⇦Chinese Physics Letters, 2020, Vol. 37, No. 11, Article code 110302 Mutual Restriction between Concurrence and Intrinsic Concurrence for Arbitrary Two-Qubit States A-Long Zhou (周阿龙), Dong Wang (王栋)*, Xiao-Gang Fan (范小刚), Fei Ming (明飞), and Liu Ye (叶柳)* Affiliations School of Physics & Material Science, Anhui University, Hefei 230601, China Received 8 July 2020; accepted 30 September 2020; published online 8 November 2020 Supported by the National Science Foundation of China (Grant Nos. 12075001, 61601002 and 11575001), the Anhui Provincial Natural Science Foundation (Grant No. 1508085QF139), and the Fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701).
*Corresponding authors. Email: dwang@ahu.edu.cn; yeliu@ahu.edu.cn Citation Text: Zhou A L, Wang D, Fan X G, Ming F and Ye L et al. 2020 Chin. Phys. Lett. 37 110302    Abstract Concurrence is viewed as the most commonly approach for quantifying entanglement of two-qubit states, while intrinsic concurrence contains concurrence of four pure states consisting of a special pure state ensemble concerning an arbitrary two-qubit state. Thus, a natural question arises: Whether there is a specified relation between them. We firstly examine the relation between concurrence and intrinsic concurrence for the maximally nonlocal mixed state under a special unitary operation, which is not yet rigorously proved. In order to obtain a general result, we investigate the relation between concurrence and intrinsic concurrence using randomly generated two-qubit states, and derive an inequality relation between them. Finally, we take into account the relation between concurrence and intrinsic concurrence in open systems, and reveal the ratio of the two quantum resources, which is only correlated with the experiencing channels. DOI:10.1088/0256-307X/37/11/110302 PACS:03.67.-a, 03.67.Hk, 03.65.Ta © 2020 Chinese Physics Society Article Text With the development of quantum physics, the combination of quantum mechanics and classical information theory has produced a new discipline, which is called the quantum information science. Over the past few decades, quantum information has developed rapidly. Many practical applications are focusing on two vital quantum resources,[1] i.e., coherence and non-classical correlation. Typically, one of non-classical correlations can be deemed as quantum entanglement. As we know, coherence is the result of quantum states' superposition, which can be used to characterize the interference of the interaction field. For a many-body system, quantum entanglement is popularly utilized to quantify the non-classical properties.[2,3] In 1935, the concept of quantum entanglement was introduced by Einstein, Podolsky and Rosen (EPR)[4] and Schrödinger,[5] respectively. With the advent of quantum entanglement, Werner put forward a more reasonable definition of quantum entanglement in 1989.[6] Technically, the quantum states can be classified into entangled states and separable states. Suppose that the two-qubit system states $\rho _{AB}$ can be decomposed into separate direct product forms of each subsystem as $$\begin{align} \rho _{AB} = \sum\limits_i {{p_i}} \rho _i^A \otimes \rho _i^B,~~ \tag {1} \end{align} $$ where $p$ is the probability distribution and $\sum_i {{p_i}} = 1$, ${p_i} \ge 0$. Then we call the states $\rho _{AB}$ as separable states. If not, the quantum states $\rho _{AB}$ are deemed as entangled states. Essentially, entanglement plays an important role in the field of quantum information science, and is an irreplaceable part of quantum information processing protocols.