1. Introduction
Quantum entanglement is one of the most fascinating phenomena of quantum theory. It serves as a fundamental resource for applications in quantum computing, quantum communication, and quantum teleportation [1–3]. Therefore, it is crucial to distinguish entangled states from non-entangled (or separable) states. To date, a number of separable criteria have been proposed [4–6]. Nevertheless, most of them rely on quantum state tomography [7], which is highly challenging, particularly for high-dimensional quantum systems. Entanglement witnesses (EWs) [8] provide an economical means of experimentally detecting entanglement without performing state tomography.
An EW is defined as an observable whose expectation value is non-negative for all separable states, yet negative for at least one entangled state [8]. It is established in [9] that every entangled state can be detected by at least one EW. Accordingly, a variety of systematic methods for constructing EWs have been developed [10].
An EW W is said to be decomposable [11] if it can be expressed as
$\begin{eqnarray*}W=M+{N}^{{\rm{\Gamma }}},\end{eqnarray*}$
where M and N are Hermitian positive semi-definite, and Γ denotes the partial transposition. Clearly, such EWs are only capable of detecting entangled states with a non-positive partial transposition. An EW W is said to be extremal [10] if it cannot be expressed as a convex combination of two distinct EWs. It is shown in [10] that a decomposable EW W is extremal if and only if it takes the form $\begin{eqnarray*}W=| \phi \rangle {\langle \phi | }^{{\rm{\Gamma }}},\end{eqnarray*}$
where ∣φ⟩ is an entangled vector.Detecting entanglement using only a limited set of local measurements is crucial for the experimental implementation of EWs [12–14]. Gühne et al [15] proposed a general method for experimentally detecting entanglement using only a few local measurements, given some prior knowledge of the density matrix. Noise-robust EWs for detecting genuine multipartite entanglement are presented in [16], which require only two local measurement settings independent of the number of qubits.
For two-qubit systems, Riccardi et al [17] have fully characterized the extremal decomposable EWs derived from the following set of local measurements:
$\begin{eqnarray}{ \mathcal A }=\{{\sigma }_{1}\otimes {\sigma }_{1},{\sigma }_{2}\otimes {\sigma }_{2},{\sigma }_{3}\otimes {\sigma }_{3}\}.\end{eqnarray}$
The generalization to two-qudit systems has been thoroughly investigated in [18] based on the fixed set of local measurements: $\begin{eqnarray*}{ \mathcal B }=\{{G}_{k}^{D}\otimes {G}_{l}^{D},{G}_{ij}^{S}\otimes {G}_{ij}^{S},{G}_{ij}^{A}\otimes {G}_{ij}^{A}\},\end{eqnarray*}$
where ${G}_{k}^{D}$, ${G}_{ij}^{S}$ and ${G}_{ij}^{A}$ denote the generalized Gell-Mann matrices [19]. These EWs based on limited local measurements have been experimentally demonstrated in [14]. In this paper, for two-qubit systems, we completely characterize the extremal decomposable EWs from another fixed set of local measurements: $\begin{eqnarray}{ \mathcal C }=\{{\sigma }_{1}\otimes {\sigma }_{3},{\sigma }_{2}\otimes {\sigma }_{2},{\sigma }_{3}\otimes {\sigma }_{1}\}.\end{eqnarray}$
Such witnesses can be used to detect gravitational entanglement [20–22]. In addition, we extend the corresponding characterization to higher-dimensional systems, where we explicitly construct the extremal decomposable EWs.The remainder of the paper is organized as follows. Section 2 is devoted to a complete characterization of extremal decomposable EWs for two-qubit systems. In section 3 , we extend our investigation to higher-dimensional systems. Finally, concluding remarks are presented in section 4 .
2. Extremal decomposable EWs for two-qubit systems
Any two-qubit Hermitian matrix can be expressed as follows: 3 ). The complete characterization of extremal decomposable EWs with form (3 ) is given by the following theorem.
