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Remote preparation of quantum states

Charles H. Bennett, Patrick Hayden, Debbie W. Leung, Peter W. Shor, Andreas Winter

TL;DR

The paper includes an extensive discussion of the results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.

Abstract

Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a trade-off between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this "universal" formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact trade-off curve for arbitrary ensembles of pure states and pure entangled states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical trade-off for quantum data compression. The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.

Remote preparation of quantum states

TL;DR

The paper includes an extensive discussion of the results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.

Abstract

Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a trade-off between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this "universal" formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact trade-off curve for arbitrary ensembles of pure states and pure entangled states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical trade-off for quantum data compression. The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.

Paper Structure

This paper contains 15 sections, 20 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Universal r.s.p.: 1 cbit + 1 ebit $\mathbf{\succ}$ 1 qubit
  3. Optimality of cbit and ebit resources
  4. Ensemble trade--off curve
  5. Preparation of entangled states
  6. Discussion
  7. Models and resources
  8. Approximate obliviousness
  9. Further applications of randomisation and trade--off r.s.p.
  10. Gaussian distributed vectors
  11. State randomisation
  12. Universal quantum--classical state description
  13. Typical subspaces
  14. A possible operational reduction of r.s.p. to q.c.t.
  15. Miscellaneous proofs

Key Result

Theorem 2

For Hilbert space ${\cal H}$ of dimension $D$ and $\epsilon>0$ there exist unitaries $U_k$ on ${\cal H}$ such that for every state $\varphi$, where the closed interval to the right refers to the operator order.

Figures (2)

  • Figure 1: The q.c.t. trade--off curve for qubits vs. cbits according to Devetak and Berger Devetak:Berger (solid) and the implied r.s.p. trade--off for ebits vs. cbits (broken).
  • Figure 2: Schematic of the trade--off curve for an ensemble of entangled states. The shaded area is forbidden by causality and the curve begins at the point $\left(\chi\bigl(\{p_i,\varphi_i^B\}\bigr),S\left(\sum_i p_i\varphi_i^B\right) \right)$, due to the protocol of proposition \ref{['prop:peters:protocol']}. It can never go below $E=\sum_i p_i S(\varphi_i^B)$, which is reached at cbit rate $R=H(p)$, as this is the very amount of entanglement in the ensemble.

Theorems & Definitions (27)

  • Example 1: Column method BDSSTW
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Theorem 8
  • Theorem 9: Hayden, Jozsa and Winter HJW
  • Remark 10
  • ...and 17 more