Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .
Abstract Many works have been devoted to determining the volume of the set of separable states and the separability probability both analytically and numerically. However, computing the exact separability probability remains a highly nontrivial open problem. In these notes, we examine this problem in the simplest context of two-qubit states. A novel aspect of our method is the use of the Duistermaat-Heckman measure, which is the push-forward of Liouville measure on the coadjoint orbit along the moment map. On each coadjoint orbit, the Hilbert-Schmidt measure differs from the Duistermaat-Heckman measure by a constant. The latter measure can be explicitly computed by using tools from symplectic geometry. We confirm the conjecture that the separability probability of two-qubit states with Hilbert-Schmidt measure is 8 33 .
Abstract While the exact separability probability of 8/33 for two-qubit states under the Hilbert-Schmidt measure has been reported by Huong and Khoi, detailed derivations remain inaccessible for general audiences. This paper provides a comprehensive, self-contained derivation of this result, elucidating the underlying geometric and probabilistic structures. We achieve this by developing a framework centered on the computation of Hilbert-Schmidt volumes for key components: the quantum state space, relevant flag manifolds, and regular (co)adjoint orbits. Crucially, we establish and leverage the connection between these Hilbert-Schmidt volumes and the symplectic volumes of the corresponding regular co-adjoint orbits, formalized through the Duistermaat-Heckman measure. By meticulously synthesizing these volume computations—specifically, the ratios defining the relevant probability measures—we reconstruct and rigorously verify the 8/33 separability probability. Our approach offers a transparent pathway to this fundamental constant, detailing the interplay between symplectic geometry, representation theory, and quantum probability.
Abstract The geometric separability probability of composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit–qubit and rebit–rebit quantum states with respect to the Hilbert–Schmidt measure. A lot of numerical simulations confirm the P r e b i t − r e b i t = 29 64 and P q u b i t − q u b i t = 8 33 conjectured probabilities. We provide a rigorous proof for the separability probability in the real case and we give explicit integral formulas for the complex and quaternionic case. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubit–qubit (and qubit-qutrit) states of the form of ( D 1 C C ∗ D 2 ) depends on D = D 1 + D 2 (on single qubit subsystems), moreover it depends only on the Bloch radii ( r ) of D and it is constant in r . Using the Peres–Horodecki criterion for separability we give a mathematical proof for the 29 64 probability and we present an integral formula for the complex case which hopefully will help to prove the 8 33 probability, too. We prove Milz and Strunz’s conjecture for rebit–rebit and qubit–qubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f ( x ) =
We conduct a pair of quasirandom estimations of the separability probabilities with respect to ten measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parameterization. The measures include the (non-monotone) Hilbert-Schmidt one, plus nine based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert-Schmidt and Bures (minimal monotone) separability probabilities are 338≈0.242424 and 34125≈0.0733138, respectively, as well as suggestive of the Wigner-Yanase counterpart being 201. However, they appear inconsistent (much too small) with the additional claim that the separability probability associated with the operator monotone (geometric-mean) function x is 1−27π2256≈0.0393251. But a seeming explanation for this phenomenon is that the volume of states for the x-based measure is infinite, so the validity of the conjecture--as well as an alternative one, 91(593−60π2)≈0.0915262, we now introduce--can not be examined through our numerical approach, at least perhaps not without some truncation procedure for extreme values.
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of 338 for two-qubit systems (ρ) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius (r) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, r=1). Further, they also observed (personal communication) such separability probability r-invariance, when using, more broadly, random induced measure (K=3,4,5,…), with K=4 corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality ∣ρPT∣>∣ρ∣ and those from ∣ρ∣>∣ρPT∣>0, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar r-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with r. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii (rA,rB) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) X-states model.
Due to recent important work of \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski and Sommers [J. Phys. A 36, 10115 (2003); 36, 10083 (2003)], exact formulas are available (in terms of both the Hilbert-Schmidt and the Bures metrics) for the $({n}^{2}\ensuremath{-}1)$-dimensional and $[n(n\ensuremath{-}1)∕2\ensuremath{-}1]$-dimensional volumes of the complex and real $n\ifmmode\times\else\texttimes\fi{}n$ density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, two-qubit systems. Making use of the density matrix $(\ensuremath{\rho})$ parametrization of Bloore [J. Phys. A 9, 2059 (1976)], we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, $\ensuremath{\nu}={\ensuremath{\rho}}_{11}{\ensuremath{\rho}}_{44}∕{\ensuremath{\rho}}_{22}{\ensuremath{\rho}}_{33}$. The associated integrand in each case is the product of a known Jacobian (highly oscillatory near $\ensuremath{\nu}=1$) and a certain unknown univariate function, which our extensive numerical (quasi Monte Carlo) computations indicate is very closely proportional to an (incomplete) Beta function ${B}_{\ensuremath{\nu}}(a,b)$, with a=1∕2, b=3 in the real case, and a=26∕5,b=3∕2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.