$\begin{eqnarray*}\begin{array}{rcl}M & = & \alpha {\mathbb{I}}\displaystyle \otimes {\mathbb{I}}+\displaystyle \sum _{k=1}^{3}({a}_{k}{\mathbb{I}}\displaystyle \otimes {\sigma }_{k}+{b}_{k}{\sigma }_{k}\displaystyle \otimes {\mathbb{I}})\\ & & +\displaystyle \sum _{k,j=1}^{3}{c}_{kj}{\sigma }_{k}\displaystyle \otimes {\sigma }_{j},\end{array}\end{eqnarray*}$
where σi, i = 1, 2, 3, are Pauli matrices and $C={({c}_{kj})}_{3\times 3}$ is called the correlation matrix. In this section, we consider the EWs with anti-diagonal local correlations: $\begin{eqnarray}\begin{array}{rcl}W & = & \alpha {\mathbb{I}}\displaystyle \otimes {\mathbb{I}}+\displaystyle \sum _{k=1}^{3}({a}_{k}{\mathbb{I}}\displaystyle \otimes {\sigma }_{k}+{b}_{k}{\sigma }_{k}\displaystyle \otimes {\mathbb{I}})\\ & & +{c}_{13}{\sigma }_{1}\displaystyle \otimes {\sigma }_{3}+{c}_{22}{\sigma }_{2}\displaystyle \otimes {\sigma }_{2}+{c}_{31}{\sigma }_{3}\displaystyle \otimes {\sigma }_{1}.\end{array}\end{eqnarray}$
The following EWs [20–22] are employed to detect the entanglement between two optomechanical systems, which is generated via their mutual gravitational interaction: $\begin{eqnarray}\begin{array}{rcl}{W}_{1} & = & {\mathbb{I}}\displaystyle \otimes {\mathbb{I}}+{\sigma }_{1}\displaystyle \otimes {\sigma }_{3}+{\sigma }_{2}\displaystyle \otimes {\sigma }_{2},\\ {W}_{2} & = & {\mathbb{I}}\displaystyle \otimes {\mathbb{I}}+{\sigma }_{1}\displaystyle \otimes {\sigma }_{3}-{\sigma }_{3}\displaystyle \otimes {\sigma }_{1}+{\sigma }_{2}\displaystyle \otimes {\sigma }_{2}.\end{array}\end{eqnarray}$
It is evident that both EWs W1 and W2 take the form (For any 2 ⨂ 2 quantum system, there exist exactly six families of extremal decomposable EWs ∣φ⟩⟨φ∣Γ with the form (
$\begin{eqnarray}\begin{array}{rcl}|{\varphi }_{1}^{\pm }\rangle & = & a|{\tau }^{\pm }\rangle +b|{\sigma }^{\mp }\rangle ,|{\varphi }_{2}^{\pm }\rangle =a|{\xi }^{\pm }\rangle +b|{\mu }^{\mp }\rangle ,\\ |{\varphi }_{3}^{\pm }\rangle & = & \displaystyle \frac{1}{\sqrt{2}}\left(|{\beta }^{\pm }\rangle +{{\rm{e}}}^{{\rm{i}}c}|{\gamma }^{\mp }\rangle \right),\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rc}\left|{\tau }^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|00\right\rangle \pm \left|01\right\rangle \right),\left|{\sigma }^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|10\right\rangle \pm \left|11\right\rangle \right),\\ \left|{\xi }^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|00\right\rangle \pm \left|10\right\rangle \right),\left|{\mu }^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|01\right\rangle \pm \left|11\right\rangle \right),\\ \left|{\beta }^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|00\right\rangle \pm \left|11\right\rangle \right),\left|{\gamma }^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|01\right\rangle \pm \left|10\right\rangle \right),\end{array}\end{eqnarray*}$
and the parameters $a,b,c\in {\mathbb{R}}$ satisfy the conditions ab ≠ 0, a2 + b2 = 1, − π < c ≤ π, and $c\ne \pm \frac{\pi }{2}.$Any pure state ∣φ⟩ can be expressed as
$\begin{eqnarray}\begin{array}{rl} & | \varphi \rangle ={{\rm{e}}}^{\,\rm{i}\,{x}_{0}}m\left|00\right\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}n\left|01\right\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{2}}q\left|10\right\rangle +t\left|11\right\rangle ,\end{array}\end{eqnarray}$
where −π < xi ≤ π for i = 0, 1, 2, and the real parameters m, n, q, t satisfy the normalization condition $\begin{eqnarray}{m}^{2}+{n}^{2}+{q}^{2}+{t}^{2}=1.\end{eqnarray}$
Due to the (anti-)symmetry of the Pauli matrices, both ∣φ⟩⟨φ∣ and its partial transpose ∣φ⟩⟨φ∣Γ conform to the form in (From (
$\begin{eqnarray*}\begin{array}{rcl}| \varphi \rangle \langle \varphi | & = & {m}^{2}| 00\rangle \langle 00| +{{\rm{e}}}^{\,\rm{i}\,({x}_{0}-{x}_{1})}mn| 00\rangle \langle 01| \\ & & +{{\rm{e}}}^{\,\rm{i}\,({x}_{0}-{x}_{2})}mq| 00\rangle \langle 10| +{{\rm{e}}}^{\,\rm{i}\,{x}_{0}}mt| 00\rangle \langle 11| \\ & & +{{\rm{e}}}^{\,\rm{i}\,({x}_{1}-{x}_{0})}mn| 01\rangle \langle 00| +{n}^{2}| 01\rangle \langle 01| \\ & & +{{\rm{e}}}^{\,\rm{i}\,({x}_{1}-{x}_{2})}nq| 01\rangle \langle 10| +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}nt| 01\rangle \langle 11| \\ & & +{{\rm{e}}}^{\,\rm{i}\,({x}_{2}-{x}_{0})}mq| 10\rangle \langle 00| +{{\rm{e}}}^{\,\rm{i}\,({x}_{2}-{x}_{1})}nq| 10\rangle \langle 01| \\ & & +{q}^{2}| 10\rangle \langle 10| +{{\rm{e}}}^{\,\rm{i}\,{x}_{2}}qt| 10\rangle \langle 11| \\ & & +{{\rm{e}}}^{-\,\rm{i}\,{x}_{0}}mt| 11\rangle \langle 00| +{{\rm{e}}}^{-\,\rm{i}\,{x}_{1}}nt| 11\rangle \langle 01| \\ & & +{{\rm{e}}}^{-\,\rm{i}\,{x}_{2}}qt| 11\rangle \langle 10| +{t}^{2}| 11\rangle \langle 11| .\\ \end{array}\end{eqnarray*}$
By the properties of Pauli matrices, we can easily get $\begin{eqnarray*}\begin{array}{rlr} & \left|0\right\rangle \left\langle 0\right|=\frac{1}{2}\left({\mathbb{I}}+{\sigma }_{3}\right),\left|0\right\rangle \left\langle 1\right|=\frac{1}{2}\left({\sigma }_{1}+\,\rm{i}\,{\sigma }_{2}\right), & \\ & \left|1\right\rangle \left\langle 0\right|=\frac{1}{2}\left({\sigma }_{1}-\,\rm{i}\,{\sigma }_{2}\right),\left|1\right\rangle \langle 1| =\frac{1}{2}\left({\mathbb{I}}-{\sigma }_{3}\right).\end{array}\end{eqnarray*}$
Since ∣φ⟩⟨φ∣ takes the form in ( $\begin{eqnarray}nq\cos ({x}_{1}-{x}_{2})=-mt\cos {x}_{0},\end{eqnarray}$
$\begin{eqnarray}mn\sin ({x}_{0}-{x}_{1})=qt\sin {x}_{2},\end{eqnarray}$
$\begin{eqnarray}mq\sin ({x}_{0}-{x}_{2})=nt\sin {x}_{1},\end{eqnarray}$
$\begin{eqnarray}nq\sin ({x}_{1}-{x}_{2})=0,\end{eqnarray}$
$\begin{eqnarray}mt\sin {x}_{0}=0,\end{eqnarray}$
$\begin{eqnarray}{m}^{2}+{t}^{2}={n}^{2}+{q}^{2}.\end{eqnarray}$
We now analyze these equalities by considering the following cases.
