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Near-optimal entanglement-communication tradeoffs for remote state preparation

Srijita Kundu, Olivier Lalonde

TL;DR

This work analyzes remote state preparation of flat states $P/k$ with shared entanglement and classical communication, establishing near-optimal tradeoffs between entanglement and communication. The authors introduce a strong entanglement-min-entropy lower bound and two practical protocols that nearly match the lower bounds, including a Kraus-operator-based scheme and a refined rejection-sampling method with decoupling post-selection. They also develop an average-case to worst-case reduction, enabling worst-case guarantees from average-case performance. The results extend to mixed states via entanglement of formation and yield applications such as incompressibility of flat-state ensembles and an entanglement-efficient bound for EQ$_n$, highlighting deep connections between RSP efficiency and entanglement distillation with concrete, nearly tight resource costs. The techniques combine majorization arguments, decoupling with post-selection, epsilon-net discretization, and semidefinite programming to map protocol performance to spectral constraints, offering both theoretical insights and practical entanglement-assisted communication protocols.

Abstract

We study the following task: Alice is given a classical description of a rank-$k$ projector $P$ on $\mathbb{C}^d$, and Alice and Bob want to prepare the quantum state $P/k$ on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states $P/k$. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with $o(d)$ bits of communication, can distill $\log d$ ebits of entanglement, and conversely, any state that can distill $\log d$ ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form $P/k$, and give a new entanglement-assisted communication protocol for the equality function that uses $\frac{1}{2}\log n + O(1)$ many ebits, and $O(1)$ communication.

Near-optimal entanglement-communication tradeoffs for remote state preparation

TL;DR

This work analyzes remote state preparation of flat states with shared entanglement and classical communication, establishing near-optimal tradeoffs between entanglement and communication. The authors introduce a strong entanglement-min-entropy lower bound and two practical protocols that nearly match the lower bounds, including a Kraus-operator-based scheme and a refined rejection-sampling method with decoupling post-selection. They also develop an average-case to worst-case reduction, enabling worst-case guarantees from average-case performance. The results extend to mixed states via entanglement of formation and yield applications such as incompressibility of flat-state ensembles and an entanglement-efficient bound for EQ, highlighting deep connections between RSP efficiency and entanglement distillation with concrete, nearly tight resource costs. The techniques combine majorization arguments, decoupling with post-selection, epsilon-net discretization, and semidefinite programming to map protocol performance to spectral constraints, offering both theoretical insights and practical entanglement-assisted communication protocols.

Abstract

We study the following task: Alice is given a classical description of a rank- projector on , and Alice and Bob want to prepare the quantum state on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states . Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with bits of communication, can distill ebits of entanglement, and conversely, any state that can distill ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form , and give a new entanglement-assisted communication protocol for the equality function that uses many ebits, and communication.
Paper Structure (32 sections, 34 theorems, 148 equations, 5 figures, 1 table)

This paper contains 32 sections, 34 theorems, 148 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. Lower bound
  4. Upper bounds
  5. Applications
  6. Our techniques
  7. Lower bound
  8. Upper bound
  9. Second protocol, average-case.
  10. Average-case to worst-case reduction.
  11. Open problems
  12. Extension of lower bound to mixed states
  13. Improvement of upper bounds
  14. Preliminaries
  15. Grassmannian.
  16. ...and 17 more sections

Key Result

Theorem 1.1

For all $\gamma > 0$, any $(d,k)$-RSP protocol with relaxed average error $\varepsilon_r$, $m$ bits of communication, and initial shared entangled state $\ket{\sigma}^{AB}$, must satisfy where $\delta = F\left(\frac{k}{d}+O(\sqrt{\frac{m}{d}}), 1-\varepsilon_r\right)$, and $F$ is a truncated version of the fidelity function.See Section sec:prelim for a formal definition of $F$. What is important

Figures (5)

  • Figure 1: Formal description of a $(d,k)$-RSP protocol with pure shared entanglement
  • Figure 2: Protocol $\mathcal{P}'$ with worst-case error, given $\mathcal{P}$ with average-case error
  • Figure 3: First average-case-correct $(d,k)$-RSP protocol
  • Figure 4: Second average-case protocol for RSP of flat states
  • Figure 5: An entanglement-assisted protocol for the equality function on $n$ bits

Theorems & Definitions (69)

  • Theorem 1.1: Combined version of Theorems \ref{['thm:comm-lowerbound']} and \ref{['thm:minentropy']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Schatten $p$-norm
  • Definition 2.2: Trace distance
  • Definition 2.4: Fidelity
  • Definition 2.5: Truncated fidelity function
  • Lemma 2.6
  • ...and 59 more