1 Introduction

The use of entanglement as a resource in quantum information processing tasks has accelerated research efforts on its quantification, characterization, and control over the past two decades [1,2,3,4,5]. While multipartite entanglement poses higher level of complexity than the bipartite case, it enriches our theoretical understanding and paves way for innovative applications in distributed quantum networks [6,7,8,9,10,11,12]. It has been recognized that geometry associated with particular symmetry transformations plays a vital role in exploring multipartite entanglement—especially in the distribution of entanglement among the constituent subsystems [13,14,15,16,17,18,19]. Study of geometric invariants and canonical forms of composite quantum states under local symmetry operations on subsystems serves as a powerful tool to probe different manifestations of entanglement. To this end, considerable progress has been evinced in exploring local invariant quantities, canonical forms of equivalence classes of states under local unitary (LU) transformations, stochastic local operations and classical communication (SLOCC) associated with local SL(2,C) transformations [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40].

An essential feature of entanglement is that it remains invariant under LU operations. Any two arbitrary pure states \(\vert \psi \rangle \) and \(\vert \phi \rangle \) are LU equivalent (written symbolically as \(\vert \psi \rangle ~\sim ~\vert \phi \rangle \)) if and only if they can be transformed into each other by local unitary operations. A complete set of polynomial quantities that remain unaltered under LU operations on subsystems are used to certify LU equivalence of multipartite states. Recognizing normal/canonical form of a composite system by using LU transformations on individual subsystems is advantageous in evaluating these polynomial invariants.

Acín et al. showed that a three-qubit pure state under LU transformations can be reduced to a canonical form given by [16]:

$$\begin{aligned} \vert \psi _{ABC}\rangle =\lambda _0\vert 0,0,0\rangle +\lambda _1 e^{i\phi }\vert 1,0,0\rangle +\lambda _2\vert 1,0,1\rangle +\lambda _3\vert 1,1,0\rangle +\lambda _4\vert 1,1,1\rangle \nonumber \\ \end{aligned}$$
(1.1)

in terms of five real entanglement parameters \(\lambda _i\ge 0,\ i=0,1,2,3,4\) satisfying \(\sum _{i=0}^4\,\lambda _i^2=1\), and a phase \(\phi \) ranging between 0 and \(\pi \). This gives a minimal form of pure three-qubit states containing only five terms and is helpful for evaluating LU invariants.

We consider a set of five LU invariants [16] characterizing pure three-qubit states (apart from normalization):

$$\begin{aligned} I_1= & {} \textrm{Tr}\, [\rho _{ BC}^2]=\textrm{Tr}\, [\rho _{ A}^2]= 1-2\lambda _0^2(1-\lambda _0^2-\lambda _1^2),\nonumber \\ I_2= & {} \textrm{Tr}\, [\rho _{AC}^2]=\textrm{Tr} [\rho _{ B}^2] = 1-2\lambda _0^2(1-\lambda _0^2-\lambda _1^2-\lambda _2^2)-2\bigtriangleup , \nonumber \\ I_3= & {} \textrm{Tr}\, [\rho _{AB}^2]=\textrm{Tr}\,[\rho _{ C}^2] =1-2\lambda _0^2(1-\lambda _0^2-\lambda _1^2-\lambda _3^2)-2\bigtriangleup , \nonumber \\ I_4= & {} \textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ B})\,\rho _{AB}]= 1+\lambda _0^2\,\left( \lambda _2^2 \lambda _3^2-\lambda _1^2 \lambda _4^2-2\,\lambda _2^2 -3\,\lambda _3^2-3\,\lambda _4^2\,\right) \nonumber \\{} & {} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \qquad -(2-\lambda _0^2)\,\bigtriangleup \nonumber \\ I_5= & {} \lambda _0^4\lambda _4^4=\frac{\tau ^2}{16}, \end{aligned}$$
(1.2)

where \(\rho _A=\textrm{Tr}_\textrm{B,C}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \), \(\rho _B=\textrm{Tr}_\textrm{A,C}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \), \(\rho _C=\textrm{Tr}_\textrm{A,B}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \), \(\rho _{AB}~=~\textrm{Tr}_\textrm{C}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \), \(\rho _{BC}=\textrm{Tr}_\textrm{A}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \), \(\rho _{AC}=\textrm{Tr}_\textrm{B}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert \) and

$$\begin{aligned} \bigtriangleup \equiv \vert \lambda _1\lambda _4e^{i\phi }-\lambda _2\lambda _3\vert ^2. \end{aligned}$$
(1.3)

The first three invariants \(I_1,\, I_2,\, I_3\) are related to the squares of the three one-to-other bipartite concurrences \(C^2_{A(BC)},\ C^2_{B(AC)},\) and \(C^2_{C(AB)}\), respectively [15, 41]. The fourth one, \(I_4\), is related to the Kempe invariant [6, 17]

$$\begin{aligned} {{\mathcal {I}}}_4= & {} 3\textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ B})\,\rho _{AB}]-\textrm{Tr}\,[\rho _{A}^3]-\textrm{Tr}\,[\rho _{B}^3], \nonumber \\= & {} 3\textrm{Tr}\,[(\rho _{ B}\otimes \rho _{ C})\,\rho _{BC}]-\textrm{Tr}\,[\rho _{B}^3]-\textrm{Tr}\,[\rho _{C}^3], \nonumber \\= & {} 3\textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ C})\,\rho _{AC}]-\textrm{Tr}\,[\rho _{A}^3]-\textrm{Tr}\,[\rho _{C}^3], \end{aligned}$$
(1.4)

which is symmetric under the permutation of qubits. This quantity, while algebraically independent of the other LU invariants, has no known implication toward the classification of three-qubit entanglement [28].

Writing \(\vert \psi _\textrm{ABC}\rangle ~=~\sum _{i,j,k=0,1}\, c_{ijk}\,\vert i,j,k\rangle \) in the computational basis, the invariant \(I_5\) (Cayley’s hyperdeterminant [42, 43]) is expressed as

$$\begin{aligned} I_5=\frac{1}{4}\,\left| \, \epsilon _{i_1\,i_2}\,\epsilon _{i_3\,i_4}\,\epsilon _{j_1\,j_2}\,\epsilon _{j_3\,j_4}\, \epsilon _{k_1\,k_3}\,\epsilon _{k_2\,k_4}\, c_{i_1j_1k_1}c_{i_2j_2k_2}c_{i_3j_3k_3}c_{i_4j_4k_4}\right| ^2, \end{aligned}$$
(1.5)

where \(\epsilon _{ij}\) denote antisymmetric tensor of rank-2; repeated indices are to be summed over in (1.5). In Acín’s canonical form (1.1) of the three-qubit state, one obtains a simple form \(I_5~=~\lambda _0^4\lambda _4^4\), which is related to the three-tangle \(\tau =4\,\lambda _0^2\lambda _4^2\), a measure of three-way entanglement of three qubits in a pure state [15].

