Computational Entanglement Theory

TL;DR

This paper introduces a computational perspective on entanglement, defining new operational measures, revealing fundamental differences from classical information theory, and exploring implications for cryptography and quantum complexity.

Contribution

It presents new computational measures of entanglement, demonstrates their divergence from traditional measures, and extends the concept of pseudo-entanglement with cryptographic constructions.

Findings

Computational entanglement measures differ from information-theoretic ones.

Constructed pseudo-entangled states based on cryptographic assumptions.

Discussed links between computational entanglement, cryptography, and holography.

Abstract

We initiate a rigorous study of computational entanglement theory, inspired by the emerging usefulness of ideas from quantum information theory in computational complexity. We define new operational computational measures of entanglement -- the computational one-shot entanglement cost and distillable entanglement. We then show that the computational measures are fundamentally different from their information-theoretic counterparts by presenting gaps between them. We proceed by refining and extending the definition of pseudo-entanglement, introduced by Aaronson et al., 2022, using the new operational measures; and we present constructions of pseudo-entangled states (for our new definition) based on post-quantum cryptographic assumptions. Finally, we discuss the relations between computational entanglement theory and other topics, such as quantum cryptography and notions of pseudoentropy, as well as the relevance of our new definitions to the study of the AdS/CFT correspondence. We believe that, in addition to the contributions presented in the current manuscript, our work opens multiple research directions, of relevance both to the theoretical quantum information theory community as well as for future applications of quantum networks and cryptography.