Entanglement production in nonideal cavities and optimal opacity

Abstract

We compute analytically the distributions of concurrence C and squared norm N for the production of electronic entanglement in a chaotic quantum dot. The dot is connected to the external world via one ideal and one partially transparent lead, characterized by the opacity γ. The average concurrence increases with γ while the average squared norm of the entangled state decreases, making it less likely to be detected. When a minimal detectable norm N0 is required, the average concurrence is maximal for an optimal value of the opacity γ ⋆ (N0) which is explicitly computed as a function of N0. If N0 is larger than the critical value N ⋆ 0 ≃ 0.3693 . . . , the average entanglement production is maximal for the completely ideal case, a direct consequence of an interesting bifurcation effect. The phenomenon of entanglement between spatially separated particles is probably one of the most baffling predictions of quantum mechanics. Initially, the idea that entangled quantum states could be subject to non-classical correlations was seen as a possible evidence of the incompleteness of quantum theory [1]. Nowadays, thanks to fascinating experimental developments, the reality of quantum entanglement is no longer disputed. Besides its fundamental and intellectually challenging interest, modern research on entanglement mainly focuses on its possible applications to quantum information processing, cryptography and teleportation [2, 3]. As a consequence, it is of paramount interest to discover feasible pathways for the production, manipulation, and detection of quantum entanglement in a variety of physical devices.Among the most promising routes, the entanglement in solid-state electronic systems [4, 5] plays a prominent role. As far as the production of entangled electrons is concerned, several proposals based on interacting [6] and noninteracting [7][8][9] electron mechanisms are already available. A very interesting interactionless mechanism was recently proposed by Beenakker et al. [9] where a ballistic quantum dot (see sketch in Fig. 1) is used as an orbital entangler for pairs of noninteracting electrons [10]. A quantum dot is basically a mesoscopic electronic billiard connected to the external world by two double-channel leads and brought out of equilibrium by an applied external voltage. Assuming that the classical electron dynamics inside the cavity is chaotic, experimental observables such as conductance and shot noise fluctuate from sample to sample and their statistics (about which nearly everything is known [11]) is governed by the transmission eigenvalues of the cavity.In order to quantify the degree of entanglement produced inside the cavity, one uses a standard measure C called concurrence, which is related to the violation of a Bell inequality for current correlators. The average and variance of the concurrence for a chaotic dot with ideal leads (i.e. when the tunnelling probability across the leads is unity) was computed in [9]. It was found that these two moments are practically unaffected by the breaking of t...

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