Parity-based, bias-free optical quantum random number generation with min-entropy estimation

Mathew R. Coleman, Kaylin G. Ingalls, John T. Kavulich, Sawyer J. Kemmerly, Nicolas C. Salinas, Efrain Venegas Ramirez, and Maximilian Schlosshauer

Author Affiliations

Mathew R. Coleman,
Kaylin G. Ingalls,
John T. Kavulich,
Sawyer J. Kemmerly,
Nicolas C. Salinas,
Efrain Venegas Ramirez,
and Maximilian Schlosshauer^{*}

^{}Department of Physics, University of Portland, 5000 North Willamette Boulevard, Portland, Oregon 97203, USA

Mathew R. Coleman, Kaylin G. Ingalls, John T. Kavulich, Sawyer J. Kemmerly, Nicolas C. Salinas, Efrain Venegas Ramirez, and Maximilian Schlosshauer, "Parity-based, bias-free optical quantum random number generation with min-entropy estimation," J. Opt. Soc. Am. B 37, 2088-2094 (2020)

We describe the generation of sequences of random bits from the parity of photon counts produced by polarization measurements on a polarization-entangled state. The resulting sequences are bias free, pass the applicable tests in the NIST battery of statistical randomness tests, and are shown to be Borel normal, without the need for experimental calibration stages or postprocessing of the output. Because the photon counts are produced in the course of a measurement of the violation of the Clauser–Horne–Shimony–Holt inequality, we are able to concurrently verify the nonclassical nature of the photon statistics and estimate a lower bound on the min-entropy of the bit-generating source. The rate of bit production in our experiment is around 13 bits/s.

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Results of NIST Statistical Randomness Tests [33] Applied to the Sequences ${\mathbf{\text{x}}}_{1}$ and ${\mathbf{\text{x}}}_{2}$^{a}

Sequence ${\mathbf{x}}_{1}$

Sequence ${\mathbf{x}}_{2}$

Statistical Test

$p$-Value

Proportion

$P$-Value

$p$-Value

Proportion

$P$-Value

Frequency

0.327383

100/100

0.153763

0.90035

99/100

0.071177

Frequency within a block

0.679802

100/100

0.437274

0.263874

100/100

0.045675

Runs

0.880981

100/100

0.455937

0.237737

98/100

0.595549

Longest run within a block

0.230194

100/100

0.153763

0.06073

98/100

0.759756

Cumulative sums (forward)

0.343646

100/100

0.924076

0.813301

99/100

0.137282

Cumulative sums (backward)

0.605296

100/100

0.275709

0.914125

99/100

0.062821

Discrete Fourier transform

0.189098

100/100

0.006196

0.651679

100/100

0.304126

Serial-1

0.510447

96/100

0.637119

0.55776

100/100

0.224821

Serial-2

0.161343

98/100

0.213309

0.318324

99/100

0.383827

Binary matrix rank

0.52999

n/a

n/a

0.134194

20/20

0.534146

Template matching

0.894736

18/20

0.534146

0.399080

97/100

0.040108

Approximate entropy

0.070835

98/100

0.334538

0.712837

100/100

0.595549

Maurer

n/a

n/a

n/a

0.650933

n/a

n/a

Entries marked “n/a” indicate that a test could not be applied because of insufficient sequence length. The serial test produces two $p$-values as output, shown as “Serial-1” and “Serial-2.” For $p$-values, the following values for the block length parameter $m$ are used: frequency within a block test, $m = 2 \times {10^3}$ for ${{\bf x}_1}$ and $m = 8 \times {10^3}$ for ${{\bf x}_2}$; serial test, $m = 15$ for ${{\bf x}_1}$ and $m = 16$ for ${{\bf x}_2}$; approximate entropy test, $m = 10$. For the “Proportion” values and $P$-values, the choices for $m$ are: frequency within a block test, $m = 20$ for ${{\bf x}_1}$ and $m = 80$ for ${{\bf x}_2}$; serial test, $m = 8$ for ${{\bf x}_1}$ and $m = 10$ for ${{\bf x}_2}$; approximate entropy test, $m = 5$ for ${{\bf x}_1}$ and $m = 7$ for ${{\bf x}_2}$. For the template matching applied to ${{\bf x}_1}$ and the binary matrix rank applied to ${{\bf x}_2}$, a set of only 20 subsequences could be tested due to limited sequence lengths. The discrete Fourier transform test has been shown to have problems [49] that may render its reliability and sensitivity questionable.

