The Entanglement Hierarchy 2 x m x n Systems

Abstract

We consider three-partite pure states in the Hilbert space ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^n$ and investigate to which states a given state can be locally transformed with a non-vanishing probability. Whenever the initial and final states are elements of the same Hilbert space, the problem can be solved via the characterization of the entanglement classes which are determined via stochastic operations and classical communication (SLOCC). In general, there are infinitely many SLOCC classes. However, when considering transformations from higher- to lower-dimensional Hilbert spaces, an additional hierarchy among the classes can be found. This hierarchy of SLOCC classes coarse grains SLOCC classes which can be reached from a common resource state of higher dimension. We first show that a generic set of states in ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^n$ for $n=m$ is the union of infinitely many SLOCC classes, which can be parameterized by $m-3$ parameters. However, for $n\ne m$ there exists a single SLOCC class which is generic. Using this result, we then show that there is a full-measure set of states in ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^n$ such that any state within this set can be transformed locally to a full measure set of states in any lower-dimensional Hilbert space. We also investigate resource states, which can be transformed to any state (not excluding any zero-measure set) in the smaller-dimensional Hilbert space. We explicitly derive a state in ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^{2m-2}$ which is the optimal common resource of all states in ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^m$. We also show that for any $n<2m$ it is impossible to reach all states in ${\mathbb{C}}^2\otimes {\mathbb{C}}^m\otimes {\mathbb{C}}^{\tilde{n}}$ whenever $\tilde{n}>m$.

29 pages, 3 figures

References in corpus (4)

• Entanglement detection
• Entanglement as a resource for local state discrimination in multipartite systems
• Classification of multipartite entanglement containing infinitely many kinds of states
• Common Resource State for Preparing Multipartite Quantum Systems via Local Operations and Classical Communication

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• A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems
• Symmetries and local transformations of translationally invariant Matrix Product States
• Approximate and ensemble local entanglement transformations for multipartite states