[1–3,7] Generally, quantum entanglement can be applied to quantum information processing including quantum key distribution,[8] quantum dense coding,[9] quantum teleportation,[10,11] remote state preparation,[12,13] quantum computation,[14,15] and so forth. For two-qubit states, degree of entanglement of quantum states can be measured in the following ways: partial entropy of entanglement,[16] entanglement of formation,[17] concurrence,[18,19] relative entropy of entanglement,[20] and negativity.[21] For a multipartite system, there exists many additional condition restrictions, and it is difficult to find a universal method to quantify its entanglement. Therefore, in practice, we must consider the explicit situation and choose an appropriate method to measure the entanglement. Herein, we consider concurrence as a measurement of entanglement for two-qubit states. In 1996, to facilitate calculation about the entanglement of formation under Bell diagonal states, Bennett et al. introduced a famous method for quantum entanglement measurement, i.e., concurrence.[19] Subsequently, Wootters generalized the definition of concurrence into mixed states, and derived the analytic expression of concurrence in 1998.[18] The analytic result in Ref. [18] is suitable for estimating the entanglement degree with respect to both pure and mixed states.[22,23] Recently, Fan et al. have investigated the relationship between quantum entanglement and classical coherence, and put forward a new concept, i.e., intrinsic concurrence.[24] According to the result, it has been revealed that the concurrence is the maximum lower bound of intrinsic concurrence as to arbitrary two-qubit states. In other words, the range of intrinsic concurrence limits the result of concurrence. Then, we have to ask whether there exists a minimum upper bound of intrinsic concurrence. Herein, we focus on addressing this issue to further uncover the properties of intrinsic concurrence. In this Letter, we find out the inherent relation between concurrence and intrinsic concurrence with respect to arbitrary two-qubit states. Thus, we believe that the obtained result not only offers the inherent connection between concurrence and intrinsic concurrence, but also helps us better understand entanglement and even coherence concepts. In the following, first we briefly introduce the concurrence and intrinsic concurrence, and the inequality $C(\rho) \leq {C_{\rm I}}(\rho)$. Then, by the maximally nonlocal mixed state, we derive the upper bound of the inequality. We discuss the examples with a Bell-class state under the five typical channels, and verify our findings. Finally, we give a brief conclusion. Preliminaries. For a general two-qubit pure state $|\psi\rangle$, concurrence is defined as $$\begin{align} C({| \psi \rangle}) = {| {\langle {\psi | \tilde{\psi} \rangle } } }| = | {\langle \psi |{\sigma _2} \otimes {\sigma _2}| {{\psi ^ * }} \rangle } |,~~ \tag {2} \end{align} $$ where $|{\psi ^ * }\rangle$ is the complex conjugate of $|\psi\rangle$ under the standard basis, and Pauli matrix ${\sigma _2} = i({| 0 \rangle \langle 1 | - | 1 \rangle \langle 0 |})$.[3,25] In fact, Eq. (2) can be rewritten as $$\begin{align} C(|\psi\rangle) = \sqrt {2[{1-{\rm Tr}({\rho _A^2})}]},~~ \tag {3} \end{align} $$ where $\rho_A$ is the reduced density operator of the pure state $|\psi\rangle$. For a general two-qubit mixed state $\rho$, its spin-flipped density matrix $\tilde{\rho}$ is given by $$\begin{align} \tilde{\rho}=({{\sigma_2}\otimes {\sigma_2}}){\rho^*}({{\sigma _2}\otimes{\sigma _2}}).~~ \tag {4} \end{align} $$ Concurrence of the mixed state is defined by the convex-roof as follows:[26] $$ C(\rho) = \mathop {\min }\limits_{\{ {y_n},| {{\psi _n}} \rangle \} } \sum\limits_n {{y_n}} C(| {{\psi _n}} \rangle).