Distillation of secret-key from a class of compound memoryless quantum sources (Q2820903)

The paper addresses the issue of obtaining secret-key distillations in a secure manner using a classical-quantum-quantum (cqq) memoryless source. Three entities are considered: the sender, the legal receiver and the eavesdropper, in a scenario proposed in [\textit{R. Ahlswede} and \textit{I. Csiszàr}, IEEE Trans. Inf. Theory 39, No. 4, 1121--1132 (1993; Zbl 0802.94013)] for classical information-theoretic approach, and rebuilt in a similar way for quantum systems in [\textit{I. Devetak} and \textit{A. Winter}, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2053, 207--235 (2005; Zbl 1145.81325)]. Except for the initial input in this three-party system, all communications use quantum media. In general the class of density matrices that characterizes cqq in this paper passes a special regularity property verification. Additionally, two cases are considered and compared: (1) when the legitimate parts have an incomplete knowledge about the generating density matrix, and therefore they must develop the same protocols for obtaining the secret-key (the intruder has not this restriction), and (2) when the sender has the whole information about the marginal distribution of his systems deriving from the source statistics (variant denoted SMI -- Sender Marginal Information). The main results of the paper are Theorem 8 and 9, which can be summarized as follows: For a regular set \(\mathcal I\) of cqq density matrices on a three-party Hilbert space \({\mathcal H}_{ABE}\) NEWLINE\[NEWLINEK_{\rightarrow}({\mathcal I})=K_{\rightarrow,SMI}=\lim_{k\to\infty}\frac{1}{k}K_{\rightarrow}^{(1)}\left({\mathcal I}^{\otimes k}\right)NEWLINE\]NEWLINE where \(K_{\rightarrow}({\mathcal I})\; \left( K_{\rightarrow,SMI}({\mathcal I})\right)\) is the forward secret-key capacity of \(\mathcal I\) without (respectively with) SMI defined by \(\sup\{R\geq 0\;|\; R\) is an achievable secret-key distillation rate for \(\mathcal I\) without (with) SMI\(\}\), and \(K_{\rightarrow}^{(1)}\left({\mathcal I}^{\otimes k}\right)\) is a value based on the Markov transition matrices of the system.NEWLINENEWLINETherefore, under a special assumption of regularity, the secret-key distillation capacities are equals -- and asymptotically optimal -- whether the two parties have relatively complete knowledge of their marginal state deriving from the source statistics.NEWLINENEWLINEThe proofs of these theorems are very laborious and cover half of this very long paper (two sections and three annexes).NEWLINENEWLINEThe other two important sections (6 and 7) complete this result, by showing that for irregular sources the capacities are generally not equal (a suitable example with \(K_{\rightarrow}({\mathcal I})<K_{\rightarrow,SMI}({\mathcal I})\) is developed in the proof of Theorem 21 from section 6), and that \(K_{\rightarrow}({\mathcal I})\) remains asymptotically optimal even for a weakly regular set of cqq density matrices, based on the set-valued maps (Theorem 29 from section 7).NEWLINENEWLINEThe study of this paper (in fact a real chapter book) is continued in [\textit{N. Tavangaran}, \textit{H. Boche} and \textit{R. F. Schaefer}, ``Secret-key generation using compound sources and one-way public communication'', Preprint, \url{arXiv:1601.07513}] where Theorems 8 and 9 are completed under other weaker hypothesis.