Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.

The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\in\mathbb{C}^2\otimes\mathbb{C}^2$. We can equivalently write this condition as $\langle \mathbb{P}_u,R\rangle\ge0$, where $\mathbb{P}_u\equiv uu^\dagger\equiv |u\rangle\!\langle u|$ and $\langle \cdot,\cdot\rangle$ denotes here the trace inner product between operators (which equals the standard complex inner product between their vectorisations).

On the other hand, $R$ is an entanglement witness if $R$ is not semidefinite positive, but satisfies $\langle \mathbb{P}_u\otimes\mathbb{P}_v,R\rangle\ge0$ for all $u,v\in\mathbb{C}^2$.

Some easy consequences of this requirement are that $R$ must have some negative eigenvalue corresponding to some non-separable vector. A standard example of such an object is the Swap operator: if $$W \equiv \begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}\equiv \mathbb{P}_{00}+\mathbb{P}_{11} + E_{01}\otimes E_{01} + E_{10}\otimes E_{10} = I\otimes I - 2\mathbb{P}_{\Psi^-},$$ where $|\Psi^-\rangle\equiv\frac{1}{\sqrt2}(|01\rangle-|10\rangle)$, then $W$ is not positive semidefinite, as $\langle \mathbb{P}_{\Psi^-},W\rangle=-1$, but we nonetheless have $$\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle = |u_0 v_0|^2 + |u_1 v_1|^2 + 2\operatorname{Re}(\bar u_1 u_2 v_1 \bar v_2) = a^2 + b^2 + 2 ab \cos\alpha, $$ where $a\equiv |u_0 v_0|, b\equiv |u_1 v_1|$, and $\alpha$ is some angle depending on the relative phases of $|u\rangle$ and $|v\rangle$. We conclude that $\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle\ge (a-b)^2\ge0$, hence $W$ is a witness. We can equivalently reach the same conclusion observing that $W$ is the Choi operator of the transpose map, which is positive.

My question is: while I know that characterising witness operators (equivalently, characterising positive non-CP maps) is nontrivial, is there a "good" set of conditions that we can use to characterise two-qubit witness operators? In particular, we know how to characterise the positive semidefiniteness of Hermitian matrices, and for 2x2 matrices we know that $A\ge0$ iff diagonal elements and determinant are non-negative. Is there any similar kind of condition that can be used to characterise witnesses? Given that characterising the positive semidefiniteness of $R$ itself is relatively easy via Sylvester's criteria, I'd settle with conditions on $R$ characterising $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$.

An easy sufficient condition for $R$ to be a witness is that its top-left and bottom-right 2x2 submatrices must be positive semidefinite. These come from requiring $\langle \mathbb{P}_0\otimes \mathbb{P}_v,R\rangle,\langle \mathbb{P}_1\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v\in\mathbb{C}^2$. More generally, for any fixed $u\in\mathbb{C}^2$, requiring $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v$ amounts to the condition $$|u_0|^2 R_{00} + |u_1| R_{11} + (\bar u_0 u_1 R_{01} + u_0 \bar u_1 R_{01}^\dagger)\ge0,$$ where I'm denoting with $R_{ij}$ the $(i,j)$-th 2x2 submatrix of $R$. This seems to give a N&S set of conditions for $R$ being a witness (technically, for it having non-negative expectation value on separable states), but if we have to check the condition for all possible $u\in\mathbb{C}^2$ it's not tremendously useful.

Of course, any other kind of characterisation that is not in the form of a similar set of inequalities would also be great.

Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.

The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\in\mathbb{C}^2\otimes\mathbb{C}^2$. We can equivalently write this condition as $\langle \mathbb{P}_u,R\rangle\ge0$, where $\mathbb{P}_u\equiv uu^\dagger\equiv |u\rangle\!\langle u|$ and $\langle \cdot,\cdot\rangle$ denotes here the trace inner product between operators (which equals the standard complex inner product between their vectorisations).

On the other hand, $R$ is an entanglement witness if $R$ is not semidefinite positive, but satisfies $\langle \mathbb{P}_u\otimes\mathbb{P}_v,R\rangle\ge0$ for all $u,v\in\mathbb{C}^2$.

