Weak Locking Capacity of Quantum Channels Can be Much Larger Than Private Capacity
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We show that it is possible for the so-called weak locking capacity of a quantum channel (Guha et al. in Phys Rev X 4:011016, 2014) to be much larger than its private capacity. Both reflect different ways of capturing the notion of reliable communication via a quantum system while leaking almost no information to an eavesdropper; the difference is that the latter imposes an intrinsically quantum security criterion whereas the former requires only a weaker, classical condition. The channels for which this separation is most straightforward to establish are the complementary channels of classical-quantum (cq-)channels and, hence, a subclass of Hadamard channels. We also prove that certain symmetric channels (related to photon number splitting) have positive weak locking capacity in the presence of a vanishingly small pre-shared secret, whereas their private capacity is zero. These findings are powerful illustrations of the difference between two apparently natural notions of privacy in quantum systems, relevant also to quantum key distribution: the older, naïve one based on accessible information, contrasting with the new, composable one embracing the quantum nature of the eavesdropper’s information. Assuming an additivity conjecture for constrained minimum output Rényi entropies, the techniques of the first part demonstrate a single-letter formula for the weak locking capacity of complements to cq-channels, coinciding with a general upper bound of Guha et al. for these channels. Furthermore, still assuming this additivity conjecture, this upper bound is given an operational interpretation for general channels as the maximum weak locking capacity of the channel activated by a suitable noiseless channel.
KeywordsQuantum channel Private capacity Quantum key distribution Accessible information Composability Locking capacity
I thank Mark Wilde for enlightening discussions on information locking, for introducing me to locking capacities and for first raising the problem of separating the private capacity from the weak locking capacity. The keen interest he and Saikat Guha took in this project helped immensely to develop the ideas of the present paper. This research was supported by the European Commission (STREP “RAQUEL”), the European Research Council (Advanced Grant “IRQUAT”) and the Spanish MINECO (project FIS2008-01236) with FEDER funds. Part of this work was done during the program “Mathematical Challenges in Quantum Information” (MQI), 27/8-20/12/2013, at the Isaac Newton Institute in Cambridge, whose hospitality during the semester is gratefully acknowledged.
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