Any two pure three-qubit states \(\vert \psi \rangle \) and \(\vert \phi \rangle \) are SLOCC equivalent if and only if they are mutually interconvertible by means of local invertible transformations:

$$\begin{aligned} \vert \psi \rangle \sim A\otimes B\otimes C\, \vert \phi \rangle \end{aligned}$$
(1.6)

where \(A,\,B, \, C\in \ \)SL(2,C) denote \(2\times 2\) complex matrices with determinant unity. Because local protocols are unable to generate entanglement, invariant quantities under local SL(2,C) transformations are used for classification and also quantification of entanglement. Equivalence classes of pure three-qubit states under local invertible operations were explored in the celebrated work by Dür et al. [14], where it was shown that there exist two inequivalent tripartite entanglement classes—represented by the Greenberger–Horne–Zeilinger (GHZ) state [44]

$$\begin{aligned} \vert \textrm{GHZ}\rangle =\frac{1}{\sqrt{2}}\left( \vert 0,0,0\rangle +\vert 1,1,1\rangle \right) \end{aligned}$$
(1.7)

and the W state [14]

$$\begin{aligned} \vert \textrm{W}\rangle =\frac{1}{\sqrt{3}}\left( \vert 1,0,0\rangle +\vert 0,1,0\rangle +\vert 0,0,1\rangle \right) . \end{aligned}$$
(1.8)

There has been a large effort toward gaining deeper insight into the structure of local SL(2,C) invariant quantities, where three-qubit pure state is considered as a test bed [15, 25, 27,28,29,30,31, 33, 36].

In this paper, we extend the mathematical framework of Ref. [38] to construct Lorentz invariants of pure three-qubit states. In the following section, we describe the basic formalism of Ref. [38]. Mainly we highlight here that the transformation property of the real \(4\times 4\) matrix parametrization \(\Lambda _{AB}\) of a two-qubit density matrix \(\rho _{AB}\) paves the way to identify Lorentz invariance of the eigenvalues \(\mu ^{AB}_\alpha ,\alpha =0,1,2,3\) of the matrix \(\Gamma _{AB}=G\,\Lambda ^T_{AB}\,G\,\Lambda _{AB}\). Section 3 is devoted to explore the properties of the Lorentz invariant eigenvalues of \(\Gamma _{ij}, \ ij=AB,\ BC,\ AC\) associated with the reduced two-qubit density matrices \(\rho _{ij}\) of a pure three-qubit state. We recognize that (i) the matrices \(\Gamma _{ij}, \ ij=AB,\ BC,\ AC\) associated with a pure three-qubit state have at most two distinct Lorentz invariant eigenvalues; (ii) difference between the two eigenvalues is symmetric under the interchange of qubits and is equal to the three-tangle \(\tau \) of the three-qubit state; (iii) the smallest Lorentz invariant eigenvalue of \(\Gamma _{ij}\) is equal to the squared concurrence \(C^{2}_{ij}, \ ij=AB,\ BC,\ AC\) of the two-qubit subsystems. We explicitly illustrate these features in pure permutation symmetric three-qubit states in Subsect. 3.1. Construction of a set of five local SL(2,C) invariants, which turn out to be the algebraic analogues of corresponding set of LU invariants of the three-qubit pure state, is outlined in Subsect. 3.2. A summary of our results is given in Sect. 4.

2 Transformation of two-qubit state under local SL(2,C) operations

Let us consider an arbitrary two-qubit density matrix \(\rho _{AB}\), expanded in the Hilbert–Schmidt basis \(\{\sigma _\alpha \otimes \sigma _\beta , \alpha ,\beta =0,1,2,3\}\):

$$\begin{aligned} \rho _{AB}= & {} \frac{1}{4}\, \sum _{\alpha ,\,\beta =0}^{3}\, \left( \Lambda _{AB}\right) _{\alpha \, \beta }\, \left( \sigma _\alpha \otimes \sigma _\beta \right) , \nonumber \\{} & {} \left( \Lambda _{AB}\right) _{\alpha \, \beta }= \textrm{Tr}\,\left[ \rho _{AB}\, (\sigma _\alpha \otimes \sigma _\beta )\,\right] \end{aligned}$$
(2.1)

where

$$\begin{aligned} \sigma _0=\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array}\right) ,\ \ \sigma _1=\left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \end{array}\right) ,\ \ \sigma _2=\left( \begin{array}{cc} 0 &{} -i \\ i &{} 0 \end{array}\right) ,\ \ \sigma _3=\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \end{array}\right) . \end{aligned}$$
(2.2)

It is convenient to express the expansion coefficients \(\left( \Lambda _{AB}\right) _{\mu \, \nu }\) in (2.1) as a \(4\times 4\) real matrix:

$$\begin{aligned} \Lambda _{AB}= & {} \left( \begin{array}{llll} 1&{} s_{B1} &{} s_{B2} &{} s_{B3} \\ s_{A1} &{} t^{AB}_{11} &{} t^{AB}_{12} &{} t^{AB}_{13} \\ s_{A2} &{} t^{AB}_{21} &{} t^{AB}_{22} &{} t^{AB}_{23} \\ s_{A3} &{} t^{AB}_{31} &{} t^{AB}_{32} &{} t^{AB}_{33} \\ \end{array}\right) . \end{aligned}$$
(2.3)

Here \(\textbf{s}_{A}\), \(\textbf{s}_{B}\) are Minkowski four vectors with components \(s_{A\alpha }\), \(s_{B\alpha },\ \alpha =0,1,2,3\), respectively, and \(T^{AB}=(t^{AB}_{ij}), \ i,\,i=1,2,3\) denotes the two-qubit correlation matrix:

$$\begin{aligned} s_{A\alpha }= & {} \left( \Lambda _{AB}\right) _{\alpha \, 0}=\textrm{Tr}\,\left[ \rho _{AB}\, (\sigma _\alpha \otimes \sigma _0)\,\right] =\textrm{Tr}\,\left[ \rho _{A}\,\sigma _\alpha \right] , \end{aligned}$$
(2.4)
$$\begin{aligned} s_{B\alpha }= & {} \left( \Lambda _{AB}\right) _{0 \, \alpha }= \textrm{Tr}\,\left[ \rho _{AB}\, (\sigma _0\otimes \sigma _\alpha )\,\right] =\textrm{Tr}\,\left[ \rho _{B}\,\sigma _\alpha \right] , \ \alpha =0,1,2,3 \end{aligned}$$
(2.5)
$$\begin{aligned} t^{AB}_{ij}= & {} \left( \Lambda _{AB}\right) _{i \, j}= \textrm{Tr}\,\left[ \rho _{AB}\, (\sigma _i\otimes \sigma _j)\,\right] ,\ \ \ \ i,\,j=1,\,2,\,3. \end{aligned}$$
(2.6)

Under local SL(2,C) operations, the two-qubit state \(\rho _{AB}\) transforms as

$$\begin{aligned} \rho _{AB}\longrightarrow \bar{\rho }_{AB}= & {} \frac{(A\otimes B)\, \rho _{AB}\, (A^\dag \otimes B^\dag )}{\textrm{Tr}\left[ \rho _{AB}\, (A^\dag \, A\otimes B^\dag \, B)\right] } \end{aligned}$$
(2.7)

where \(A, B\in \mathrm{SL(2,C)}\) denote \(2\times 2\) complex matrices with unit determinant. As a result, one finds that

$$\begin{aligned} \Lambda _{AB}\longrightarrow \bar{\Lambda }_{AB}= & {} \frac{L_A\,\Lambda _{AB}\, L^T_B}{\left( L_A\,\Lambda _{AB}\, L^T_B\right) _{00}} \end{aligned}$$
(2.8)

where \(L_A,\, L_B\in SO(3,1)\) are \(4\times 4\) proper orthochronous Lorentz transformation matrices [45] corresponding to A, \(B\in \mathrm{SL(2,C)}\), respectively, and the superscript ‘T’ denotes transpose operation.