Case 1. One of parameters n, m, q, t is zero.
Case 1.1. n = 0.
From equations (
$\begin{eqnarray*}\begin{array}{r}\begin{array}{rcl}m & = & 0,{t}^{2}={q}^{2}=\frac{1}{2},\sin {x}_{2}=0\,\,\,\rm{or}\,\\ t & = & 0,{m}^{2}={q}^{2}=\frac{1}{2},\sin ({x}_{0}-{x}_{2})=0.\end{array}\end{array}\end{eqnarray*}$
Since global phases are physically irrelevant [1], the vector ∣φ⟩ reduces to
$\begin{eqnarray}\begin{array}{r}| {\phi }_{1}^{\pm }\rangle =\frac{\sqrt{2}}{2}(| 10\rangle \pm | 11\rangle )\,\rm{or}\,\\ | {\phi }_{2}^{\pm }\rangle =\frac{\sqrt{2}}{2}(| 00\rangle \pm | 10\rangle ).\end{array}\end{eqnarray}$
Case 1.2. q = 0.
From equations (
$\begin{eqnarray*}\begin{array}{r}\begin{array}{rcl}m & = & 0,{n}^{2}={t}^{2}=\frac{1}{2},\sin {x}_{1}=0\,\,\,\rm{or}\,\\ t & = & 0,{n}^{2}={m}^{2}=\frac{1}{2},\sin ({x}_{0}-{x}_{1})=0.\end{array}\end{array}\end{eqnarray*}$
Following the same reasoning as in Case 1.1, we find that ∣φ⟩ takes the form $\begin{eqnarray}\begin{array}{rcl}|{\phi }_{3}^{\pm }\rangle & = & \frac{\sqrt{2}}{2}(|01\rangle \pm |11\rangle )\,\rm{or}\\ \,|{\phi }_{4}^{\pm }\rangle & = & \frac{\sqrt{2}}{2}(|00\rangle \pm |01\rangle ).\end{array}\end{eqnarray}$
Case 1.3. m = 0.
From equations (
$\begin{eqnarray*}\begin{array}{r}\begin{array}{rcl}n & = & 0,{q}^{2}={t}^{2}=\frac{1}{2},\sin {x}_{2}=0\,\,\,\rm{or}\,\\ q & = & 0,{n}^{2}={t}^{2}=\frac{1}{2},\sin {x}_{1}=0.\end{array}\end{array}\end{eqnarray*}$
By the same argument as in Case 1.1, we recover the states $| {\phi }_{1}^{\pm }\rangle $ and $| {\phi }_{3}^{\pm }\rangle $ in equations (Case 1.4. t = 0.
From equations (
$\begin{eqnarray*}\begin{array}{r}\begin{array}{rcl}n & = & 0,{m}^{2}={q}^{2}=\frac{1}{2},\sin ({x}_{0}-{x}_{2})=0\,\,\,\rm{or}\,\\ q & = & 0,{m}^{2}={n}^{2}=\frac{1}{2},\sin ({x}_{0}-{x}_{1})=0.\end{array}\end{array}\end{eqnarray*}$
Similar to the proof of Case 1.1, we obtain the states $| {\phi }_{2}^{\pm }\rangle $ and $| {\phi }_{4}^{\pm }\rangle $ in equations (However, ${| {\phi }_{i}^{\pm }\rangle \langle {\phi }_{i}^{\pm }| }^{{\rm{\Gamma }}},i=1,2,3,4,$ are not EWs, since the states $| {\phi }_{i}^{\pm }\rangle ,i=1,2,3,4,$ are all separable.
Case 2. All parameters m, n, q, t are non-zero.
From equations (
$\begin{eqnarray*}\begin{array}{rl} & \sin ({x}_{1}-{x}_{2})=0\,\,\rm{and}\,\,\sin {x}_{0}=0.\end{array}\end{eqnarray*}$
We consider the following subcases.
