1 Introduction

Exploiting and exploring the quantum world, in recent few decades, have been drawing more attention. With the deepening of research, quantum world has shown many internal differences from the classical world, and entanglement which was proposed by Einstein, Podolsky, and Rosen (EPR) dates back as early as the 1930s is one of the most profound of these differences [1,2,3,4,5]. It is not only an inherent quantum phenomenon that has gained prominence, but it has also become a useful source of quantum information theory in developing quantum information. Entanglement is regarded as only quantum correlations in early studies [6,7,8,9,10,11].

In 2001, however, Ollivier and Zurek [12], Henderson and Vedral [13] showed the notion of discord as a different quantum correlation with entanglement. It is more general than entanglement that separable states may also have nonvanishing discord. Furthermore, considerable attention has been devoted to study discord in the last decade since it has proved to be a useful resource in quantum information [14,15,16,17,18,19,20,21,22,23,24].

Despite the striking resemblance between entanglement and discord, in recent years, relationships between entanglement and discord are shown in the framework of comparing entanglement between two parties with the discord involving a third party and incorporating entanglement directly into general measurement-induced disturbance [25,26,27,28]. In this paper, we study the entanglement measurement proposed in [28] to analyze the relation between entanglement and discord, however, based on Tsallis q-discord [29, 30], which considers that entanglement may be a special discord. This paper is organized as follows: In the Section 2, we recall some primary knowledge such as state extensions, Tsallis entropy, Tsallis q-discord and so on; In the Section 3, an entanglement measurement based on Tsallis q-discord is established and some properties are showed. There is a conclusion in the Section 4.

2 Primary Knowledge

At the beginning, we recall a typical scheme for classifying quantum correlation: separable-entangled. If a quantum state ρAB from two parties (A and B) system can be decomposed as

$$\begin{array}{@{}rcl@{}} \rho_{AB}=\sum\limits_{i}p_{i}\rho_{A}^{i} \otimes \rho_{B}^{i}. \end{array}$$
(1)

with local states \({\rho _{A}^{i}}\) and \({\rho _{B}^{i}}\) for subsystems A and B and probabilities pi is called separable quantum state. Otherwise, it is called entangled [31]. As such phenomenon is an inherent difference between quantum world and classical world, entanglement states are important resources for quantum information processing. A great deal of research has been done on the detection and quantification of entanglement. In general, however, entanglement is extremely difficult and complex. Entanglement measurement is an effective quantitative method, which can quantitatively estimate the entanglement of formation, the entanglement cost, the distillable entanglement, the relative entropy of entanglement, the robustness of entanglement, and the squashed entanglement. It is a strange phenomenon that entanglement is a unique property of the quantum world, however, the “separable” does not mean the “classically correlated”. This is to say, there are separable states that have quantum. In fact, from a classical world point of view, we can measure without interfering with the measured system to obtain information. A bipartite state ρAB is classically correlated if it satisfies

$$\begin{array}{@{}rcl@{}} \rho_{AB}=\sum\limits_{i,j}\left( {\Pi}_{i}^{A} \otimes {\Pi}_{j}^{B}\right)\rho_{AB}\left( {\Pi}_{i}^{A} \otimes {\Pi}_{j}^{B}\right) \end{array}$$
(2)

where \(\left \{{{\Pi }_{i}^{A}} \right \}\) and \(\left \{{{\Pi }_{j}^{B}} \right \}\) are von Neumann measurements on parties A and B, respectively. Although they are not equally, in Ref. [32], authors show a separable state ρAB can be embedded into a larger classical state \(\rho _{A^{\prime }A|BB^{\prime }}\) that

$$\begin{array}{@{}rcl@{}} \rho_{AB}=\text{Tr}_{A^{\prime}B^{\prime}}(\rho_{A^{\prime}A|BB^{\prime}}), \end{array}$$
(3)

where \(A^{\prime }\) and \(B^{\prime }\) are two ancillary systems pertinent to parties A and B, respectively. Any entangled state does not admit such an extension.