We construct a \(4\times 4\) real matrix

$$\begin{aligned} \Gamma _{AB}=G\,\Lambda ^T_{AB}\, G\, \Lambda _{AB}, \end{aligned}$$
(2.9)

where \(G=\textrm{diag}\,(1,-1,-1,-1)\) denotes the Minkowski metric [45]. It is readily identified that the matrix \(\Gamma _{AB}\) undergoes a similarity transformation (up to an overall factor) [38]:

$$\begin{aligned} \Gamma _{AB}\rightarrow \bar{\Gamma }_{AB}= & {} G\,\bar{\Lambda }_{AB}^{T}\, G\, \bar{\Lambda }_{AB} \nonumber \\= & {} G\,\left( L_{A}\,\Lambda _{AB}\, L_{B}^{T}\right) ^T\,G\,\, L_{A}\, \Lambda _{AB} L_{B}^T\nonumber \\= & {} G\,L_{B}\, \Lambda ^T_{AB}\, L_{A}^T\, G \, L_{A}\, \Lambda _{AB} L_{B}^T \nonumber \\= & {} \left( G\,L_{B}\,G\right) \, G\,\Lambda ^T_{AB}\, \left( L_{A}^T\, G \, L_{A}\right) \, \Lambda _{AB} L_{B}^T \nonumber \\= & {} \left( L_{B}^T\right) ^{-1}\, \Gamma _{AB}\, L_{B}^T \end{aligned}$$
(2.10)

where the defining property [45] \(L^T\,G\,L=G\) of Lorentz transformation is used.

The matrix \(\Gamma _{AB}\), constructed using the real matrix parametrization \(\Lambda _{AB}\) of the two-qubit density matrix \(\rho _{AB}\) (see (2.1), (2.3)), exhibits the following important properties (see Theorem of Ref. [38] on the nature of eigenvalues and eigenvectors of the matrix \(\Gamma _{AB}\)):

  1. (i)

    It possesses non-negative eigenvalues \(\mu ^{AB}_0\ge \mu ^{AB}_1\ge \mu ^{AB}_2\ge \mu ^{AB}_3\ge 0\).

  2. (ii)

    Four-eigenvector X associated with the highest eigenvalue \(\mu ^{AB}_0\) of the matrix \(\Gamma _{AB}\) satisfies one of the following Lorentz invariant properties:

    $$\begin{aligned} X^T\, G\, X>0 \end{aligned}$$
    (2.11)

    or

    $$\begin{aligned} X^T\, G\, X =0. \end{aligned}$$
    (2.12)

    The condition (2.12) is accompanied by the observation that the matrix \(\Gamma _{AB}\) has only two eigenvalues \(\mu ^{AB}_0\), \(\mu ^{AB}_2\) with \( \mu ^{AB}_0\ge \mu ^{AB}_2\), both of which are doubly degenerate.

  3. (iii)

    Suppose the eigenvector X satisfies the Lorentz invariant condition (2.11). Then there exist suitable SL(2,C) transformations \(A_1,\, B_1\in \) SL(2,C) (with corresponding Lorentz transformations \(L_{A_1},\, L_{B_1}\in \) SO(3,1), respectively) such that the matrix \(\Gamma _{AB}\) assumes a diagonal canonical form:

    $$\begin{aligned} \bar{\Gamma }_{AB}^{(I_c)}= & {} \left( L^T_{B_{I_c}}\right) ^{-1}\, \Gamma _{AB}\, L^T_{B_{I_c}} =\textrm{diag} \, \left( \mu ^{AB}_0,\, \mu ^{AB}_1,\, \mu ^{AB}_2,\, \mu ^{AB}_3\right) . \end{aligned}$$
    (2.13)
  4. (iv)

    Associated with the standard form \(\bar{\Gamma }_{AB}^{(I_c)}\), it is seen that [38]

    $$\begin{aligned} \rho _{AB}\longrightarrow \bar{\rho }^{\,I_c}_{AB}= & {} \frac{(A_{I_c}\otimes B_{I_c})\, \rho _{AB}\, (A_{I_c}^\dag \otimes B_{I_c}^\dag )}{\textrm{Tr}\left[ \rho _{AB}\, (A_{I_c}^\dag \, A_{I_c} \otimes B_{I_c}^\dag \, B_{I_c})\right] } \end{aligned}$$

    reduces to the Bell diagonal form

    $$\begin{aligned} \bar{\rho }^{\,I_c}_{AB}= & {} \frac{1}{4}\, \left( \sigma _0\otimes \sigma _0 + \sum _{i=1,2}\, \sqrt{\frac{\mu ^{AB}_i}{\mu ^{AB}_0}}\, \sigma _i\otimes \sigma _i \pm \sqrt{\frac{\mu ^{AB}_3}{\mu ^{AB}_0}}\, \sigma _3\otimes \sigma _3 \right) \end{aligned}$$
    (2.14)

    under local SL(2,C) operations. Here the sign ± is chosen based on sgn\([\det (\Lambda _{AB})]=\pm \).

  5. (v)

    Whenever the eigenvector X obeys the condition (2.12), suitable local SL(2,C) transformations \(A_{II_c},\, B_{II_c}\in \) SL(2,C) (associated Lorentz transformations denoted, respectively, by \(L_{A_{II_c}},\, L_{B_{II_c}}\in \) SO(3,1)) exist such that the real symmetric matrix \(\Gamma _{AB}\) takes the following canonical form:

    $$\begin{aligned} \bar{\Gamma }_{AB}^{(II_c)}= & {} \left( L^T_{B_{II_c}}\right) ^{-1}\, \Gamma _{AB}\, L^T_{B_{II_c}} = \begin{pmatrix} \phi ^{AB}_0 &{} 0 &{} 0 &{} \phi ^{AB}_0-\mu ^{AB}_0 \\ 0 &{} \mu ^{AB}_2 &{} 0 &{} 0 \\ 0 &{} 0 &{} \mu ^{AB}_2 &{} 0 \\ \mu _0^{AB}-\phi ^{AB}_0 &{} 0 &{} 0 &{} 2\,\mu ^{AB}_0-\phi ^{AB}_0, \end{pmatrix}\nonumber \\ \end{aligned}$$
    (2.15)

    where

    $$\begin{aligned} \phi ^{AB}_0=\left( L_{B_{II_c}}\,\Gamma _{AB}\,L_{B_{II_c}}^T\right) _{00}. \end{aligned}$$
    (2.16)
  6. (vi)

    Consequently, the canonical form of the two-qubit density matrix is given by

    $$\begin{aligned} \rho _{AB}\longrightarrow \bar{\rho }^{\,II_c}_{AB}= & {} \frac{(A_{II_c}\otimes B_{II_c})\, \rho _{AB}\, (A_{II_c}^\dag \otimes B_{II_c}^\dag )}{\textrm{Tr}\left[ \rho _{AB}\, (A_{II_c}^\dag \, A_{II_c} \otimes B_{II_c}^\dag \, B_{II_c})\right] }, \nonumber \\= & {} \frac{1}{4}\, \left[ \, \sigma _0\otimes \sigma _0 + (1-\gamma ^{AB}_0)\,\sigma _0\otimes \sigma _3 + \gamma ^{AB}_2 \,(\sigma _1\otimes \sigma _1 - \sigma _2\otimes \sigma _2)\right. \nonumber \\{} & {} \quad \left. +\,\gamma ^{AB}_0\, \sigma _3\otimes \sigma _3\right] \end{aligned}$$
    (2.17)

    where

    $$\begin{aligned} \gamma ^{AB}_0=\frac{\mu ^{AB}_0}{\phi ^{AB}_0},\ \ \ \ \ \gamma ^{AB}_2=\sqrt{\frac{\mu ^{AB}_2}{\phi ^{AB}_0}}, \ \ \ 0\le \left( \gamma ^{AB}_2\right) ^2\le \gamma ^{AB}_0\le 1. \end{aligned}$$
    (2.18)
  7. (vii)

