Pauli-ball discrepancy and stability of quantum codes
The Pauli-ball discrepancy, a quadratic discrepancy summed over all error radii, measures how uniformly the Pauli error images of a $K$-dimensional quantum code cover the $d$-dimensional Hilbert space. At parameters attaining equality in the quantum Singleton bound with minimum code distance $\geq3$, or in the one-error Hamming bound with prime-power local dimension, the discrepancy of every projector of the prescribed rank equals an explicit benchmark plus an exact nonnegative remainder, which vanishes precisely for pure quantum maximum-distance-separable codes and nondegenerate perfect codes, respectively. For a Pauli ball of radius $e$ below half the target distance and size $V_e$, this remainder linearly bounds the squared Hilbert--Schmidt distance of the error-image Gram matrix from the identity. When $KV_e\leq d$, it linearly bounds the diamond error and entanglement infidelity of a noise-independent polar recovery under any noise in that ball's span. For five qubits, we obtain the sharp rank-two Gram constant and exact logical channels of rotated codes attaining this linear order in diamond error. For pure states, the discrepancy equals a pure-state benchmark plus a positive weighted sum of marginal-purity defects, so absolutely maximally entangled states, when they exist, are exactly the minimizers.