Case 2.1. $\cos ({x}_{1}-{x}_{2})=1,\cos {x}_{0}=1.$
In this subcase, x1 = x2, x0 = 0. Substituting these into equations (
$\begin{eqnarray*}\left\{\begin{array}{l}nq=-mt,\quad \\ -mn\sin {x}_{1}=qt\sin {x}_{1},\quad \\ -mq\sin {x}_{1}=nt\sin {x}_{1},\quad \\ {m}^{2}+{t}^{2}={n}^{2}+{q}^{2}.\quad \end{array}\right.\end{eqnarray*}$
If $\sin {x}_{1}=0$, then it is easy to get n = ± m,q = ∓ t or n = ± t, q = ∓ m. Combining with equations (
$\begin{eqnarray*}\begin{array}{r}m| 00\rangle +m| 01\rangle -t| 10\rangle +t| 11\rangle ,\\ m| 00\rangle -m| 01\rangle +t| 10\rangle +t| 11\rangle ,\\ m| 00\rangle +t| 01\rangle -m| 10\rangle +t| 11\rangle ,\\ m| 00\rangle -t| 01\rangle +m| 10\rangle +t| 11\rangle ,\end{array}\end{eqnarray*}$
which reduce to $| {\varphi }_{1}^{+}\rangle ,| {\varphi }_{1}^{-}\rangle ,| {\varphi }_{2}^{-}\rangle ,| {\varphi }_{2}^{+}\rangle $ in equation (If $\sin {x}_{1}\ne 0$, then direct calculation results in four additional states
$\begin{eqnarray*}\begin{array}{rc}| {\phi }_{1}^{+}\rangle = & \frac{1}{2}(| 00\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 01\rangle -{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 10\rangle +| 11\rangle ,\\ | {\phi }_{2}^{+}\rangle = & \frac{1}{2}(| 00\rangle -{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 01\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 10\rangle +| 11\rangle ,\\ | {\phi }_{1}^{-}\rangle = & \frac{1}{2}(| 00\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 01\rangle +{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 10\rangle -| 11\rangle ),\\ | {\phi }_{2}^{-}\rangle = & \frac{1}{2}(| 00\rangle -{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 01\rangle -{{\rm{e}}}^{\,\rm{i}\,{x}_{1}}| 10\rangle -| 11\rangle .\end{array}\end{eqnarray*}$
The states $| {\phi }_{1}^{+}\rangle $ and $| {\phi }_{2}^{+}\rangle $ belong to the same family as $| {\varphi }_{3}^{+}\rangle $, while the states $| {\phi }_{1}^{-}\rangle $ and $| {\phi }_{2}^{-}\rangle $ belong to the same family as $| {\varphi }_{3}^{-}\rangle $. Case 2.2. $\cos ({x}_{1}-{x}_{2})=1,\cos {x}_{0}=-1.$
Here, x1 = x2, x0 = π. The equalities (
$\begin{eqnarray*}\left\{\begin{array}{l}nq=mt,\quad \\ mn\sin {x}_{1}=qt\sin {x}_{1},\quad \\ mq\sin {x}_{1}=nt\sin {x}_{1},\quad \\ {m}^{2}+{t}^{2}={n}^{2}+{q}^{2}.\quad \end{array}\right.\end{eqnarray*}$
Following the same reasoning as in Case 2.1, if $\sin {x}_{1}=0$, then we obtain n = ± m, q = ± t or n = ± t, q = ± m, and ∣φ⟩ reduces to $| {\varphi }_{1}^{\pm }\rangle $ or $| {\varphi }_{2}^{\pm }\rangle $. If $\sin {x}_{1}\ne 0$, then ∣φ⟩ reduces to $| {\varphi }_{3}^{\pm }\rangle $. Case 2.3. $\cos ({x}_{1}-{x}_{2})=-1,\cos {x}_{0}=1.$
In this subcase, x1 − x2 = π or −π, x0 = 0. The equalities (
$\begin{eqnarray*}\left\{\begin{array}{l}nq=mt,\quad \\ mn\sin {x}_{1}=qt\sin {x}_{1},\quad \\ mq\sin {x}_{1}=nt\sin {x}_{1},\quad \\ {m}^{2}+{t}^{2}={n}^{2}+{q}^{2}.\quad \end{array}\right.\end{eqnarray*}$
Analogous to Case 2.2, if $\sin {x}_{1}=0$, then we can get n = ± m, q = ± t or n = ± t, q = ± m. The state ∣φ⟩ reduces to $| {\varphi }_{1}^{\pm }\rangle $ or $| {\varphi }_{2}^{\pm }\rangle $. If $\sin {x}_{1}\ne 0$, then ∣φ⟩ reduces to $| {\varphi }_{3}^{\pm }\rangle $.
Case 2.4. $\cos ({x}_{1}-{x}_{2})=-1,\cos {x}_{0}=-1.$
Here, x1 − x2 = π or −π, x0 = π. The equalities (
$\begin{eqnarray*}\left\{\begin{array}{l}nq=-mt,\quad \\ -mn\sin {x}_{1}=qt\sin {x}_{1},\quad \\ -mq\sin {x}_{1}=nt\sin {x}_{1},\quad \\ {m}^{2}+{t}^{2}={n}^{2}+{q}^{2}.\quad \end{array}\right.\end{eqnarray*}$
If $\sin {x}_{1}=0$, then we obtain n = ± m, q = ∓ t or n = ± t, q = ∓ m. The state ∣φ⟩ reduces to $| {\varphi }_{1}^{\pm }\rangle $ or $| {\varphi }_{2}^{\pm }\rangle $. If $\sin {x}_{1}\ne 0$, then ∣φ⟩ reduces to $| {\varphi }_{3}^{\pm }\rangle $.
