It is shown that every entangled state in an infinite-dimensional composite system has a simple entanglement witness of the form $\ensuremath{\alpha}I+T$ with $\ensuremath{\alpha}$ a non-negative number and
T a finite rank self-adjoint operator. We also provide two methods of constructing an entanglement witness and apply them to obtain some entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion.
We study the structure of k-Schmidt witnesses in infinite dimensions. We show that for any states with Schmidt number greater than , there exists a -Schmidt witnesses of the form to detect it, where and is a finite rank self-adjoint operator. We provide a method to construct -Schmidt witnesses with these forms. An elementary operator criterion to determine the lower bound of the Schmidt number of a state is obtained. We also construct a class of 2-Schmidt witnesses, from which we can detect the entanglement of states by entries of their matrix representations.
k-entanglement provides a hierarchical characterization of multipartite correlations according to their entanglement structure, ranging from multipartite entanglement (
k=n) to genuine multipartite entanglement (
k=2). Its quantitative treatment for mixed states faces three related obstacles: the standard axioms for such quantifications does not incorporate the requirements from many-body resource theory, as it
does not regulate how to quantify the entanglement assigned to part systems of the same multipartite state and how to describe the rule that the entanglement possessed by a part system never exceeds the entanglement possed by whole system;
no explicit measures are known that satisfy the above requirements from many-body resource theory; and the extreme difficulty in numerical evaluation at mixed states of known quantifications. In this work, we first formulate an axiomatic framework for \emph{true}
k-entanglement measures that obeys the many-body resource theory, by supplementing usual faithfulness and LOCC monotonicity with two new ones: unification condition and hierarchy condition. Then, for every
2≤k≤n, we construct a witness-based true
k-entanglement measure
Ew(k,n) without the convex-roof extension, which is valid for both finite- and infinite-dimensional
n-partite systems. Finally, we derive a method of constructing finite-dimensional
k-entanglement witnesses from positive operator and
k-separable-state optimizations, design an evaluation scheme of
Ew(k,n) based on pre-established database of
k-entanglement witnesses sampled by positive operators. To check the feasibility of this computing scheme, we develop a software tool, which implements the computation of the approximation of
Ew(k,n) for systems up to four qubits. Numerical tests show that the software
is sufficiently accurate and reliable.