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arXiv:1907.09744 · Physical Review X

Connector Tensor Networks: A Renormalization-Type Approach to Quantum Certification

Miguel Navascués, Sukhbinder Singh, Antonio Acín

Published Jun 19, 2020• 8 citations
arXivHTMLPDF

Abstract

Using insights from statistical physics, a new approach to detecting quantum properties such as entanglement works for large systems and can be tailored to identify a wide variety of global collective properties.
DOI
10.1103/physrevx.10.021064

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As quantum technologies develop, we acquire control of an ever-growing number of quantum systems. Unfortunately, current tools to detect relevant quantum properties of quantum states, such as entanglement and Bell nonlocality, suffer from severe scalability issues and can only be computed for systems of a very modest size, of around 666 sites. In order to address large many-body systems, we propose a renormalisation-type approach based on a class of local linear transformations, called connectors, which can be used to coarse-grain the system in a way that preserves the property under investigation. Repeated coarse-graining produces a system of manageable size, whose properties can then be explored by means of usual techniques for small systems. In case of a successful detection of the desired property, the method outputs a linear witness which admits an exact tensor network representation, composed of connectors. We demonstrate the power of our method by certifying using a normal desktop computer entanglement, Bell nonlocality and supra-quantum Bell nonlocality in systems with hundreds of sites.
Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.
The exotic properties which appear in the quantum setting, mainly manifested in strongly-correlated systems, oer potential applications in future technologies. For instance, high-precision measurementsor the new paradigm of a quantum computer. One of the most prominent features of quantum physics is entanglement: the correlationsbetween the parties of a system that cannot be described classically. This property is believed to be the one endowing quantum mechanics its complexity. Therefore, characterizing the entanglement properties of strongly-correlated systems plays a fundamental role for condensed-matter physics. However, this complexity comes hand in hand with a challenge: the number of parameters needed to describe a system grows exponentially with the number of parties in the system. This challengelies at the heart of the mathematical description of quantum mechanics. Such a situation happens naturally in many-body systems and in particular, in condensed-matter physics where the relevant physics appears when considering large systems. Then, how can we deal with this diculty? Thekey observation here is that realistic physical systems are the ones whose parties interact locally and this restricts the entanglement pattern in the low-energy sector (zero temperature). So thequestion is shifted to: Is there a framework that captures states with such entanglement pattern? The answer is yes: tensor network states...
Attempts to understand multipartite quantum nonlocality are thwarted by the difficulty of devising quantum Bell inequalities (QBI) for systems composed of more than a few separate parties. In this Letter, we introduce the notion of tight connectors, a class of tensors which, if contracted according to some simple rules, result in tight QBIs. The new inequalities are saturated by tensor network states whose structure mimics the corresponding network of connectors. We further formulate the connector property of "full self-testing," which singles out a subset of tight connectors whose associated QBIs can only be saturated (modulo local isometries) by probing the corresponding tensor network state with specific measurement operators. We provide large analytic families of tight, full self-testing connectors that allow for the generation, via contraction, of N-partite QBIs with a ratio between the maximum quantum and classical values that grows exponentially with N. In turn, our method provides a modular recipe for the generation of extremal quantum behaviors and associated tight bounds in arbitrarily large systems.