SPDCStats.jl

---

Julia package to compute statistics arising from a SPDC source and NPRPDs

                                            Spatial mode number:
										  A_k         (1)
                                          |
			 ↗ a_k , a_k⟂  → ( α_x,ϕ_αx ) ▩ ─ A_k⟂    (2)
    SPDC ⇉⇉⇉⇉
			 ↘ b_k , b_k⟂  → ( β_y,ϕ_βy ) ▩ ─ B_k     (3)
                                          |  
										  B_k⟂        (4)  

System parameters are:

• X,Y : Number of measurement for Alice/Bob
• N : Number of modes (k∈[1,N])
• ηA,ηB : Detector efficieny of Alice/Bob
• pdcA,pdcB : Probability of dark count (thermal noise) in Alices/Bob's detectors

Circuit parameters are:

• g,g_ : Squeezing parameters
• α_x,ϕ_αx : Alice setting for measurement A_x
• β_y,ϕ_βy : Alice setting for measurement B_y

Usage

---

This package export 4 functions:

param : translate a vector of Float64 to a NamedTuple. Contains both circuit paramaters and system parameters (as keyword arguments) with the exception of X and Y. Default efficiencies are 1.0, default number of mode is 1 and default dark count probability is 0.0. Note that the two first circuit parameters (squeezing) are stored as Tg = tanh(g) and Tg_= tanh(g_)

# Construct the NamedTuple from random parametersin a X,Y=(2,3) scenario with detection efficiency 0.8
> p = param(rand(12),2,3;ηA=.8,ηB=.8)
(Tg = 0.2494803207411917, Tg_ = 0.7424573267165877, α1 = 0.010129492977825283, ϕ_α1 = 0.2957634323161866, α2 = 0.8837522471034687, ϕ_α2 = 0.71578185734515, β1 = 0.48774641481304837, ϕ_β1 = 0.1872266067465822, β2 = 0.1703390158609357, ϕ_β2 = 0.8110150871030339, β3 = 0.4627290061607028, ϕ_β3 = 0.5435762981679317, N = 1.0, pdcA = 0.0, pdcB = 0.0, ηA = 0.8, ηB = 0.8)

p_ab : Compute the set of probabilities p(ab|xy) for all a,b∈[-1,+1]. Return a matrix with element [a,b] corresponding to p(ab|xy).

# Probabilites for x=1,y=1
> p_11 = p_ab(p,1,1)
2×2 Matrix{Float64}:
 0.00437022  0.021084
 0.31208     0.662466

correlator : Compute the correlator <A_x,A_y> either from a given Matrix of the form arising from p_ab, or directly from the NamedTuple.

# Correlator <A_1 B_1> from the probabilities
> A1B1 = correlator(p_11)
0.33367240161689893
# Correlator <A_1 B_1> from the parameters
> A1B1 = correlator(p,1,1)
0.33367240161689893

spdc_correlators : Return all the X*Y correlators. The output is a matrix with element [x,y] equal to <A_x B_y>.

> spdc_correlators(p,2,3)
2×3 Matrix{Float64}:
 0.333672  0.058213  0.307882
 0.286395  0.260977  0.263226

Examples and data

---

Examples are available in the example folder.

• chsh.jl : show how to compute and optimize the CHSH value from the SPDC setup.
• chsh.py : same as chsh.jl but as an import in Python.
• key_rate_with_noisypreprocessing.jl : show how to compute and optimize the DIQKD key rate, obtained using noisy preprocessing, from the SPDC setup.

We provide some data

• key_rate_noisypp_with_loss.csv : Contains the optimized key rate as well as the circuit parameters (X=2,Y=3) for different value of efficiency.
• key_rate_noisypp_with_loss.jld2 : Same as above but in HDF5.

Installation

---

Install SPDCStats.jl using Julia Pkg package as follow

> using Pkg
> Pkg.add(url="/~https://github.com/xvalcarce/SPDCStats.jl")