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Entanglement and complexity of purification in (1+1)-dimensional free conformal field theories

Hugo A. Camargo, Lucas Hackl, Michał P. Heller, A. Jahn, Tadashi Takayanagi, Bennet Windt

Published Jan 1, 2021• 25 citations
Publisher

Abstract

Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using the most general Gaussian purifications. We provide a comprehensive comparison with existing results and identify universal properties. We further discuss important subtleties in our setup: the massless limit of the free bosonic theory and the corresponding behavior of the mutual information, as well as the Hilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Ising conformal field theory.

DOI
10.17169/refubium-30902

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Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using the most general Gaussian purifications. We provide a comprehensive comparison with existing results and identify universal properties. We further discuss important subtleties in our setup: the massless limit of the free bosonic theory and the corresponding behavior of the mutual information, as well as the Hilbert space structure under the Jordan-Wigner mapping in the spin chain model of the Ising conformal field theory.
Quantifying entanglement properties of mixed states in quantum field theory via entanglement of purification and reflected entropy is a new and challenging subject. In this work, we study both quantities for two spherical subregions far away from each other in the vacuum of a conformal field theory in any number of dimensions. Using lattice techniques, we find an elementary proof that the decay of both the entanglement of purification and reflected entropy is enhanced with respect to the mutual information behavior by a logarithm of the distance between the subregions. In the case of the Ising spin chain at criticality and the related free fermion conformal field theory, we compute also the overall coefficients numerically for the both quantities of interest.
Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all these models, and we support our conjectures with a variety of numerical and analytical results. We find that such minimally entangled purifications have a number of applications, from enhancing entanglement-based tensor network methods for describing mixed states to elucidating novel aspects of the emergence of geometry from entanglement in the AdS/CFT correspondence.
In quantum systems, purification can map mixed states into pure states and a nonunitary evolution into a unitary one by enlarging the Hilbert space. We establish a connection between the complexities of mixed quantum states and their purification, proposing new inequalities among these complexities. By examining single qubits, two-qubit Werner states, eight-dimensional Gaussian random unitary ensembles, and infinite-dimensional systems, we demonstrate how these relationships manifest across a broad class of systems. We find that the spread complexity of purification of a vacuum state evolving into a thermal state equals the average number of Rindler particles. This complexity is also shown to adhere to the Lloyd-like bound, indicating a further relation to the quantum speed limit. Finally, using mutual Krylov complexity, we observe subadditivity of the Krylov complexities, which contrasts with known results from holographic volume complexity. We put forward Krylov mutual complexity as a diagnosis of a potential gravity dual of Krylov complexities.
Quantum information has produced fresh insights into foundational questions about the AdS/CFT correspondence. One fascinating concept, which has captured increasing attention, is quantum circuit complexity. As a natural generalization for the complexity of pure states, we investigate the circuit complexity of mixed states in this thesis. \n \nFirst of all, we explore the so-called purification complexity which is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We focus on studying the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of ‘mode-by-mode purifications’ where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. In order to compare with the results from using the various holographic proposals for the complexity of subregions, we explore the purification complexity for thermal states of a free scalar QFT, and for subregions of the vacuum state in two dimensions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the ‘mutual complexity’ in the various cases studied in this thesis. \n \nIn addition, we propose to generalize the Fubini-Study method for pure-state complexity to generic quantum states including mixed states by taking Bures metric or quantum Fisher information metric on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purifications. We also find that the purification complexity is nonincreasing under any trace-preserving quantum operations. As an illustration, we study the mixed Gaussian states as an example to explicitly show our conclusions for purification complexity.
Abstract We study the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of ‘mode-by- mode purifications’ where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. We explore the purification complexity for thermal states of a free scalar QFT in any number of dimensions, and for subregions of the vacuum state in two dimensions. We compare our results to those found using the various holographic proposals for the complexity of subregions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the ‘mutual complexity’ in the various cases studied in this paper.