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Register a new userConstructions of k-uniform and absolutely maximally entangled states beyond maximum distance codes
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Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes. To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from the so-called maximum distance separable error correction codes. Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.
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We propose a scheme for long-distance quantum communication where the elementary entanglement is generated through two-photon interference and quantum swapping is performed through one-photon interference. Local polarization maximally entangled states of atomic ensembles are generated by absorbing a single photon from on-demand single-photon sources. This scheme is robust against phase fluctuations in the quantum channels, moreover speeds up long-distance high-fidelity entanglement generation rate.
A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a $2 times3 times 3 times 3$ system that shows maximal entanglement across every bipartition.
A set of quantum states is said to be absolutely entangled, when at least one state in the set remains entangled for any definition of subsystems, i.e. for any choice of the global reference frame. In this work we investigate the properties of absolutey entangled sets (AES) of pure quantum states. For the case of a two-qubit system, we present a sufficient condition to detect an AES, and use it to construct families of $N$ states such that $N-3$ (the maximal possible number) remain entangled for any definition of subsystems. For a general bipartition $d=d_1d_2$, we prove that sets of $N>leftlfloor{(d_{1}+1)(d_{2}+1)/2}right rfloor$ states are AES with Haar measure 1. Then, we define AES for multipartitions. We derive a general lower bound on the number of states in an AES for a given multipartition, and also construct explicit examples. In particular, we exhibit an AES with respect to any possible multi-partitioning of the total system.
By incorporating the asymmetry of local protocols, i.e., some party has to start with a nontrivial measurement, into an operational method of detecting the local indistinguishability proposed by Horodecki {it et al.} [Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to efficiently detect the local indistinguishability of maximally entangled states. Locally indistinguishable sets of $d$ maximally entangled states in a $dotimes d$ system are systematically constructed for all $dge 4$ as an application. Furthermore, by exploiting the fact that local protocols are necessarily separable, we explicitly construct small sets of $k$ locally indistinguishable maximally entangled states with the ratio $k/d$ approaching 3/4. In particular, in a $dotimes d$ system with even $dge 6$, there always exist $d-1$ maximally entangled states that are locally indistinguishable by separable measurements.
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.
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