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Register a new userEntangled Entanglement: The Geometry of GHZ States
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Added by
Beatrix Hiesmayr C.
Publication date
2014
fields
Physics
and research's language is
English
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The familiar Greenberger-Horne-Zeilinger (GHZ) states can be rewritten by entangling the Bell states for two qubits with a state of the third qubit, which is dubbed entangled entanglement. We show that in this way we obtain all 8 independent GHZ states that form the simplex of entangled entanglement, the magic simplex. The construction procedure allows a generalization to higher dimensions both, in the degrees of freedom (considering qudits) as well as in the number of particles (considering n-partite states). Such bases of GHZ-type states exhibit a certain geometry that is relevant for experimental and quantum information theoretic applications. Furthermore, we study the geometry of these particular state spaces, the inherent symmetries, the cyclicity of the phase operations, and the regions of (genuine multi-partite) entanglement and the several classes of separability. We find non-trivial geometrical properties and a conceptually clear procedure to compare state spaces of different dimensions and number of particles.
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Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.
The ability to generate and verify multipartite entanglement is an important benchmark for near-term quantum devices devices. We develop a scalable entanglement metric based on multiple quantum coherences, and demonstrate experimentally on a 20-qubit superconducting device - the IBM Q System One. We report a state fidelity of 0.5165$pm$0.0036 for an 18-qubit GHZ state, indicating multipartite entanglement across all 18 qubits. Our entanglement metric is robust to noise and only requires measuring the population in the ground state; it can be readily applied to other quantum devices to verify multipartite entanglement.
We show that not all 4-party pure states are GHZ reducible (i.e., can be generated reversibly from a combination of 2-, 3- and 4-party maximally entangled states by local quantum operations and classical communication asymptotically) through an example, we also present some properties of the relative entropy of entanglement for those 3-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.
Entanglement swapping has played an important role in quantum information processing, and become one of the necessary core technologies in the future quantum network. In this paper, we study entanglement swapping for multi-particle pure states and maximally entangled states in qudit systems. We generalize the entanglement swapping of two pure states from the case where each quantum system contains two particles to the case of containing any number of particles, and consider the entanglement swapping between any number of systems. We also generalize the entanglement swapping chain of bipartite pure states to the one of multi-particle pure states. In addition, we consider the entanglement swapping chains for maximally entangled states.
A comparison is made of various searching procedures, based upon different entanglement measures or entanglement indicators, for highly entangled multi-qubits states. In particular, our present results are compared with those recently reported by Brown et al. [J. Phys. A: Math. Gen. 38 (2005) 1119]. The statistical distribution of entanglement values for the aforementioned multi-qubit systems is also explored.
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