The bound is given by $\sqrt {\mathop {\log}\nolimits_2 | {\bf x} |/| {\bf x} |}$, where $| {\bf x} |$ is the sequence length, with $| {{{\bf x}_1}} | = 2 \times {10^5}$ and $| {{{\bf x}_2}} | = 8 \times {10^5}$.

Tables (2)

Table 1.

Results of NIST Statistical Randomness Tests [33] Applied to the Sequences ${\mathbf{\text{x}}}_{1}$ and ${\mathbf{\text{x}}}_{2}$^{a}

Sequence ${\mathbf{x}}_{1}$

Sequence ${\mathbf{x}}_{2}$

Statistical Test

$p$-Value

Proportion

$P$-Value

$p$-Value

Proportion

$P$-Value

Frequency

0.327383

100/100

0.153763

0.90035

99/100

0.071177

Frequency within a block

0.679802

100/100

0.437274

0.263874

100/100

0.045675

Runs

0.880981

100/100

0.455937

0.237737

98/100

0.595549

Longest run within a block

0.230194

100/100

0.153763

0.06073

98/100

0.759756

Cumulative sums (forward)

0.343646

100/100

0.924076

0.813301

99/100

0.137282

Cumulative sums (backward)

0.605296

100/100

0.275709

0.914125

99/100

0.062821

Discrete Fourier transform

0.189098

100/100

0.006196

0.651679

100/100

0.304126

Serial-1

0.510447

96/100

0.637119

0.55776

100/100

0.224821

Serial-2

0.161343

98/100

0.213309

0.318324

99/100

0.383827

Binary matrix rank

0.52999

n/a

n/a

0.134194

20/20

0.534146

Template matching

0.894736

18/20

0.534146

0.399080

97/100

0.040108

Approximate entropy

0.070835

98/100

0.334538

0.712837

100/100

0.595549

Maurer

n/a

n/a

n/a

0.650933

n/a

n/a

Entries marked “n/a” indicate that a test could not be applied because of insufficient sequence length. The serial test produces two $p$-values as output, shown as “Serial-1” and “Serial-2.” For $p$-values, the following values for the block length parameter $m$ are used: frequency within a block test, $m = 2 \times {10^3}$ for ${{\bf x}_1}$ and $m = 8 \times {10^3}$ for ${{\bf x}_2}$; serial test, $m = 15$ for ${{\bf x}_1}$ and $m = 16$ for ${{\bf x}_2}$; approximate entropy test, $m = 10$. For the “Proportion” values and $P$-values, the choices for $m$ are: frequency within a block test, $m = 20$ for ${{\bf x}_1}$ and $m = 80$ for ${{\bf x}_2}$; serial test, $m = 8$ for ${{\bf x}_1}$ and $m = 10$ for ${{\bf x}_2}$; approximate entropy test, $m = 5$ for ${{\bf x}_1}$ and $m = 7$ for ${{\bf x}_2}$. For the template matching applied to ${{\bf x}_1}$ and the binary matrix rank applied to ${{\bf x}_2}$, a set of only 20 subsequences could be tested due to limited sequence lengths. The discrete Fourier transform test has been shown to have problems [49] that may render its reliability and sensitivity questionable.

The bound is given by $\sqrt {\mathop {\log}\nolimits_2 | {\bf x} |/| {\bf x} |}$, where $| {\bf x} |$ is the sequence length, with $| {{{\bf x}_1}} | = 2 \times {10^5}$ and $| {{{\bf x}_2}} | = 8 \times {10^5}$.