Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables

Abstract

We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. %while other ones require that the line transmission is larger than 50%. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be implemented very simply by transmitting coherent states and performing homodyne detection. Here, we show that entanglement nevertheless plays a crucial role in the security analysis of coherent state protocols. Every cryptographic protocol based on displaced gaussian states turns out to be equivalent to an entanglement-based protocol, even though no entanglement is actually present. This equivalence even holds in the absence of squeezing, for coherent state protocols. This ``virtual'' entanglement is important to assess the security of these protocols as it provides an upper bound on the mutual information between Alice and Bob if they had used entanglement. The resulting security criteria are compared to the separability criterion for bipartite gaussian variables. It appears that the security thresholds are well within the entanglement region. This supports the idea that coherent state quantum cryptography may be unconditionally secure.

Explicit asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation

“…It is therefore necessary to find some indirect approach in order to get some bounds on Z = tr(ρ C). Protocols with a Gaussian modulation (of Gaussian states) are an exception: in this case, one can easily compute this covariance matrix, and in particular the value of Z = tr(ρ C) from the data observed in the PM protocol [14]. The reason for this is that the measurement performed by Alice in the EB protocol is a Gaussian measurement, and therefore the observed statistics are sufficient to infer the covariance matrix.…”

“…In this section, we show that the formula from Eqn. (14) gives the standard value for a Gaussian modulation [10]. Let us consider a modulation such that τ G has n photons on average:…”

“…The goal of this section is to provide an explicit expression for the value of Z * of Eqn. (14) corresponding to the case of a lossy and noisy Gaussian channel:…”

“…The main lesson one can draw from the formula obtained in Eqn. (14) for Z is that the key rate will increase when the modulation scheme gets closer to a Gaussian distribution, and this is mainly quantified by the value of…”

“…(3), where we compute the symplectic eigenvalues for the covariance matrix Γ = V 1 2 Z * σ Z Z * σ Z W 1 2 with V given by the modulation scheme, W computed from the value of n B and Z * computed from the values of c 1 , c 2 , n B by the formula given in Eqn. (13). We note that the function f depends implicitly on the modulation scheme, for example via the value of w appearing in the expression of Z * .…”