~~ \tag {5} $$ The minimization is taken over all possible decompositions $\varrho$ into pure states, and $\rho = \sum\nolimits_n {{y_n}} | {{\psi _n}} \rangle \langle {{\psi _n}} |$, ${y_n} \ge 0$, $\sum\nolimits_n {{y_n}} = 1$. With regard to Eq. (5), it can be expressed as $$ C(\rho) = \max \big\{ 0,\sqrt {{\lambda _1}} - \sqrt {{\lambda _2}} - \sqrt {{\lambda _3}} - \sqrt {{\lambda _4}} \big\},~~ \tag {6} $$ where ${\lambda_n}~({n \in \{ 1,2,3,4\}})$ are the eigenvalues of the non-Hermitian matrix $\rho\tilde{\rho}$, and the relation ${\lambda _1} \ge {\lambda _2} \ge {\lambda _3} \ge {\lambda _4}$. Intrinsic concurrence contains concurrence of four pure states, which are members of a special pure state ensemble for an arbitrary two-qubit state. For a general two-qubit state $\rho$, intrinsic concurrence is defined as $$ {C_{\rm I}}(\rho)=\sqrt {\sum\limits_{n=1}^4 {{x_n^2}{C^2}({|{{\phi_n}}\rangle})}},~~ \tag {7} $$ where $\{{{x_n},|{{\phi_n}}\rangle}\}$ represents a special pure state ensemble of $\rho$ and these pure states $| {{\phi _n}}\rangle$ satisfy the tilde orthogonal relation $\langle {{\phi _n}} | {{\tilde{\phi} _m}} \rangle = {\delta _{nm}}\langle {{\phi _n}} | {{\tilde{\phi} _n}} \rangle $. For intrinsic concurrence, we can interpret it as the inner product of the quantum state $\rho$ and its spin flipped state $\tilde{\rho}$, which reads $$\begin{align} {C_{\rm I}}(\rho)= \sqrt {{\rm Tr}({\rho \tilde{\rho}})}.~~ \tag {8} \end{align} $$ According to Eqs. (6) and (8), the relation between concurrence and intrinsic concurrence can be derived as $$\begin{align} C(\rho) \leq {C_{\rm I}}(\rho).~~ \tag {9} \end{align} $$ Note that the relation will become $C({| \phi \rangle }) = {C_{\rm I}}({| \phi \rangle}$) in the case of an arbitrary two-qubit pure state $| \phi\rangle$. As a matter of fact, intrinsic concurrence is not used to measure entanglement of two particles, and it can be viewed as a statistical interpretation of a special pure state ensemble for an arbitrary two-qubit state. Herein, we discuss the relation between concurrence and intrinsic concurrence, and the results would benefit to further understand and clarify the meaning of concurrence and intrinsic concurrence. Improving Relation Between Concurrence and Intrinsic Concurrence. Here, we focus on examining an explicit and inherent relation between concurrence and intrinsic concurrence, by means of considering a situation with a quantum system which is with a maximally nonlocal mixed state (MNMS). In a nonlinear process designed to generate entanglement with polarization, the MNMS generated from nonlinear crystal can be usually expressed in the computational basis $\{|{00}\rangle,|{01} \rangle,|{10}\rangle,|{11}\rangle \}$ as[27,28] $$\begin{align} {\rho _M} = \frac{1}{2}\begin{pmatrix} 1&0&0&\varepsilon \\ 0&0&0&0\\ 0&0&0&0\\ \varepsilon &0&0&1 \end{pmatrix},~~ \tag {10} \end{align} $$ where $\varepsilon \in [0,1]$. Obviously, the state ${\rho_M}$ belongs to Bell diagonal states, and its decomposition can be written as $$ {\rho _M} = \frac{{1 + \varepsilon }}{2}|{{\beta_1}}\rangle \langle {{\beta_1}}|+\frac{{1-\varepsilon}}{2}|{{\beta _2}}\rangle \langle{{\beta_2}}|,~~ \tag {11} $$ where $|{{\beta_1}}\rangle = \frac{1}{{\sqrt 2}}({|{00}\rangle+|{11}\rangle})$ and $|{{\beta_2}}\rangle = \frac{1}{{\sqrt 2 }}({| {00}\rangle-|{11}\rangle})$ are Bell states. Moreover, the concurrence and intrinsic concurrence of the state ${\rho_M}$ can be calculated as $$\begin{align} &C({{\rho _M}}) = \varepsilon, \\ &{C_{\rm I}}({{\rho_M}})=\sqrt{({1+{\varepsilon^2}})/2}~.~~ \tag {12} \end{align} $$ Hence, one obtains easily $$ C({{\rho _M}}) \leq {C_{\rm I}}({\rho _M}) = \sqrt {[ {1 + {C^2}({\rho _M})}]/2}~.