On the other hand, contrast with entanglement which reflects some specific structure of quantum state to classifying quantum correlation, another classification scheme called discord which measures the amount of quantum correlations between subsystems in terms of quantum mutual information. In detail, the quantum mutual information between subsystem A and B is characterized as

$$\begin{array}{@{}rcl@{}} I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}) \end{array}$$
(4)

as long as the von Neumann entropy function \(S(\rho _{A})= -\text {Tr}\rho _{A}\ln \rho _{A}\) is used. What’s more, global discord represent change of quantum mutual information under local measurement that

$$\begin{array}{@{}rcl@{}} D(\rho_{AB})=I(\rho_{AB})-\underset{\Pi}{\max} I({\Pi}(\rho_{AB})) \end{array}$$
(5)

where the maximum runs amount of all local von Neumann measurement \({\Pi }=\left \{{{\Pi }_{i}^{A}} \otimes {{\Pi }_{j}^{B}}\right \}\) on ρAB and \({\Pi }(\rho _{AB})=\sum \limits _{i,j}\left ({{\Pi }_{i}^{A}} \otimes {{\Pi }_{j}^{B}}\right )\rho _{AB}\left ({{\Pi }_{i}^{A}} \otimes {{\Pi }_{j}^{B}}\right )\) is the post measurement state [33, 34]. Considering the generalizations of Shannon entropy and von Neumann entropy such as Tsallis entropy, the Tsallis q-global quantum discord is given that

$$\begin{array}{@{}rcl@{}} D_{q}(\rho_{AB})=I_{q}(\rho_{AB})-\underset{\Pi}{\max} I_{q}({\Pi}(\rho_{AB})) \end{array}$$
(6)

where Iq = Sq(ρA) + Sq(ρB) − Sq(ρAB) with \(S_{q}(\rho )=\frac {1}{q-1}(1-\text {Tr}(\rho ^{q}))\) and \(\underset {\Pi }{\max \limits } I_{q}({\Pi }(\rho _{AB}))\) also using Tsallis entropy instead of von Neumann ones in definition of discord. Some basic properties of Dq are discussed in [30], such as an important fact that its non-negative for 0 < q ≤ 1.

3 Entanglement Based on Tsallis Q-discord

In this section, taking use of the above concepts and the Tsallis q-global quantum discord is written as Tsallis q-discord briefly for convenience, we introduce a measurement of entanglement based on Tsallis q-discord and show some fundamental properties of it as the main results in this paper. Entanglement, in this circumstance, can be regarded as a special quantum discord. In fact, here we restrict ourselves to the symmetric case, global discord, with the understanding that the nonsymmetric scenario can be treated similarly. At the beginning, we define that

$$\begin{array}{@{}rcl@{}} E_{q}(\rho_{AB})= \underset{\rho_{AB}= \text{Tr}_{A^{\prime}B^{\prime}}(\rho_{A^{\prime}A|BB^{\prime}})}{\min} D_{q}(\rho_{A^{\prime}A|BB^{\prime}}) \end{array}$$
(7)

where Dq is taken with respect to the bipartition \(A^{\prime }A|BB^{\prime }\) Tsallis q-discord with \(\rho _{A^{\prime }A|BB^{\prime }}\) is an extension of ρAB in a four-partite system. We conclude the properties of Eq under the case of 0 < q ≤ 1 in following theorem.

Theorem 1

The Eq (0 < q ≤ 1) has the following ideal and remarkable characteristics:

  1. (1)

    For any separable states the entanglement is vanish, i.e. Eq(ρAB) = 0 if ρAB is separable.

  2. (2)

    If ρAB = |ψAB〉〈ψAB| is a pure state, then Eq(ρAB) = Dq(ρAB) = Sq(ρA). That is to say, the entanglement Eq(ρAB) coincides with Tsallis q-discord Dq(ρAB) and also coincides with Tsallis q-entropy of reduced state ρA = TrB|ψAB〉〈ψAB| for any pure state.

  3. (3)

    Eq is dominated by the Tsallis q-discord for any ρAB, i.e. Eq(ρAB) ≤ Dq(ρAB).

