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A Computational Separation Between Quantum No-cloning and No-telegraphing

Barak Nehoran, Mark Zhandry

TL;DR

A collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable is given, suggesting that this is not the case when computational efficiency is considered.

Abstract

Two of the fundamental no-go theorems of quantum information are the no-cloning theorem (that it is impossible to make copies of general quantum states) and the no-teleportation theorem (the prohibition on telegraphing, or sending quantum states over classical channels without pre-shared entanglement). They are known to be equivalent, in the sense that a collection of quantum states is telegraphable if and only if it is clonable. Our main result suggests that this is not the case when computational efficiency is considered. We give a collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable. Given that the opposite scenario is impossible (states that can be telegraphed can always trivially be cloned), this gives the most complete quantum oracle separation possible between these two important no-go properties. We additionally study the complexity class clonableQMA, a subset of QMA whose witnesses are efficiently clonable. As a consequence of our main result, we give a quantum oracle separation between clonableQMA and the class QCMA, whose witnesses are restricted to classical strings. We also propose a candidate oracle-free promise problem separating these classes. We finally demonstrate an application of clonable-but-not-telegraphable states to cryptography, by showing how such states can be used to protect against key exfiltration.

A Computational Separation Between Quantum No-cloning and No-telegraphing

TL;DR

A collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable is given, suggesting that this is not the case when computational efficiency is considered.

Abstract

Two of the fundamental no-go theorems of quantum information are the no-cloning theorem (that it is impossible to make copies of general quantum states) and the no-teleportation theorem (the prohibition on telegraphing, or sending quantum states over classical channels without pre-shared entanglement). They are known to be equivalent, in the sense that a collection of quantum states is telegraphable if and only if it is clonable. Our main result suggests that this is not the case when computational efficiency is considered. We give a collection of quantum states and quantum oracles relative to which these states are efficiently clonable but not efficiently telegraphable. Given that the opposite scenario is impossible (states that can be telegraphed can always trivially be cloned), this gives the most complete quantum oracle separation possible between these two important no-go properties. We additionally study the complexity class clonableQMA, a subset of QMA whose witnesses are efficiently clonable. As a consequence of our main result, we give a quantum oracle separation between clonableQMA and the class QCMA, whose witnesses are restricted to classical strings. We also propose a candidate oracle-free promise problem separating these classes. We finally demonstrate an application of clonable-but-not-telegraphable states to cryptography, by showing how such states can be used to protect against key exfiltration.
Paper Structure (58 sections, 28 theorems, 30 equations)

This paper contains 58 sections, 28 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Introducing Computational Constraints.
  3. Our Results
  4. Complexity Theory.
  5. Cryptography.
  6. Motivation
  7. Cryptographic Applications.
  8. On Oracles.
  9. Other Related Work
  10. Technical Overview
  11. Open Problems and Future Directions
  12. Variations on Telegraphing and Cloning.
  13. Cloning and Telegraphing the Functionality of a State.
  14. No-go Properties Exclusive to Efficient Computation.
  15. Separation by a Classical Oracle.
  16. ...and 43 more sections

Key Result

Theorem 1.1

There exists a quantum oracle $\mathcal{O}$ and a set of quantum states $S$ such that $S$ can be efficiently cloned relative to $\mathcal{O}$, but there is no efficient telegraphing procedure relative to $\mathcal{O}$. Even more, there is no telegraphing procedure where the reconstruction is efficie

Theorems & Definitions (74)

  • Theorem 1.1: Informal presentation of Theorem \ref{['thm:clonable_untelegraphable']}
  • Theorem 1.2: Informal presentation of Theorem \ref{['thm:clonableqma-qcma-oracle-separation']}
  • Theorem 1.3: Informal presentation of Theorem \ref{['thm:non-exfiltration-sufficient-2']}
  • Corollary 1.4
  • Remark 1.5
  • Definition 2.1: $\QCMA$
  • Definition 2.2: $\QMA$
  • Theorem 2.3: Theorem 3.1 from Bennett1997-xg
  • Definition 2.4: Definition 3.2 from Bennett1997-xg
  • Theorem 2.5: Theorem 3.3 from Bennett1997-xg
  • ...and 64 more