~~ \tag {13} $$ Next, we transform the state ${\rho_M}$ to a new state ${\overline{\rho} _M}$ using an unitary operation ${U_0}$, and the state ${\overline{\rho} _M } = {U_0}{\rho _M}U_0^†$.[28,29] The unitary operation can be expressed as ${U_0}= {M_0} V^†$, where $$\begin{align} {M_0} = \frac{{1}}{2}\begin{pmatrix} {\sqrt 2 }&1&0&{ - 1}\\ 0&1&{\sqrt 2 }&1\\ 0&{ - 1}&{\sqrt 2 }&{ - 1}\\ {\sqrt 2 }&{ - 1}&0&1 \end{pmatrix},~~ \tag {14} \end{align} $$ and $V$ consists of the corresponding eigenvectors of the state $\rho_M$. The state $\overline{\rho} _M$ can be written as $$ \overline{\rho} _M = \frac{1}{8}\begin{pmatrix} {3 + \varepsilon }&{1 - \varepsilon }&{\varepsilon - 1}&{1 + 3\varepsilon }\\ {1 - \varepsilon }&{1 - \varepsilon }&{\varepsilon - 1}&{\varepsilon - 1}\\ {\varepsilon - 1}&{\varepsilon - 1}&{1 - \varepsilon }&{1 - \varepsilon }\\ {1 + 3\varepsilon }&{\varepsilon - 1}&{1 - \varepsilon }&{3 + \varepsilon } \end{pmatrix}.~~ \tag {15} $$ In addition, the state $\overline{\rho} _M$ can be decomposed into $$ \overline{\rho} _M = \frac{{1 + \varepsilon }}{2}|{{\beta _1}}\rangle\langle{{\beta_1}}|+\frac{{1-\varepsilon}}{2}|{{\beta _3}} \rangle\langle {{\beta_3}}|,~~ \tag {16} $$ where $|{{\beta_3}}\rangle=\frac{1}{2}({|{00}\rangle+|{01}\rangle-|{10}\rangle-|{11}\rangle})$ is an unentangled state. As a result, the concurrence and intrinsic concurrence of the state $\overline{\rho} _M$ can be given by $$ C({\overline{\rho}_M})={C_{\rm I}}({\overline{\rho}_M})=\frac{1}{2}({1+\varepsilon}).~~ \tag {17} $$ Hence, we can obtain the relation $C({\overline{\rho}_M})={C_{\rm I}}({\overline{\rho} _M}) < \sqrt {\frac{1}{2}[ {1 + {C^2}({\overline{\rho} _M})}]}$. From the above analysis, we can attain the inequality between concurrence and intrinsic concurrence as $$ C(\rho) \leq {C_{\rm I}}(\rho),~~ \tag {18} $$ what's more, we have realized that the concurrence and intrinsic concurrence satisfy the following relation $$ {C_{\rm I}}({\rho}) \leq \sqrt {\frac{1}{2}[ {1 + {C^2}({\rho})}]}.~~ \tag {19} $$ Meanwhile, we also find that for the case of quantum state $R({\rho }) = 2$, concurrence and intrinsic concurrence have both $C({\rho}) \leq {C_{\rm I}}({\rho})$ and ${C_{\rm I}}({\rho }) \leq \sqrt {\frac{1}{2}[ {1 + {C^2}({\rho })}]}$ cases. Thus, one can say that when the rank of the system state is $R({\rho }) = 2$, the relation between concurrence and intrinsic concurrence can fully reflect the maximum lower bound and the minimum upper bound of intrinsic concurrence. For the purpose of verifying our obtained result, we choose $1.5 \times {10^5}$ randomly generated two-qubit states. The concurrence and intrinsic concurrence of random states are plotted in Fig. 1. It is clearly shown that the relation between concurrence and intrinsic concurrence is held as $$ C(\rho) \leq {C_{\rm I}}(\rho) \le \sqrt {\frac{{1+{C^2}(\rho)}}{2}}.~~ \tag {20} $$ This inequality is universal with respect to arbitrary two-qubit states. From Fig. 1, we have the following conclusion: If concurrence $C(\rho) = 0$, then we can obtain that the maximum of intrinsic concurrence is $\frac{1}{{\sqrt 2 }}$ by Eq. (20). As we know, when the entanglement of system is zero, the entanglement of four pure states decomposed by the system state may not be zero.
cpl-37-11-110302-fig1.png
Fig. 1. The intrinsic concurrence ${C_{\rm I}} (\rho)$ versus the concurrence $C(\rho)$, for $1.5 \times {10^5}$ randomly generated two-qubit states [there are $5 \times {10^4}$ generated states of rank $R(\rho) = 2, 3, 4,$ respectively]. The red lines represent the minimum upper bound $\sqrt {\frac{1}{2}[ {1 + {C^2}(\rho)}]}$ and the maximum lower bound $C(\rho)$ of ${C_{\rm I}}(\rho)$, respectively.
