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arXiv:1807.09187 · npj Quantum Information

A spacetime area law bound on quantum correlations

Ilya Kull, Philippe Allard Guérin, Časlav Brukner

Published Jul 8, 2019• 5 citations
arXivHTMLPDF

Abstract

Abstract Area laws are a far-reaching consequence of the locality of physical interactions, and they are relevant in a range of systems, from black holes to quantum many-body systems. Typically, these laws concern the entanglement entropy or the quantum mutual information of a subsystem at a single time. However, when considering information propagating in spacetime, while carried by a physical system with local interactions, it is intuitive to expect area laws to hold for spacetime regions. In this work, we prove such a law for quantum lattice systems. We consider two agents interacting in disjoint spacetime regions with a spin-lattice system that evolves in time according to a local Hamiltonian. In their respective spacetime regions, the two agents apply quantum instruments to the spins. By considering a purification of the quantum instruments, and analyzing the quantum mutual information between the ancillas used to implement them, we obtain a spacetime area law bound on the amount of correlation between the agents’ measurement outcomes. Furthermore, this bound applies both to signaling correlations between the choice of operations on the side of one agent, and the measurement outcomes on the side of the other; as well as to the entanglement they can harvest from the spins by coupling detectors to them.

DOI
10.1038/s41534-019-0171-x

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Abstract Area laws are a far-reaching consequence of the locality of physical interactions, and they are relevant in a range of systems, from black holes to quantum many-body systems. Typically, these laws concern the entanglement entropy or the quantum mutual information of a subsystem at a single time. However, when considering information propagating in spacetime, while carried by a physical system with local interactions, it is intuitive to expect area laws to hold for spacetime regions. In this work, we prove such a law for quantum lattice systems. We consider two agents interacting in disjoint spacetime regions with a spin-lattice system that evolves in time according to a local Hamiltonian. In their respective spacetime regions, the two agents apply quantum instruments to the spins. By considering a purification of the quantum instruments, and analyzing the quantum mutual information between the ancillas used to implement them, we obtain a spacetime area law bound on the amount of correlation between the agents’ measurement outcomes. Furthermore, this bound applies both to signaling correlations between the choice of operations on the side of one agent, and the measurement outcomes on the side of the other; as well as to the entanglement they can harvest from the spins by coupling detectors to them.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
Information shared between parties quantifies their correlation. The encoding of correlations across space and time characterises the structure, history, and interactions of systems. One of the most fundamental properties that emerges from studies of information is the area law, which states that information shared between spatial subregions typically scales with the area of their boundary rather than their volume. In non-interacting, quantum many-body systems, where Gaussian statistics apply, the scaling of information measures is well understood. Within interacting systems, the readout of information measures is impeded by the complexity of state reconstruction. As such, no measurements beyond small quantum systems (e.g., composed of few, localised particles) have been made. Here, we fill this gap by experimentally demonstrating the area law of mutual information in an ultra-cold atom simulator of quantum fields with tuneable interaction strength. Our results detail the scaling of mutual information with subsystem volume, boundary area, and separation between spatial regions at finite temperature. Moreover, we quantify the total effect of non-Gaussian correlations using an information-theoretic measure - relative entropy. Our presented approach is data-driven, model agnostic, and readily applicable to other platforms and observables, thus constituting a universal toolkit for probing information in high-dimensional quantum systems and its role in shaping quantum matter and spacetime.
Abstract For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states’ entanglement entropy. The condition is also shown to imply exponential decay of the state’s mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model’s subcritical ground states, valid in all dimensions and up to the model’s quantum phase transition.
A remarkable feature of typical ground states of strongly-correlated many-body systems is that the entanglement entropy is not an extensive quantity. In one dimension, there exists a proof that a finite correlation length sets a constant upper-bound on the entanglement entropy, called the area law. However, the known bound exists only in a hypothetical limit, rendering its physical relevance highly questionable. In this paper, we give a simple proof of the area law for entanglement entropy in one dimension under the condition of exponentially decaying correlations. Our proof dramatically reduces the previously known bound on the entanglement entropy, bringing it, for the first time, into a realistic regime. The proof is composed of several simple and straightforward steps based on elementary quantum information tools. We discuss the underlying physical picture, based on a renormalization-like construction underpinning the proof, which transforms the entanglement entropy of a continuous region into a sum of mutual informations in different length scales and the entanglement entropy at the boundary.
Abstract A more complete understanding of entanglement entropy in a covariant manner could inform the search for quantum gravity. We build on work in this direction by extending previous results to disjoint regions in 1 + 1D. We investigate the entanglement entropy of a scalar field in disjoint intervals within the causal set framework, using the spacetime commutator and correlator, i Δ and W (or the Pauli–Jordan and Wightman functions). A new truncation scheme for disjoint causal diamonds is presented, which follows from the single diamond truncation scheme. We investigate setups including two and three disjoint causal diamonds, as well as a single causal diamond that shares a boundary with a larger global causal diamond. In all the cases that we study, our results agree with the expected area laws. In addition, we study the mutual information in the two disjoint diamond setup. The ease of our calculations indicate our methods to be a useful tool for numerically studying such systems. We end with a discussion of some of the strengths and future applications of the spacetime formulation we use in our entanglement entropy computations, both in causal set theory and in the continuum.
We prove an upper bound on the maximal rate at which a Hamiltonian interaction can generate entanglement in a bipartite system. The scaling of this bound as a function of the subsystem dimension on which the Hamiltonian acts nontrivially is optimal and is exponentially improved over previously known bounds. As an application, we show that a gapped quantum many-body spin system on an arbitrary lattice satisfies an area law for the entanglement entropy if and only if any other state with which it is adiabatically connected (i.e., any state in the same phase) also satisfies an area law.