  4. (4)

    Eq is convex, this is \(E_{q}\left (\sum \limits _{i}p_{i} \rho _{AB}^{i}\right ) \leq \sum \limits _{i}p_{i} E_{q}\left (\rho _{AB}^{i}\right )\), where \(\sum \limits _{i}p_{i} E_{q}\left (\rho _{AB}^{i}\right )\) is convex combination of bipartite states \(\rho _{AB}^{i}\) shared by parties A and B.

  5. (5)

    Considering the Tsallis q-entropy entanglement Tq that Tq(|ψAB〉) = Sq(ρA) for any pure state |ψAB〉 Eq and \(T_{q}(\rho _{AB})=\min \limits \sum \limits _{i} p_{i} T_{q}(|\psi _{AB}^{i}\rangle )\), for a mixed state ρAB with the minimum is taken over all possible pure state decompositions \(\{p_{i},\psi _{AB}^{i}\}\) of ρAB, we have Eq(ρAB) ≤ Tq(ρAB).

  6. (6)

    Eq is locally unitary invariant in the sense that Eq(UA ⊗ UB)ρAB(UA ⊗ UB)‡ = Eq(ρAB) for any unitary operators UA and UB on parties A and B, respective.

  7. (7)

    Eq is not increasing under local partial trace for state extension \(\rho _{A^{\prime }A|BB^{\prime }}\) of ρAB, this is to say \(E_{q}(\rho _{AB})\leq E_{q}(\rho _{A^{\prime }A|BB^{\prime }})\).

  8. (8)

    For any local channels ΛA and ΛB on parties A and B, it holds Eq((ΛA ⊗ΛB)ρAB) ≤ Eq(ρAB).

Proof

Here, we point out that we abbreviate the extension state \(\rho _{A^{\prime }A|BB^{\prime }}\) of ρAB as \(\bar {\rho }_{AB}\) for convenience.

  1. (1)

    ρAB is separable, according to [32], ρAB can be extend to a certain classical state \(\bar {\rho }_{AB}\), this is to say \(D_{q}(\bar {\rho }_{AB})=0\). Combine with \(D_{q}(\bar {\rho }_{AB})\geq 0\) we have \(\underset {\rho _{AB}= \text {Tr}_{A^{\prime }B^{\prime }}\bar {\rho }_{AB}}{\min \limits } D_{q}(\bar {\rho }_{AB})=0\). Finally, we have Eq(ρAB) = 0.

  2. (2)

    From the Schmidt decomposition, if \(|\psi \rangle\) is a pure state, there exist orthonormal states \(|i \rangle _{A}\) for system A and orthonormal states \(|i \rangle _{B}\) for system B, such that \(|\psi \rangle = \sum \limits _{i}\lambda _{i} |i \rangle _{A} |i \rangle _{B}\) where λi ≥ 0 and \(\sum \limits _{i}{\lambda _{i}^{2}}=1\).

    Then we have

    $$\begin{array}{@{}rcl@{}} \rho_{A} &=& \text{Tr}_{B} \sum\limits_{ij}\lambda_{i} \lambda_{j} (|i \rangle_{A}|i \rangle_{B})(\langle j|_{A} \langle j|_{B}) \\ &=& \text{Tr}_{B} \left( \sum\limits_{ij}\lambda_{i}\lambda_{j} |i \rangle_{A} \langle j|_{A} \otimes |i \rangle_{B} \langle j|_{B} \right) \\ &=& \sum\limits_{ij}\lambda_{i}\lambda_{j} |i \rangle_{A} \langle j|_{A} \text{Tr} (|i \rangle_{B} \langle j|_{B})\\ &=& \sum\limits_{ij}\lambda_{i}\lambda_{j} |i \rangle_{A} \langle j|_{A} \delta_{ij}\\ &=& \sum\limits_{i}{\lambda_{i}^{2}} |i \rangle_{A} \langle i|_{A}. \end{array}$$

    As the same reason, \(\rho _{B} =\sum \limits _{i}{\lambda _{i}^{2}} |i \rangle _{B} \langle i|_{B}\). So Sq(ρA) = Sq(ρB). Combining with ρAB is a pure state, we can get Iq(ρAB) = 2Sq(ρA).