    Corresponding to the canonical form \(\bar{\Gamma }_{AB}^{(I_c)}\) (see (2.13)), an elegant geometric visualization in terms of an ellipsoid inscribed inside the Bloch sphere [38], with semiaxes lengths (i) \(\left( \sqrt{\frac{\mu ^{AB}_1}{\mu ^{AB}_0}},\sqrt{\frac{\mu ^{AB}_2}{\mu ^{AB}_0}}, \sqrt{\frac{\mu ^{AB}_3}{\mu ^{AB}_0}}\right) \) and with center coinciding with that of the Bloch sphere, obeying the equation

    $$\begin{aligned} \left( \frac{\mu ^{AB}_0}{\mu ^{AB}_1}\right) \, x^2+ \left( \frac{\mu ^{AB}_0}{\mu ^{AB}_2}\right) \, y^2+ \left( \frac{\mu ^{AB}_0}{\mu ^{AB}_3}\right) \, z^2=1. \end{aligned}$$
    (2.19)

    Associated with the Lorentz canonical structure \(\bar{\Gamma }_{AB}^{(II_c)}\) (see (2.15)), a shifted spheroid [38], with semiaxes lengths \((\sqrt{\gamma ^{AB}_1}, \sqrt{\gamma ^{AB}_1}, \sqrt{\gamma ^{AB}_0})\) (see (2.18)) and center \((0,\,0,\, (1-\gamma ^{AB}_0))\) inside the Bloch sphere, satisfying the equation

    $$\begin{aligned} \frac{ (x^2+y^2)}{\gamma ^{AB}_1}+ \frac{\left( z- (1-\gamma ^{AB}_0)\right) ^2}{\gamma ^{AB}_0}=1. \end{aligned}$$
    (2.20)

    represents the set of all two-qubit states on the SL(2,C) orbit of \(\bar{\rho }^{\,II_c}_{AB}\) (see (2.17)).

  8. (viii)

    The eigenvalues \(\mu ^{AB}_\alpha ,\ \alpha =0,1,2,3\) of the \(4\times 4\) matrix \(\Gamma _{AB}\) are Lorentz invariant, i.e., they are unchanged under local SL(2,C) operations.

In the following section, we construct local SL(2,C) invariants, by exploiting the Lorentz transformation properties of the real matrices \(\Lambda _{AB},\ \Lambda _{BC}\), and \(\Lambda _{AC}\) characterizing the two-qubit subsystems of a pure three-qubit state.

3 Lorentz invariants of pure three-qubit state

Let us write the two-qubit reduced density matrices \(\rho _{AB}\), \(\rho _{BC}\) and \(\rho _{AC}\) of a pure three-qubit state as

$$\begin{aligned} \rho _{AB}= & {} \textrm{Tr}_\textrm{C}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert = \frac{1}{4}\, \sum _{\alpha ,\,\beta =0}^{3}\, \left( \Lambda _{AB}\right) _{\alpha \, \beta }\, \left( \sigma _\alpha \otimes \sigma _\beta \right) , \end{aligned}$$
(3.1)
$$\begin{aligned} \rho _{BC}= & {} \textrm{Tr}_\textrm{A}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert = \frac{1}{4}\, \sum _{\alpha ,\,\beta =0}^{3}\, \left( \Lambda _{BC}\right) _{\alpha \, \beta }\, \left( \sigma _\alpha \otimes \sigma _\beta \right) , \end{aligned}$$
(3.2)
$$\begin{aligned} \rho _{AC}= & {} \textrm{Tr}_\textrm{B}\, \vert \psi _\textrm{ABC}\rangle \langle \psi _\textrm{ABC}\vert = \frac{1}{4}\, \sum _{\alpha ,\,\beta =0}^{3}\, \left( \Lambda _{AC}\right) _{\alpha \, \beta }\, \left( \sigma _\alpha \otimes \sigma _\beta \right) . \end{aligned}$$
(3.3)

We employ Acín’s canonical form (1.1) of the pure three-qubit state, for evaluating the \(4\times 4\) real matrices \(\Lambda _{AB},\ \Lambda _{BC}\) and \(\Lambda _{AC}\) explicitly:

$$\begin{aligned} \Lambda _{AB}= & {} \left( \begin{array}{cccc} 1&{} 2\,(\lambda _2\,\lambda _4+\lambda _1\lambda _3\,\cos \phi ) &{} -2\, \lambda _1\,\lambda _3\,\sin \phi &{} 1-2\,(\lambda ^2_3+\lambda ^2_4) \\ 2\,\lambda _0\,\lambda _1\,\cos \phi &{} 2\, \lambda _0\,\lambda _3 &{} 0 &{} 2\,\lambda _0\,\lambda _1\,\cos \phi \\ 2\,\lambda _0\,\lambda _1\,\sin \phi &{} 0 &{} -2\, \lambda _0\,\lambda _3 &{} 2\,\lambda _0\,\lambda _1\,\sin \phi \\ 2\, \lambda _0^2-1 &{} -2\,(\lambda _2\,\lambda _4+\lambda _1\lambda _3\,\cos \phi ) &{} 2\,\lambda _1\,\lambda _3\,\sin \phi &{} 1-2\,(\lambda ^2_1+\lambda ^2_2) \end{array}\right) , \end{aligned}$$
(3.4)
$$\begin{aligned} \Lambda _{BC}= & {} \left( \begin{array}{cccc} 1&{} 2\,(\lambda _3\,\lambda _4+\lambda _1\lambda _2\,\cos \phi ) &{} -2\, \lambda _1\,\lambda _2\, \sin \phi \, &{} 1-2\,(\lambda ^2_2+\lambda ^2_4) \\ 2\,(\lambda _2\,\lambda _4+\lambda _1\lambda _3\,\cos \phi ) &{} 2\,(\lambda _2\,\lambda _3+\lambda _1\lambda _4\,\cos \phi ) &{} -2\, \lambda _1\,\lambda _4\, \sin \phi &{} -2\,(\lambda _2\,\lambda _4-\lambda _1\lambda _3\,\cos \phi ) \\ -2\, \lambda _1\,\lambda _3\, \sin \phi &{} -2\, \lambda _1\,\lambda _4\, \sin \phi &{} 2\,(\lambda _2\,\lambda _3-\lambda _1\lambda _4\,\cos \phi ) &{} -2\,\lambda _1\,\lambda _3\,\sin \phi \\ 1-2\,(\lambda ^2_3+\lambda ^2_4) &{} -2\,(\lambda _3\,\lambda _4-\lambda _1\lambda _2\,\cos \phi ) &{} -2\,\lambda _1\,\lambda _2\,\sin \phi &{} 1-2\,(\lambda ^2_2+\lambda ^2_3) \end{array}\right) ,\nonumber \\ \end{aligned}$$
(3.5)
$$\begin{aligned} \Lambda _{AC}= & {} \left( \begin{array}{cccc} 1&{} 2\,(\lambda _3\,\lambda _4+\lambda _1\lambda _2\,\cos \phi ) &{} -2\, \lambda _1\,\lambda _2\,\sin \phi &{} 1-2\,(\lambda ^2_2+\lambda ^2_4) \\ 2\,\lambda _0\,\lambda _1\,\cos \phi &{} 2\, \lambda _0\,\lambda _2 &{} 0 &{} 2\,\lambda _0\,\lambda _1\,\cos \phi \\ 2\,\lambda _0\,\lambda _1\,\sin \phi &{} 0 &{} -2\, \lambda _0\,\lambda _2 &{} 2\,\lambda _0\,\lambda _1\,\sin \phi \\ 2\, \lambda _0^2-1 &{} -2\,(\lambda _3\,\lambda _4+\lambda _1\lambda _2\,\cos \phi ) &{} 2\,\lambda _1\,\lambda _2\,\sin \phi &{} 1-2\,(\lambda ^2_1+\lambda ^2_3) \end{array}\right) . \end{aligned}$$
(3.6)

