# Is the Hilbert-Schmidt probability simply zero that a generic rank-2 two-qubit ("pseudo-pure") density matrix is separable?

The multifacted evidence is very compelling--although not yet presented in a formal proof--that the Hilbert-Schmidt probability that a generic (full rank/rank-4) two-qubit density matrix is separable is $$\frac{8}{33}$$ (MasterLovas-AndaiFormula)

Assuming this proposition, it follows from the interesting 2005 analysis of Szarek, Bengtsson and Zyczkowski structure of the body of states with positive partial transpose that the Hilbert-Schmidt probability that a generic boundary (rank-3) two-qubit density matrix is separable/PPT is simply one-half, that is $$\frac{4}{33}$$.

So, what can be said in such regards for generic rank-2 two-qubit density matrices? (I suspect the associated separability probability is zero--as some current numerical analyses of mine appear to indicate--but also surmise that there is a manner in which to demonstrate such a proposition formally. Perhaps there is some literature to this effect that I would appreciate having indicated.)

Of course, one can pose parallel questions--which I am also investigating--for higher-dimensional qubit-qutrit, two-qutrit,...states (for which the Szarek, Bengtsson, Zyczkowski boundary-state PPT result still holds). The Hilbert-Schmidt separability/PPT probability for generic (full rank) qubit-qutrit states has been conjectured to be $$\frac{27}{1000}$$. NumericalExact (It was also speculated there that the Hilbert-Schmidt PPT-probability for generic two-qutrit states might be $$\frac{323}{3161088}=\frac{17 \cdot 19}{2^{10} \cdot 3^2 \cdot 7^3} \approx 0.000102180009$$ or $$\frac{11}{107653} = \frac{11}{7^2 \cdot 13^3} \approx 0.000102180153$$.)

Rank-2 two-qubit states have been studied in considerable depth by Horia Scutaru in "On the pseudo-pure states of two qubits." Proceedings of the Romanian Academy. Series A. Mathematics, Physics, Technical Sciences, Information Science 5.2 (2004): 136-140. pseudo-pure state article (I considered sending him this question, but found that he is deceased.)

Let us also point out that in our 2005 paper qubit-qutrit ratios a value (33.9982) of close to 34 was reported for the ratio of the Hilbert-Schmidt separability probability of rank-6 to rank-4 qubit-qutrit states. This would appear to be a further topic for updated analyses.

Theorem 1 of the 2001 paper LowRankSeparable, "Low Rank Separable States Are A Set Of Measure Zero Within The Set of Low Rank States" of R. B. Lockhart deals with general cases of the type raised here, but appears to apply only to rank-1 (pure) two-qubit states and not to rank-2 such states, so leaving the question put forth here still not apparently answered.

Apparently, the specific question posed here has been answered in the affirmative--at least (first, we point out) through numerical means--by Arsen Khvedelidze and Ilya Rogojin in Table 2 of their 2018 paper, "On the Generation of Random Ensembles of Qubits and Qutrits: Computing Separability Probabilities for Fixed Rank States" ArsenIlya

They report a Hilbert-Schmidt separability probability of zero for the rank-2 two-qubit states--based on the complex Ginibre-ensemble randomization procedures they detail in the paper. Also, in Table 1, they give a full-rank two-qubit HS separability probability of 0.2424, agreeing to the given number of places with the well-supported, presumed exact value of $$\frac{8}{33} \approx 0.24242424...$$.

In Table 2, however, they give for the rank-3 two-qubit states, a HS separability probability of 0.1652, which seems in rather substantial disagreement with a value of $$\frac{4}{33} \approx 0.121212...$$, based on the application of the noted theorem of Szarek, Bengtsson and Zyczkowski to the $$\frac{8}{33}$$ assertion.

A formalized theorem regarding this rank-2 two-qubit question would still seem of significant interest. Khvedelidze and Rogojin state that their result is consistent with the assertions in RuskaiWerner. Upon the first submission of this answer, I had not perceived that the specific question posed here was fully addressed there.

However, I now see that their

$$\bf{Corollary}$$ $${4}$$. If a state $$\gamma_{AB}$$ on $$\bf{C}_2 \otimes \bf{C}_2$$ has rank 2, then $$γ_{AB}$$ is almost surely entangled

$$\bf{Theorem}$$ $$\bf{9}$$. Assume $$d_A \geq 􏰅d_B \geq 􏰅2$$. If a state $$\gamma_{AB}$$ on $$M_{d_A} \otimes M_{d_B}$$ has rank $$\gamma_{AB} 􏰄\leq d_{A}$$, then $$\gamma_{AB}$$ is almost surely entangled.