It is straightforward to verify that $| {\varphi }_{1}^{\pm }\rangle $ and $| {\varphi }_{2}^{\pm }\rangle $ are entangled if and only if ab ≠ 0, while $| {\varphi }_{3}^{\pm }\rangle $ is entangled if and only if $c\ne \pm \frac{\pi }{2}$. Thus, we have obtained all EWs $| {\varphi }_{i}^{\pm }\rangle {\langle {\varphi }_{i}^{\pm }| }^{{\rm{\Gamma }}},i=1,2,3,$ of the form in equation (
From theorem 2.1 , all EWs ${W}_{i}^{\pm }=| {\varphi }_{i}^{\pm }\rangle {\langle {\varphi }_{i}^{\pm }| }^{{\rm{\Gamma }}}$ for i = 1, 2, 3, are constructed from the set ${ \mathcal C }$ in equation (2 ) of local measurements. Concretely, 4 ). This particular witness was adopted in [22] to witness gravitational entanglement within the quantum state
$\begin{eqnarray*}\begin{array}{rcl}| {\varphi }_{1}^{\pm }\rangle \langle {\varphi }_{1}^{\pm }{| }^{{\rm{\Gamma }}} & = & \frac{1}{4}({\mathbb{I}}\displaystyle \otimes {\mathbb{I}}\pm {\sigma }_{3}\displaystyle \otimes {\sigma }_{1})\\ & & +\frac{{a}^{2}-{b}^{2}}{4}({\sigma }_{3}\displaystyle \otimes {\mathbb{I}}\pm {\mathbb{I}}\displaystyle \otimes {\sigma }_{1})\\ & & +\frac{ab}{2}({\sigma }_{1}\displaystyle \otimes {\sigma }_{3}\mp {\sigma }_{2}\displaystyle \otimes {\sigma }_{2}),\\ | {\varphi }_{2}^{\pm }\rangle \langle {\varphi }_{2}^{\pm }{| }^{T} & = & \frac{1}{4}({\mathbb{I}}\displaystyle \otimes {\mathbb{I}}\pm {\sigma }_{1}\displaystyle \otimes {\sigma }_{3})\\ & & +\frac{{a}^{2}-{b}^{2}}{4}({\mathbb{I}}\displaystyle \otimes {\sigma }_{3}\pm {\sigma }_{1}\displaystyle \otimes {\mathbb{I}})\\ & & +\frac{ab}{2}({\sigma }_{3}\displaystyle \otimes {\sigma }_{1}\mp {\sigma }_{2}\displaystyle \otimes {\sigma }_{2}),\\ | {\varphi }_{3}^{\pm }\rangle \langle {\varphi }_{3}^{\pm }{| }^{{\rm{\Gamma }}} & = & \frac{1}{4}\left({\mathbb{I}}\displaystyle \otimes {\mathbb{I}}\pm {\sigma }_{2}\displaystyle \otimes {\sigma }_{2}\right.\\ & & +\cos c\,({\sigma }_{3}\displaystyle \otimes {\sigma }_{1}\mp {\sigma }_{1}\displaystyle \otimes {\sigma }_{3})\\ & & \mp \sin c\,({\sigma }_{2}\displaystyle \otimes {\mathbb{I}}\pm \left.{\mathbb{I}}\displaystyle \otimes {\sigma }_{2})\right).\end{array}\end{eqnarray*}$
This makes these EWs particularly suitable for experimental implementation, as they bypass the need for complete quantum state reconstruction. Furthermore, these EWs can be utilized for the detection of gravitational entanglement [20, 21]. Specifically, for the parameter choice $a=b=\frac{\sqrt{2}}{2}$, the EW ${W}_{1}^{-}=| {\varphi }_{1}^{-}\rangle \langle {\varphi }_{1}^{-}{| }^{{\rm{\Gamma }}}$ reduces to W2 in equation ( $\begin{eqnarray*}| \psi ({\theta }_{1},{\theta }_{2})\rangle =\frac{1}{2}\left(| 00\rangle +{{\rm{e}}}^{\,\rm{i}\,{\theta }_{1}}| 01\rangle +{{\rm{e}}}^{\,\rm{i}\,{\theta }_{2}}| 10\rangle +| 11\rangle \right),\end{eqnarray*}$
where θ1, θ2 are real-valued parameters. Consider the mixture of the state ∣ψ(θ1, θ2)⟩ with white noise $\begin{eqnarray*}\begin{array}{rc}{\rho }_{p} & =p| \psi ({\theta }_{1},{\theta }_{2})\rangle \langle \psi ({\theta }_{1},{\theta }_{2})| +\frac{1-p}{4}{\mathbb{I}}\displaystyle \otimes {\mathbb{I}},\end{array}\end{eqnarray*}$
where 0 ≤ p ≤ 1. Table 1 shows the intervals of p for detectable entanglement of ρp from the EWs ${W}_{i}^{\pm },i=1,2,3.$ It can be found that, apart from the EW ${W}_{1}^{-}$, both ${W}_{2}^{-}$ and ${W}_{3}^{+}$ can also detect gravitational entanglement.Table 1. The intervals of p for detectable entanglement of ρp. |
| ${W}_{1}^{+}$ | ${W}_{1}^{-}$ | ${W}_{2}^{+}$ | ${W}_{2}^{-}$ | ${W}_{3}^{+}$ | ${W}_{3}^{-}$ | |
|---|---|---|---|---|---|---|
| ${\theta }_{1}=\frac{8}{7}\pi ,{\theta }_{2}=\frac{13}{7}\pi $ | ${\rm{\varnothing }}$ | ${\rm{\varnothing }}$ | ${\rm{\varnothing }}$ | [0.3827, 1] | [0.4795, 1] | ${\rm{\varnothing }}$ |
| ${\theta }_{1}=\frac{1}{4}\pi ,{\theta }_{2}=\frac{11}{3}\pi $ | ${\rm{\varnothing }}$ | [0.5225, 1] | ${\rm{\varnothing }}$ | ${\rm{\varnothing }}$ | ${\rm{\varnothing }}$ | ${\rm{\varnothing }}$ |
From the unitary equivalence relations 1 ). However, these EWs are generally not extremal decomposable. Therefore, the extremal decomposable EWs derived from equation (1 ) are generally not locally unitarily equivalent to those obtained from equation (2 ).
