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arXiv:2206.09929

Locality and error correction in quantum dynamics with measurement

Aaron J. Friedman, Chao Nan Yin, Yifan Hong, Andrew Lucas

Published Jun 20, 2022• 3 citations
arXivHTMLPDF

Abstract

The speed of light $c$ sets a strict upper bound on the speed of information transfer in both classical and quantum systems. In nonrelativistic quantum systems, the Lieb-Robinson Theorem imposes an emergent speed limit $v \hspace{-0.2mm} \ll \hspace{-0.2mm} c$, establishing locality under unitary evolution and constraining the time needed to perform useful quantum tasks. We extend the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the expectation that measurements can arbitrarily violate spatial locality, we find at most an $(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)$-fold enhancement to the speed $v$ of quantum information, provided the outcomes of measurements in $M$ local regions are known. This holds even when classical communication is instantaneous, and extends beyond projective measurements to weak measurements and other nonunitary channels. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols. We tightly constrain the resource requirements for quantum computation, error correction, teleportation, and generating entangled resource states (Bell, GHZ, quantum-critical, Dicke, W, and spin-squeezed states) from short-range-entangled initial states. Our results impose limits on the use of measurements and active feedback to speed up quantum information processing, resolve fundamental questions about the nature of measurements in quantum dynamics, and constrain the scalability of a wide range of proposed quantum technologies.

DOI
10.48550/arxiv.2206.09929

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The speed of light ccc sets a strict upper bound on the speed of information transfer in both classical and quantum systems. In nonrelativistic quantum systems, the Lieb-Robinson Theorem imposes an emergent speed limit v≪cv \hspace{-0.2mm} \ll \hspace{-0.2mm} cv≪c, establishing locality under unitary evolution and constraining the time needed to perform useful quantum tasks. We extend the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the expectation that measurements can arbitrarily violate spatial locality, we find at most an (M+1)(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)(M+1)-fold enhancement to the speed vvv of quantum information, provided the outcomes of measurements in MMM local regions are known. This holds even when classical communication is instantaneous, and extends beyond projective measurements to weak measurements and other nonunitary channels. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols. We tightly constrain the resource requirements for quantum computation, error correction, teleportation, and generating entangled resource states (Bell, GHZ, quantum-critical, Dicke, W, and spin-squeezed states) from short-range-entangled initial states. Our results impose limits on the use of measurements and active feedback to speed up quantum information processing, resolve fundamental questions about the nature of measurements in quantum dynamics, and constrain the scalability of a wide range of proposed quantum technologies.
We review the mathematical speed limits on quantum information processing in many-body systems. After the proof of the Lieb-Robinson Theorem in 1972, the past two decades have seen substantial developments in its application to other questions, such as the simulatability of quantum systems on classical or quantum computers, the generation of entanglement, and even the properties of ground states of gapped systems. Moreover, Lieb-Robinson bounds have been extended in non-trivial ways, to demonstrate speed limits in systems with power-law interactions or interacting bosons, and even to prove notions of locality that arise in cartoon models for quantum gravity with all-to-all interactions. We overview the progress which has occurred, highlight the most promising results and techniques, and discuss some central outstanding questions which remain open. To help bring newcomers to the field up to speed, we provide self-contained proofs of the field's most essential results.
The speed of information propagation in long-range interacting quantum systems is limited by Lieb-Robinson-type bounds, whose tightness can be established by finding specific quantum state-transfer protocols. Previous works have given quantum state-transfer protocols that saturate the corresponding Lieb-Robinson bounds using time-dependent Hamiltonians. Are speed limits for quantum information propagation different for time-independent Hamiltonians? In a step towards addressing this question, we present and analyze two optimal time-independent state-transfer protocols for free-particle systems, which utilize continuous-time single-particle quantum walks with hopping strength decaying as a power law. We rigorously prove and numerically confirm that our protocols achieve quantum state transfer, with controllable error over an arbitrarily long distance in any spatial dimension, at the speed limits set by the free-particle Lieb-Robinson bounds. This shows that time independence does not limit information flow for long-range free-particle Hamiltonians.
The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.
We investigate the fundamental time complexity, as constrained by Lieb-Robinson bounds, for preparing entangled states useful in quantum metrology. We relate the minimum time to the Quantum Fisher Information (FQF_QFQ​) for a system of NNN quantum spins on a ddd-dimensional lattice with 1/rα1/r^α1/rα interactions with rrr being the distance between two interacting spins. We focus on states with FQ∼N1+γF_Q \sim N^{1+γ}FQ​∼N1+γ where γ∈(0,1]γ\in (0,1]γ∈(0,1], i.e., scaling from the standard quantum limit to the Heisenberg limit. For short-range interactions (α>2d+1α> 2d+1α>2d+1), we prove the minimum time ttt scales as t≳Lγt \gtrsim L^γt≳Lγ, where L∼N1/dL \sim N^{1/d}L∼N1/d. For long-range interactions, we find a hierarchy of possible speedups: t≳Lγ(α−2d)t \gtrsim L^{γ(α-2d)}t≳Lγ(α−2d) for 2d<α<2d+12d < α< 2d+12d<α<2d+1, t≳log⁡Lt \gtrsim \log Lt≳logL for (2−γ)d<α<2d(2-γ)d < α< 2d(2−γ)d<α<2d, and ttt may even vanish algebraically in 1/L1/L1/L for α<(2−γ)dα< (2-γ)dα<(2−γ)d. These bounds extend to the minimum circuit depth required for state preparation, assuming two-qubit gate speeds scale as 1/rα1/r^α1/rα. We further show that these bounds are saturable, up to sub-polynomial corrections, for all ααα at the Heisenberg limit (γ=1γ=1γ=1) and for α>(2−γ)dα> (2-γ)dα>(2−γ)d when γ<1γ<1γ<1. Our results establish a benchmark for the time-optimality of protocols that prepare metrologically useful quantum states.
The Lieb-Robinson bound (LRB) states that the range and strength of interactions between the constituents of a complex many-body system impose upper limits to how fast the signal can propagate. It manifests in a light conelike growth of correlation function connecting two distant subsystems. Here, we employ the techniques of quantum information to demonstrate that the LRB can be determined from local measurements performed on a single qubit that is connected to a many-body system. This formulation provides an operational recipe for estimating the LRB in complex systems, replacing the measurement of the correlation function with simple single-particle manipulations. We demonstrate the potency of this approach by deriving the upper limit to the speed of signal propagation in the XY spin chain.
Any protocol to process quantum information has to conclude with a measurement, aimed at producing a specific set of probabilities of measurement outcomes. In this work, we investigate the time, energy and importantly the genuine quantum resources necessary for transforming a set of measurement probabilities generated by a positive-operator-valued measure (POVM), to a target set of measurement probabilities. To this end, we first show that the speed of measurement probabilities, defined as the average rate of the surprisal of measurement outcomes, is constrained by the genuine quantum fluctuations contained in the measurement probabilities. Interestingly, this quantum speed limit can act as a witness for bipartite quantum correlations by selecting an optimal local projective measurement. Furthermore, we obtain a minimum time to transform an initial measurement probabilities to a target measurement probabilities, and apply this result to analyzing the cost of generating a local athermality in terms of genuine quantum uncertainty.