Examples—Quantum States with Rank-2. In order to better reflect the performance of the relation between concurrence and intrinsic concurrence, we can resort to the Bell-class state of rank $R(\rho) = 1$ as the initial state, which interacts with different channels to evolve as different quantum states with different ranks, and discuss the relation between the concurrence and intrinsic concurrence. A Bell-class state can be written as $$ {\rho_{AB}}=|\psi\rangle\langle\psi|,~~ \tag {21} $$ where ${|\psi\rangle=\sin \theta |{11}\rangle+\cos \theta|{00}\rangle}$. According to Eqs. (6) and (8), we have $$ C({\rho _{AB}}) = {C_{\rm I}}({\rho _{AB}}) = |\sin({2\theta})|.~~ \tag {22} $$ It is well known that the rank of Bell-class states is $R(\rho _{AB}) = 1$, and we can obtain $C(\rho_{AB}) = {C_{\rm I}}(\rho _{AB})$. Next, let us discuss the relation between the concurrence and intrinsic concurrence for the quantum states with the different rank $R$. As we all know, any quantum system in practice inevitably interacts with its surrounding environment, which causes the phenomenon of decoherence of the quantum system. This is due to the fact that quantum entanglement of the quantum system will decrease exponentially or even completely disappear, which is called the entanglement sudden death (ESD).[30–32] At the same time, quantum decoherence effects have largely prevented applications of quantum entanglement in quantum communication. Therefore, it is necessary to explore the nature between concurrence and intrinsic concurrence in the phase damped (PD) channel, amplitude damped (AD) channel, bit flip (BF) channel, bit-phase flip (BPF) channel and phase flip (PF) channel, respectively.[33–35] Herein, we consider Bell-class states as the initial state, separately interacting with five different channels, i.e., PD, AD, BF, BPF and PF channels, respectively. As an illustration, we observe the PD channel to illustrate the relation of interest. For the process of a bipartite quantum system passing through the channels, we can describe it as the game between Alice and Bob.[36–40] Suppose that a bipartite quantum system is composed of particle $A$ and particle $B$, Alice has $A$ and Bob owns $B$. Then Alice let $A$ go through the channel and $B$ do not perform any operation. After that, we can compare the corresponding concurrence and intrinsic concurrence under such a scenario with the system consisting of $A$ and $B$. The Kraus operators $E_i$ of PD channel can be given by $$\begin{align} &{E_1} =\begin{pmatrix} 1&0\\ 0&{\sqrt {1 -p} } \end{pmatrix}, \\ &{E_2} = \begin{pmatrix} 0&0\\ 0&{\sqrt p } \end{pmatrix},~~ \tag {23} \end{align} $$ where $0 \le p \le 1$ is expressed as the probability of quantum system decay. The evolved state of a composite system composed of two qubits can independently be expressed as $$ {\rho}=\sum\limits_{i= 1}^2 {({{E_i} \otimes {𝟙_B}})}{\rho _{AB}}{({{E_i} \otimes {𝟙_B}})^† }.~~ \tag {24} $$ According to Eqs. (4) and (24), one can obtain the non-Hermitian matrix $$ \rho \tilde{\rho} =\begin{pmatrix} {(2 - p){{\cos }^2}\theta {{\sin }^2}\theta }&0&0&{2\sqrt {1 - p} \cos \theta {{\sin }^3}\theta }\\ 0&0&0&0\\ 0&0&0&0\\ {2\sqrt {1 - p}{{\cos }^3}\theta \sin \theta }&0&0&{(2 - p){{\cos }^2}\theta {{\sin }^2}\theta } \end{pmatrix}.\\~~ \tag {25} $$ Combining Eqs. (6) and (8), we can obtain concurrence $C(\rho)$ and intrinsic concurrence ${C_{\rm I}}(\rho)$, respectively, which can be expressed as $$\begin{align} &C(\rho) = |\sin(2\theta)|\sqrt {1 - p}, \\ &{C_{\rm I}}(\rho) = |\sin (2\theta)|\sqrt {\frac{{2 - p}}{2}}.