Global symmetry violation from non-isometric codes

We study the no-global-symmetry conjecture in quantum gravity by modeling black holes as non-isometric codes that encode the interior states with global charges into the fundamental states. The fluctuation in the inner products of the charged states can be larger compared to the case without charges. The non-isometric map causes states with different charges to have non-zero overlaps, signaling symmetry violation. The Renyi entropies of the radiation with global charges are found to be consistent with the quantum extremal surface formula. We compute various forms of Renyi relative entropy, as well as the fidelity, to quantify the degree of a global symmetry violation in Hawking radiation, demonstrating that global symmetries are indeed violated. We also comment on the instability of the black hole remnant.

💡 Research Summary

The paper tackles the long‑standing “no‑global‑symmetry” conjecture in quantum gravity by explicitly constructing a model in which a black hole is represented as a non‑isometric quantum error‑correcting code. In the usual holographic picture bulk degrees of freedom are encoded isometrically into the boundary Hilbert space, preserving inner products. By contrast, the authors argue that the effective interior of a black hole contains far more degrees of freedom than can be accommodated by the finite‑dimensional Hilbert space of the fundamental description. Consequently the encoding map must be non‑isometric: many interior states are projected out (or “annihilated”) when mapped to the fundamental description, while a small subset survives with a non‑trivial overlap structure.

The authors first set up the non‑isometric maps V and W that encode the left‑moving sector ℓ and the right‑moving sector r (each carrying a global U(1) charge) into two auxiliary fundamental systems B and C. The maps are defined by Haar‑random unitaries together with fixed reference states on auxiliary systems f, P and f_G, P_G, ensuring that the total dimensions match: |ℓ|·|f|·|r| = |B|·|P| and |Q_ℓ|·|f_G|·|Q_r| = |C|·|P_G|. Averaging over the random unitaries reproduces the identity on average, but the fluctuations of the inner product are bounded by a term proportional to |B|·|C| (eq. 9). Importantly, when global charges are present the bound is larger than in the charge‑free case, reflecting a larger typical overlap between charged states after the non‑isometric projection.

Next the paper constructs a Hawking‑type state |ψ_H⟩ that entangles a reference system L with the left‑moving interior ℓ and a radiation system R with the right‑moving interior r, with opposite charges so that the total charge vanishes. The non‑isometric map X = V⊗W is applied to this state, producing a density matrix for the radiation ρ_{R Q_R}. Because X is non‑isometric, the reduced density matrix acquires off‑diagonal components in the charge sector: matrix elements with q_R ≠ q_R′ are non‑zero. These off‑diagonal terms are precisely the signal of global‑symmetry violation: a global‑symmetry transformation U_{Q_R}(a)=e^{i a Q_R} acts non‑trivially on the radiation, and the relative entropy S(ρ̃_{R Q_R}(a)‖ρ_{R Q_R}) is no longer zero.

The authors compute the second Renyi entropy S_2(ρ_{R Q_R}) and show that it reproduces the quantum‑extremal‑surface (QES) formula. The dominant contributions come from two pieces: the ordinary Hawking radiation entropy χ_{out R Q_R} and an “area‑like’’ term log(|B||C|) plus the entropy of the interior left‑moving sector χ_{in ℓ Q_ℓ}. The result can be written as
 S_2(ρ_{R Q_R}) ≈ min{ S_2(χ_{out R Q_R}), log(|B||C|)+S_2(χ_{in ℓ Q_ℓ) } ,
which is exactly the QES prescription for generalized entropy. At early times the first term dominates, reproducing the semiclassical Hawking entropy and preserving the global charge. At later times, when the black‑hole Hilbert space shrinks (|B|,|C| become small), the second term dominates and the off‑diagonal charge contributions become important, signalling the breakdown of the global symmetry.

To quantify the symmetry breaking the paper evaluates the Renyi relative entropy between the symmetry‑transformed radiation state ρ̃_{R Q_R}(a)=U_{Q_R}(a) ρ_{R Q_R} U_{Q_R}†(a) and the original ρ_{R Q_R}. In the effective description the two states are identical, so the relative entropy vanishes. After the non‑isometric map, however, only the off‑diagonal charge sectors contribute, giving a non‑zero result. For small transformation parameter a the Renyi relative entropy behaves as
 S_2 ≈ a² (⟨q_R²⟩ − ⟨q_R⟩²) / (|B||C|) e^{‑S_2(χ_{in ℓ Q_ℓ)} ,
showing that the amount of symmetry violation is suppressed by the product of the black‑hole and charge Hilbert‑space dimensions, but is nevertheless finite. This provides a concrete order‑parameter for the conjectured absence of global symmetries in quantum gravity.

The paper also discusses the fate of possible charged remnants. A charged remnant would contribute to the entropy at early stages, but as the black hole evaporates the effective dimensions |B| and |C| grow relative to the remaining charge sector, causing the remnant’s contribution to be exponentially suppressed. Hence any would‑be‑stable charged remnant is unlikely to survive the evaporation process, in line with expectations from the holographic and “no‑remnant” arguments.

Two technical appendices support the main results. Appendix A derives the bound on the fluctuation of the inner product, while Appendix B presents an alternative single‑unitary model of the non‑isometric map, confirming that the conclusions do not depend on the specific circuit implementation.

In summary, the authors demonstrate that a non‑isometric encoding of black‑hole interior states inevitably leads to non‑zero overlaps between states of different global charge, which manifests as a non‑vanishing relative entropy between symmetry‑related radiation states. This provides a quantitative, information‑theoretic proof of global‑symmetry violation in a concrete holographic‑inspired model, thereby lending strong support to the no‑global‑symmetry conjecture in quantum gravity.