Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes andk-uniform states. In this work, we establish a connection between quantum weight enumerators and then-qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using then-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine thek-uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.

Do N-partite k-uniform states always exist when$k\leq \left \lfloor {{\frac {N}{2}}}\right \rfloor -1$? In this work, we provide new upper bounds on the parameter k for the existence of k-uniform states in$(\mathbb {C}^{d})^{\otimes N}$when$d=3,4,5$, which extend Rains’ bound in 1999 and improve Scott’s bound in 2004. Since a k-uniform state in$(\mathbb {C}^{d})^{\otimes N}$corresponds to a pure$((N,1,k+1))_{d}$quantum error-correcting code, we also give new upper bounds on the minimum distance$k+1$of pure$((N,1,k+1))_{d}$quantum error-correcting codes. Furthermore, we generalize Scott’s bound to heterogeneous systems, and show some non-existence results of absolutely maximally entangled states in$\mathbb {C}^{d_{1}}\otimes (\mathbb {C}^{d_{2}})^{\otimes 2n}$.

The resource framework of quantum coherence was introduced by Baumgratz, Cramer, and Plenio [Phys. Rev. Lett. 113, 140401 (2014)] and further developed by Winter and Yang [Phys. Rev. Lett. 116, 120404 (2016)]. We consider the one-shot problem of distilling pure coherence from a single instance of a given resource state. Specifically, we determine the distillable coherence with a given fidelity under incoherent operations (IO) through a generalization of the Winter-Yang protocol. This is compared to the distillable coherence under maximal incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO), which can be cast as a semidefinite programme, that has been presented previously by Regula et al. [Phys. Rev. Lett. 121, 010401 (2018)]. Our results are given in terms of a smoothed min-relative entropy distance from the incoherent set of states, and a variant of the hypothesis-testing relative entropy distance, respectively. The one-shot distillable coherence is also related to one-shot randomness extraction. Moreover, from the one-shot formulas under IO, MIO, and DIO, we can recover the optimal distillable rate in the many-copy asymptotics, yielding the relative entropy of coherence. These results can be compared with previous work by some of the present authors [Zhao et al., Phys. Rev. Lett. 120, 070403 (2018)] on one-shot coherence formation under IO, MIO, DIO and also SIO. This shows that the amount of distillable coherence is essentially the same for IO, DIO, and MIO, despite the fact that the three classes of operations are very different. We also relate the distillable coherence under strictly incoherent operations (SIO) to a constrained hypothesis testing problem and explicitly show the existence of bound coherence under SIO in the asymptotic regime.