Table 2 Critical visibilities for classical simulation of quantum sets

From: Operationally classical simulation of quantum states

d

N_U

Set

Numerical

Result 1

Result 3

3

3000

\({{{{\mathcal{E}}}}}_{1}\)

0.6122

0.4167

0.5000

3

3000

\({{{{\mathcal{E}}}}}_{2}\)

0.5257

0.4167

0.3333

3

3000

\({{{{\mathcal{E}}}}}_{3}\)

0.4567

0.4167

0.2500

3

20

\({{{{\mathcal{E}}}}}_{4}\)

0.6722

0.4167

0.5000

3

100

\({{{{\mathcal{E}}}}}_{5}\)

0.4290

0.4167

0.1111

4

1000

\({{{{\mathcal{E}}}}}_{6}\)

0.3785

0.3611

0.0625

  1. Classical simulation thresholds obtained from Equation (11) for six noisy quantum sets. The sets \({{{{\mathcal{E}}}}}_{1}\), \({{{{\mathcal{E}}}}}_{2}\) and \({{{{\mathcal{E}}}}}_{3}\) correspond to all states in 2, 3 and 4 MUBs, respectively, in dimension d = 3. The set \({{{{\mathcal{E}}}}}_{4}\) is just two states: \(\left|0\right\rangle\) and \(\frac{\left|0\right\rangle+\left|1\right\rangle+\left|2\right\rangle }{\sqrt{3}}\) in dimension d = 3. \({{{{\mathcal{E}}}}}_{5}\) and \({{{{\mathcal{E}}}}}_{6}\) are the nine and sixteen states forming a SIC-set in dimension d = 3 and d = 4 respectively. Analytical models based on Results 1 and 3 are included for comparison. NU is the number of classical devices used in the simulation.
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