$\begin{eqnarray*}{U}^{\dagger }{\sigma }_{1}U={\sigma }_{3},{U}^{\dagger }{\sigma }_{3}U={\sigma }_{1},{U}^{\dagger }{\sigma }_{2}U=-{\sigma }_{2},\end{eqnarray*}$
with $\begin{eqnarray*}U=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1 & 1\\ 1 & -1\end{array}\right),\end{eqnarray*}$
all EWs of the form ${\mathbb{I}}\otimes {U}^{\dagger }| {\varphi }_{i}^{\pm }\rangle {\langle {\varphi }_{i}^{\pm }| }^{{\rm{\Gamma }}}{\mathbb{I}}\otimes U$ for i = 1, 2, 3 can be constructed from the local measurement set ${ \mathcal A }$ in equation (3. Extremal decomposable EWs for d ⨂ d (d ≥ 3) systems
For d ⨂ d (d ≥ 3) quantum systems, any Hermitian matrix R can be expressed as 2.1 , we now consider the following extremal decomposable EWs acting on ${{\mathbb{C}}}^{d}\otimes {{\mathbb{C}}}^{d}$:
$\begin{eqnarray*}\begin{array}{rcl}R & = & \gamma {\mathbb{I}}\otimes {\mathbb{I}}+\displaystyle \sum _{k=1}^{{d}^{2}-1}\left({a}_{k}{{\rm{\Lambda }}}_{k}\otimes {\mathbb{I}}+{b}_{k}{\mathbb{I}}\otimes {{\rm{\Lambda }}}_{k}\right)\\ & & +\displaystyle \sum _{j,k=1}^{{d}^{2}-1}{c}_{jk}{{\rm{\Lambda }}}_{j}\otimes {{\rm{\Lambda }}}_{k},\end{array}\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{rcl}{\left\{{{\rm{\Lambda }}}_{i}\right\}}_{i=1}^{{d}^{2}-1} & = & {\left\{{G}_{m}^{D}\right\}}_{m=1}^{d-1}\cup \left\{{G}_{{jk}}^{S}|1\leqslant j\lt k\leqslant d\right\}\\ & & \cup \left\{{G}_{{jk}}^{A}|1\leqslant j\lt k\leqslant d\right\}\end{array}\end{eqnarray*}$
denotes the set of generalized Gell-Mann matrices [19] defined by $\begin{eqnarray}{G}_{jk}^{S}=| j\rangle \langle k| +| k\rangle \langle j| ,\quad 1\leqslant j\lt k\leqslant d,\end{eqnarray}$
$\begin{eqnarray}{G}_{jk}^{A}=\,\rm{-i}\,\left|j\right\rangle \left\langle k\right|+\,\rm{i}\,\left|k\right\rangle \left\langle j\right|,\,1\leqslant j\lt k\leqslant d,\end{eqnarray}$
$\begin{eqnarray}{G}_{l}^{D}=\sqrt{\frac{2}{l\left(l+1\right)}}\left(\displaystyle \sum _{j=1}^{l}| j\rangle \left\langle j\right|-l\left|l+1\right\rangle \left\langle l+1\right|\right),\,1\leqslant l\leqslant d-1.\end{eqnarray}$
As a generalization of the extremal decomposable EWs presented in theorem $\begin{eqnarray}{| {\varphi }_{i}^{\pm }\rangle }_{jk}{{\langle {\varphi }_{i}^{\pm }| }_{jk}}^{{\rm{\Gamma }}},i=1,2,3,1\leqslant j\lt k\leqslant d,\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{|{\varphi }_{1}^{\pm }\rangle }_{{jk}} & = & a|{\tau }_{{jk}}^{\pm }\rangle +b|{\sigma }_{{jk}}^{\mp }\rangle ,{|{\varphi }_{2}^{\pm }\rangle }_{{jk}}=a|{\xi }_{{jk}}^{\pm }\rangle +b|{\mu }_{{jk}}^{\mp }\rangle ,\\ {|{\varphi }_{3}^{\pm }\rangle }_{{jk}} & = & \frac{1}{\sqrt{2}}\left(|{\beta }_{{jk}}^{\pm }\rangle +{{\rm{e}}}^{\rm{i}c}|{\gamma }_{{jk}}^{\mp }\rangle \right).\end{array}\end{eqnarray*}$
Here, the basis vectors are given by
$\begin{eqnarray*}\begin{array}{rc}\left|{\tau }_{jk}^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|jj\right\rangle \pm \left|jk\right\rangle \right),\left|{\sigma }_{jk}^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|kj\right\rangle \pm \left|kk\right\rangle \right),\\ \left|{\xi }_{jk}^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|jj\right\rangle \pm \left|kj\right\rangle \right),\left|{\mu }_{jk}^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|jk\right\rangle \pm \left|kk\right\rangle \right),\\ \left|{\beta }_{jk}^{\pm }\right\rangle & =\frac{1}{\sqrt{2}}\left(\left|jj\right\rangle \pm \left|kk\right\rangle \right),\left|{\gamma }_{jk}^{\pm }\right\rangle =\frac{1}{\sqrt{2}}\left(\left|jk\right\rangle \pm \left|kj\right\rangle \right),\end{array}\end{eqnarray*}$
and the parameters $a,b,c\in {\mathbb{R}}$ satisfy ab ≠ 0, a2 + b2 =1, − π < c≤π, and $c\ne \pm \frac{\pi }{2}.$Our main result is summarized as follows.
