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. − √ − H H 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 A and B have dimension two. On the other hand, the same is also true for the cor- responding classical 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). ♦ ≤ ≤ ≤ 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 allows classical key agreement exactly for α > 4. However, this does not imply (although it appears very plausible) that no classical protocol exists at all for the case α ≤ 4. ∈ { } ∈ { } { } Let α > 4. We consider the following protocol for classical key agreement. First of all, ▇▇▇▇▇ and ▇▇▇ both restrict their ranges to 1, 2 (i.e., publicly reject a realization unless X 1, 2 and Y 1, 2 ). The resulting distribution is as follows (to be normalized): X Y (Z) 1 2 1 (0) 2 (4) 5 − α 2 (2) α (0) 2 Then, ▇▇▇▇▇ and ▇▇▇ both send their bits locally over channels PX|X and PY |Y , respectively, such that the resulting bitsX and Y are symmetric. The channel PX|X [PY |Y ] sends X = 0 [Y = 1] to X = 1 [Y = 0] with probability (2α −

Examples II. The following examples support Conjectures 1 and 2 and illustrate their conse- quencesconsequences. 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. − √ − H H Example 1 (cont’d). We have shown in Section 3.2 that the resulting quantum quan- tum state is entangled if and only if the intrinsic information of the corresponding cor- responding classical situation (with respect to the standard bases) is non-non- zero. Such Here, we show that such a corre- spondence correspondence also holds on the protocol pro- tocol level. First of all, it is clear for the quantum state that QPA is possible whenever the state is entangled because both A HA and B HB have dimension two. On the other hand, the same is also true for the cor- responding classical 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. The necessary This is shown in Appendix C. There we describe the required protocol, more precisely, the advantage-distillation phase (called repeat-code protocol includes an interactive phase, called advantage distillation, based on a repeat code or on parity checks (see [20] or [29]), in which ▇▇▇▇▇ and ▇▇▇ 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 ▇▇▇ in terms of the (▇▇▇▇▇▇▇) 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 entangledentan- 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↓Z) > 0, but nevertheless, all the known classical advantage-distillation protocols [20], [22] fail for this distributiondistri- 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 allows classical key agreement exactly for α > 4. However, this does not imply (although it appears very plausible) that no classical protocol exists at all for the case α ≤ 4. ∈ { } ∈ { } { } Let α > 4. We consider the following protocol for classical key agreementagree- ment. First of all, ▇▇▇▇▇ and ▇▇▇ both restrict their ranges to {1, 2 2} (i.e., publicly reject a realization unless X ∈ {1, 2 2} and Y ∈ {1, 2 2}). The resulting distribution is as follows (to be normalized): X Y (Z) 1 2 1 (0) 2 (4) 5 − α 2 (2) α (0) 2 Then, ▇▇▇▇▇ and ▇▇▇ both send their bits locally over channels PX|X and PY |Y , respectively, such that the resulting bitsX bits X and Y are symmetric. The channel PX|X [PY |Y ] sends X = 0 [Y = 1] to X = 1 [Y = 0] with probability (2α −− 5)/(2α + 4), and leaves X [Y ] unchanged otherwise. The distribution PXY Z is then X Y (Z) 1 2 1 9 (0) 2 · 2α+4 9 2α−5 (2) α · 2α+4 · 2α+4 (1) 5 − α 2 (2) α 2α−5 2α+4 2α−5 (0) 2 · 2 · 2α+4 It is not difficult to see that for α > 4, we have Prob [X = Y ] > 1/2 and that, given that X = Y holds, ▇▇▇ has no information at all about what this bit is. Thus the repeat-code protocol described in Appendix C allows for classical key agreement in this situation. For α ≤ 4 however, classical key agreement, like quantum key agreement, seems impossible. The results of Example 3 are illustrated in Figure 3 in Appendix A. ♦

Examples II. The following examples support Conjectures 1 and 2 and illustrate their conse- quencesconsequences. 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. − √ − H H Example 1 (cont’d). We have shown in Section 3.2 that the resulting quantum quan- tum state is entangled if and only if the intrinsic information of the corresponding cor- responding classical situation (with respect to the standard bases) is non-non- zero. Such Here, we show that such a corre- spondence correspondence also holds on the protocol pro- tocol level. First of all, it is clear for the quantum state that QPA is possible whenever the state is entangled because both A HA and B HB have dimension two. On the other hand, the same is also true for the cor- responding classical 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. The necessary This is shown in Appendix C. There we describe the required protocol, more precisely, the advantage-distillation phase (called repeat-code protocol includes an interactive phase, called advantage distillation, based on a repeat code or on parity checks (see [20] or [29]), in which ▇▇▇▇▇ and ▇▇▇ 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 Eve in terms of the (▇▇▇▇▇▇▇) 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 entangledentan- 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↓Z) > 0, but nevertheless, all the known classical advantage-distillation protocols [20], [22] fail for this distributiondistri- 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 allows classical key agreement exactly for α > 4. However, this does not imply (although it appears very plausible) that no classical protocol exists at all for the case α ≤ 4. ∈ { } ∈ { } { } Let α > 4. We consider the following protocol for classical key agreementagree- ment. First of all, ▇▇▇▇▇ and ▇▇▇ both restrict their ranges to {1, 2 2} (i.e., publicly reject a realization unless X ∈ {1, 2 2} and Y ∈ {1, 2 2}). The resulting distribution is as follows (to be normalized): X Y (Z) 1 2 1 (0) 2 (4) 5 − α 2 (2) α (0) 2 Then, ▇▇▇▇▇ and ▇▇▇ both send their bits locally over channels PX|X and PY |Y , respectively, such that the resulting bitsX bits X and Y are symmetric. The channel PX|X [PY |Y ] sends X = 0 [Y = 1] to X = 1 [Y = 0] with probability (2α −− 5)/(2α + 4), and leaves X [Y ] unchanged otherwise. The distribution PXY Z is then X Y (Z) 1 2 1 9 (0) 2 · 2α+4 9 2α−5 (2) α · 2α+4 · 2α+4 . Σ (1) 5 − α . Σ2 (2) α 2α−5 2α+4 2α−5 (0) 2 · 2 · 2α+4 It is not difficult to see that for α > 4, we have Prob [X = Y ] > 1/2 and that, given that X = Y holds, Eve has no information at all about what this bit is. Thus the repeat-code protocol described in Appendix C allows for classical key agreement in this situation. For α ≤ 4 however, classical key agreement, like quantum key agreement, seems impossible. The results of Example 3 are illustrated in Figure 3 in Appendix A. ♦