Quantum Link-Up Cuts the Cost of Remote State Transfer Dramatically

Researchers investigated the fundamental limits of remote state preparation, a crucial task in quantum information theory where Alice and Bob aim to recreate a quantum state on Bob’s side using shared entanglement and classical communication. Srijita Kundu from the Quantum Computing Research Centre Hon Hai (Foxconn) Research Institute and Olivier Lalonde from the Institute for Quantum Computing University of Waterloo, alongside Kundu et al., present nearly-matching upper and lower bounds on the entanglement and communication costs required for preparing arbitrary quantum states. This work represents a significant advance as it establishes the first nearly-matching bounds for mixed states and, importantly, improves upon existing limits for pure states. Their findings demonstrate a direct link between the ability to perform remote state preparation and the distillability of entanglement, offering insights into the inherent resources needed for various quantum communication protocols and re-establishing known results for state incompressibility and providing a new entanglement-assisted protocol for the equality function.

This work focuses on the efficient transfer of quantum states between parties sharing entanglement and utilising classical communication channels.

Researchers have derived matching upper and lower bounds for the resources required to remotely prepare states described by rank-k projectors on d-dimensional quantum systems. These findings represent the first nearly matching bounds for mixed states, and in the specific case of pure states, the new lower bound surpasses previously established limits.

The study demonstrates a fundamental connection between the ability to perform remote state preparation with limited communication and the capacity to distill maximally entangled states, known as ebits. Any pure entangled state enabling remote state preparation with fewer than o(d) bits of communication can also generate log d ebits of entanglement, and conversely, any state capable of distilling log d ebits can facilitate efficient remote state preparation.

This relationship clarifies the interplay between these two essential quantum information processing tasks. As a direct application, the research successfully re-establishes a known incompressibility result for states of the form P/k, and introduces a novel entanglement-assisted protocol for the equality function.

This new protocol requires 1/2 log n + O ebits and O communication, highlighting a practical application of the theoretical advancements. The investigation centres on ‘flat states’ , mixed states with a uniform spectral distribution, which are particularly relevant to remote state preparation protocols.

These states arise naturally in scenarios involving projective measurements and shared entanglement, making the results broadly applicable to various quantum communication schemes. The work provides a deeper understanding of the fundamental limits governing quantum state transfer and opens avenues for optimising quantum communication protocols.

Entanglement and communication cost analysis via decoupling of iteratively refined states

Researchers investigated remote state preparation, focusing on the entanglement cost and communication cost required to transfer an unknown quantum state between two parties, Alice and Bob. The study centres on preparing a rank-1 projector on a d-dimensional Hilbert space, where Alice receives a classical description and both parties share pre-existing entanglement and classical communication channels.

Work presented nearly-matching lower and upper bounds for both entanglement and communication costs associated with preparing these states, representing a significant advance for mixed states and improving upon existing bounds for pure states. Central to the methodology is the application of decoupling techniques to analyse residual states after each iteration of the protocol.

Alice and Bob repeatedly apply random unitaries, Ui, and through post-selection on successful measurements, the resulting state on Bob’s side, σi, is shown to approximate the target state, Pk. A novel version of the Decoupling Theorem with post-selection was proven, demonstrating that σi closely resembles Pk, accounting for the post-selection process and the expected success probability, k d.

This allowed for bounding the distance between Bob’s final state and the target state, thereby establishing an upper limit on the protocol’s error. To reduce average-case errors to worst-case scenarios, the research employed a probabilistic selection of unitaries, U1 through UN, sampled independently and identically distributed from the unitary group U(d).

The protocol approximates the minimum error over the entire unitary group by sampling from a finite set, leveraging concentration of measure to ensure the selected unitaries fall within an acceptable error range with high probability. A union bound over a δ/4-net on G(d, k) then establishes the worst-case error, achieving a protocol with error ε + δ. The study further extends these findings to lower bound the entanglement of formation for mixed states, connecting efficient remote state preparation with distillable entanglement, a non-trivial result given that not all mixed entangled states possess distillable entanglement.

Entanglement and communication costs for remote state preparation of P/k states

Researchers established nearly-matching lower and upper bounds for both the entanglement cost and communication cost associated with remote state preparation (RSP) of states P/k. These bounds represent the first nearly-matching results achieved for RSP involving mixed states, and in the case of pure states, the lower bound presented surpasses previously known limitations.

The work demonstrates that any pure entangled state capable of facilitating RSP of these states with o(d) bits of communication can distill log d ebits of entanglement, and conversely, any state capable of distilling log d ebits of entanglement can efficiently enable RSP of these states. As an application of these findings, a previously established incompressibility result for states of the form P/k was successfully rederived.

Furthermore, a novel entanglement-assisted communication protocol for the equality function was developed, requiring 12 log n + O ebits and O communication. The study focused on the task of remotely preparing the quantum state P/k on Bob’s side, utilising shared entanglement and classical communication as resources.

Investigations revealed that for RSP protocols employing arbitrary pure states as the shared entanglement resource, a connection exists between the ability to perform RSP with minimal communication and the capacity to distill ebits from the same shared state. Specifically, the research highlights that efficient RSP implies efficient entanglement distillation, and vice versa.

The established bounds provide a fundamental understanding of the resource trade-offs inherent in RSP, particularly for mixed states where such analysis was previously lacking. These results contribute to a deeper understanding of quantum information transmission and its associated resource requirements.

Entanglement costs and communication bounds for mixed quantum state transfer

Researchers have established nearly-matching upper and lower bounds for the entanglement cost and communication cost associated with remote state preparation of mixed quantum states. This work builds upon existing knowledge of remote state preparation, a task where Alice and Bob utilise shared entanglement and classical communication to recreate a quantum state on Bob’s side, given Alice possesses a classical description of the state.

The findings demonstrate a fundamental connection between the ability to perform this remote state preparation and the capacity to distill entanglement, indicating that a pure entangled state enabling efficient remote state preparation can also generate a substantial amount of entanglement. These results clarify the resources required for remote state preparation and provide a deeper understanding of the relationship between entanglement and communication.

Specifically, the established bounds are the first of their kind to nearly match for mixed states, and improve upon previous lower bounds for pure states. As an application of these findings, the researchers have successfully rederived a known result concerning the incompressibility of certain quantum states and developed a new entanglement-assisted protocol for the equality function, requiring a significant amount of entanglement and classical communication.

The authors acknowledge that their analysis relies on certain approximations and assumptions regarding the properties of the quantum states and communication channels involved. Furthermore, they highlight the need for future research to explore the practical implications of these theoretical bounds in real-world quantum communication systems. Investigating the feasibility of implementing these protocols with imperfect entanglement and noisy communication channels represents a crucial next step in advancing the field of quantum information processing.