~~ \tag {26} \end{align} $$ Then the ratio ${\cal K}$ about concurrence $C(\rho)$ and intrinsic concurrence ${C_{\rm I}}(\rho)$ can be given by $$\begin{align} {\cal K} &= \frac{ {C(\rho)}} {{C_{\rm I}}(\rho)} =\sqrt {\frac{{2 - 2p}}{{2 - p}}},~~ \tag {27} \end{align} $$ where $0 \leq p \leq 1$. Therefore, we have the relation $$\begin{align} C(\rho) \leq {C_{\rm I}}(\rho).~~ \tag {28} \end{align} $$ Similarly, we can obtain the difference $$\begin{align} {\cal Q} &= {C_{\rm I}}^2(\rho)- \frac{{1 + {C^2}(\rho)}}{2} \\ &=\frac{{{{\sin }^2}(2\theta) - 1}}{2},~~ \tag {29} \end{align} $$ where $0 \le {\sin ^2}(2\theta) \le 1$. Obviously, we can summarize the relation as $$\begin{align} {C_{\rm I}}(\rho)\leq \sqrt{ \frac{{1 + {C^2}(\rho)}}{2}}.~~ \tag {30} \end{align} $$
cpl-37-11-110302-fig2.png
Fig. 2. For Bell-class states to interact with PD channel: (a) the red lines represent the ratio ${\cal K }=\sqrt {\frac{{2 - 2p}}{{2 - p}}} $ between concurrence and intrinsic concurrence, (b) the blue lines represent the difference ${\cal Q }=\frac{1}{2}[{{{\sin }^2}(2\theta) - 1}]$ about concurrence and intrinsic concurrence.
Herein, it is worth noting that when the state of the system is the maximally entanglement states with $\theta = \frac {{\pi}}{4}$, and ${C_{\rm I}}(\rho)=\sqrt {\frac{{1 + {C^2}(\rho)}}{2}}$. In Fig. 2, we show the range of ratio ${\cal K}$ and difference ${\cal Q}$. Clearly, one can conclude the concurrence and intrinsic concurrence meets $$\begin{align} C(\rho) \leq {C_{\rm I}}(\rho) \le \sqrt {\frac{{1 + {C^2}(\rho)}}{2}}~.~~ \tag {31} \end{align} $$ As shown in Table 1, we give the corresponding details of the Kraus operators $E_i$, the rank $R(\rho)$ of new quantum state, concurrence $C(\rho)$, intrinsic concurrence ${C_{\rm I}}(\rho)$, the ratio ${\cal K} = \frac{ {C(\rho)}} {{C_{\rm I}}(\rho)}$ and the difference ${\cal Q} = {C_{\rm I}}^2(\rho)- \frac{{1 + {C^2}(\rho)}}{2}$, respectively. When the Bell-class state interacts with the AD channel, we can find that the rank of the new state will change to $R(\rho) = 2$. For the case of the rank $R(\rho) = 2$ of new quantum state, we can conclude that concurrence and intrinsic concurrence are equal to each other. When the Bell-class state interacts with the BF channel, BPF channel and PF channel respectively, we obtain that concurrence $C(\rho)$, intrinsic concurrence ${C_{\rm I}}(\rho)$, the ratio ${\cal K}$ and the difference ${\cal Q}$ are the same. We explain this phenomenon to the Kraus operators of these three channels constructed by the Pauli operators, thus their results are consistent. For the definition of intrinsic concurrence, we interpret it to the weight of the four pure states' concurrence in a special pure state ensemble $\{{{x_n},|{{\phi _n}}\rangle}\}$. Then the ratio ${\cal K} = \frac{ {C(\rho)}} {{C_{\rm I}}(\rho)} \leq 1$ can be regarded such that the intrinsic concurrence contains concurrence. In other words, the ratio is the weight of system states' concurrence in intrinsic concurrence. Therefore, the relation between concurrence and intrinsic concurrence depends on the ratio ${\cal K} = \frac{ {C(\rho)}} {{C_{\rm I}}(\rho)}$, and the ratio ${\cal K}$ is only related to the probability of quantum system decay $p$. Simultaneously, in order to verify the relation between ${C_{\rm I}}(\rho)$ and $\sqrt {\frac{{1 + {C^2}(\rho)}}{2}}$, we calculate the value range of difference ${\cal Q}$.
