[Paper Review] A Four-Round LOCC Protocol Outperforms All Two-Round Protocols in Reducing the Entanglement Cost for A Distributed Quantum Information Processing

Eyuri Wakakuwa, Akihito Soeda+1 more|arXiv (Cornell University)|Aug 26, 2016

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper demonstrates that a four-round LOCC protocol outperforms all two-round protocols in reducing entanglement cost for implementing a class of two-qubit controlled-unitary gates via local operations and classical communication. By proving that two-round protocols require at least one ebit of entanglement per gate, while a four-round protocol achieves strictly less than one ebit, the work establishes the first example of a genuinely bidirectional quantum task where increased communication rounds reduce entanglement cost.

ABSTRACT

We prove that there is a trade-off relation between the entanglement cost and the number of rounds of communication, for two distant parties to accomplish a bidirectional quantum information task by local operations and classical communication (LOCC). We consider an implementation of a class of two-qubit controlled-unitary gate by LOCC assisted by shared entanglement, in an information theoretical scenario of asymptotically many input pairs and vanishingly small error. We prove the trade-off relation by showing that one ebit of entanglement per pair is necessary to be consumed for implementing the unitary by any two-round protocol, whereas the entanglement cost by a four-round protocol is strictly smaller than one ebit per pair.

Motivation & Objective

• To investigate the trade-off between entanglement cost and the number of communication rounds in distributed quantum information processing.
• To determine whether protocols with more rounds can achieve lower entanglement costs than those with fewer rounds for genuinely bidirectional tasks.
• To establish a formal trade-off relation between communication rounds and entanglement cost for a class of two-qubit controlled-unitary gates.
• To provide the first example of a task where a four-round protocol outperforms all two-round protocols in entanglement efficiency.

Proposed method

• Analyzes the implementation of a two-qubit controlled-unitary gate using LOCC with shared entanglement in an asymptotic, i.i.d. scenario with vanishing error.
• Uses an information-theoretic framework where Alice and Bob apply the gate on multiple copies of a completely mixed state without prior knowledge of the ensemble.
• Proves that any two-round LOCC protocol requires at least one ebit of entanglement per gate, using trace distance and fidelity bounds.
• Constructs a four-round protocol that achieves an entanglement cost strictly less than one ebit per gate, leveraging typical subspace projections and LOCC transformations.
• Applies the gentle measurement lemma and majorization techniques to show that $ n(H(λ) + \delta) $ Bell pairs can be deterministically transformed into a typical state.
• Uses the trace distance monotonicity and triangle inequality to bound the error in state transformation, ensuring asymptotic fidelity approaches one.

Experimental results

Research questions

• RQ1 Can a four-round LOCC protocol achieve a lower entanglement cost than any two-round protocol for implementing a controlled-unitary gate?
• RQ2 Is there a non-trivial trade-off between the number of communication rounds and entanglement cost in genuinely bidirectional quantum tasks?
• RQ3 Does increasing the number of rounds from two to four provide a strict advantage in entanglement efficiency for distributed quantum operations?
• RQ4 Can the entanglement cost be reduced below one ebit per gate using more rounds of communication in a controlled-unitary implementation?
• RQ5 What is the minimal entanglement cost achievable for two-qubit controlled-unitary gates using LOCC with a fixed number of communication rounds?

Key findings

• Any two-round LOCC protocol requires at least one ebit of entanglement per gate to implement the controlled-unitary operation with vanishing error.
• A four-round LOCC protocol achieves an entanglement cost strictly less than one ebit per gate, demonstrating a quantitative advantage over two-round protocols.
• The four-round protocol leverages typical subspaces and LOCC transformations to achieve a lower entanglement cost by exploiting the structure of the input state and measurement outcomes.
• The proof establishes that the entanglement cost is bounded below by the von Neumann entropy of the state distribution, which is less than one bit for certain parameter choices.
• The fidelity of the protocol approaches one in the asymptotic limit, with error terms decaying exponentially in the number of input pairs.
• This work provides the first example of a genuinely bidirectional quantum task where more communication rounds lead to a strict reduction in entanglement cost.