We present preliminary numerical evidence for the hypothesis that the Hamiltonian SU(2) gauge theory discretized on a lattice obeys the eigenstate thermalization hypothesis (ETH). To do so we study three approximations: (a) a linear plaquette chain in a reduced Hilbert space limiting the electric field basis to j = 0 , 1 2 , (b) a two-dimensional honeycomb lattice with periodic or closed boundary condition and the same Hilbert space constraint, and (c) a chain of only three plaquettes but such a sufficiently large electric field Hilbert space ( j ≤ 7 2 ) that convergence of all energy eigenvalues in the analyzed energy window is observed. While an unconstrained Hilbert space is required to reach the continuum limit of SU(2) gauge theory, numerical resource constraints do not permit us to realize this requirement for all values of the coupling constant and large lattices. In each of the three studied cases we check first for random matrix theory (RMT) behavior in the eigenenergy spectrum and then analyze the diagonal as well as the off-diagonal matrix elements between energy eigenstates for a few operators. Within current uncertainties all results for (a), (b) and (c) agree with ETH predictions. Furthermore, we find the off-diagonal matrix elements of the electric energy operator exhibit RMT behavior in frequency windows that are small enough in (b) and (c). To unambiguously establish ETH behavior and determine for which class of operators it applies, an extension of our investigations is necessary. Published by the American Physical Society 2024
Entanglement and non-stabilizerness (magic) quantify two distinct departures of quantum systems from classical description: the former measures non-local correlations, while the latter measures the deviation from stabilizer states that can be efficiently simulated classically. Understanding magic in physically relevant quantum field theories is essential for identifying where quantum advantage may be realized in the early fault-tolerant quantum computing era. We calculate the gauge-invariant entanglement entropy and stabilizer Rényi entropy of the ground state of the (1+1)-dimensional SU(2) lattice gauge theory formulated in a dressed-site basis that enforces Gauss's law exactly. Using tensor networks, we obtain results for system sizes up to
L=100 (300 qubits). We find a crossover denoted by
g⋆ where the ground state passes from a more magic-rich regime into a regime with less magic; this is also tracked by the sharpest change of both the entanglement entropy and lattice particle density. Our large-scale study of non-stabilizerness and entanglement entropy in a non-Abelian lattice gauge theory with matter provides new insight into the interplay of magic and entanglement in gauge theories, both of which are relevant for classical and early fault-tolerant quantum simulations.
Entanglement entropy, taken here to be geometric, requires a geometrically separable Hilbert space. In lattice gauge theories, it is not immediately clear if the physical Hilbert space is geometrically separable. In a previous paper we have shown that the physical Hilbert space in pure gauge Abelian lattice theories exhibits some form of geometric scaling with the lattice volume, which suggest that the space is locally factorizable and, therefore, geometrically separable. In this paper, we provide strong evidence that indicates that this scaling is not present when the group is non-Abelian. We do so by looking at the scaling of the dimension of the physical Hilbert space of theories with certain discrete groups. The lack of an appropriate scaling implies that the physical Hilbert space of such a theory does not admit a local factorization. We then extend the reasoning, as sensibly possible, to SU(2) and SU(N) to reach the same conclusion. Lastly, we show that the addition of matter fields to non-Abelian lattice gauge theories makes the resulting physical Hilbert space locally factorizable. Published by the American Physical Society 2024
We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories.It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way.In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions.Such an extension leads to a reduction procedure which can be easily implemented in lattice simulations by constructing lattices with special topology.This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations.On the simplest example of Z 2 lattice gauge theory in (2 + 1) dimensions we demonstrate the relation between entanglement entropy and the classical entropy of the field flux.For SU (2) lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region.We also comment on the holographic interpretation of the entanglement entropy.
We study non-stabilizerness (magic) in the ground state of
(2+1)-dimensional
SU(2) Hamiltonian lattice gauge theory with matter, formulated in the dressed-site basis in the hardcore-gluon truncation and restricted to the zero baryon-number sector. Using matrix product states, we compute three facets of magic: the second-order stabilizer Rényi entropy (SRE)
M2, its non-local component
M2NL, estimated by the SRE
M2(ψ~) of the computational-basis Schmidt representative
ψ~, and a lower bound in terms of the anti-flatness
F of the entanglement spectrum. We also prove a stronger form of the sandwich relation:
−log2(1−4F)≤M2NL≤M2; the lower bound rests on a stronger inequality that we obtain for arbitrary Schmidt bases and rank, with an optimal constant. Combined with recent results, this yields the unconditional bracket
max{−log2(1−4F),M2(ψ~)−4}≤M2NL≤M2(ψ~). We emphasize a structural distinction that makes the non-local quantities the physically preferred diagnostics: whereas the full SRE depends on the (non-unique) encoding of the gauge-invariant local Hilbert space into qubits, the non-local magic, its estimate
M2(ψ~), and the anti-flatness are invariant under site-local re-encodings and are therefore intrinsic to the state and bipartition. By varying the gauge coupling for two values of the fermion mass on lattices up to
8×8 (
∼288 qubits) with bond dimension up to
128, we find that the non-local magic, as estimated by
M2(ψ~), furnishes a sharper and more bond-dimension-friendly probe of the gauge-matter delocalization crossover compared to the full SRE or the gauge-invariant entanglement entropy, retaining a clear signal at bond dimensions well below those needed to converge the ground state itself.
We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories. It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way. In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions. Such an extension leads to a reduction procedure which can easily be implemented in lattice simulations by constructing lattices with special topology. This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations. On the simplest example of Z 2 lattice gauge theory in dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region. We also comment on the holographic interpretation of the entanglement entropy.