Table 1. The Kraus operators ${E_i}$, rank $R(\rho)$, concurrence $C(\rho)$, intrinsic concurrence ${C_{\rm I}}(\rho)$, ratio $\cal K$ and difference $\cal Q$ corresponding to the AD channel, BF channel, BPF channel and PF channel, respectively.
Channels AD channel BF channel BPF channel PF channel
The Kraus
operators		 $\begin{array}{l} {E_1} = \begin{pmatrix} 1&0\\ 0&{\sqrt {1 - p} } \end{pmatrix}\end{array}$ $\begin{array}{l} {E_1} = \begin{pmatrix} {\sqrt {1 - p} }&0\\ 0&{\sqrt {1 - p} } \end{pmatrix}\end{array}$ $\begin{array}{l} {E_1} =\begin{pmatrix} {\sqrt {1 - p} }&0\\ 0&{\sqrt {1 - p} } \end{pmatrix}\end{array}$ $\begin{array}{l} {E_1} = \begin{pmatrix} {\sqrt {1 - p} }&0\\ 0&{\sqrt {1 - p} } \end{pmatrix}\end{array}$
$\begin{array}{l}{E_2} = \begin{pmatrix} 0&0\\ {\sqrt p }&0 \end{pmatrix} \end{array}$ $\begin{array}{l} {E_2} = \begin{pmatrix} 0&{\sqrt p }\\ {\sqrt p }&0 \end{pmatrix} \end{array}$ $\begin{array}{l} {E_2} = \begin{pmatrix} 0&{i\sqrt p }\\ { - i\sqrt p }&0 \end{pmatrix} \end{array}$ $\begin{array}{l} {E_2} = \begin{pmatrix}{\sqrt p }&0\\ 0&{ - \sqrt p } \end{pmatrix} \end{array}$
$R(\rho)$ 2 2 2 2
$C(\rho)$ $\sqrt{1 - p}|\sin ({2\theta})|$ $|({1 - 2p})\sin ({2\theta})| $ $|({1 - 2p})\sin ({2\theta})| $ $|({1 - 2p})\sin ({2\theta})| $
${C_{\rm I}}(\rho)$ $\sqrt{1 - p}|\sin({2\theta})|$ $\sqrt{1 - 2p +2{p^2}} |\sin ({2\theta})|$ $\sqrt{1 - 2p +2{p^2}} |\sin ({2\theta})|$ $\sqrt{1 - 2p +2{p^2}} |\sin ({2\theta})|$
Ratio ${\cal K}$ ${\cal K} = 1$ ${\cal K} \leq 1$ ${\cal K} \leq 1$ ${\cal K} \leq 1$
Difference ${\cal Q}$ ${\cal Q} \leq 0$ ${\cal Q} \leq 0$ ${\cal Q} \leq 0$ ${\cal Q} \leq 0$
cpl-37-11-110302-fig3.png
Fig. 3. (a) The difference ${\cal M}=C(\rho)-{C_{\rm I}}(\rho)$ between concurrence and intrinsic concurrence. (b) The difference ${\cal N}={C_{\rm I}}^2(\rho)- \frac{{1 + {C^2}(\rho)}}{2}$ about concurrence and intrinsic concurrence.