Let us recall the formula for the concurrence \(C_{AB}\) of an arbitrary two-qubit state \(\rho _{AB}\) introduced by Wootters [41]:

$$\begin{aligned} C_{AB}=\textrm{max}\,\{0,\nu ^{AB}_1-\nu ^{AB}_2-\nu ^{AB}_3-\nu ^{AB}_4\} \end{aligned}$$
(3.7)

where \(\nu ^{AB}_i,\ i=1,2,3,4\) are the square roots of the eigenvalues of

$$\begin{aligned} \rho _{AB}\,\widetilde{\rho }_{AB}= \rho _{AB}\,(\sigma _2\otimes \sigma _2)\, \rho _{AB}^T\, (\sigma _2\otimes \sigma _2) \end{aligned}$$
(3.8)

in decreasing order. While the matrix \(\rho _{AB}\,\widetilde{\rho }_{AB}\) is non-hermitian, it has only real and positive eigenvalues [41].

To gain further insight into the structure of the non-hermitian matrix \(\rho _{AB}\,\widetilde{\rho }_{AB}\), we express the spin flipped two-qubit density matrix \(\widetilde{\rho }_{AB}=(\sigma _2\otimes \sigma _2)\, \rho _{AB}^T\, (\sigma _2\otimes \sigma _2)\) in the basis \(\{\sigma _\alpha \otimes \sigma _\beta , \alpha ,\beta =0,1,2,3\}\) to obtain

$$\begin{aligned} \widetilde{\rho }_{AB}= \frac{1}{4}\, \sum _{\alpha ,\beta =0}^3\, \left( \Lambda ^{'}_{AB}\right) _{\alpha \,\beta }\, \sigma _\alpha \otimes \sigma _\beta \end{aligned}$$
(3.9)

where the \(4\times 4\) real matrix \(\Lambda ^{'}_{AB}\) characterizing the spin flipped two-qubit density matrix \(\widetilde{\rho }_{AB}\) is found to be [46]

$$\begin{aligned} \Lambda ^{'}_{AB}=G\,\Lambda _{AB}\,G. \end{aligned}$$
(3.10)

We thus recognize (see (2.1),(3.9), (3.10) and (2.9)) that

$$\begin{aligned} \textrm{Tr}[\rho _{AB}\,\widetilde{\rho }_{AB}]= \frac{1}{4}\, \textrm{Tr}\,[G\,\Lambda ^T_{AB}\,G\,\Lambda _{AB}]=\frac{1}{4}\, \textrm{Tr}\,[\Gamma _{AB}]. \end{aligned}$$
(3.11)

Evidently, \(\textrm{Tr}\,[\Gamma _{AB}]\ge 0\) as the \(4\times 4\) real matrix \(\Gamma _{AB}=G\,\Lambda ^T_{AB}\,G\,\Lambda _{AB}\) is non-negative [38] and this, in turn, justifies that trace of the non-hermitian matrix \(\rho _{AB}\,\widetilde{\rho }_{AB}\) (LHS of (3.11)) is also positive.

In a pure entangled three-qubit state, every pair of qubits are entangled with the remaining qubit. Thus, the two-qubit subsystem density matrix of a pure three-qubit state has at most two nonzero eigenvalues. As a result, the matrix \(\rho _{ij}\,\widetilde{\rho }_{ij}\) has only two nonzero eigenvalues \(\left( \nu ^{ij}_1\right) ^2,\left( \nu ^{ij}_2\right) ^2,\ ij=AB, BC,AC\). Thus, the squared concurrence \(C^2_{ij}\) of two-qubit subsystem state \(\rho _{ij}\) of a three-qubit pure state simplifies to

$$\begin{aligned} C^2_{ij}=(\nu ^{ij}_1-\nu ^{ij}_2)^2, \ \ ij=AB, BC,AC. \end{aligned}$$
(3.12)

We are interested in recognizing local SL(2,C) invariants of three-qubit pure state, which are useful in determining the Lorentz invariant eigenvalues of the matrices \(\Gamma _{AB}, \ \Gamma _{BC}\), and \(\Gamma _{AC}.\) In Acín’s canonical form (1.1), the matrices \(\Gamma _{AB}, \ \Gamma _{BC}\) and \(\Gamma _{AC}\) have the following explicit structure:

$$\begin{aligned} \Gamma _{AB}= & {} \left( \begin{array}{cccc} 4\, \lambda _0^2 (\lambda _2^2+\lambda _3^2+\lambda _4^2) &{} 4\, \lambda _0^2\, \lambda _2\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2\, \lambda _2^2 \\ -4\, \lambda _0^2\, \lambda _2\, \lambda _4 &{} 4\, \lambda _0^2\, \lambda _3^2 &{} 0 &{} -4\, \lambda _0^2\, \lambda _2\, \lambda _4 \\ 0 &{} 0 &{} 4\, \lambda _0^2\, \lambda _3^2 &{} 0 \\ -4\, \lambda _0^2\, \lambda _2^2 &{} -4\, \lambda _0^2\, \lambda _2\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2 (\lambda _3^2-\lambda _2^2+\lambda _4^2) \end{array}\right) \nonumber \\ \end{aligned}$$
(3.13)
$$\begin{aligned} \Gamma _{BC}= & {} \left( \begin{array}{cccc} 4\, \lambda _0^2\,(\lambda _3^2+\lambda _4^2)+4\,\bigtriangleup ) &{} 4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2\, \lambda _3^2 \\ -4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 4\, \bigtriangleup &{} 0 &{} -4\, \lambda _0^2\, \lambda _3\, \lambda _4 \\ 0 &{} 0 &{} 4\,\bigtriangleup &{} 0 \\ -4\, \lambda _0^2\, \lambda _3^2 &{} -4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2\,(\lambda _4^2-\lambda _3^2)+4\,\bigtriangleup ) \end{array}\right) \nonumber \\ \end{aligned}$$
(3.14)
$$\begin{aligned} \Gamma _{AC}= & {} \left( \begin{array}{cccc} 4\, \lambda _0^2 (\lambda _2^2+\lambda _3^2+\lambda _4^2) &{} 4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2\, \lambda _3^2 \\ -4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 4\, \lambda _0^2\, \lambda _2^2 &{} 0 &{} -4\, \lambda _0^2\, \lambda _3\, \lambda _4 \\ 0 &{} 0 &{} 4\, \lambda _0^2\, \lambda _2^2 &{} 0 \\ -4\, \lambda _0^2\, \lambda _3^2 &{} -4\, \lambda _0^2\, \lambda _3\, \lambda _4 &{} 0 &{} 4\, \lambda _0^2 (\lambda _2^2-\lambda _3^2+\lambda _4^2) \end{array}\right) .\nonumber \\ \end{aligned}$$
(3.15)

