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arXiv:2310.02783

Computational Entanglement Theory

Rotem Arnon-Friedman, Zvika Brakerski, Thomas Vidick

Published Oct 4, 2023• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2310.02783

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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.
The advent of pseudoentanglement and computational entanglement theory bootstrapped a wave of research at the intersection of computer science and information theory. In parallel, computational cryptography has undergone substantial development, prompted by the introduction of pseudorandom states and followed by the establishment of a baseline for the computational hardness required for quantum cryptography, from which EFI pairs emerge as a central primitive. We study the connection between pseudoentanglement and computational cryptography through EFI pairs. Our goal is to enable the use of resources arising from computational entanglement theory in the field of cryptography. For this, we establish the relation between operational instances of pseudoentanglement and the hierarchy of minimal assumptions for computational cryptography. We show that the existence of pseudoentanglement under two different operational definitions, with efficient state generation, is a sufficient condition for the existence of EFI pairs. Combined with a previously established result that the converse also holds under the second definition, this allows us to also demonstrate their equivalence. This places pseudoentanglement alongside other minimal assumptions in cryptography, not only offering an alternative perspective on this fundamental problem, but also building a bridge that allows insights from either area to inform the other. While proving these theorems, we introduce and demonstrate technical lemmas in quantum information and computational entanglement theory, relating the computational entanglement measures to the distance between states, establishing distinguishing conditions for mixtures of two families given pairwise distances between their states, and demonstrating the first continuity relation for a computational entanglement measure.
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Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication.
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states -- first defined by Ji, Liu and Song -- which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to log⁡n\log nlogn across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.
Abstract The precise quantification of the limits to manipulating quantum resources lies at the core of quantum information theory. However, standard information-theoretic analyses do not consider the actual computational complexity involved in performing certain tasks. Here we address this issue within the realm of entanglement theory, finding that accounting for computational efficiency substantially changes what can be achieved using entangled resources. We consider two key figures of merit: the computational distillable entanglement and the computational entanglement cost. These measures quantify the optimal rates of entangled bits that can be extracted from or used to dilute many identical copies of n -qubit bipartite pure states, using computationally efficient local operations and classical communication. We demonstrate that computational entanglement measures diverge considerably from their information-theoretic counterparts. Whereas the information-theoretic distillable entanglement is determined by the von Neumann entropy of the reduced state, we show that the min-entropy governs the computationally efficient setting. On the other hand, computationally efficient entanglement dilution requires maximal consumption of entangled bits, even for nearly unentangled states. Furthermore, in the worst-case scenario, even when an efficient description of the state exists and is fully known, one gains no advantage over state-agnostic protocols. Our findings establish sample-complexity bounds for measuring and testing the von Neumann entropy, fundamental limitations on efficient state compression and efficient local tomography protocols.
Quantum entanglement is a key physical resource in quantum information processing that allows for performing basic quantum tasks such as teleportation and quantum key distribution, which are impossible in the classical world. Ever since the rise of quantum information theory, it has been an open problem to quantify entanglement in an information-theoretically meaningful way. In particular, every previously defined entanglement measure bearing a precise information-theoretic meaning is not known to be efficiently computable, or if it is efficiently computable, then it is not known to have a precise information-theoretic meaning. In this Letter, we meet this challenge by introducing an entanglement measure that has a precise information-theoretic meaning as the exact cost required to prepare an entangled state when two distant parties are allowed to perform quantum operations that completely preserve the positivity of the partial transpose. Additionally, this entanglement measure is efficiently computable by means of a semidefinite program, and it bears a number of useful properties such as additivity and faithfulness. Our results bring key insights into the fundamental entanglement structure of arbitrary quantum states, and they can be used directly to assess and quantify the entanglement produced in quantum-physical experiments.