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Register a new userLocal numerical range for a class of $2otimes d$ hermitian operators
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A local numerical range is analyzed for a family of circulant observables and states of composite $2 otimes d$ systems. It is shown that for any $2otimes d$ circulant operator $cal O$ there exists a basis giving rise to the matrix representation with real non-negative off-diagonal elements. In this basis the problem of finding extremum of $cal O$ on product vectors $ket{x}otimes ket{y} in mathbb{C}^2otimes mathbb{C}^d$ reduces to the corresponding problem in $mathbb{R}^2otimes mathbb{R}^d$. The final analytical result for $d=2$ is presented.
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Absolute separable states is a kind of separable state that remain separable under the action of any global unitary transformation. These states may or may not have quantum correlation and these correlations can be measured by quantum discord. We find that the absolute separable states are useful in quantum computation even if it contains infinitesimal quantum correlation in it. Thus to search for the class of two-qubit absolute separable states with zero discord, we have derived an upper bound for $Tr(varrho^{2})$, where $varrho$ denoting all zero discord states. In general, the upper bound depends on the state under consideration but if the state belong to some particular class of zero discord states then we found that the upper bound is state independent. Later, it is shown that among these particular classes of zero discord states, there exist sub-classes which are absolutely separable. Furthermore, we have derived necessary conditions for the separability of a given qubit-qudit states. Then we used the derived conditions to construct a ball for $2otimes d$ quantum system described by $Tr(rho^{2})leq Tr(X^{2})+2Tr(XZ)+Tr(Z^{2})$, where the $2otimes d$ quantum system is described by the density operator $rho$ which can be expressed by block matrices $X,Y$ and $Z$ with $X,Zgeq 0$. In particular, for qubit-qubit system, we show that the newly constructed ball contain larger class of absolute separable states compared to the ball described by $Tr(rho^{2})leq frac{1}{3}$. Lastly, we have derived the necessary condition in terms of purity for the absolute separability of a qubit-qudit system under investigation.
There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {bf 61}, 052306 (2000); Phys. Rev. Lett. {bf 96}, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, $mathcal{C}_{A(BC)}^2=mathcal{C}_{AB}^2+(mathcal{C}_{AC}^a)^2$, in the three-qubit system. In this paper, we consider the monogamy equality in $2otimes 2 otimes d$ quantum systems. We show that $mathcal{C}_{A(BC)}=mathcal{C}_{AB}$ if and only if $mathcal{C}_{AC}^a=0$, and also show that if $mathcal{C}_{A(BC)}=mathcal{C}_{AC}^a$ then $mathcal{C}_{AB}=0$, while there exists a state in a $2otimes 2 otimes d$ system such that $mathcal{C}_{AB}=0$ but $mathcal{C}_{A(BC)}>mathcal{C}_{AC}^a$.
We introduce the concept of essential numerical range $W_{!e}(T)$ for unbounded Hilbert space operators $T$ and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do emph{not} carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range $W_{!e}(T)$ is that it captures spectral pollution in a unified and minimal way when approximating $T$ by projection methods or domain truncation methods for PDEs.
We prove a necessary and sufficient condition for the Liouville and strong Liouville properties of the infinitesimal generator of a Levy process and subordinate Levy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of) the set of zeros of the characteristic exponent of the Levy process.
Let $mathfrak{n}$ be a nonempty, proper, convex subset of $mathbb{C}$. The $mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.
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