Entanglement concentration by ordinary linear optical devices without post - selection

Recently, entanglement concentrations have been experimently demonstrated by post-selection ( T. Yamamoto et al, Nature, 421, 343(2003), and Z. Zhao et al, Phys. Rev. Lett., 90,207901(2003) ), i.e., to each individual outcome state, one has to destroy it to know whether it has been purified. Here we give proposal for entanglement concentration without any post selection by using only practically existing linear optical devices. In particular, a sophisticated photon detector to distinguish one photon or two photons is not required. The resource of maximally entangled state(EPR state) plays a fundamentally important role in testing the quantum laws related to the non-locality [1] and in many tasks of quantum information processing [2,6] such as the quantum teleportation [3,4], quantum dense coding [3], entanglement based quantum key distribution [5] and quantum computation [6]. So far, it is generally believed that the two-photon polarized EPR state is particularly useful in quantum information processing. ∗email: [email protected] †email: [email protected] 1 If certain type of non-maximally entangled states are shared by distant two parties Alice and Bob initially, the raw states can be distilled into highly entangled states by local quantum operation and classical communication through the entanglement concentration scheme [7]. Although research on such issues have been extensively done theoretically, the feasible experimental schemes and experimental demonstrations of the entanglement concentration are rare. So far, some schemes by linear optical devices have been proposed [8,9] and some experimental demonstrations have also been reported [10,12,13]. In all these schemes [8–12], one has to verify that each of the indicator light beams contains exactly one photon. However, by our current technology, it’s not likely to really implement a sophisticated detector [19] in the scheme to distinguish one photon or two photon in a light beam. What we can see from a normal photon detector is that whether it is clicked or not. When it is clicked, the measured light beam could contain either exactly one photon or 2 photons in the previously proposed entanglement concentration schemes [8–10]. In a recent experimental report [10], the unwanted events that one indicator beam contain two photons are excluded by post selection: Both the two indicator beams and the two outcome beams are measured. If all the 4 detectors are clicked, the maximally entangled state must have been created on the two outcome beams. Such a post selection destroies the outcome itself. That is to say, limited by the available technology of photon detector, one can only verify a maximally entangled state by totally destroying that state. This means, without a sophisticated photon detector, the set-up [10] is not supposed to really produce any maximally entangled state through the entanglement concentration, even though the requested raw states are supplied deterministically. A similar drawback also appears in [8,11,13] . The very recent experiment [12] also relies on post-selection. So far non-post-selection entanglement concentration in polarization space with linear optical devices has never been proposed, though there are some studies on the possibility of the entanglement concentration to continuous variable states through the Gaussification scheme [14,15]. In the situation of Ref. [10,8,12], the entanglement is in the two level polarization space 2 and all pairs are equally shared by two remotely separated parties. This case is rather important because the polarization entanglement is easy manipulate, e.g., the local rotation operation. Also the assumption of two pair states with unknown identical parameters is reasonable in cases such as that Alice sends two halves of EPR pairs to Bob through the same dephasing channel. Note that here each group contains two identical pairs with unknown parameters, the parameters for the pair states in different groups are different. Now we go to the main result of this Letter. We will show that the following raw state |r, φ〉 = 1 1 + r2 (|HH〉+ re|V V 〉)⊗ (|HH〉+ re|V V 〉) (1) can be probabilistically distilled into maximally entangled state without post selection, even though only normal photon detectors are used. Note that in Eq.(1) r > 0, both r and φ are unknown parameters. In particular, the special case r = 1 is just the one treated in the recent experiment [10]. Since the raw state contains unknown parameters, our purification(concentration) scheme is actually to distill the maximum entanglement from the un-normalized mixed state ∫ ∞ 0 ∫ π 0 |r, φ〉〈r, φ|drdφ. (2) However, for simplicity we shall use the pure state notation |r, φ〉, keeping in the mind that both parameters are totally unknown. Consider the schematic diagram, figure 1. This diagram is a modified scheme from the one given by Pan et al in Ref. [8]. However, as we shall see, the modification leads to a totally different result: Our scheme uniquely produces the even-ready maximally entangled state through entanglement purification, provided that the requested state in Eq.(2) is supplied. Our scheme requires the two fold coincidence event as the indication that the maximally entangled state has been produced on the outcome beams 2’,3’, i.e. whenever both photon detectors Dx, Dwclick, the two outcome beams, beam 2’ and beam 3’ must be in the maximally entangled state: 3 |Φ〉2′,3′ = 1 √ 2 (|H〉2′|H〉3′|+ V 〉2′ |V 〉3′). (3) Now we show the mathematical details for the claim above. The polarizing beam splitters transmit the horizontally polarized photons and reflect the vertically polarized photons. For clarity, we use the Schrodinger picture. And we assume the non-trivial time evolutions to the light beams only takes place in passing through the optical devices. Consider Fig.(1). Suppose initially two remote parties Alice and Bob share two pairs of non-maximally entangled photons as defined by Eq.(1), denoted by photon pair 1,2 and photon 3,4 respectively. The half wave plate HWP1 here is to change the polarization between the horizontal and the vertical. After photon 3 and 4 each pass through HWP1, the state is evolved to: 1 1 + r2 (|HH〉12 + re|V V 〉12)⊗ (|V V 〉34 + re|HH〉34). (4) Furthermore, after the beams pass through the two horizontal polarizing beam splitters( denoted by PBS1), with perfect synchronization [16], the state is |χ′〉 = 1 1 + r2 (|H〉1′|H〉2′ + re|V 〉3′ |V 〉4′)⊗ (|V 〉1|V 〉2 + re|H〉3|H〉4). (5) This can be recast to the summation of three orthogonal terms: |χ′〉 = 1 1 + r2 (|A〉+ |B〉+ |C〉) (6)