Quantum States with Rank-4. It is well known that a two-qubit state with the rank $R(\rho) = 4$ contains that with $R(\rho) = 3$. In this sense, we here only need to discuss the case of the quantum states with the rank $R(\rho) = 4$. As an illustration, we resort to a Werner-class state with $R(\rho) = 4$, which can be expressed as $$\begin{align} {\rho} = r{|\psi \rangle}\langle \psi|+\frac{{1-r}}{4}{𝟙_A} \otimes {𝟙_B}~~ \tag {32} \end{align} $$ where $0\leq r \leq1$ is the probability of Bell-class state $|\psi\rangle=\cos\theta|{00}\rangle+\sin \theta |{11}\rangle $. According to Eqs. (6) and (8), we can analytically obtain the corresponding concurrence $C(\rho)$ and intrinsic concurrence ${C_{\rm I}}(\rho)$ as $$\begin{align} &C({{\rho}}) =\frac{1}{2} {\rm{{\max}}} \{0, r -1 + 2r\sin ({2\theta}) \}, \\ &{C_{\rm I}}({{\rho}})=\frac{1}{2}\sqrt {1 + {r^2} - 2{r^2}\cos({4\theta})}~.~~ \tag {33} \end{align} $$ To prove our result in Eq. (31), we make use of the difference method to show the performance. The differences are denoted as ${\cal M}=C(\rho)-{C_{\rm I}}(\rho)$ and ${\cal N}={C_{\rm I}}^2(\rho)- \frac{{1 + {C^2}(\rho)}}{2}$, respectively. As shown in Fig. 3, we find out that the relation $C(\rho) \leq {C_{\rm I}}(\rho)\le \sqrt {\frac{{1+{C^2}(\rho)}}{2}}$ is satisfied between concurrence and intrinsic concurrence with regard to the arbitrary Werner-class state, which directly shows that the relation (31) is held for the quantum states with the rank $R (\rho)= 4$. In summary, we have derived the relation between the concurrence and intrinsic concurrence. It is obtained that the intrinsic concurrence has not only a maximum lower bound, but also a minimum upper bound by analyzing the concurrence and intrinsic concurrence for the maximally nonlocal mixed state. In order to better prove the minimum upper bound of intrinsic concurrence, we choose randomly generated two-qubit states as illustrations. Meanwhile, as examples we analyze the case in the Bell-class state interacting with the different channels, for the quantum states with $R (\rho)=2$, and the Werner-class states with rank $R (\rho)= 4$. We prove that the relation between concurrence and intrinsic concurrence in open systems is satisfied. Further, this supposes our results. Thereby, we believe that our findings would be beneficial to understand the relationship between different quantum resources, and be of importance to realistic quantum information processing. References Quantum coherence and geometric quantum discordQuantum entanglementCan Quantum-Mechanical Description of Physical Reality Be Considered Complete?Die gegenwärtige Situation in der QuantenmechanikQuantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable modelEntanglement detectionThree-intensity decoy-state method for device-independent quantum key distribution with basis-dependent errorsCommunication via one- and two-particle operators on Einstein-Podolsky-Rosen statesTeleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channelsAsymptotically limitless quantum energy teleportation via qudit probesMinimum classical bit for remote preparation and measurement of a qubitRemote State PreparationA One-Way Quantum ComputerA scheme for efficient quantum computation with linear opticsQuantifying EntanglementA comparison of entanglement measuresEntanglement of Formation of an Arbitrary State of Two QubitsMixed-state entanglement and quantum error correctionRelative Entropy of Entanglement and Restricted MeasurementsOrdering two-qubit states with concurrence and negativityVariational characterizations of separability and entanglement of formationConcurrence in arbitrary dimensionsUniversal complementarity between coherence and intrinsic concurrence for two-qubit statesEntanglement, coherence, and redistribution of quantum resources in double spontaneous down-conversion processesMonotones and invariants for multi-particle quantum statesNew High-Intensity Source of Polarization-Entangled Photon PairsMaximally entangled mixed states of two qubitsRevealing Hidden Coherence in Partially Coherent LightSudden Death of EntanglementEvolution equation for geometric quantum correlation measuresQuantum decoherence in noninertial framesComparative investigation of the freezing phenomena for quantum correlations under nondissipative decoherenceCharacterizing the dynamics of entropic uncertainty for multi-measurementDynamical characteristic of measurement uncertainty under Heisenberg spin models with Dzyaloshinskii–Moriya interactionsCharacterization of dynamical measurement's uncertainty in a two-qubit system coupled with bosonic reservoirsDynamics of the measurement uncertainty in an open system and its controllingImproved tripartite uncertainty relation with quantum memory
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