Quantum / CONCEPT

The minimum of purity plus time-reversal overlap

For every bipartition of N qubits into a part A of k qubits and its non-empty complement, the minimum over pure states of the purity of the reduced state plus its overlap with the time-reversed reduced state is 2^(k-N) when 2k > N and 2^(1-k) when 2k <= N. This proves Conjecture 2 of E. Serrano-Ensastiga, O. Giraud and J. Martin (arXiv:2507.12680), who proved the case k = 1 and found the other values numerically for up to ten qubits.