Common use of Examples II Clause in Contracts

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 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 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) = ||Z)= 0 holds (although it is not clear how this fact could be rigorously proven). ♦ ≤ ≤