    Let us consider \({\Pi }=\left \{{\pi _{i}^{A}}\otimes {\pi _{j}^{B}}\right \}_{ij}\) is a local von Neumann measurement and \(|m \rangle _{A}\) and \(|m \rangle _{B}\) are the orthonormal state in systems A and B. Taking the similar processing as above, \(\tilde {\rho }_{A}=\text {Tr}_{B}({\Pi }(\rho _{AB})) =\sum \limits _{m}{\lambda _{i}^{2}} |m \rangle _{A} \langle m|_{A}\) and \(\tilde {\rho }_{B}=\text {Tr}_{A}({\Pi }(\rho _{AB})) =\sum \limits _{m}{\lambda _{i}^{2}} |m \rangle _{B} \langle m|_{B}\), then Sq(ρA) = Sq(ρB) = Sq(π(ρAB)). Also, we have \(\underset {\Pi }{\max \limits } I_{q}({\Pi }(\rho _{AB}))=S_{q}(\tilde {\rho }_{A})+S_{q}(\tilde {\rho }_{B})-S_{q}({\Pi }(\rho )_{AB})=S_{q}({\rho }_{A})+S_{q}({\rho }_{A})-S_{q}(\rho _{A})= S_{q}(\rho _{A})\). Furthermore, \(D_{q}(\rho _{AB})=I_{q}(\rho _{AB})-\underset {\Pi }{\max \limits } I_{q}({\Pi }(\rho )_{AB})=2S_{q}(\rho _{A})- S_{q}(\rho _{A})=S_{q}(\rho _{A})\). From all the equation relations we can conclude that Eq(ρAB) ≤ Dq(ρAB).

  3. (3)

    Combine the definition of Eq with the fact that when both \(A^{\prime }\) and \(B^{\prime }\) are one-dimensional ancillary systems ρAB can be regarded as a trivial extension of itself, we obtain \(E_{q}(\rho _{AB})=\underset {\rho _{AB}= \text {Tr}_{A^{\prime }B^{\prime }}\bar {\rho }_{AB}}{\min \limits } D_{q}(\bar {\rho }_{AB})\leq D_{q}(\bar {\rho }_{AB})=D_{q}(\bar {\rho }_{AB})\).

  4. (4)

    Taking two anciliary systems C and D of the same dimension, with orthonormal bases {|i〉C} and {|i〉D} respectively. Noting that \(\rho _{CA^{\prime }A|BB^{\prime }D}= \sum \limits _{i} p_{i} |i \rangle _{C} \langle i|\otimes \bar {\rho }_{AB}^{i} \otimes |i \rangle _{D} \langle i|\), here \(\bar {\rho }_{AB}^{i}\) are extension of \(\rho _{AB}^{i}\). We can get \(\rho _{AB}= \text {Tr}_{A^{\prime }B^{\prime }}\bar {\rho }_{AB}\) directly. Then \(E_{q}(\rho _{AB})\leq D_{q}(\rho _{CA^{\prime }A|BB^{\prime }D})= D_{q}\left (\sum \limits _{i} p_{i} |i \rangle _{C} \langle i|\otimes \bar {\rho }_{AB}^{i} \otimes |i \rangle _{D}\langle i|\right )\). As follows, we will show that \(D_{q}\left (\sum \limits _{i} p_{i} |i \rangle _{C} \langle i|\otimes \bar {\rho }_{AB}^{i} \otimes |i \rangle _{D}\langle i|) \leq \sum \limits _{i}p_{i} E_{q}(\bar {\rho }_{AB}^{i}\right ).\) Firstly, let \({\Pi }=\left \{ {\Pi }^{ij}_{A^{\prime }A} \otimes {\Pi }^{ik}_{B^{\prime }B}\right \}\) and \(\tilde {\Pi }\) be the local von Neumann measurement on \(\bar {\rho }_{AB}^{i}\) and \(\rho _{CA^{\prime }A|BB^{\prime }D}\), moreover, π satisfies maximal of \(\underset {\Pi }{\max \limits } I_{q}({\Pi }\left (\bar {\rho }_{AB}^{i}\right ))\) is achieved. Then we have

    $$\begin{array}{@{}rcl@{}} &&D_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\\ &=&I_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)-\underset{\tilde{\Pi}}{\max}, I_{q}\left( \tilde{\Pi}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\right)\\ &\leq& I_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)- I_{q}\left( \left( |i\rangle_{C} \langle i| \otimes {\Pi} \otimes |j\rangle_{D} \langle j|\right)\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\right) \end{array}$$

