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Quantum Information and Computation

General fixed points of quasi-local frustration-free quantum semigroups

D JohnsonPeter, TicozziFrancesco, ViolaLorenza

Published May 1, 2016• 0 citations
Publisher

Abstract

We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject...
DOI
10.5555/3179466.3179471

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"Characterizing and quantifying quantum frustration: exact bounds to entanglement and correlations, and local and global observables"

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We discuss a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems with Hamiltonians made of sums of local two-body contributions in terms of the overlap between the unfrustrated ground state of the pair Hamiltonians and the frustrated ground state of the total Hamiltonian. We derive an exact lower bound this overlap measure of frutration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration. The mathematical scheme can be put to test by relating the overlap measure and the inequalities it obeys to global and local observables in terms of the static structure factor and transverse n-point correlation functions.

•Jan 1, 2013•0 citations
We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization results yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information processing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question.
We focus on two classes of pure states for a finite-dimensional multipartite quantum system: those that can be identified as unique ground states (UGS) of a locally-constrained Hamiltonian, and those that can be obtained as unique, asymptotically stable equilibria of a Markov quantum dynamics under the same quasi-locality constraints (QLS). While there is an equivalence between (strict) subsets of two classes, namely, states that are UGS of frustration-free Hamiltonians and unique equilibria of purely dissipative dynamics, two paradigmatic entangled states, GHZ and W, offer counterexamples to either implication in the general setting. The intuition that every UGS state can be obtained as the exact steady state of a Markovian cooling process is salvaged only if suitable non-local resources are added. We show that stochastic stabilization of the target UGS state can be achieved if we perform continuous measurement of the global Hamiltonian, while controlling the dynamics with a switching, filtering-based quantum feedback strategy.
We discuss a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems with Hamiltonians made of sums of local two-body contributions in terms of the overlap between the unfrustrated ground state of the pair Hamiltonians and the frustrated ground state of the total Hamiltonian. We derive an exact lower bound this overlap measure of frutration in degenerate ground states of quantum many-body systems. The bound results in the sum of two contributions: entanglement and classical correlations arising from local measurements. We show that average frustration properties are completely determined by the behavior of the maximally mixed ground state. We identify sufficient conditions for a quantum spin system to saturate the bound, and for models with twofold degeneracy we prove that average and local frustration coincide. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration. The mathematical scheme can be put to test by relating the overlap measure and the inequalities it obeys to global and local observables in terms of the static structure factor and transverse n-point correlation functions.