On the complexity of search for keys in quantum cryptography

The trace distance is used as a security criterion in proofs of security of keys in quantum cryptography. Some authors doubted that this criterion can be reduced to criteria used in classical cryptography. The following question has been answered in this work. Let a quantum cryptography system provide an $\varepsilon$-secure key such that $\frac{1}{2}|| \rho_{XE}-\rho_U\otimes \rho_E ||_1<\varepsilon$, which will be repeatedly used in classical encryption algorithms. To what extent does the $\varepsilon$-secure key reduce the number of search steps (guesswork) as compared to the use of ideal keys? A direct relation has been demonstrated between the complexity of the complete consideration of keys, which is one of the main security criteria in classical systems, and the trace distance used in quantum cryptography. Bounds for the minimum and maximum numbers of search steps for the determination of the actual key have been presented.