    The first and second terms on the right side of the inequality have the following relations respectively by using the joint entropy theorem discuessed in [35]:

    $$\begin{array}{@{}rcl@{}} &&I_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\\ &=&S_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \rho_{A^{\prime}A}^{i}\right)+S_{q}\left( \sum\limits_{i} p_{i} \rho_{B^{\prime}B}^{i} \otimes |i \rangle_{D} \langle i| \right)-S_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|)\right)\\ &=&H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \rho_{A^{\prime}A}^{i}\right)+H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \rho_{B^{\prime}B}^{i}\right)-\left( H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \bar{\rho}_{AB}^{i}\right)\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \rho_{A^{\prime}A}^{i}\right)+\sum\limits_{i} p_{i}^{q} S_{q}\left( \rho_{B^{\prime}B}^{i}\right)-\sum\limits_{i} p_{i}^{q} S_{q}\left( \bar{\rho}_{AB}^{i}\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} \left( S_{q}\left( \rho_{A^{\prime}A}^{i}\right)+S_{q}\left( \rho_{B^{\prime}B}^{i}\right)-S_{q}\left( \bar{\rho}_{AB}^{i}\right)\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} I_{q}\left( \bar{\rho}_{AB}^{i}\right) \end{array}$$

    where \(\vec {p}\) is the probability distributions composed by pi and \(H_{q}(\vec {p})\) is entropy function.

    $$\begin{array}{@{}rcl@{}} &&I_{q}\left( \left( |i\rangle_{C} \langle i| \otimes {\Pi} \otimes |j\rangle_{D} \langle j|\right)\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\right) \\ &=& I_{q}\left( \sum\limits_{i,j,k} p_{i} |i \rangle_{C} \langle i|\otimes \left( \left( {\Pi}_{A^{\prime}A}^{ij} \otimes {\Pi}_{B^{\prime}B}^{ik} \right)\bar{\rho}_{AB}^{i} \left( {\Pi}_{A^{\prime}A}^{ij} \otimes {\Pi}_{B^{\prime}B}^{ik}\right )\right)\otimes |i \rangle_{D}\langle i|\right)\\ &=& S_{q}\left( \tilde{\rho}_{CA^{\prime}A}\right)+S_{q}\left( \tilde{\rho}_{BB^{\prime}D}\right)-S_{q}(\tilde{\rho}_{CA^{\prime}A|BB^{\prime}D})\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \tilde{\rho}_{A^{\prime}A}^{i}\right)+H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \tilde{\rho}_{BB^{\prime}}^{i}\right)-\left( H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} S_{q}\left( \tilde{\rho}_{A^{\prime}A|BB^{\prime}}^{i}\right)\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} \left( S_{q}\left( \tilde{\rho}_{A^{\prime}A}^{i}\right)+S_{q}\left( \tilde{\rho}_{B^{\prime}B}^{i}\right)-S_{q}\left( \tilde{\rho}_{A^{\prime}A|BB^{\prime}}^{i}\right)\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} I_{q}\left( {\Pi}\left( \bar{\rho}_{AB}^{i}\right)\right) \end{array}$$

    where \(\tilde {\rho }\) means post measured state of ρ. From above calculation, there is

