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We show that it is possible for the so-called weak locking capacity of aquantum channel [Guha et al., PRX 4:011016, 2014] to be much larger than itsprivate capacity. Both reflect different ways of capturing the notion ofreliable communication via a quantum system while leaking almost no informationto an eavesdropper; the difference is that the latter imposes an intrinsicallyquantum security criterion whereas the former requires only a weaker, classicalcondition. The channels for which this separation is most straightforward toestablish are the complementary channels of classical-quantum (cq-)channels,and hence a subclass of Hadamard channels. We also prove that certain symmetricchannels (related to photon number splitting) have positive weak lockingcapacity in the presence of a vanishingly small pre-shared secret, whereastheir private capacity is zero.  These findings are powerful illustrations of the difference between twoapparently natural notions of privacy in quantum systems, relevant also toquantum key distribution (QKD): the older, naive one based on accessibleinformation, contrasting with the new, composable one embracing the quantumnature of the eavesdropper's information.  Assuming an additivity conjecture for constrained minimum output Renyientropies, the techniques of the first part demonstrate a single-letter formulafor the weak locking capacity of complements to cq-channels, coinciding with ageneral upper bound of Guha et al. for these channels. Furthermore, stillassuming this additivity conjecture, this upper bound is given an operationalinterpretation for general channels as the maximum weak locking capacity of thechannel activated by a suitable noiseless channel.

Weak locking capacity of quantum channels can be much larger than private capacity