The Round Complexity of Local Operations and Classical Communication (LOCC) in Random-Party Entanglement Distillation

Guangkuo Liu1,2,3, Ian George4,5, and Eric Chitambar4,5

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801,USA
2JILA, University of Colorado/NIST, Boulder, CO, 80309, USA
3Department of Physics, University of Colorado, Boulder CO 80309, USA
4Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
5Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois Urbana-Champaign, Urbana, IL 61801

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A powerful operational paradigm for distributed quantum information processing involves manipulating pre-shared entanglement by local operations and classical communication (LOCC). The LOCC round complexity of a given task describes how many rounds of classical communication are needed to complete the task. Despite some results separating one-round versus two-round protocols, very little is known about higher round complexities. In this paper, we revisit the task of one-shot random-party entanglement distillation as a way to highlight some interesting features of LOCC round complexity. We first show that for random-party distillation in three qubits, the number of communication rounds needed in an optimal protocol depends on the entanglement measure used; for the same fixed state some entanglement measures need only two rounds to maximize whereas others need an unbounded number of rounds. In doing so, we construct a family of LOCC instruments that require an unbounded number of rounds to implement. We then prove explicit tight lower bounds on the LOCC round number as a function of distillation success probability. Our calculations show that the original W-state random distillation protocol by Fortescue and Lo is essentially optimal in terms of round complexity.

Entanglement distillation allows us to take a collection of weakly entangled multipartite states and transform them into fewer-party, highly entangled states. These highly entangled states can then be used in various quantum information processing tasks, such as quantum teleportation, quantum error correction, and quantum key distribution. Distillation procedures typically require multiple rounds of operations. For a goal of distilling a certain total amount of entanglement, minimizing the required number of rounds in the distillation protocol becomes an important practical task.

In this work, we design and prove the optimality of distillation protocols, focusing on pure-state distillation from tripartite W-class states to bipartite entangled states using local operations and classical communication (LOCC) operations. We consider two different ways of measuring the resulting bipartite entanglement: EPR distillation and concurrence distillation, each with its own optimal protocol. We accept the outcoming state regardless of which two parties share the bipartite entanglement, therefore the name “random-party”. We also identify a family of separable operations, parametrized by a real variable, which lies on the boundary of the set of LOCC protocols. Finally, we find numerically a clear gap in the distillation capability between LOCC and positive partial transpose (PPT ) operations.

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