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For two-qubit states, represented by a $4\times 4$ density matrix, the generic state is described by 15 real parameters. For ease of calculation, it can help to consider restricted families of states, such as the "$X$"-states, where any matrix elements not on either the main diagonal or anti-diagonal are 0 (requiring 7 real parameters), or rebits, where the matrix elements are all real (requiring 9 real parameters).

For any given density matrix of two qubits, it is easy to tell if it's entangled: we just test the partial transpose criterion, and the presence of negative eigenvalues. One might like to measure the fraction of the space that is entangled, and for that, one must pick a particular measure.

The probability with respect to Hilbert-Schmidt measure that generic two-qubit $X$-states are separable has been shown to be $\frac{2}{5}$ (arXiv:1408.3666v2, arXiv:1501.02289v2). Additionally, Lovas and Andai have demonstrated that the corresponding probability for the two-rebit density matrices is $\frac{29}{64}$ (https://arxiv.org/abs/1610.01410). Additionally, a strong body of various forms of evidence (though yet no formal proof) has been developed that the probabilities for the arbitrary two-qubit and (27-dimensional) two-``quater”[nionic]bit'' density matrices are $\frac{8}{33}$ and $\frac{26}{323}$, respectively (arXiv:1701.01973).

However, analogous results with respect to the important Bures (minimal monotone) measure are presently unknown.

Now, in what manner, if any, might these known Hilbert-Schmidt results be employed to assist in the further estimation/determination of their Bures counterparts?

Perhaps useful in such an undertaking would be the procedures for the generation of random density matrices with respect to Bures and Hilbert-Schmidt measure outlined in arXiv:0909.5094v1. Further, Chapter 14 of "Geometry of Quantum States" of Bengtsson and Zyczkowski presents formulas for the two measures, among a wide literature of related analyses.

It seems a particularly compelling conjecture that the two-qubit Bures separability probability assumes some yet unknown simple, elegant form ($\approx 0.073321$), as has been demonstrated do its counterparts, also based on fundamental quantum information-theoretic measures. A value of $\frac{11}{150) =\frac{11}{2 \cdot 3 cdot 5^2} \approx 0.07333...$ is an interesting candidate in this matter.

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  • $\begingroup$ Could you clarify the question a bit? How can you have 2 qubits with either dimension 7 or 15? 2 qubits have dimension 4. $\endgroup$ – DaftWullie Jul 14 '18 at 5:10
  • $\begingroup$ I introduced density-matrix terminology in response to DaftWullie clarification request. $\endgroup$ – Paul B. Slater Jul 14 '18 at 18:12
  • $\begingroup$ @PaulB.Slater Hi and welcome to Quantum Computing Stack Exchange! I have replaced the .pdf links in your post with abs links. As a rule of thumb, always link to the abstract rather than the PDF version (since, not everybody would want to download a large PDF without knowing what it is about). $\endgroup$ – Blue Jul 16 '18 at 15:56
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    $\begingroup$ I have seen some of your contributions to this subject. Could you please explain in a few words why the analytical computations are harder to obtain in the Bures case? Thank you. $\endgroup$ – David Bar Moshe Aug 6 '18 at 7:04
  • $\begingroup$ Thanks, DBM! The HS formula is given in eq. (3.11) in arxiv.org/pdf/quant-ph/0302197.pdf and the Bures formula in eq. (3.18) in arxiv.org/pdf/quant-ph/0304041.pdf (Notation is different in the two papers--with $\Lambda$ and $\rho$ both referring to the eigenvalues.) The notation is consistent, however, in eqs. (14.35) and (14.46) in Bengtsson and Zyczkowski's "Geometry of Quantum States". Obviously, one has additional (denominator!) factors in the Bures case. $\endgroup$ – Paul B. Slater Aug 6 '18 at 11:43

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