[Paper Review] Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation

Eyuri Wakakuwa, Mio Murao|arXiv (Cornell University)|Oct 15, 2013

Quantum Information and Cryptography42 references3 citations

TL;DR

This paper investigates the asymptotic compressibility of entanglement and classical communication costs in entanglement-assisted local operations and classical communication (EALOCC) for implementing bipartite unitaries on many unknown input states. By leveraging quantum state merging and the decoupling theorem, it establishes that the Schmidt strength of the unitary provides a tight lower bound on the minimal resource costs, which is achieved exactly for generalized Clifford operators.

ABSTRACT

We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.

Motivation & Objective

• To analyze the compressibility of entanglement and classical communication costs in EALOCC protocols for implementing bipartite unitaries on many unknown input states.
• To determine the minimal asymptotic rates of entanglement, forward, and backward classical communication required for implementing $(U^{AB})^{ imes n}$ with vanishing error as $n \to \infty$.
• To connect distributed quantum computation to quantum Shannon theory by applying the decoupling theorem and quantum state merging.
• To derive tight bounds on resource costs for generalized Clifford operators, where the lower and upper bounds coincide.
• To apply the results to distributed quantum state compression, deriving achievable rates for tripartite quantum state compression.

Proposed method

• Formulates the problem as simulating a pure bidirectional quantum channel using shared entanglement and classical communication.
• Applies the decoupling theorem to relate resource costs to the degree of correlation between the system and reference states.
• Uses quantum state merging as a core primitive to construct a protocol that achieves the upper bounds on communication and entanglement costs.
• Defines the Schmidt strength of a unitary as a lower bound on the minimal entanglement and classical communication costs.
• Constructs a protocol that splits shared entanglement into components for state merging and classical communication, optimizing resource usage.
• Generalizes the protocol to arbitrary $ r \geq \frac{3}{2}R $, where $ R $ is the randomness required to partially decouple the tripartite state.

Experimental results

Research questions

• RQ1 What is the minimal asymptotic rate of entanglement required to implement a bipartite unitary on $ n $ pairs of unknown input states with vanishing error?
• RQ2 Can the forward and backward classical communication costs be compressed below their single-shot limits in the asymptotic regime?
• RQ3 How do the Schmidt strength and the randomness required for partial decoupling relate to the minimal resource costs in EALOCC?
• RQ4 For which classes of unitaries do the lower and upper bounds on resource costs coincide?
• RQ5 What are the achievable quantum communication rates for distributed compression of tripartite entangled states derived from unitaries?

Key findings

• The Schmidt strength of a bipartite unitary provides a lower bound on the minimal entanglement cost, forward classical communication cost, and backward classical communication cost per pair in the asymptotic limit.
• An upper bound on these three resource costs is given by the amount of randomness required to partially decouple the tripartite quantum state associated with the unitary.
• For generalized Clifford operators, the lower and upper bounds on resource costs coincide, meaning the bounds are tight and optimal.
• A protocol based on quantum state merging achieves the upper bounds, demonstrating that the derived rates are asymptotically achievable.
• The rate triplet $(Q_{\tilde{A}}, Q_{\tilde{B}}, Q_C) = (\frac{1}{2}K(U_{\rm Cl}), \frac{3}{2}K(U_{\rm Cl}) + \log d, 0)$ is achievable for distributed compression of $|\tilde{\Psi}_{3R/2}(U_{\rm Cl}^\dagger)\rangle$.
• For the two-qubit controlled-Z gate, the achievable rate triplet is $(\frac{1}{2}, \frac{5}{2}, 0)$, showing that $Q_{\tilde{A}} + Q_C$ can be reduced by exploiting classical correlation even without entanglement in the compressed state.