  • We find that these matrices \(\Gamma _{AB}\), \(\Gamma _{BC}\), and \(\Gamma _{AC}\) have at most two distinct eigenvalues:

    $$\begin{aligned} \mu _0^{AB}= & {} \mu _1^{AB}=4\, \lambda _0^2\,(\lambda _3^2+\lambda _4^2),\ \mu _2^{AB}=\mu _3^{AB}=4\, \lambda _0^2\,\lambda _3^2, \end{aligned}$$
    (3.16)
    $$\begin{aligned} \mu _0^{BC}= & {} \mu _1^{BC}=4\, (\bigtriangleup + \lambda _0^2\,\lambda _4^2), \ \mu _2^{BC}=\mu _3^{BC}=4\, \bigtriangleup , \end{aligned}$$
    (3.17)
    $$\begin{aligned} \mu _0^{AC}= & {} \mu _1^{AC}=4\, \lambda _0^2\,(\lambda _2^2+\lambda _4^2),\ \mu _2^{AC}=\mu _3^{AC}=4\, \lambda _0^2\,\lambda _2^2. \end{aligned}$$
    (3.18)

  • We notice an interesting feature that the differences between the largest and the smallest eigenvalues of \(\Gamma _{AB}\), \(\Gamma _{BC}\), \(\Gamma _{AC}\) are identically equal to the three-tangle \(\tau \) of the three-qubit state:

    $$\begin{aligned} \mu _0^{ij}-\mu _2^{ij}= 4\, \lambda _0^2\,\lambda _4^2=\tau , \ \ \ \ ij=AB, BC,AC. \end{aligned}$$
    (3.19)

    This reveals the fact that the LU invariant \(I_5=\frac{\tau ^2}{16}\) of the three-qubit state (see last line of (1.2)) is a permutation symmetric local SL(2,C) invariant. It is worth noting that \(\sqrt{I_5}=\lambda _0^2\,\lambda _4^2\) is equal to the product [15] \(\nu ^{AB}_1\,\nu ^{AB}_2=\nu ^{BC}_1\,\nu ^{BC}_2=\nu ^{AC}_1\,\nu ^{AC}_2\) of the square root of the eigenvalues of \(\rho _{ij}\,\widetilde{\rho }_{ij},\ ij=AB, BC,AC\). Thus,

    $$\begin{aligned} \mu _0^{ij}-\mu _2^{ij}= 4\, \nu ^{ij}_1\,\nu ^{ij}_2. \end{aligned}$$
    (3.20)

  • The Lorentz invariant eigenvalues of \(\Gamma _{AB}\), \(\Gamma _{BC}\) and \(\Gamma _{AC}\) can be determined using \(I_5\) along with three more invariants given by

    $$\begin{aligned} {{\mathcal {K}}}_1= & {} \frac{1}{4}\,\textrm{Tr}[\, \Gamma _{AB}] =\frac{1}{2}\,(\mu _0^{AB}+\mu _2^{AB})\nonumber \\ {{\mathcal {K}}}_2= & {} \frac{1}{4}\,\textrm{Tr}[\, \Gamma _{BC}]=\frac{1}{2}\,(\mu _0^{BC}+\mu _2^{BC}) \nonumber \\ {{\mathcal {K}}}_3= & {} \frac{1}{4}\,\textrm{Tr}[\, \Gamma _{AC}]= \frac{1}{2}\, (\mu _0^{AC}+\mu _2^{AC}). \end{aligned}$$
    (3.21)

    Substituting (3.11) in (3.21), we obtain

    $$\begin{aligned} {{\mathcal {K}}}_1= & {} \textrm{Tr}[\rho _{AB}\,\widetilde{\rho }_{AB}]=(\nu ^{AB}_1)^2+(\nu ^{AB}_2)^2, \nonumber \\ {{\mathcal {K}}}_2= & {} \textrm{Tr}[\rho _{BC}\,\widetilde{\rho }_{BC}]= (\nu ^{BC}_1)^2+(\nu ^{BC}_2)^2, \nonumber \\ {{\mathcal {K}}}_3= & {} \textrm{Tr}[\rho _{AC}\,\widetilde{\rho }_{AC}]= (\nu ^{AC}_1)^2+(\nu ^{AC}_2)^2. \end{aligned}$$
    (3.22)

We proceed to prove the following theorem:

Theorem 1

The squared concurrence \(C^2_{ij}\) of the two-qubit subsystem \(\rho _{ij}\) of a pure three-qubit state is equal to the smallest Lorentz invariant eigenvalue \(\mu ^{ij}_{2}\) of the \(4\times 4\) matrix \(\Gamma _{ij}~=~G\,\Lambda ^T_{ij}\,G\,\Lambda _{ij},\ \ \ ij=AB,BC, AC.\)

Proof

Using (3.20), (3.21) and (3.22), we connect the eigenvalues of \(\Gamma _{AB},\ \Gamma _{BC},\ \Gamma _{AC}\) with those of \(\rho _{AB}\,\widetilde{\rho }_{AB},\ \rho _{BC}\,\widetilde{\rho }_{BC},\ \rho _{AC}\,\widetilde{\rho }_{AC}\), respectively:

$$\begin{aligned} 4\, \nu ^{ij}_1 \nu ^{ij}_2= & {} \mu _0^{ij}-\mu _2^{ij}, \nonumber \\ \left[ \,(\nu ^{ij}_1)^2+(\nu ^{ij}_2)^2\right]= & {} \frac{1}{2}\,(\mu _0^{ij}+\mu _2^{ij}), \ \ \ \ ij=AB, BC,AC. \end{aligned}$$
(3.23)

We thus obtain (see (3.12))

$$\begin{aligned} \mu _2^{ij}=\left( \nu ^{ij}_1-\nu ^{ij}_2\right) ^2=C^2_{ij},\ \ \ ij=AB, BC,AC. \end{aligned}$$
(3.24)

The above theorem offers an interesting alternate method to evaluate concurrences of two-qubit subsystems \(\rho _{ij}, \ ij=AB,BC,AC\) of a pure three-qubit state, in terms of the smallest Lorentz invariant eigenvalues of \(\Gamma _{ij}\). \(\square \)

  • With the help of the following Lorentz transformations

    $$\begin{aligned} L_B=\left( \begin{array}{cccc} \frac{\tau + 2\, C^2_{AC}}{2\,C_{AC}\,\sqrt{\tau }} &{} \frac{C_{AC}}{\sqrt{\tau }} &{} 0 &{} \frac{\sqrt{\tau }}{2\,C_{AC}} \\ -1 &{} -1 &{} 0 &{} -1 \\ 0 &{} 0 &{} -1 &{} 0 \\ \frac{\tau - 2\, C^2_{AC}}{2\,C_{AC}\,\sqrt{\tau }} &{} -\frac{C_{AC}}{\sqrt{\tau }} &{} 0 &{} \frac{\sqrt{\tau }}{2\,C_{AC}} \end{array}\right) \end{aligned}$$
    (3.25)

    and

    $$\begin{aligned} L_C=\left( \begin{array}{cccc} \frac{\tau + 2\, C^2_{AB}}{2\,C_{AB}\,\sqrt{\tau }} &{} \frac{C_{AB}}{\sqrt{\tau }} &{} 0 &{} \frac{\sqrt{\tau }}{2\,C_{AB}} \\ -1 &{} -1 &{} 0 &{} -1 \\ 0 &{} 0 &{} -1 &{} 0 \\ \frac{\tau - 2\, C^2_{AB}}{2\,C_{AB}\,\sqrt{\tau }} &{} -\frac{C_{AB}}{\sqrt{\tau }} &{} 0 &{} \frac{\sqrt{\tau }}{2\,C_{AB}} \end{array}\right) \end{aligned}$$
    (3.26)

