Key distillation from quantum channels using two-way communication protocols

We provide a general formalism to characterize the cryptographic properties of quantum channels in the realistic scenario where the two honest parties employ prepare and measure protocols and the known two-way communication reconciliation techniques. We obtain a necessary and sufficient condition to distill a secret key using this type of schemes for Pauli qubit channels and generalized Pauli channels in higher dimension. Our results can be applied to standard protocols such as Bennett-Brassard 1984 or six-state, giving a critical error rate of 20% and 27.6%, respectively. We explore several possibilities to enlarge these bounds, without any improvement. These results suggest that there may exist weakly entangling channels useless for key distribution using prepare and measure schemes.

Figure 1

Schematic diagram for key distillation from quantum states: a secret key can be distilled either by entanglement distillation plus measurement, which is an experimentally challenging process, or by measurement plus classical processing of the outcomes, whose implementation is much more feasible.

Figure 2

A tripartite pure state is prepared by Eve, who send two of the particles to Alice and Bob and keeps one. From the Alice and Bob viewpoint the situation resembles a standard noisy channel. The honest parties perform measurements at the single copy level, possibly with some preliminary filtering step. Eve keeps her quantum states and can arbitrarily delay her collective measurement.

Figure 3

Security bounds for QKD protocols using key distillation techniques with one-way communication: based on the analogy between these techniques and quantum error correction, the security bounds for the BB84 and the six-state protocols are 11% and 12.7%, respectively. These bounds have later been improved by information-theoretic considerations up to 12.4% and 14.1%. The improvement is achieved using some classical preprocessing by one of the parties.

Figure 4

Security bounds for QKD protocols using two-way followed by one-way communication techniques: based on the analogy between the two-way plus one-way communication and two-way entanglement distillation protocol, general security bounds of the BB84 and the six-state protocols are given by 18.9% and 26.4%, respectively [29]. Later, Chau improved the error correction method and the bounds are moved to 20.0% and 27.6% [30]. In Secs. 4, 5, we show that those bounds are tight. Note that the key distillability condition is stronger than the entanglement condition, which is 25.0% and 33.3% for the BB84 and the six-state protocols.

Figure 5

Graphical depiction of the security condition (26): the security region is defined by the intersection of the entanglement condition ${\lambda }_1>1∕2$, the normalization condition ${\lambda }_1+{\lambda }_2<1$, and the security condition (26).

Figure 6

Security bounds of the BB84 and the six-state protocols against individual and collective attacks: When Eve is supposed to apply individual attacks, all entangled states are distillable to a secret key. Assuming general attacks, security bounds are 20.0 % and 27.6%, respectively, for the BB84 and the six-state protocols. This means that nondistillable secret correlations may exist (see, Sec. 6).

Figure 7

Considered classical preprocessing: Alice introduces some extra noise by permuting her classical variable with probability $q$.

Figure 8

Quantum advantage distillation protocol: Alice performs single-copy measurement and processes the obtained classical outcomes. Bob keeps his quantum states on a quantum memory and performs coherent quantum operations.

Figure 9

Comparison of the security bounds and the entanglement condition. The security condition against collective attacks requires stronger correlation than the entanglement limit. Again, there may exist some entangled states that are useless for key distillation with the considered techniques.