    $$\begin{array}{@{}rcl@{}} &&D_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\\ &\leq& I_{q}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)-I_{q}\left( {\Pi}\left( \sum\limits_{i} p_{i} |i \rangle_{C} \langle i|\otimes \bar{\rho}_{AB}^{i} \otimes |i \rangle_{D}\langle i|\right)\right)\\ &=& H_{q}(\vec{p})+\sum\limits_{i} p_{i}^{q} I_{q}\left( \bar{\rho}_{AB}^{i}\right)- H_{q}(\vec{p})-\sum\limits_{i} p_{i}^{q} I\left( {\Pi}\left( \bar{\rho}_{AB}^{i}\right)\right)\\ &=&\sum\limits_{i} p_{i}^{q} I_{q}\left( \bar{\rho}_{AB}^{i}\right)-\sum\limits_{i} p_{i}^{q} I\left( {\Pi}\left( \bar{\rho}_{AB}^{i}\right)\right)\\ &=&\sum\limits_{i} p_{i}^{q} \left( I_{q}\left( \bar{\rho}_{AB}^{i}\right)- I\left( {\Pi}\left( \bar{\rho}_{AB}^{i}\right)\right)\right)=\sum\limits_{i} p_{i}^{q} D_{q}\left( \bar{\rho}_{AB}^{i}\right)\leq \sum\limits_{i} p_{i} D_{q}\left( \bar{\rho}_{AB}^{i}\right)\\ &\leq& \sum\limits_{i} p_{i} E_{q}\left( \rho_{AB}^{i}\right) \end{array}$$

    the second inequality from the fact that 1 > pi > 0 and q > 0.

  5. (5)

    For any pure state decomposition \(\rho _{AB}=\sum \limits _{i} p_{i}|\psi _{i}\rangle _{AB} \langle \psi _{i}|\), there is

    $$\begin{array}{@{}rcl@{}} E(\rho_{AB})\leq \sum\limits_{i} p_{i} E_{q}(|\psi_i\rangle_{AB} \langle\psi_i|)=\sum\limits_{i} p_{i} T_{q}(|\psi_i\rangle_{AB} \langle\psi_i|) \end{array}$$

    So, we have Eq(ρAB) ≤ Tq(ρAB).

  6. (6)

    From Ref. [30], we know Dq is invariant under local unitary transformations, that is, Dq(UA ⊗ UB)ρAB(UA ⊗ UB)‡ = Dq(ρAB). On the other hand, from definition \(E_{q}(\rho _{AB})= \underset {\rho _{AB}= \text {Tr}_{A^{\prime }B^{\prime }}\bar {\rho }_{AB}}{\min \limits } D_{q}(\bar {\rho }_{AB})\), we can directly get Eq(UA ⊗ UB)ρAB(UA ⊗ UB)‡ = Eq(ρAB).

  7. (7)

    Since any state extension \(\rho _{A^{\prime \prime }A^{\prime }A|BB^{\prime }B^{\prime \prime }}\) of \(\bar {\rho }_{AB}\) is necessarily a state extension of the reduced state \(\rho _{AB}=\text {Tr}_{A^{\prime }B^{\prime }}\bar {\rho }_{AB}\), we have

    $$\begin{array}{@{}rcl@{}} E_{q}(\rho_{AB})&=& \underset{\rho_{AB}= \text{Tr}_{A^{\prime}B^{\prime}}\bar{\rho}_{AB}}{\min} D_{q}(\bar{\rho}_{AB})\\ &\leq& \underset{\rho_{AB}= \text{Tr}_{A^{\prime\prime}A^{\prime}B^{\prime}B^{\prime\prime}}\rho_{A^{\prime\prime}A^{\prime}A|BB^{\prime}B^{\prime\prime}}}{\min} D_{q}(\rho_{A^{\prime\prime}A^{\prime}A|BB^{\prime}B^{\prime\prime}})\\ &\leq& \underset{\bar{\rho}_{AB}= \text{Tr}_{A^{\prime\prime}B^{\prime\prime}}\rho_{A^{\prime\prime}A^{\prime}A|BB^{\prime}B^{\prime\prime}}}{\min} D_{q}(\rho_{A^{\prime\prime}A^{\prime}A|BB^{\prime}B^{\prime\prime}})\\ &=& E_{q}(\bar{\rho}_{AB}) \end{array}$$
  8. (8)

