Non-stabilizerness versus entanglement in matrix product states

Non-stabilizerness versus entanglement in matrix product states

16 Jul 2024 | M. Frau, P. S. Tarabunga, M. Collura, M. Dalmonte, E. Tirrito
This paper investigates the relationship between entanglement and non-stabilizerness (magic) in matrix product states (MPSs). The authors study the connection between magic and the bond dimension used to approximate the ground state of a many-body system in two contexts: full-state magic and mutual magic (the non-stabilizer analogue of mutual information). They find that obtaining converged results for non-stabilizerness is typically easier than for entanglement. For full-state magic at critical points and sufficiently large volumes, they observe convergence with \(1/\chi^2\), where \(\chi\) is the MPS bond dimension. At small volumes, magic saturation is rapid, making it difficult to appreciate any finite-\(\chi\) correction. Mutual magic also shows fast convergence with bond dimension, though its specific functional form is hindered by sampling errors. The study also demonstrates how Pauli-Markov chains, originally formulated to evaluate magic, can reset the state of the art in computing mutual information for MPS. By verifying the logarithmic increase of mutual information between connected partitions at critical points, the authors show that mutual information and mutual magic exhibit different scaling behaviors. Mutual information scales with the partition size, while mutual magic is typically constant in size for connected partitions and scales much slower for disconnected partitions. Overall, the findings highlight the strong connection between magic and entanglement within the context of MPS, suggesting a robust relationship across different classes of criticality. The paper provides numerical evidence that stabilizer Renyi entropies, a measure of non-stabilizerness, converge rapidly with a scaling of \(1/\chi^2\), significantly faster than entanglement. Additionally, the efficiency of Pauli-Markov chains in estimating mutual information of disconnected subsystems is noted, offering a promising approach for experimental protocols.This paper investigates the relationship between entanglement and non-stabilizerness (magic) in matrix product states (MPSs). The authors study the connection between magic and the bond dimension used to approximate the ground state of a many-body system in two contexts: full-state magic and mutual magic (the non-stabilizer analogue of mutual information). They find that obtaining converged results for non-stabilizerness is typically easier than for entanglement. For full-state magic at critical points and sufficiently large volumes, they observe convergence with \(1/\chi^2\), where \(\chi\) is the MPS bond dimension. At small volumes, magic saturation is rapid, making it difficult to appreciate any finite-\(\chi\) correction. Mutual magic also shows fast convergence with bond dimension, though its specific functional form is hindered by sampling errors. The study also demonstrates how Pauli-Markov chains, originally formulated to evaluate magic, can reset the state of the art in computing mutual information for MPS. By verifying the logarithmic increase of mutual information between connected partitions at critical points, the authors show that mutual information and mutual magic exhibit different scaling behaviors. Mutual information scales with the partition size, while mutual magic is typically constant in size for connected partitions and scales much slower for disconnected partitions. Overall, the findings highlight the strong connection between magic and entanglement within the context of MPS, suggesting a robust relationship across different classes of criticality. The paper provides numerical evidence that stabilizer Renyi entropies, a measure of non-stabilizerness, converge rapidly with a scaling of \(1/\chi^2\), significantly faster than entanglement. Additionally, the efficiency of Pauli-Markov chains in estimating mutual information of disconnected subsystems is noted, offering a promising approach for experimental protocols.
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