Some easy consequences of this requirement are that $R$ must have some negative eigenvalue corresponding to some non-separable vector. A standard example of such an object is the Swap operator: if $$W \equiv \begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}\equiv \mathbb{P}_{00}+\mathbb{P}_{11} + E_{01}\otimes E_{10} + E_{10}\otimes E_{01} = I\otimes I - 2\mathbb{P}_{\Psi^-},$$ where $|\Psi^-\rangle\equiv\frac{1}{\sqrt2}(|01\rangle-|10\rangle)$, then $W$ is not positive semidefinite, as $\langle \mathbb{P}_{\Psi^-},W\rangle=-1$, but we nonetheless have $$\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle = |u_0 v_0|^2 + |u_1 v_1|^2 + 2\operatorname{Re}(\bar u_1 u_2 v_1 \bar v_2) = a^2 + b^2 + 2 ab \cos\alpha, $$ where $a\equiv |u_0 v_0|, b\equiv |u_1 v_1|$, and $\alpha$ is some angle depending on the relative phases of $|u\rangle$ and $|v\rangle$. We conclude that $\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle\ge (a-b)^2\ge0$, hence $W$ is a witness. We can equivalently reach the same conclusion observing that $W$ is the Choi operator of the transpose map, which is positive.

My question is: while I know that characterising witness operators (equivalently, characterising positive non-CP maps) is nontrivial, is there a "good" set of conditions that we can use to characterise two-qubit witness operators? In particular, we know how to characterise the positive semidefiniteness of Hermitian matrices, and for 2x2 matrices we know that $A\ge0$ iff diagonal elements and determinant are non-negative. Is there any similar kind of condition that can be used to characterise witnesses? Given that characterising the positive semidefiniteness of $R$ itself is relatively easy via Sylvester's criteria, I'd settle with conditions on $R$ characterising $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$.

An easy necessary condition for $R$ to be a witness is that its top-left and bottom-right 2x2 submatrices must be positive semidefinite. These come from requiring $\langle \mathbb{P}_0\otimes \mathbb{P}_v,R\rangle,\langle \mathbb{P}_1\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v\in\mathbb{C}^2$. More generally, for any fixed $u\in\mathbb{C}^2$, requiring $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v$ amounts to the condition $$|u_0|^2 R_{00} + |u_1| R_{11} + (\bar u_0 u_1 R_{01} + u_0 \bar u_1 R_{01}^\dagger)\ge0,$$ where I'm denoting with $R_{ij}$ the $(i,j)$-th 2x2 submatrix of $R$. This seems to give a N&S set of conditions for $R$ being a witness (technically, for it having non-negative expectation value on separable states), but if we have to check the condition for all possible $u\in\mathbb{C}^2$ it's not tremendously useful.

Of course, any other kind of characterisation that is not in the form of a similar set of inequalities would also be great.

Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.

The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\in\mathbb{C}^2\otimes\mathbb{C}^2$. We can equivalently write this condition as $\langle \mathbb{P}_u,R\rangle\ge0$, where $\mathbb{P}_u\equiv uu^\dagger\equiv |u\rangle\!\langle u|$ and $\langle \cdot,\cdot\rangle$ denotes here the trace inner product between operators (which equals the standard complex inner product between their vectorisations).

On the other hand, $R$ is an entanglement witness if $R$ is not semidefinite positive, but satisfies $\langle \mathbb{P}_u\otimes\mathbb{P}_v,R\rangle\ge0$ for all $u,v\in\mathbb{C}^2$.

Some easy consequences of this requirement are that $R$ must have some negative eigenvalue corresponding to some non-separable vector. A standard example of such an object is the Swap operator: if $$W \equiv \begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}\equiv \mathbb{P}_{00}+\mathbb{P}_{11} + E_{01}\otimes E_{01} + E_{10}\otimes E_{10} = I\otimes I - 2\mathbb{P}_{\Psi^-},$$ where $|\Psi^-\rangle\equiv\frac{1}{\sqrt2}(|01\rangle-|10\rangle)$, then $W$ is not positive semidefinite, as $\langle \mathbb{P}_{\Psi^-},W\rangle=-1$, but we nonetheless have $$\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle = |u_0 v_0|^2 + |u_1 v_1|^2 + 2\operatorname{Re}(\bar u_1 u_2 v_1 \bar v_2) = a^2 + b^2 + 2 ab \cos\alpha, $$ where $a\equiv |u_0 v_0|, b\equiv |u_1 v_1|$, and $\alpha$ is some angle depending on the relative phases of $|u\rangle$ and $|v\rangle$. We conclude that $\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle\ge (a-b)^2\ge0$, hence $W$ is a witness. We can equivalently reach the same conclusion observing that $W$ is the Choi operator of the transpose map, which is positive.

My question is: while I know that characterising witness operators (equivalently, characterising positive non-CP maps) is nontrivial, is there a "good" set of conditions that we can use to characterise two-qubit witness operators? In particular, we know how to characterise the positive semidefiniteness of Hermitian matrices, and for 2x2 matrices we know that $A\ge0$ iff diagonal elements and determinant are non-negative. Is there any similar kind of condition that can be used to characterise witnesses? Given that characterising the positive semidefiniteness of $R$ itself, I'd settle with conditions on $R$ characterising $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$.

