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Received Apr 18, 2017; Revised Jul 31, 2017; Accepted Aug 10, 2017

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1. Introduction

In a previous paper [1], a family of formulas was obtained for the (total) separability probabilities P(k,α) of generalized two-qubit states (N=4) endowed with Hilbert-Schmidt (k=0) [2] or more generally random induced measure [3, 4]. In this regard, we note that the natural, rotationally invariant measure on the set of all pure states of a N×K composite system (k=K-N) induces a unique measure in the space of N×N mixed states [3, eq. (3.6)]. Further, α serves as a Dyson-index-like parameter [5, 6], assuming the values 1/2,1,2 for the two-rebit (N=4), two-qubit (standard/complex), and two-quaterbit states, respectively.

The concept itself of a “separability probability,” apparently first (implicitly) introduced by Życzkowski et al. in their much cited 1998 paper [7], entails computing the ratio of the volume—in terms of a given measure [8]—of the separable quantum states to all quantum states. Here, we first examine a certain component Q(k,α) of P(k,α). This informs us of that portion—equalling simply P(k,α)/2 in the Hilbert-Schmidt (k=0) case [9]—for which the determinantal inequality ρPT>ρ holds, with ρ denoting a 4×4 density matrix and ρPT denoting its partial transpose. By consequence [10] of the Peres-Horodecki conditions [11, 12], a necessary and sufficient condition for separability in this 4×4 setting is that ρPT>0. The nonnegativity condition ρ≥0 itself certainly holds, independently of any separability considerations. So, the total separability probability can clearly be expressed as the sum of that part for which ρPT>ρ and that for which ρ>ρPT≥0. The former quantity will be the one...

  

  

  

  

            

  

 

 

 

 

 

 

 

 

 

           

 

 

 

 

 

 

     

 

 

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

  

 

 

 

 

 

 

 

        

 

 

 

   

 

 

 

   

 

           

 

 

 

 

 

   

 

           

 

 

                                                                                                                           

                                                                         

                         

 

 

 

   

 

         

 

    

 

 

  

 

       

 

      

 

 

   

 

    

 

    

 

 

   

 

 

 

 

 

  

 

 

 

     

 

   

 

 

 

 

      

 

 

 

                                               

     

              

           

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