    it is seen that

    $$\begin{aligned} \left( L^T_B\right) ^{-1}\Gamma _{AB}L^T_B=\bar{\Gamma }_{AB}^{(II_c)}= & {} \left( \begin{array}{cccc} C^2_{AB}+\tau &{} 0 &{} 0 &{} 0 \\ 0 &{} C^2_{AB} &{} 0 &{} 0 \\ 0 &{} 0 &{} C^2_{AB} &{} 0 \\ 0 &{} 0 &{} 0 &{} C^2_{AB}+\tau \end{array}\right) \end{aligned}$$
    (3.27)
    $$\begin{aligned} \left( L^T_C\right) ^{-1}\Gamma _{BC}L^T_C=\bar{\Gamma }_{BC}^{(II_c)}= & {} \left( \begin{array}{cccc} C^2_{BC}+\tau &{} 0 &{} 0 &{} 0 \\ 0 &{} C^2_{BC} &{} 0 &{} 0 \\ 0 &{} 0 &{} C^2_{BC} &{} 0 \\ 0 &{} 0 &{} 0 &{} C^2_{BC}+\tau \end{array}\right) \end{aligned}$$
    (3.28)
    $$\begin{aligned} \left( L^T_C\right) ^{-1}\Gamma _{AC}L^T_C=\bar{\Gamma }_{AC}^{(II_c)}= & {} \left( \begin{array}{cccc} C^2_{AC}+\tau &{} 0 &{} 0 &{} 0 \\ 0 &{} C^2_{AC} &{} 0 &{} 0 \\ 0 &{} 0 &{} C^2_{AC} &{} 0 \\ 0 &{} 0 &{} 0 &{} C^2_{AC}+\tau \end{array}\right) . \end{aligned}$$
    (3.29)

    It follows that (see (2.16),(2.18))

    $$\begin{aligned} \gamma ^{ij}_0=1,\ \ \gamma ^{ij}_2=\frac{C_{ij}}{\sqrt{C^2_{ij}+\tau }}, \ ij=AB,\, BC,\, AC. \end{aligned}$$
    (3.30)

    A spheroid inside the Bloch sphere (see Fig. 1) with semiaxes lengths \(\left( \frac{C_{ij}}{\sqrt{C^2_{ij}+\tau }}, \frac{C_{ij}}{\sqrt{C^2_{ij}+\tau }}, 1\right) \) and origin (0,0,0) (see (2.20), (3.30) offers geometrical visualization of the reduced two-qubit systems of the three-qubit pure state \(\vert \psi _{ABC}\rangle \) given in (1.1).

Fig. 1
figure 1

Spheroid centered at the origin of the Bloch sphere with semiaxes lengths \(\left( \frac{C}{\sqrt{C^2+\tau }}, \frac{C}{\sqrt{C^2+\tau }},1\right) \), for \(C=0.8\) and \(\tau =0.5\) (Color figure online)

Full size image

3.1 Permutation symmetric three-qubit pure states

It is well known that permutation symmetric states offer conceptual clarity and computational simplicity in the analysis of local invariants [34, 39, 47,48,49]. In this subsection, we illustrate the effectiveness of our framework to evaluate concurrence and tangle in pure three-qubit permutation symmetric states, where we make use of the explicit parametrization given by Meill and Meyer [34] recently.

Consider a one-parameter family of three-qubit permutation symmetric state [34, 39]:

$$\begin{aligned} \vert \psi _\textrm{sym}(\beta )\rangle = \frac{1}{\sqrt{2+\cos \beta }}\, \left( \sqrt{3}\,\cos \frac{\beta }{2}\,\left| 0_A\,0_B\, 0_C\,\right\rangle + \sin \frac{\beta }{2}\,\left| \textrm{W}\,\right\rangle \right) \end{aligned}$$
(3.31)

where \(0<\beta \le \,\pi \). We find that [39]

$$\begin{aligned} \Gamma _\textrm{sym}(\beta )= & {} \Gamma _{AB}(\beta )=\Gamma (\beta )_{BC}=\Gamma _{AC}(\beta )\nonumber \\= & {} \left( \begin{array}{cccc} 2\,u(\beta ) &{} 0 &{}0 &{} u(\beta ) \\ 0&{} u(\beta ) &{} 0 &{} 0 \\ 0 &{} 0 &{} u(\beta ) &{} 0 \\ -u(\beta ) &{} 0 &{} 0 &{} 0\end{array} \right) ,\ \ \ \ u(\beta )=\left[ \frac{1-\cos \beta }{3(2+\cos \beta )}\right] ^2. \end{aligned}$$
(3.32)

The Lorentz invariant eigenvalues of \(\Gamma _\textrm{sym}(\beta )\) are equal, i.e.,

$$\begin{aligned} \mu _{0}(\beta )=\mu _{2}(\beta )=\left[ \frac{1-\cos \beta }{3(2+\cos \beta )}\right] ^2, \end{aligned}$$
(3.33)

and hence, the concurrence of \(\rho _\textrm{sym}(\beta )=\rho _{AB}(\beta )=\rho _{BC}(\beta )=\rho _{AC}(\beta )\) is given by

$$\begin{aligned} C(\beta )=\frac{1-\cos \beta }{3(2+\cos \beta )}, \end{aligned}$$
(3.34)

in perfect agreement with the result given by Meill and Meyer [34]. Substitution of (3.33) in the LHS of (3.19) confirms that the three-tangle \(\tau (\beta )\) for the state (3.31) is zero.

We proceed further with a three-parameter family of pure three-qubit permutation symmetric state [34]

$$\begin{aligned} \vert \psi _\textrm{sym}(y,\beta ,\phi )\rangle= & {} N \left( \vert 0\,\rangle ^{\otimes \,3} + y\, e^{i\, \phi } \, \vert \beta \rangle ^{\otimes \,3} \right) , \end{aligned}$$
(3.35)

where \(\vert \beta \rangle =\cos \,\frac{\beta }{2}\, \vert 0\rangle +\sin \,\frac{\beta }{2}\,\vert 1\rangle ,\ 0<y\le 1, \ 0\le \phi \le 2\pi ,\ 0<\beta \le \pi .\) We evaluate the matrix \(\Gamma _\textrm{sym}(y,\beta ,\phi )\equiv \Gamma _{AB}(y,\beta ,\phi )=\Gamma _{BC}(y,\beta ,\phi )=\Gamma _{AC}(y,\beta ,\phi )\) in the state (3.35) (see Ref. [39] for details):

$$\begin{aligned} \Gamma _\textrm{sym}(y,\beta ,\phi )= & {} {{\mathcal {B}}}(y,\beta ,\phi )\left( \begin{array}{cccc} 3+\cos \beta &{} \sin \beta &{} 0 &{} 1+\cos \beta \\ -\sin \beta &{} (1+\cos \beta ) &{} 0 &{} -\sin \beta \\ 0 &{} 0 &{} (1+\cos \beta ) &{} 0 \\ -(1+\cos \beta ) &{} -\sin \beta &{} 0 &{} (1-\cos \beta ) \end{array}\right) \nonumber \\ \end{aligned}$$
(3.36)

where

$$\begin{aligned} {{\mathcal {B}}}(y,\beta ,\phi )=\frac{y^2 (1-\cos \beta )^2}{2\left( 1+y^2+2\,y\cos \,\phi \cos ^3 \frac{\beta }{2} \right) ^{2}}. \end{aligned}$$
(3.37)