An easy sufficient condition for $R$ to be a witness is that its top-left and bottom-right 2x2 submatrices must be positive semidefinite. These come from requiring $\langle \mathbb{P}_0\otimes \mathbb{P}_v,R\rangle,\langle \mathbb{P}_1\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v\in\mathbb{C}^2$. More generally, for any fixed $u\in\mathbb{C}^2$, requiring $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v$ amounts to the condition $$|u_0|^2 R_{00} + |u_1| R_{11} + (\bar u_0 u_1 R_{01} + u_0 \bar u_1 R_{01}^\dagger)\ge0,$$ where I'm denoting with $R_{ij}$ the $(i,j)$-th 2x2 submatrix of $R$. This seems to give a N&S set of conditions for $R$ being a witness (technically, for it having non-negative expectation value on separable states), but if we have to check the condition for all possible $u\in\mathbb{C}^2$ it's not tremendously useful.

Of course, any other kind of characterisation that is not in the form of a similar set of inequalities would also be great.

Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.

The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\in\mathbb{C}^2\otimes\mathbb{C}^2$. We can equivalently write this condition as $\langle \mathbb{P}_u,R\rangle\ge0$, where $\mathbb{P}_u\equiv uu^\dagger\equiv |u\rangle\!\langle u|$ and $\langle \cdot,\cdot\rangle$ denotes here the trace inner product between operators (which equals the standard complex inner product between their vectorisations).

On the other hand, $R$ is an entanglement witness if $R$ is not semidefinite positive, but satisfies $\langle \mathbb{P}_u\otimes\mathbb{P}_v,R\rangle\ge0$ for all $u,v\in\mathbb{C}^2$.

Some easy consequences of this requirement are that $R$ must have some negative eigenvalue corresponding to some non-separable vector. A standard example of such an object is the Swap operator: if $$W \equiv \begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}\equiv \mathbb{P}_{00}+\mathbb{P}_{11} + E_{01}\otimes E_{01} + E_{10}\otimes E_{10} = I\otimes I - 2\mathbb{P}_{\Psi^-},$$ where $|\Psi^-\rangle\equiv\frac{1}{\sqrt2}(|01\rangle-|10\rangle)$, then $W$ is not positive semidefinite, as $\langle \mathbb{P}_{\Psi^-},W\rangle=-1$, but we nonetheless have $$\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle = |u_0 v_0|^2 + |u_1 v_1|^2 + 2\operatorname{Re}(\bar u_1 u_2 v_1 \bar v_2) = a^2 + b^2 + 2 ab \cos\alpha, $$ where $a\equiv |u_0 v_0|, b\equiv |u_1 v_1|$, and $\alpha$ is some angle depending on the relative phases of $|u\rangle$ and $|v\rangle$. We conclude that $\langle \mathbb{P}_u\otimes \mathbb{P}_v,W\rangle\ge (a-b)^2\ge0$, hence $W$ is a witness. We can equivalently reach the same conclusion observing that $W$ is the Choi operator of the transpose map, which is positive.

My question is: while I know that characterising witness operators (equivalently, characterising positive non-CP maps) is nontrivial, is there a "good" set of conditions that we can use to characterise two-qubit witness operators? In particular, we know how to characterise the positive semidefiniteness of Hermitian matrices, and for 2x2 matrices we know that $A\ge0$ iff diagonal elements and determinant are non-negative. Is there any similar kind of condition that can be used to characterise witnesses? Given that characterising the positive semidefiniteness of $R$ itself is relatively easy via Sylvester's criteria, I'd settle with conditions on $R$ characterising $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$.

An easy sufficient condition for $R$ to be a witness is that its top-left and bottom-right 2x2 submatrices must be positive semidefinite. These come from requiring $\langle \mathbb{P}_0\otimes \mathbb{P}_v,R\rangle,\langle \mathbb{P}_1\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v\in\mathbb{C}^2$. More generally, for any fixed $u\in\mathbb{C}^2$, requiring $\langle \mathbb{P}_u\otimes \mathbb{P}_v,R\rangle\ge0$ for all $v$ amounts to the condition $$|u_0|^2 R_{00} + |u_1| R_{11} + (\bar u_0 u_1 R_{01} + u_0 \bar u_1 R_{01}^\dagger)\ge0,$$ where I'm denoting with $R_{ij}$ the $(i,j)$-th 2x2 submatrix of $R$. This seems to give a N&S set of conditions for $R$ being a witness (technically, for it having non-negative expectation value on separable states), but if we have to check the condition for all possible $u\in\mathbb{C}^2$ it's not tremendously useful.

Of course, any other kind of characterisation that is not in the form of a similar set of inequalities would also be great.