Improving the lower bound to the secret-key capacity of the thermal amplifier channel

Abstract

We consider the noisy thermal amplifier channel, where signal modes are amplified together with environmental thermal modes. We focus on the secret-key capacity of this channel, which is the maximum amount of secret bits that two remote parties can generate by means of the most general adaptive protocol, assisted by unlimited and two-way classical communication. For this channel only upper and lower bounds are known, and in this work we improve the lower bound. We consider a protocol based on squeezed states and homodyne detections, in both direct and reverse reconciliation. In particular, we assume that trusted thermal noise is mixed on beam splitters controlled by the parties in a way to assist their homodyne detections. The new improved lower bounds to the secret-key capacity are obtained by optimizing the key rates over the variance of the trusted noise injected, and the transmissivity of the parties' beam splitters. Our results confirm that there is a separation between the coherent information of the thermal amplifier channel and its secret key capacity.We now present a QKD protocol whose key rate in DR and RR improves the lower bound in Eq. (4). Even though the improvement found is small, it is meaningful because it shows that the coherent information of the thermal amplifier channel is cannot be its secret key capacity. First we derive the new achievable rates in subsections 3.1 and 3.2. Then we numercally compare the results in subsection 3.3.

Advances in quantum cryptography

“…[384][385][386] and later in Refs. [387][388][389] as a tool to increase the lower bound to the secret key capacity of the thermal-loss and amplifier channels.…”

“…[34,387]. Similarly, for the noisy amplifier, we know [388] trusted-noise protocols that are able to beat the CI of the channel, which is therefore not tight. The non-tightness of the CI (and RCI) is also a feature in the computation of energy-constrained quantum capacities of bosonic Gaussian channels [626].…”

Maximum tolerable excess noise in CV-QKD and improved lower bound on two-way capacities

“…In this paper, we find a new lower bound on all twoway capacities of piBGCs, which constitutes a significant improvement with respect the state-of-the-art lower bounds [24][25][26][27][28] in many parameter regions. In particular, we completely solve the above open problem, quantitatively establishing that piBGCs can be used to distribute pure entanglement -and hence also secret key -whenever they are not entanglement breaking.…”

“…Although the right-hand sides of ( 18)-( 20) also lower bound the secret-key capacity K, improved estimates of K(E λ,ν ) and K(Φ g,ν ) have been put forth by Ottaviani et al [26] (see also [27,Sec. VII]) and by Wong, Ottaviani, Guo, and Pirandola (WOGP) [28], respectively. Lower bounds on the EC two-way capacities with energy constraint N s of piBGCs are the coherent information and the reverse coherent information evaluated on the state obtained by sending the subsystem A of |Ψ Ns AA through the channel.…”

“…The two-way capacities of the thermal attenuator and thermal amplifier have been determined for all λ and g in the case in which ν = 0 and there is no energy constraint [18]. Except for this very special case, the twoway capacities of piBGCs are not known, although upper [15,[18][19][20][21][22][23] and lower bounds [21,[24][25][26][27][28] have been established. For nonzero values of ν, the gap between such bounds is however relatively large for many values of λ and g. In particular, for a certain range of values of λ and g all lower bounds available prior to our work vanish, while the upper bounds do not.…”