The Lorentz invariant eigenvalues of \(\Gamma _\textrm{sym}(y,\beta ,\phi )\) are found to be

$$\begin{aligned} \mu _{0}(y,\beta ,\phi )= 2\,{{\mathcal {B}}}(y,\beta ,\phi ), \ \mu _{2}(y,\beta ,\phi )={{\mathcal {B}}}(y,\beta ,\phi )\,\,(1+\cos \beta ). \end{aligned}$$
(3.38)

The concurrence for the two-qubit subsystem density matrices \(\rho _\textrm{sym}(y,\beta ,\phi )=\rho _{AB}(y,\beta ,\phi )=\rho _{BC}(y,\beta ,\phi )=\rho _{AC}(y,\beta ,\phi )\) drawn from the three-qubit pure symmetric state (3.35) is thus given by

$$\begin{aligned} C(y,\beta ,\phi )= & {} \sqrt{{{\mathcal {B}}}(y,\beta ,\phi )\,\,(1+\cos \beta )}\nonumber \\= & {} \frac{2\,y\, \sin \beta \, \sin \frac{\beta }{2} }{\left( 1+y^2+2\,y\cos \,\phi \cos ^3 \frac{\beta }{2} \right) } \end{aligned}$$
(3.39)

which matches exactly with the formula derived in Ref. [34].

Substituting (3.38) in (3.19), we evaluate the three-tangle \(\tau (y,\beta ,\phi )\) in (3.35) to obtain

$$\begin{aligned} \tau (y,\beta ,\phi )= & {} \mu _0(y,\beta ,\phi )-\mu _{2}(y,\beta ,\phi ) \nonumber \\= & {} {{\mathcal {B}}}(y,\beta ,\phi )(1-\cos \beta ) \nonumber \\= & {} \left( \frac{2\,y\, \sin ^3\frac{\beta }{2} }{1+y^2+2\,y\cos \,\phi \cos ^3 \frac{\beta }{2}}\right) ^2, \end{aligned}$$
(3.40)

in agreement with the expression for \(\tau ^2\) given in Ref. [34] for the state (3.35).

3.2 LU versus local SL(2,C) invariants of pure three-qubit state

Taking a closer look at the set of five LU invariants (1.2) of pure three-qubit states, it is seen that \(I_1=\textrm{Tr}[\rho _{AB}^2]\), \(I_2=\textrm{Tr}[\rho _{AC}^2]\), \(I_3=\textrm{Tr}[\rho _{AB}^2]\) get replaced by their local SL(2,C) counterparts (see (3.22)) \({{\mathcal {K}}}_1=\textrm{Tr}[\rho _{AB}\,\widetilde{\rho }_{AB}]\), \({{\mathcal {K}}}_2=\textrm{Tr}[\rho _{BC}\,\widetilde{\rho }_{BC}]\), \({{\mathcal {K}}}_3=\textrm{Tr}[\rho _{AC}\,\widetilde{\rho }_{AC}]\). It is seen that the LU invariant \(I_5=\tau ^2/16\) enjoys a higher level of invariance, by remaining unchanged when a three-qubit pure state undergoes local SL(2,C) transformation. Thus, we have four local SL(2,C) invariants, which encode information about the entanglement content in the three-qubit pure state since it is possible to reconstruct concurrences \(C_{AB},\ C_{BC},\ C_{AC}\) and three-tangle \(\tau \) using them. On the other hand, the Kempe invariant \({{\mathcal {I}}}_4\) given by (1.4) (which is a permutation symmetric extension of the LU invariant \(I_4\) listed in (1.2)) is known to be algebraically independent of concurrences and three tangles [28]. In order to complete the set of SLOCC invariants, we study the structure of \(I_4\) with an intention to find its Lorentz invariant analogue. Using (2.1), (2.4), (2.5), we obtain

$$\begin{aligned} I_4=\textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ B})\,\rho _{AB}]=\frac{1}{4}\, \left( \textbf{s}_{A}^T\,\Lambda _{AB}\, \textbf{s}_B\right) . \end{aligned}$$
(3.41)

We consider [50]

$$\begin{aligned} {{\mathcal {K}}}_4=\textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ B})\,\widetilde{\rho }_{AB}]=\frac{1}{4}\, \left( \textbf{s}_{A}^T\,G\,\Lambda _{AB}\,G\,\textbf{s}_B\right) . \end{aligned}$$
(3.42)

to be the Lorentz invariant analogue replacing \(I_4\). Thus, we have the following set of five local SL(2,C) invariants

$$\begin{aligned} {{\mathcal {K}}}_1= & {} 2\, \textrm{Tr}[\rho _{AB}\,\widetilde{\rho }_{AB}]= 4\,\lambda _0^2\,\lambda _3^2+2\, \lambda _0^2\lambda _4^2= C^2_{AB}+\frac{\tau }{2}, \nonumber \\ {{\mathcal {K}}}_2= & {} 2\, \textrm{Tr}[\rho _{BC}\,\widetilde{\rho }_{BC}]=4\,\bigtriangleup +2\, \lambda _0^2\lambda _4^2=C^2_{BC}+\frac{\tau }{2}, \nonumber \\ {{\mathcal {K}}}_3= & {} 2\, \textrm{Tr}[\rho _{AC}\,\widetilde{\rho }_{AC}]=4\,\lambda _0^2\,\lambda _2^2+2\,\lambda _0^2\,\lambda _4^2=C^2_{AC}+\frac{\tau }{2}, \nonumber \\ {{\mathcal {K}}}_4= & {} \textrm{Tr}\,[(\rho _{ A}\otimes \rho _{ B})\,\widetilde{\rho }_{AB}]= \lambda _0^2\,\left( \bigtriangleup +\lambda _2^2 \lambda _3^2-\lambda _1^2 \lambda _4^2+ \lambda _3^2 +\lambda _4^2\right) , \nonumber \\ {{\mathcal {K}}}_5\equiv & {} I_5= \lambda _0^4\,\lambda _4^4=\frac{\tau ^2}{16}, \end{aligned}$$
(3.43)

which are the algebraic counterparts of the LU invariants of the three-qubit pure state.

4 Summary

In this paper, we have extended the mathematical framework of Ref. [38] to explore local SL(2,C) invariants of pure three-qubit states. This method enables one to evaluate concurrences and tangle in terms of the Lorentz invariant eigenvalues of the \(4\times 4\) real positive matrices \(\Gamma _{ij}=G\,\Lambda ^T_{ij}\, G\, \Lambda _{ij},\) constructed from the real parametrizations \(\Lambda _{ij}\) of the two-qubit subsystem density matrices \(\rho _{ij},\ \ ij=AB, BC,CA\) of a pure state of three qubits. In particular, we have shown that (i) the matrices \(\Gamma _{ij}\) evaluated in a pure three-qubit state have at most two distinct eigenvalues, (ii) the squared concurrence \(C^2_{ij}\) is equal to the least eigenvalue of \(\Gamma _{ij}\), and (iii) the three-tangle \(\tau \) is equal to the difference between the highest and the smallest eigenvalues of \(\Gamma _{ij}\). This is illustrated in the example of permutation symmetric three-qubit pure states. Finally, we have given a set of five local SL(2,C) invariants \(\{ {{\mathcal {K}}}_1,\, {{\mathcal {K}}}_2,\, {{\mathcal {K}}}_3,\, {{\mathcal {K}}}_4,\,{{\mathcal {K}}}_5\}\), which are the natural algebraic generalizations of Acín’s LU invariants \(\{ I_1,\, I_2,\, I_3,\, I_4,\,I_5\}\).