Abstract The ‘operator entanglement’ of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in Rath et al (2023 PRX Quantum 4 010318) and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρ β = e − β H and of charged local operators evolving in Heisenberg picture O = e i t H O e − i t H . Our main results are: i) the SROE of ρ β obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.
Abstract The ‘operator entanglement’ of a quantum operator O is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global U(1) conservation law, and on operators O with a well-defined U(1) charge, for which it is possible to resolve the operator entanglement of O according to the U(1) symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in Rath et al (2023 PRX Quantum 4 010318) and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ρ β = e − β H and of charged local operators evolving in Heisenberg picture O = e i t H O e − i t H . Our main results are: i) the SROE of ρ β obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.
The operator entanglement (OE) is a key quantifier of the complexity of a reduced density matrix. In out-of-equilibrium situations, e.g. after a quantum quench of a product state, it is expected to exhibit an entanglement barrier. The OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom, it then reaches a maximum, and ultimately decays to a small finite value as the reduced density matrix converges to a simple stationary state through standard thermalization mechanisms. Here, by performing a new data analysis of the published experimental results of [Brydges et al., Science 364, 260 (2019)], we obtain the first experimental measurement of the OE of a subsystem reduced density matrix in a quantum many-body system. We employ the randomized measurements toolbox and we introduce and develop a new efficient method to post-process experimental data in order to extract higher-order density matrix functionals and access the OE. The OE thus obtained displays the expected barrier as long as the experimental system is large enough. For smaller systems, we observe a barrier with a double-peak structure, whose origin can be interpreted in terms of pairs of quasi-particles being reflected at the boundary of the qubit chain. As U(1) symmetry plays a key role in our analysis, we introduce the notion of symmetry resolved operator entanglement (SROE), in addition to the total OE. To gain further insights into the SROE, we provide a thorough theoretical analysis of this new quantity in chains of non-interacting fermions, which, in spite of their simplicity, capture most of the main features of OE and SROE. In particular, we uncover three main physical effects: the presence of a barrier in any charge sector, a time delay for the onset of the growth of SROE, and an effective equipartition between charge sectors.
Abstract The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix ρ ρ . This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given byNmassless Dirac fermions. For the density matrix, ρA ρA , of subsystems ofLneighbouring sites we calculate the leading terms in the largeLasymptotic expansion of the symmetry-resolved Rényi entanglement entropies. This follows from a largeLexpansion of the charged moments of ρA ρA ; we derive tr(eiαQAρAn)=aeiα⟨QA⟩(σL)−x(1+O(L−μ)) tr(eiαQAρAn)=aeiα⟨QA⟩(σL)-x(1+O(L-μ)) , wherea, xand μ μ are universal and σ σ depends only on theNFermi momenta. We show that the exponentxcorresponds to the expectation from CFT analysis. The error term O(L−μ) O(L-μ) is consistent with but weaker than the field theory prediction O(L−2μ) O(L-2μ) . However, using further results and conjectures for the relevant Toeplitz determinant, we find excellent agreement with the expansion over CFT operators.
Abstract In one dimension, the area law and its implications for the approximability by matrix product states are the key to efficient numerical simulations involving quantum states. Similarly, in simulations involving quantum operators, the approximability by matrix product operators (in Hilbert–Schmidt norm) is tied to an operator area law , namely the fact that the operator space entanglement entropy (OSEE)—the natural analog of entanglement entropy for operators, investigated by Zanardi (2001 Phys. Rev . A 63 040304) and by Prosen and Pižorn (2007 Phys. Rev. A 76 032316)—is bounded. In the present paper, it is shown that the OSEE can be calculated in two-dimensional conformal field theory, in a number of situations that are relevant to questions of simulability of long-time dynamics in one spatial dimension. It is argued that: (i) thermal density matrices ρ ∝ e − β H and generalized Gibbs ensemble density matrices ρ ∝ e − H G G E with local H GGE generically obey the operator area law; (ii) after a global quench, the OSEE first grows linearly with time, then decreases back to its thermal or GGE saturation value, implying that, while the operator area law is satisfied both in the initial state and in the asymptotic stationary state at large time, it is strongly violated in the transient regime; (iii) the OSEE of the evolution operator U ( t ) = e − i H t increases linearly with t , unless the Hamiltonian is in a localized phase; (iv) local operators in Heisenberg picture, ϕ ( t ) = e i H t ϕ e − i H t , have an OSEE that grows sublinearly in time (perhaps logarithmically), however it is unclear whether this effect can be captured in a traditional CFT framework, as the free fermion case hints at an unexpected breakdown of conformal invariance.
Abstract The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in Parez et al (2021 Phys. Rev. B 103 L041104), for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. The entanglement entropies display a time delay that depends on the charge sector that we characterise exactly. Both entanglement entropies and mutual information show effective equipartition in the scaling limit of large time and subsystem size. Furthermore, we argue that the behaviour of the charged entropies can be quantitatively understood in the framework of the quasiparticle picture for the spreading of entanglement, and hence we expect that a proper adaptation of our results should apply to a large class of integrable systems. We also find that the number entropy grows logarithmically with time before saturating to a value proportional to the logarithm of the subsystem size.
Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a 1-dimensional system of free fermions hopping on a lattice, in which the conserved U(1) charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in 1-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.
Abstract In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem’s reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher–Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement (‘flux-resolved entanglement’) for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.