## 4 Beyond Memory-Restricted Scenario  
If [PERSON] is not forced to conduct her measurements right after the error-correction stage, it is more beneficial for her to measure the states later--when she will be able to take into account the information disclosed during the privacy amplification procedure. In this case, an observable that was optimal when measuring the original states can become non-optimal for measuring the states after information processing. In [32], an explicit example was provided, which shows that the strategy yields gain for [PERSON], i.e., that classical processing of states of a quantum ensemble changes the set of observables providing accessible information. The example is based on considering a quantum ensemble \(\mathcal{E}_{\text{init}}\) obtained as the result of a simple two-letter classical-quantum channel utilized twice (the lower index \"init\" indicated that the ensemble is obtained before the classical information processing). \[\mathcal{E}_{\text{init}}=\bigg{\{}\bigg{(}\frac{1}{4},\;\sigma_{0}\otimes \sigma_{0}\bigg{)},\;\bigg{(}\frac{1}{4},\;\sigma_{0}\otimes\sigma_{1}\bigg{)},\;\bigg{(}\frac{1}{4},\;\sigma_{1}\otimes\sigma_{0}\bigg{)},\;\bigg{(}\frac{ 1}{4},\;\sigma_{1}\otimes\sigma_{1}\bigg{)}\bigg{\}},\]where the equiprobable letter states (described by density operators on two-dimensional Hilbert space \(\mathcal{H}\)) \(\sigma_{0}\) and \(\sigma_{1}\) are pure and can be represented as real vectors in some orthonormal basis \(\{\ket{0},\ket{1}\}\subset\mathcal{H}\):  
\[\forall x\in\{0,1\}:\;\sigma_{x}=\ket{\psi_{x}}\bra{\psi_{x}},\quad\ket{\psi_{x }}=\cos\alpha\ket{0}+(-1)^{x}\sin\alpha\ket{1}.\]  
According to [25], an optimal strategy for extracting the maximal amount of information from the quantum ensemble \(\mathcal{E}_{\rm init}\) consists in conducting two independent local measurements (measurements in the Hadamard basis). Then, a simple classical data processing corresponding to an XOR operation can be considered. It merges some states and transforms \(\mathcal{E}_{\rm init}\) into an ensemble  
\[\mathcal{E}=\bigg{\{}\bigg{(}\frac{1}{2},\,\frac{1}{2}\sigma_{0}\otimes\sigma_ {0}+\frac{1}{2}\sigma_{1}\otimes\sigma_{1}\bigg{)},\,\,\bigg{(}\frac{1}{2},\, \frac{1}{2}\sigma_{0}\otimes\sigma_{1}+\frac{1}{2}\sigma_{1}\otimes\sigma_{0 }\bigg{)}\bigg{\}}.\]  
It was shown in [32] that there exists such a range of \(\alpha\) values for which it is true that any observable providing \(I_{\rm acc}(\mathcal{E})\) has to include entangled operators. Moreover, the measurement in the Bell basis is always the optimal measurement strategy for \(\mathcal{E}\). Thus, classical information processing can significantly change the structure of the optimal observable. However, the question of the existence of classical data processing operations preserving an optimal observable remains, to our knowledge, open. Privacy amplification in QKD is an important special case of classical data processing. In particular, the considered XOR operation can be an element of some universal hash functions family used for privacy amplification. This explains the significance of the example in the context of our study: it proves that there exist privacy amplification procedures turning the disclosure of privacy amplification-related information into QDL-type communication between legitimate users and an eavesdropper. Notably, if the opposite statement was true and any observable that was optimal before classical data processing remained optimal after the operation, then it would not have been important whether an eavesdropper conducted their measurement before or after obtaining privacy amplification-related information (this would not influence the efficiency of their attack). In this _imaginary_ situation, we could have constructed a statement about our method's applicability while leaving privacy amplification data exchange completely unencrypted.