For 1 ≤ j < k ≤ d (d ≥ 3), the extremal decomposable EWs ${| {\varphi }_{1}^{\pm }\rangle }_{jk}{\langle {\varphi }_{1}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$ and ${| {\varphi }_{2}^{\pm }\rangle }_{jk}{\langle {\varphi }_{2}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$, as defined in equation (
$\begin{eqnarray*}\begin{array}{ccc}{{ \mathcal F }}_{jk} & = & {\left\{{G}_{p}^{D}\displaystyle \otimes {G}_{l}^{D}\right\}}_{p,l=j-1}^{d-1}\cup {\left\{{G}_{p}^{D}\displaystyle \otimes {G}_{jk}^{S}\right\}}_{p=j-1}^{d-1}\\ & & \cup {\left\{{G}_{jk}^{S}\displaystyle \otimes {G}_{l}^{D}\right\}}_{l=j-1}^{d-1}\cup \left\{{G}_{jk}^{A}\displaystyle \otimes {G}_{jk}^{A}\right\}.\end{array}\end{eqnarray*}$
Similarly, the extremal decomposable EWs ${| {\varphi }_{3}^{\pm }\rangle }_{jk}{\langle {\varphi }_{3}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$, as defined in equation ( $\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal G }}_{{jk}} & = & {\left\{{G}_{p}^{D}\displaystyle \otimes {G}_{l}^{D}\right\}}_{p,l=j-1}^{d-1}\cup {\left\{{G}_{p}^{D}\displaystyle \otimes {G}_{{jk}}^{S}\right\}}_{p=j-1}^{d-1}\cup {\left\{{G}_{p}^{D}\displaystyle \otimes {G}_{{jk}}^{A}\right\}}_{p=j-1}^{d-1}\\ & & \cup {\left\{{G}_{{jk}}^{S}\displaystyle \otimes {G}_{l}^{D}\right\}}_{l=j-1}^{d-1}\cup {\left\{{G}_{{jk}}^{A}\displaystyle \otimes {G}_{l}^{D}\right\}}_{l=j-1}^{d-1}\cup \left\{{G}_{{jk}}^{A}\displaystyle \otimes {G}_{{jk}}^{A}\right\}.\end{array}\end{eqnarray*}$
We first consider the EW ${| {\varphi }_{1}^{+}\rangle }_{jk}{\langle {\varphi }_{1}^{+}| }_{jk}^{{\rm{\Gamma }}}$. Using the definitions of $| {\tau }_{jk}^{\pm }\rangle $ and $| {\sigma }_{jk}^{\mp }\rangle $, it is straightforward to derive
$\begin{eqnarray*}\begin{array}{rc}{| {\varphi }_{1}^{+}\rangle }_{jk}{\langle {\varphi }_{1}^{+}| }_{jk} & =\frac{{a}^{2}}{2}| j\rangle \langle j| \displaystyle \otimes \left(| j\rangle \langle j| +| j\rangle \langle k| +| k\rangle \langle j| +| k\rangle \langle k| \right)\\ & +\frac{ab}{2}| j\rangle \langle k| \displaystyle \otimes \left(| j\rangle \langle j| -| j\rangle \langle k| +| k\rangle \langle j| -| k\rangle \langle k| \right)\\ & +\frac{ab}{2}| k\rangle \langle j| \displaystyle \otimes \left(| j\rangle \langle j| +| j\rangle \langle k| -| k\rangle \langle j| -| k\rangle \langle k| \right)\\ & +\frac{{b}^{2}}{2}| k\rangle \langle k| \displaystyle \otimes \left(| j\rangle \langle j| -| j\rangle \langle k| -| k\rangle \langle j| +| k\rangle \langle k| \right).\end{array}\end{eqnarray*}$
Using $\begin{eqnarray}\begin{array}{r}| j\rangle \langle k| =\left\{\begin{array}{ll}\frac{1}{2}\left({G}_{jk}^{S}+{\rm{i}}{G}_{jk}^{A}\right)\quad & \,\rm{for}\,\,j\lt k,\\ \frac{1}{2}\left({G}_{kj}^{S}-{\rm{i}}{G}_{kj}^{A}\right)\quad & \,\rm{for}\,\,j\gt k,\\ -\sqrt{\frac{j-1}{2j}}{G}_{j-1}^{D}+\displaystyle \sum _{n=0}^{d-j-1}\frac{1}{\sqrt{2(j+n)(j+n+1)}}{G}_{j+n}^{D}+\frac{1}{d}{\mathbb{I}}\quad & \,\rm{for}\,\,j=k,\end{array}\right.\end{array}\end{eqnarray}$
we obtain $\begin{eqnarray*}\begin{array}{rc}{| {\varphi }_{1}^{+}\rangle }_{jk}{\langle {\varphi }_{1}^{+}| }_{jk} & =\frac{{a}^{2}}{2}| j\rangle \langle j| \displaystyle \otimes \left(| j\rangle \langle j| +| k\rangle \langle k| +{G}_{jk}^{S}\right)\\ & +\frac{ab}{2}{G}_{jk}^{S}\displaystyle \otimes \left(| j\rangle \langle j| -| k\rangle \langle k| \right)\\ & +\frac{ab}{2}{G}_{jk}^{A}\displaystyle \otimes {G}_{jk}^{A}+\frac{{b}^{2}}{2}| k\rangle \langle k| \\ & \displaystyle \otimes \left(| j\rangle \langle j| +| k\rangle \langle k| -{G}_{jk}^{S}\right).\end{array}\end{eqnarray*}$
