Lieb-Robinson bounds and the generation of correlations and topological quantum order.

By Sergey Bravyi, M. B. Hastings, Frank Verstraete. Physical Review Letters, 2006

608 citations recorded by the source corpus.

The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/7a29619b37b081351c4cb18491babc574d41e704

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Citations in the corpus

Spreading of correlations and entanglement after a quench in the one-dimensional Bose–Hubbard model (2008, methodology): “This shows that the boundary law for the dynamics of entanglement, which has been proven mostly for 1D spin systems [17, 18], seems to be valid in more general systems such as the Bose–Hubbard system considered here.”
The future of quantum computing with superconducting qubits (2022, background): “Theknown classical algorithms for simulating the coherent time evolution of a quantum spin chain have run-time min (2 O ( n ) , 2 O ( vt ) ), where v ∼ max j (cid:107) H j,j +1 (cid:107) is the Lieb-Robinson velocity, which controls how fast information propagates through the system 34 .”
Quench dynamics and relaxation in isolated integrable quantum spin chains (2016, background): “181 S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order, Phys.”
Valence Bonds in Random Quantum Magnets: Theory and Application to YbMgGaO4 (2017, background): “An example of a state which is not constructible is the ground state of a quantum spin liquid: this requires a quantum circuit of depth proportional to the system size.[104, 105] Another example is a random singlet ground state in 1d, which”
Complexity and entanglement for thermofield double states (2018, background): “[95]), since any information propagation in a lattice model is upper bounded by a Lieb-Robinson bound [99,100].”
Non-local propagation of correlations in quantum systems with long-range interactions (2014, neutral): “As a result of (2), the third and fourth lines in equation (8) vanish.”
Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases (2021, methodology): “In fact if we try to construct an SPT class with this property using the Atiyah-Hirzebruch spectral sequence, we find that the relevant class in H 2 ( Z 2 [1] , Ω ) has a nonzero differential.”
Local random quantum circuits are approximate polynomial-designs: numerical results (2012, neutral): “[16, 21, 44]).”
Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems (2011, neutral): “If dimH2 <∞, one can simply take A′ = 1 dimH2 TrH2A, as is done in [9, 36] (or see (i) in the proof of Lemma 3.”
Quantum Information Meets Quantum Matter (2015, methodology): “itrary single-qubit unitary operators and a non-trivial two-qubit unitary operator (e.g. the controlled-NOT) are enough to construct any N-qubit unitary operator. Historically, this was ﬁrst shown in [5], which is extended and simpliﬁed in the follow-up paper [1]. Our treatment in Section 2.3 starti”
Efficient tomography of a quantum many-body system (2016, background): “For finite range interactions, the correlation length L can increase at most linearly in time as entanglement grows and spreads out in the system, demanding exponentially growing number of measurements to estimate each local reduction [21, 22].”
Efficient Long-Range Entanglement Using Dynamic Circuits (2023, background): “Given that the entanglement cannot spread faster than its information light cone [14, 15], entangling two qubits that are a distance n apart requires a minimum two-qubit gate-depth that scales as O(n), and even when assuming all-to-all connectivity, the generation of entanglement over n qubits neces”
Chaos, complexity, and random matrices (2017, background): “In recent years, the notion of quantum complexity has attracted significant attention in the study of quantum many-body systems [49, 50, 51].”