Distribution ofGconcurrence of random pure states

Abstract

Average entanglement of random pure states of an N × N composite system is analyzed. We compute the average value of the determinant D of the reduced state, which forms an entanglement monotone. Calculating higher moments of the determinant we characterize the probability distribution P(D). Similar results are obtained for the rescaled N th root of the determinant, called G-concurrence. We show that in the limit N → ∞ this quantity becomes concentrated at a single point G ⋆ = 1/e. The position of the concentration point changes if one consider an arbitrary N × K bipartite system, in the joint limit N, K → ∞, K/N fixed.

Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities

“…Let us begin our investigation into the indicated statistical aspects of the "geometry of quantum states" [4,14] by noting the two following special cases-which will be extended in certain bivariate directions-of the (univariate determinantal moment) formulas [15][eq. …”

“…These last three figures are based on Hibert-Schmidt sampling (utilizing Ginibre ensembles [15]) of random density matrices, using 10, 000 = 100 2 bins. In regard to the two-qubit plot, K.Żyzckowski informally wrote: "A high peak in the upper corner means that: a) a majority of the entangled states is 'little entangled' (small det(ρ T )) or rather, they are 'close'…”

Optimal two-qubit gate for generation of random bipartite entanglement

“…For instance, the transformation U CNOT x y U CNOT † = y z can be simply stated by x = 6 going to xЈ = ␤ j Ј+4·␤ i Ј= 11. All 16 xЈ for CNOT can be written in a vector xЈ = ͑0, 1,14,15,5,4,11,10,9,8,7,6,12,13,2,3͒, denoting the transformation x = ͑0,1,2, ... ,14,15͒ → xЈ. It turns out that XY gate also preserves the Pauli group.…”

Average entanglement of spin 1 and 1/2 pair

“…The average entanglement of pure states in a two-qubit was computed in [12]. Later, the average entanglement of random pure states of an N × N composite system was analyzed [13]. In this paper, we focus on the effect of an external magnetic field upon the average entanglement of randomly distributed ground state.…”

“…All is known about the entanglement properties of this system, namely that PPT is sufficient and necessary entanglement criterion in this case [10]. In order to investigate the properties of the set of density matrices of a finite size, the notion of average entanglement was introduced [11][12][13][14]. O. Giraud obtained the analytic expressions for the probability density distribution of the linear entropy and the purity for bipartite pure random quantum states [15].…”