By further utilizing equation (We next consider the EW ${| {\varphi }_{3}^{+}\rangle }_{jk}{\langle {\varphi }_{3}^{+}| }_{jk}^{{\rm{\Gamma }}}$. Similar to the proof above, we have
$\begin{eqnarray*}\begin{array}{rcl}{|{\varphi }_{3}^{+}\rangle }_{{jk}}{\langle {\varphi }_{3}^{+}|}_{{jk}} & = & \displaystyle \frac{1}{4}\left(|j\rangle \langle j|+|k\rangle \langle k|\right)\otimes \left(|j\rangle \langle j|+|k\rangle \langle k|\right)\\ & & +\displaystyle \frac{1}{4}\cos c\left(|j\rangle \langle j|-|k\rangle \langle k|\right)\otimes {G}_{{jk}}^{S}\\ & & +\displaystyle \frac{1}{4}\sin c\left(|j\rangle \langle j|+|k\rangle \langle k|\right)\otimes {G}_{{jk}}^{A}\\ & & +\displaystyle \frac{1}{4}\cos c\,{G}_{{jk}}^{S}\otimes \left(|k\rangle \langle k|-|j\rangle \langle j|\right)\\ & & -\displaystyle \frac{1}{4}\sin c\,{G}_{{jk}}^{A}\otimes \left(|j\rangle \langle j|+|k\rangle \langle k|\right)-\displaystyle \frac{1}{4}{G}_{{jk}}^{A}\otimes {G}_{{jk}}^{A}.\end{array}\end{eqnarray*}$
Due to equation (The construction of EWs in theorem 3.1 significantly reduces experimental complexity compared to full state tomography, as it requires only a limited set of local measurements rather than full reconstruction of the density matrix.
From theorem 3.1 , it can be found that the number of local measurements required for ${| {\varphi }_{3}^{\pm }\rangle }_{jk}{\langle {\varphi }_{3}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$ is greater than those of ${| {\varphi }_{1}^{\pm }\rangle }_{jk}{\langle {\varphi }_{1}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$ and ${| {\varphi }_{2}^{\pm }\rangle }_{jk}{\langle {\varphi }_{2}^{\pm }| }_{jk}^{{\rm{\Gamma }}}$. Specifically, the d ⨂ d local measurement sets ${{ \mathcal F }}_{jk},{{ \mathcal G }}_{jk}$ contain more elements than 2 ⨂ 2 set ${ \mathcal C }$ in (2 ). The main reason is $| j\rangle \langle j| +| k\rangle \langle k| ={\mathbb{I}}$ (j ≠ k) in two-qubit systems, but $| j\rangle \langle j| +| k\rangle \langle k| \ne {\mathbb{I}}$ (j ≠ k) for higher dimensional systems.
Notably, theorem 2.1 fully characterizes all extremal decomposable EWs from the local measurement set ${ \mathcal C }$ for two-qubit systems, while theorem 3.1 only yields a partial characterization for general d ⨂ d systems.
4. Conclusion
This work presents a comprehensive characterization of extremal decomposable EWs built from anti-diagonal local correlations {σ1 ⨂ σ3, σ2 ⨂ σ2, σ3 ⨂ σ1} in two-qubit systems. Such EWs are generated by exactly six families of entangled pure states, and can be applied to the detection of gravitational entanglement. We further generalize this construction to higher-dimensional systems via generalized Gell-Mann matrices, and explicitly construct the associated extremal decomposable EWs. However, this characterization is incomplete. Accordingly, a complete characterization of extremal decomposable EWs in d ⨂ d (d ≥ 3) systems requires further investigation. In addition, the exploration of EWs constructed from alternative sets of local measurements remains an interesting direction. Furthermore, the design and construction of multipartite EWs via local measurements is also a promising direction for future research.