Hilbert-Schmidt Separability Probabilities from Bures Ensembles and vice versa: Applications to Quantum Steering Ellipsoids and Monotone Metrics

Abstract

We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of 8 33 and 25 341 , respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the Π1...4 j<k(λj − λk) term of the HS volume element (the λ’s being the four eigenvalues of the associated 4× 4 density matrix ρ). This measure is applied to the members of a HS ensemble of random two-qubit states, yielding a QES separability probability estimate of 0.105458. Alternatively, weighting members of a Bures ensemble by the QES volume divided by the eigenvalue part 1 √ detρ Π1...4 j<k (λj−λk) λj+λk of the Bures volume element, gives a close estimate of 0.100223. We also weight members of a HS ensemble by the QES volume divided not only by the indicated HS eigenvalue term, but also by the unitary component |Π1...4 j<kRe(U)Im(U)| of the volume element. For one hundred thirty (rather variable) independent separability probability estimates, we, then, obtain median and mean estimates of 0.0447729 and 0.117485 with variance 0.0381468. PACS numbers: Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w ∗Electronic address: paulslater@ucsb.edu 1 ar X iv :2 10 1. 07 71 6v 1 [ qu an tph ] 1 9 Ja n 20 21 In a recent study [1], we reported an estimate of 0.0286 for the probability that a two-qubit state is separable, if one applies to each state, as measure, the volume VA of the corresponding quantum steering ellipsoid (QES) of (say) Alice. We have that [2, eq. (5)] VA = 64π 3 ∣∣det ρ− det ρB ∣∣ (1− b2) . (1) The two-qubit 4× 4 density matrix is denoted by ρ and its partial transpose with respect to Bob’s qubit by ρB (obtained by transposing in place the four 2 × 2 blocks of ρ [3, eq. (16.52)]). b is the norm of Bob’s Bloch vector. The volume VB of EB (the set of states to which Alice can steer Bob, forming an ellipsoid in Bob’s Bloch sphere) can be computed from VA via the relation VB = (1−b2)2 (1−a2)2VA, where a is the norm of Alice’s Bloch vector. Though not yet presented as a formalized proof, multifaceted numerical and analytical evidence strongly indicates that the probability with respect to Hilbert-Schmidt (HS) measure [4] that a member of the fifteen-dimensional convex set of two-qubit systems is separable is 8 33 = 2 3 3·11 ≈ 0.242424 [5, 6] [3, p. 468]. (In fact, Lovas and Andai have given a formal proof that the HS separability probability for two-re[al]bit systems is 29 64 [7, Thm. 2]. Though clearly not so compelling, a highly extensive numerical analysis points to the corresponding separability probability based on the Bures (minimal monotone [8]) measure being 25 341 = 5 2 11·31 ≈ 0.0733138 [9, sec. II.C.1]. (The HS qubit-qutrit separability probability has been conjectured to equal 27 1000 = 3 3 23·53 = 0.027 [9, sec. III.A] [10, Tab. 1]. [11, eq.(33)]. A two-qubit separability probability of 1− 256 27]pi2 based on the operator monotone function √ x has been reported in [5, eq. (87)]–while Lovas and Andai have given an integral formula for the two-rebit case [7, Thm. 4]. Any ball with respect to the Bures distance of a fixed radius in the space of quantum states has the same measure [12].) Let us now see if we can estimate these conjectured HS and Bures two-qubit separability probabilities– 8 33 and 25 341 –in a rather unusual, somehat indirect fashion. The lessons we apparently learn from these exercises, will lead us to reappraise the findings in [1], suggesting possible alternatives to the estimate of 0.0286 for the QES-based separability probability. To begin we note that procedures for generating ensembles of quantum states random with respect to the HS and Bures measures have been given in [12]. To generate a HS random N × N density matrix, one takes a square complex random matrix A of size N pertaining to the Ginibre ensemble [13, 14] (with real and imaginary parts of each element