Linking Classical and Quantum Key Agreement. In this section we derive a close connection between the possibilities offered by classical and quantum protocols for key agreement. The intuition is as follows. As described in Section 2.2, there is a very natural connection between quantum states Ψ and classical distributions PXY Z which can be thought of as arising 3 The term “quantum privacy amplification” is somewhat unfortunate since it does not correspond to classical privacy amplification, but includes advantage distillation and error correction. from Ψ by measuring in a certain basis, e.g., the standard basis4. (Note however that the connection is not unique even for fixed bases: For a given distribution PXY Z, there are many states Ψ leading to PXY Z by carrying out measurements.) When given a state Ψ between three parties Alice, Bob, and Eve, and if ρAB denotes the resulting mixed state after Eve is traced out, then the corresponding classical distribution PXY Z has positive intrinsic information if and only if ρAB is entangled. However, this correspondence clearly depends on the measurement bases used by Alice, Bob, and Eve. If for instance ρAB is entangled, but Xxxxx and Xxx do very unclever measurements, then the intrinsic information may vanish. If on the other hand ρAB is separable, Eve may do such bad measurements that the intrinsic information becomes positive, despite the fact that ρAB could have been established by public discussion without any prior correlation (see Example 4). Consequently, the correspondence between intrinsic information and entanglement must involve some optimization over all possible measurements on all sides. A similar correspondence on the protocol level is supported by many exam- ples, but not rigorously proven: The distribution PXY Z allows for classical key agreement if and only if quantum key agreement is possible starting from the state ρAB. We show how these parallels allow for addressing problems of purely classical information-theoretic nature with the methods of quantum information theory, and vice versa.
Linking Classical and Quantum Key Agreement. In this section we derive a close connection between the possibilities offered by classical and quantum protocols for key agreement. The intuition is as follows. As described in Section 2.2, there is a very natural connection between quantum states Ψ and classical distributions PXY Z which can be thought of as arising from Ψ by measuring in a certain basis, e.g., the xxxx- dard basis6. (Note however that the connection is not unique even for fixed bases: For a given distribution PXY Z, there are many states Ψ leading to PXY Z by carrying out measurements.) When given a state Ψ between three parties Xxxxx, Xxx, and Xxx, and if ρAB denotes the resulting mixed state after tracing out Eve, then the corresponding classical distribution PXY Z has positive intrinsic information if and only if ρAB is entangled. However, this correspondence clearly depends on the measurement bases used by Al- ice, Xxx, and Xxx. If for instance ρAB is entangled, but Xxxxx and Xxx do very unclever measurements, then the intrinsic information may vanish (see Example 7 in Appendix B). If on the other hand ρAB is separable, Eve may do such bad measurements that the intrinsic information becomes positive, despite the fact that ρAB could have been established by public discussion without any prior correlation (see Example 6 in Appendix B). Consequently, the correspondence between intrinsic information and entan- 6A priori, there is no privileged basis. However, physicists often write states like ρAB in a basis which seems to be more natural than others. We refer to this as the standard basis. Somewhat surprisingly, this basis is generally easy to identify, though not precisely defined. One could characterize the standard basis as the basis for which as many coefficients as possible of Ψ are real and positive. We usually represent quantum states with respect to the standard basis. glement must involve some optimization over all possible measurements on all sides.
