Distance (m)

\(\boldsymbol{\Delta\xi^{\prime2}}\)

\(\boldsymbol{A_{J}^{\mathrm{max}}}\)


a1

≪1 
0.482

0.164

1

0.500

0.155

10

0.734

0.046

20

0.867

unf.

b1

≪1 
0.476

0.166

1

0.486

0.162

10

0.623

0.098

20

0.737

0.045

a2

≪1 
0.789

0.020

1

0.802

0.014

10

0.895

unf.

b2

≪1 
0.666

0.078

1

0.677

0.073

10

0.767

0.031

 The abbreviation ‘unf.’ means ‘unfeasible’. We assume an EPR state with the values discussed in Section 3, i.e. \(\Delta\xi^{2}\simeq0.47\), \(\Delta\xi_{\perp}^{2}\simeq16.77\), and typical values for connector losses leading to α ≃ 0.933 (\(A_{\alpha}\simeq0.036\)), β ≃ 0.891 (\(A_{\beta}\simeq0.061\)). Moreover, we assume a JPA gain \(g_{J}\simeq180\), and HEMT noise \(A_{H} \simeq7\). The noise parameter \(A_{J}^{\mathrm{max}}\) is estimated from the formulas \(\Xi\equiv(1+\Delta\xi^{\prime2} +A)^{2}\leq4\), which defines the quantum regime. Here, \(\Delta\xi^{\prime2}\) is defined in Eq. (24) and A is introduced in Eq. (27). We assume Alice and Bob symmetrically situated with respect the EPR sources. The distance is referred to the cable length from the EPR sources to Alice (Bob). The estimations take into account of the feedforward, and \(A_{J}^{\mathrm{max}}\) is evaluated for various distances and in four different situations. In a1 we assume cable power losses of 0.1 dB per meter and zero time measurement. In a2 we assume cable power losses of 0.1 dB per meter and 200 ns for measuring and processing the information in Alice. These two tables give an insight on how much the measurement duration, which result in a delay line in Bob, affects the quality of the protocol. In b1 we assume a more optimistic value for cable power losses, i.e. 0.05 dB per meter, and zero time measurement. In b2 we assume 0.05 dB of power losses per meter and 200 ns for measuring the processing the information in Alice. In all the tables, when Eq. (25) holds, we have applied a proper attenuator in Alice in order to optimise \(\Delta\xi^{\prime2}\).