# Extensions of generalized two-qubit separability probability analyses to higher dimensions, additional measures and new methodologies

• Paul B. Slater
Article

## Abstract

We first seek the rebit–retrit counterpart to the (formally proven by Lovas and Andai) two-rebit Hilbert–Schmidt separability probability of $$\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$$ and the qubit–qutrit analogue of the (strongly supported) value of $$\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$$. We advance the possibilities of a rebit–retrit value of $$\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$$ and a qubit–qutrit one of $$\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$$. These four values for $$2 \times m$$ systems ($$m=2,3$$) suggest certain numerator/denominator sequences involving powers of m, which we further investigate for $$m>3$$. Additionally, we find that the Hilbert–Schmidt separability/PPT-probabilities for the two-rebit, rebit–retrit and two-retrit X-states all equal $$\frac{16}{3 \pi ^2} \approx 0.54038$$, as well as more generally, that the probabilities based on induced measures are equal across these three sets. Then, we extend the master Lovas–Andai formula to induced measures. For instance, the two-qubit function ($$k=0$$) is $$\tilde{\chi }_{2,0}(\varepsilon )=\frac{1}{3} \varepsilon ^2 (4 -\varepsilon ^2)$$, yielding $$\frac{8}{33}$$, while its $$k=1$$ induced measure counterpart is $$\tilde{\chi }_{2,1}(\varepsilon )=\frac{1}{4} \varepsilon ^2 \left( 3-\varepsilon ^2\right) ^2$$, yielding $$\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$$, where $$\varepsilon$$ is a singular-value ratio. Interpolations between Hilbert–Schmidt and operator monotone (Bures, $$\sqrt{x}$$) measures are also studied. Using a recently-developed golden-ratio-related (quasirandom sequence) approach, current (significant digits) estimates of the two-rebit and two-qubit Bures separability probabilities are 0.15709 and 0.07331, respectively–with an additional indicator that the latter probability may be $$\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.07331378$$.

## Keywords

Separability probabilities Qubit–qudit Two-qubits Two-rebits Hilbert–Schmidt measure Random matrix theory Rebit–retrits Qubit–qutrits Quaternions PPT-probabilities Operator monotone functions Bures measure Induced measure Lovas–Andai functions Quasirandom sequences Golden ratio

## Notes

### Acknowledgements

My considerable thanks to Charles Dunkl for contributing the appendices, and his general support and advice. This research was supported by the National Science Foundation under Grant No. NSF PHY-1748958.

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## Authors and Affiliations

1. 1.Kavli Institute for Theoretical PhysicsUniversity of California, Santa BarbaraSanta BarbaraUSA