Asymptotic correctability of Bell-diagonal quantum states and maximum tolerable bit-error rates

Abstract

The general conditions are discussed which quantum state purification protocols have to fulfill in order to be capable of purifying Bell-diagonal qubit-pair states, provided they consist of steps that map Bell-diagonal states to Bell-diagonal states and they finally apply a suitably chosen Calderbank-Shor-Steane code to the outcome of such steps. As a main result a necessary and a sufficient condition on asymptotic correctability are presented, which relate this problem to the magnitude of a characteristic exponent governing the relation between bit and phase errors under the purification steps. These conditions allow a straightforward determination of maximum tolerable bit error rates of quantum key distribution protocols whose security analysis can be reduced to the purification of Bell-diagonal states.

Asymptotic correctability of Bell-diagonal qudit states and lower bounds on tolerable error probabilities in quantum cryptography

“…Using α > β, we regain the result for isotropic channels of our previous work [9]. In the case d = 2, this state reduces to the general mixture of qubit Bell states as considered in [4]. Further note, that in the case of generalised isotropic channels we could have done the calculation for r (d) without the use of the mixing operation.…”

“…We now want to generalise the criterion for asymptotic correctability of [4] to qudits. It turns out that this generalisation is straightforward and essentially is a reformulation of the previous result.…”

“…Theorem 1 (Quantum Shannon Bound) Let d be a prime number and consider a state ρ Using this bound, we can now define the notion of asymptotic correctability; due to the use of that theorem, in the following we consider d always to be prime. For d = 2, this definition reduces to the one given in [4].…”

Phase-remapping attack in practical quantum-key-distribution systems

“…4 is that there is a range of phase differences δ that causes the QBER to go below 20%, which is shown in Ref. [20,21] to be a tolerable QBER in BB84 when Eve does not have the ability to change the δ. This proves that Eve's ability to change the phase difference between Alice's states is helpful to Eve in breaking the security of BB84.…”

Postponement of dark-count effects in practical quantum key-distribution by two-way post-processing

“…Indeed, all our numerical simulations demonstrate that a few applications of B-steps are sufficient to bring the maximal secure distances very close to the upper bound. Recently, this inessentiality of P-steps has been pointed out by other authors as well [32,33,34].…”