    For any local channels ΛA and ΛB on parties A and B, respectively, using the Stinespring dilation representation, we can write

    $$\begin{array}{@{}rcl@{}} {\Lambda}_{A}(\rho_{A})&=&\text{Tr}_{A^{\prime}}U_{A^{\prime}A}\left( \sigma^{A^{\prime}}\otimes \rho_{A}\right)(U_{A^{\prime}A})^{\dag},\\ {\Lambda}_{B}(\rho_{B})&=&\text{Tr}_{B^{\prime}}U_{B^{\prime}B}\left( \sigma^{B^{\prime}}\otimes \rho_{B}\right)(U_{B^{\prime}B})^{\dag}, \end{array}$$

    where \(\sigma ^{A^{\prime }}\) and \(\sigma ^{B^{\prime }}\) are states on the ancillary systems \(A^{\prime }\) and \(B^{\prime }\) and \(U_{A^{\prime }A}\) and \(U_{B^{\prime }B}\) are unitary operators on the composite systems \(A^{\prime }A\) and \(B^{\prime }B\), respectively.

    $$\begin{array}{@{}rcl@{}} &&E_{q}(({\Lambda}_{A} \otimes {\Lambda}_{B})\rho_{AB}) \\ &=& E_{q}\left( \text{Tr}_{A^{\prime}B^{\prime}}\left( U_{A^{\prime}A} \otimes U_{B^{\prime}B}\right)\left( \sigma^{A^{\prime}}\otimes \rho_{AB}\otimes \sigma^{B^{\prime}}\right)\left( U_{A^{\prime}A} \otimes U_{B^{\prime}B}\right)^{\dag}\right)\\ &\leq& E_{q}\left( (U_{A^{\prime}A} \otimes U_{B^{\prime}B})\left( \sigma^{A^{\prime}}\otimes \rho_{AB}\otimes \sigma^{B^{\prime}}\right)(U_{A^{\prime}A} \otimes U_{B^{\prime}B})^{\dag}\right)\\ &=&E_{q}\left( \sigma^{A^{\prime}}\otimes \rho_{AB}\otimes \sigma^{B^{\prime}}\right)\\ &=& E_{q}(\rho_{AB}). \end{array}$$

    We point out that Eq is decreasing local channels based on the locally unitary invariant and the nonincreasing under local partial trace as the inequality and the second equality from properties (6) and (7).

Here, the non-negativity of Tsallis q-discord assured for a selective choice of the parameter 0 < q ≤ 1 is a fundamental restriction, however, this property is not clear for the case of 1 < q which makes puzzle of above theorem at all range of q. From Theorem 1, we have established a rigorous framework for the link between two quantum resources, entanglement on one hand, and discord on the other hand, via state extensions. It demonstrates the usefulness of discord to obtain and validate measurement of entanglement. This result provides evidence that the entanglement measurement Eq is equivalent to the minimal Tsallis q-discord over state extensions. We consider the defining a discord via Tsallis q-entropy which different to discord based on von Neumann measurement, q-discord corresponding to different values of q measure different aspects of the non-classicality of correlations.

Although entanglement measurement is an effective method to describe entangled states, its computation is extremely difficult and seems to be difficult to deal with at present. What’s more, calculation of Tsallis q-discord is a hard and complicated problem, and analytical formulas are rare, such as Werner states, isotropic states and some special cases. It is evident that the evaluation of the entanglement measurement via Tsallis q-discord is even harder since the definition of Eq is over both state extensions, it needs to seek some explicit formulas. However, the relationship between entanglement and q-discord is an interesting problem.

4 Conclusion

From the theoretical and experimental point of view, quantum entanglement, as an outstanding quantum relation, lies at the core of quantum mechanics and is still a mystical and versatile concept with far-reaching application and significance. In recent decades, considerable efforts have been made to solve the problem of entanglement. In addition, the notion of discord is considered to be a more common type of quantum correlations than entanglement, as there is growing interest in the possibility that a separated state may have a nonvanishing discord. Entanglement and discord are two typical correlations classification schemes. Because entanglement and discord are intrinsically similar, and there is a significant relationship and interaction between them, we have established a connection among them. An entanglement measurement based on Tsallis q-discord is present and some fundamental properties of it are revealed and proved. From this view, entanglement can be regarded as a special discord.