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arXiv:quant-ph/0306049

Characterizing the combinatorics of distributed EPR pairs for multi-partite entanglement

Sudhir Kumar Singh, Somesh Kumar, Sudebkumar Prasant Pal

Published Jun 6, 2003• 8 citations
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Abstract

We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite entanglement states amongst subsets of $N$. We propose protocols for establishing multi-partite entanglement in the above cases.

DOI
10.48550/arxiv.quant-ph/0306049

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We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set NNN of nnn agents in a network using apriori multi-partite entanglement states amongst subsets of NNN. We propose protocols for establishing multi-partite entanglement in the above cases.
We investigate the exploitation of various combinatorial properties of graphs and set systems to study several issues in quantum information theory. We characterize the combinatorics of distributed EPR pairs for preparing multi-partite entanglement in a real communication network. This combinatorics helps in the study of various problems in multi-party case by just reducing to the two-party case. Particularly, we use this combinatorics to (1) study various possible and impossible transformations of multi-partite states under LOCC, thus presenting an entirely new approach, not based on entropic criterion, to study such state transformations. (2) present a protocol and proof of its unconditional security for quantum key distribution amongst several trusted parties. (3) propose an idea to combine the features of quantum key distribution and quantum secret sharing. We investigate all the above issues in great detail and finally conclude briefly with some open research directions based on our research.
We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among nnn agents. Further, we discuss a new graph theoretic metric on a class of multi-partite states, and its implications.
In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of O(log⁡n)O(\log n)O(logn) distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, nnn is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number O(n/log⁡n)O(n/\log n)O(n/logn) of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number O(log⁡n)O(\log n)O(logn) of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions.
Abstract Multipartite entangled states, such as Greenberger–Horne–Zeilinger (GHZ) states, are important resources in multiparty quantum networking tasks. We consider protocols for generating such states from networks of Bell pairs and local operations and classical communication. We present a computationally-efficient protocol for generating GHZ states that is also efficient with respect to the number of consumed Bell pairs, (local) gates, and Bell-pair sources. Our protocol: (1) requires O ( N ) gates in a network with N nodes, independent of the network topology; (2) has time complexity O ( N 2 ), avoiding the Steiner tree and any other computationally-hard problem; (3) maintains a near-optimal number of consumed Bell pairs. Numerically, our protocol outperforms those based on (approximate) Steiner trees with respect to the number of gates and Bell-pair sources. We prove that the minimal Bell-pair source cost is given by solving the graph-theoretic dominating set problem, and we demonstrate numerically that our protocol is nearly optimal for this quantity. Finally, we analytically characterize the impact of noisy Bell pairs and gates on the fidelity of the distributed GHZ states.
Entanglement distribution is essential for unlocking the potential of distributed quantum information processing. We consider an N-partite network where entanglement is distributed via a central source over lossy channels, and network participants cooperate to establish entanglement between any two chosen parties under local operations and classical communication (LOCC). We develop a general mathematical framework to assess the average bipartite entanglement shared in a lossy distribution, and introduce a tractable lower bound by optimizing over a subset of single-parameter LOCC transformations. Our results show that probabilistically extracting Bell pairs from W states is more advantageous than deterministically extracting them from GHZ-like states in lossy networks, with this advantage increasing with network size. We further extend our analysis analytically, proving that W states remain more effective in large-scale networks. These findings offer valuable insights into the practical deployment of near-term networks, and corroborate a trade-off relationship between the success conversion probability of entanglement distribution protocols and their robustness to loss.
We investigate entanglement distribution in pure-state quantum networks. We consider the case when nonmaximally entangled two-qubit pure states are shared by neighboring nodes of the network. For a given pair of nodes, we investigate how to generate the maximal entanglement between them by performing local measurements, assisted by classical communication, on the other nodes. We find optimal measurement protocols for both small and large one-dimensional networks. Quite surprisingly, we prove that Bell measurements are not always the optimal ones to perform in such networks. We generalize then the results to simple small two-dimensional (2D) networks, finding again counterintuitive optimal measurement strategies. Finally, we consider large networks with hierarchical lattice geometries and 2D networks. We prove that perfect entanglement can be established on large distances with probability one in a finite number of steps, provided the initial entanglement shared by neighboring nodes is large enough. We discuss also various protocols of entanglement distribution in 2D networks employing classical and quantum percolation strategies.