ConicQKD

Implementation of the convex cones introduced in the papers

Finite-size quantum key distribution rates from Renyi entropies using conic optimization

Mariana Navarro, Andrés González Lorente, Pablo V. Parellada, Carlos Pascual-García, and Mateus Araújo

and

Quantum key distribution rates from non-symmetric conic optimization

Andrés González Lorente, Pablo V. Parellada, Miguel Castillo-Celeita, and Mateus Araújo

Installation

First you need to install Julia. From within Julia, enter the package manager by typing ]. Then install ConicQKD:

pkg> add https://github.com/araujoms/ConicQKD.jl

This will automatically install all dependencies. The main one is the solver Hypatia, which this package extends.

Usage

Simplified examples that demonstrate how to use the cones are available in the examples folder. They can also be used to reproduce the results of "Quantum key distribution rates from non-symmetric conic optimization". To reproduce the results of "Finite-size quantum key distribution rates from Renyi entropies using conic optimization" use the code available in MarianaNvrr/Renyi-ConicQKD.

The interface is formulated using the modeller JuMP.

To constraint a quantum state ρ to belong to the von Neumann QKD cone with CP maps Ghat and ZGhat the syntax is

@constraint(model, [h; ρ_vec] in EpiQKDTriCone{T,R}(Ghat, ZGhat, 1 + length(ρ_vec); blocks))

To constraint a quantum state ρ to belong to the FastRényiQKD cone with CP maps Ghat, ZGhat, isometry S, and Rényi parameter β = 1/α the syntax is

@constraint(model, [h; ρ_vec] in EpiFastRenyiQKDTriCone{T,R}(β, Ghat, ZGhat, 1 + length(ρ_vec); S, blocks))

To constraint quantum states ρ, σ to belong to the RényiQKD cone with CP maps Ghat, Zhat, isometry S, and Rényi parameter γ = α/(2α - 1) the syntax is

@constraint(model, [h; ρ_vec; σ_vec] in EpiRenyiQKDTriCone{T,R}(γ, Ghat, Zhat, 1 + length(ρ_vec) + length(σ_vec); S, blocks))

• model is the JuMP optimization model being used.
• h is a variable which will have the conditional entropy in base e.
• ρ_vec, σ_vec are vectorizations of ρ, σ in the svec format (we provided a function svec to compute it).
• T is the floating point type to be used (e.g. Float64, Double64, Float128, BigFloat, etc.).
• R is either equal to T, in order to optimize over real matrices, or equal to Complex{T} in order to optimize over complex matrices.
• Ghat, ZGhat, and Zhat encode the CP maps as vectors of Kraus operators.
• blocks is an optional keyword argument specifying the block structure of Zhat as a vector of vectors. For example, if Zhat maps a 4x4 ρ to a matrix M such that only M[1:2,1:2] and M[3:4,3:4] are nonzero, then blocks should be [1:2, 3:4]. If this argument is omitted the computation will be considerably slower.
• S is an optional keyword argument, assumed to be identity if ommitted.