A new approximate Eastin-Knill theorem Alexander-Turner, Rhea Transversal encoded gatesets are highly desirable for fault tolerant quantum computing. However, a quantum error correcting code which exactly corrects for local erasure noise and supports a universal set of transversal gates is ruled out by the Eastin-Knill theorem. Here, we provide a new approximate Eastin-Knill theorem for the single-shot regime when we allow for some probability of error in the decoding. In particular, we show that a quantum error correcting code can support a universal set of transversal gates and approximately correct for local erasure if and only if the conditional min-entropy of the Choi state of the encoding and noise channel is upper bounded by a simple function of the worstcase error probability. Our no-go theorem can be computed by solving a semidefinite program, and, in the spirit of the original Eastin-Knill theorem, is formulated in terms of a condition that is both necessary and sufficient, ensuring achievability whenever it is passed. As an example, we find that with n = 100 physical qutrits we can encode k = 1 logical qubit in the W-state code, which admits a universal transversal set of gates and corrects for single subsystem erasure with error probability of ε = 0.005. To establish our no-go result, we leverage tools from the resource theory of asymmetry, where, in the single-shot regime, a single (output state-dependent) resource monotone governs all state purifications. 2026-02-04T09:44:31Z 2026-02-04T09:44:31Z 2025-12-20 journal article Alexander, R. A new approximate Eastin-Knill theorem. npj Quantum Inf 12, 15 (2026). [https://doi.org/10.1038/s41534-025-01156-0] https://hdl.handle.net/10481/110641 10.1038/s41534-025-01156-0 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional Springer Nature