) channel. The Kraus operators E k and unitary evolution operator U corresponding to these channels are given in the table 1, respectively. Here, the parameter = -g p 1 e t is the strength of the noise, where γ is the decay factor and t represents the time the particle A is in the environment. Therefore one can obtain the dynamics of coherence, concurrence and intrinsic concurrence for the evolution states ρ of these quantum systems (see the figure 1). From these figures, we find that the concurrence r ( ) C of the composite system ρ will be increased from the decrease of its coherence r ( ) D . For the state ρ , we obtain that the ratio r = The above equation (40) expresses that when the state ρ A is coupled with these channels (AD, BF, BPF and PF) respectively, the decrease of the first-order coherence function r ( ) D 2 can be converted to the concurrence function r ( ) C 2 with a conversion efficiency r ( ) S 1 2 .