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Alpha-bit state merging

Jessica Yeh, Jinzhao Wang, Patrick Hayden

TL;DR

The paper develops a formal framework for state merging using $\alpha$-bits, showing a fundamental gap between catalytic and non-catalytic merging rates and providing both achievability and optimality results. It clarifies how $\alpha$-bits interpolate between qubits and zero-bits and connects these protocols to entanglement wedge reconstruction in AdS/CFT, revealing when catalytic entanglement is essential versus when non-catalytic schemes suffice. The results sharpen operational meanings of $\alpha$-bits, and suggest that EWR may correspond to situation-specific non-catalytic merging in holographic models rather than a universal catalytic protocol. Overall, the work deepens the link between quantum information resources and holographic duality, with implications for how information is reconstructed across spacetime regions.

Abstract

State merging is a fundamental protocol in quantum information theory that generalizes quantum teleportation. Traditionally, it is achieved by local operations on shared entanglement and classical communication. In this work, we study state merging done with alpha-bits, a versatile quantum communication resource weaker than qubits. We study alpha-bit state merging with and without catalytic entanglement, and we find a potential gap between the rates of alpha-bits consumed. In light of our result, we discuss how to interpret entanglement wedge reconstruction in AdS/CFT in terms of alpha-bit state merging

Alpha-bit state merging

TL;DR

The paper develops a formal framework for state merging using -bits, showing a fundamental gap between catalytic and non-catalytic merging rates and providing both achievability and optimality results. It clarifies how -bits interpolate between qubits and zero-bits and connects these protocols to entanglement wedge reconstruction in AdS/CFT, revealing when catalytic entanglement is essential versus when non-catalytic schemes suffice. The results sharpen operational meanings of -bits, and suggest that EWR may correspond to situation-specific non-catalytic merging in holographic models rather than a universal catalytic protocol. Overall, the work deepens the link between quantum information resources and holographic duality, with implications for how information is reconstructed across spacetime regions.

Abstract

State merging is a fundamental protocol in quantum information theory that generalizes quantum teleportation. Traditionally, it is achieved by local operations on shared entanglement and classical communication. In this work, we study state merging done with alpha-bits, a versatile quantum communication resource weaker than qubits. We study alpha-bit state merging with and without catalytic entanglement, and we find a potential gap between the rates of alpha-bits consumed. In light of our result, we discuss how to interpret entanglement wedge reconstruction in AdS/CFT in terms of alpha-bit state merging

Paper Structure

This paper contains 22 sections, 4 theorems, 44 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Quantum communication with $\alpha$-bits
  4. $\alpha$-bits
  5. Quantum protocols
  6. Mother protocol
  7. Coherent dense coding/teleportation
  8. $\alpha$-bit dense coding/teleportation
  9. Relating $\alpha$-bits and zero-bits
  10. $\alpha$-bit state merging with catalytic entanglement
  11. Achievability
  12. Optimality
  13. $\alpha$-bit state merging without catalytic entanglement
  14. Achievability
  15. Optimality
  16. ...and 7 more sections

Key Result

Theorem 1

(Subspace decoupling duality HP2017) The following two statements are equivalent with some dimension-independent universal relation between $\epsilon$ and $\delta$: 1. For all subspaces $\tilde{A} \subset A$ of dimension less than $k$, there exists a decoding channel $\mathcal{\tilde{D}}^{B\rightarr where $\mathcal{\tilde{N}}$ is the restriction of the channel $\mathcal{N}^{A\rightarrow B}$ to $\t

Figures (8)

  • Figure 1: Circuit diagram for the $\alpha$-bit channel. $V_{\varepsilon}$ is the dilation of Alice's encoding channel, $U_{\mathcal{{N}}}$ is the dilation of the transmitting channel, and $V_D$ is the dilation of Bob's decoding channel. The decoding is possible if $|\tilde{A}|\leq |A|^{\alpha}$.
  • Figure 2: Circuit diagram for converting $\alpha$-bits into cobits (the forward direction of \ref{['eqn:alpha-dense-coding']}).
  • Figure 3: Circuit diagram for $\alpha$-bit state merging with catalytic entanglement.
  • Figure 4: Circuit diagram for the $\alpha$-bit state merging protocol without catalytic entanglement. The line with the $\alpha$ indicates the $\alpha$-bit channel in Fig. \ref{['fig:alpha']}. $U$ is the Haar random applied by Alice and $D$ is the final isometry used by Bob to recover the state.
  • Figure 5: Circuit diagram for state merging starting with some initial state $|\psi\rangle_{ABR}$. $B$ can achieve state merging if and only if $R$ and $E$ decouple at the dashed line. In the context of mother protocol, this can be realized by choosing $V_A$ as Haar random. In the context of proving the optimality of $\alpha$-bit state merging without catalytic entanglement, the box $V_A$ represents all possible isometries $A$ can perform, including the encoding and the Stinespring dilation of the $\alpha$-bit channel.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3