Examples II. The following examples support Conjectures 1 and 2 and illustrate their conse- quences. We consider mainly the same distributions as in Section 3.2, but this time under the aspect of the existence of classical and quantum key-agreement protocols. Example 1 (cont’d). We have shown in Section 3.2 that the resulting quantum state is entangled if and only if the intrinsic information of the corresponding classical situation (with respect to the standard bases) is non-zero. Such a corre- spondence also holds on the protocol level. First of all, it is clear for the quantum state that QPA is possible whenever the state is entangled because both HA and HB have dimension two. On the other hand, the same is also true for the cor- responding clas√sical situation, i.e., secret-key agreement is possible whenever D/(1 − D) < 2 (1 − δ)δ holds, i.e., if the intrinsic information is positive. The necessary protocol includes an interactive phase, called advantage distillation, based on a repeat code or on parity checks (see [20] or [29]). ♦ ↓ Example 2 (cont’d). The quantum state ρAB in this example is bound entangled, meaning that the entanglement cannot be used for QPA. Interestingly, but not surprisingly given the discussion above, the corresponding classical distribution has the property that I(X; Y Z) > 0, but nevertheless, all the known classical advantage-distillation protocols [20], [22] fail for this distribution! It seems that S(X; Y ||Z) = 0 holds (although it is not clear how this fact could be rigorously proven). ♦ ≤ ≤
Examples II. The following examples support Conjectures 1 and 2 and illustrate their consequences. We consider mainly the same distributions as in Section 3.2, but this time under the aspect of the existence of classical and quantum key-agreement protocols. Example 1 (cont’d). We have shown in Section 3.2 that the resulting quan- tum state is entangled if and only if the intrinsic information of the cor- responding classical situation (with respect to the standard bases) is non- zero. Here, we show that such a correspondence also holds on the pro- tocol level. First of all, it is clear for the quantum state that QPA is possible whenever the state is entangled because both HA and HB have dimension two. On the other hand, the same is also true for the corre- sponding classic√al situation, i.e., secret-key agreement is possible whenever D/(1 − D) < 2 (1 − δ)δ holds, i.e., if the intrinsic information is positive. This is shown in Appendix C. There we describe the required protocol, more precisely, the advantage-distillation phase (called repeat-code protocol [20]), in which Xxxxx and Xxx use their advantage given by the authenticity of the public-discussion channel for generating new random variables for which the legitimate partners have an advantage over Xxx in terms of the (Xxxxxxx) information about each other’s new random variables. For a further discus- sion of this example, see also [15]. ♦ Example 2 (cont’d). The quantum state ρAB in this example is bound entan- gled, meaning that the entanglement cannot be used for QPA. Interestingly, but not surprisingly given the discussion above, the corresponding classical distribution has the property that I(X; Y ↓Z) > 0, but nevertheless, all the known classical advantage-distillation protocols [20], [22] fail for this distri- bution! It seems that S(X; Y ||Z) = 0 holds (although it is not clear how this fact could be rigorously proven, except by proving Conjecture 1 directly). ♦ Example 3 (cont’d). We have seen already that for 2 ≤ α ≤ 3, the quantum state is separable and the corresponding classical distribution (with respect to the standard bases) has vanishing intrinsic information. Moreover, it has been shown that for the quantum situation, 3 < α ≤ 4 corresponds to bound entanglement, whereas for α > 4, QPA is possible and allows for generating a secret key [18]. We describe a classical protocol here which suggests that the situation for the classical translation of the scenario is totally analogous: The protocol allow...
Examples II. The following examples support Conjectures 1 and 2 and illustrate their conse- quences. We consider mainly the same distributions as in Section 3.2, but this time under the aspect of the existence of classical and quantum key-agreement protocols. Example 1 (cont'd). We have shown in Section 3.2 that the resulting quantum state is entangled if and only if the intrinsic information of the corresponding classical situation (with respect to the standard bases) is non-zero. Such a corre- spondence also holds on the protocol level. First of all, it is clear for the quantum state that QPA is possible whenever the state is entangled because both HA and H B have dimension two. On the other hand, the same is also true for the cor- p responding classical situation, i.e., secret-key agreement is possible whenever D=(1 D) < 2 (1 ) holds, i.e., if the intrinsic information is positive. The necessary protocol includes an interactive phase, called advantage distillation, based on a repeat code or on parity checks (see [20] or [29]). } #
Examples II. The following examples support Conjectures 1 and 2 and illustrate their consequences. We consider mainly the same distributions as in Section 3.2, but this time under the aspect of the existence of classical and quantum key-agreement protocols.
