Realistic teleportation with linear optical elements

We calculate the highest possible information gain in a measurement of entangled states when employing a beamsplitter. The result is used to evaluate the fidelity, averaged over all unknown inputs, in a realistic teleportation protocol that takes account of the imperfect detection of Bell states. Finally, we introduce a probabilistic teleportation scheme, where measurements are made in a partially entangled basis.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.67.-a; 03.65.-w In the recent years some striking features of quantum entanglement have been unveiled in the context of quantum information [1]. One of them is the possibility of teleporting an unknown state between two distant locations [2]. Teleportation has been realised experimentally for qubits [3,4]. A fundamental ingredient in the teleportation protocol is a Bell measurement, i.e., a projection onto the four Bell states ∣∣ψ±〉= 1 √ 2 (|01〉 ± |10〉), (1) ∣∣φ±〉= 1 √ 2 (|00〉 ± |11〉). In this Letter we study teleportation and Bell measurements with realistic resources. The teleportation channel, which consists ideally of a maximally entangled state as in (1), can nowadays be readily provided experimentally by entangled photons [5–7]. Implementing teleportation with photons means having to per* Corresponding author. E-mail address: [email protected] (D. Bruß). form a Bell measurement, i.e., a joint measurement, on two photons. Using non-linear devices this would be in principle easy to do, but the efficiency of present day non-linear optical elements is not sufficient to implement Bell measurements. Standard optical tools, like, e.g., a beamsplitter, act in a linear way on photons. It has been shown in [8] that it is impossible to distinguish the four Bell states unambiguously with linear optical elements. In the case of a beamsplitter with 50% transmission this is easy to understand, as both |φ+〉 and |φ−〉 as input states will lead to two identically polarised photons in one of the outgoing paths. The topics we address in this Letter are outlined as follows: First, we ask what is the highest information one can retrieve from a Bell measurement with a linear optical element, namely a beamsplitter? This question has also been discussed recently in [9] for a maximally entangled basis. We will in addition consider the case of measurements in a partially entangled basis, which can be more successful than in the case of maximal entanglement, depending on the parameters of the beamsplitter. We will then explain a strategy for teleportation, using such imperfect measurements, and 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0375-9601(00) 00 78 42 8 C. Trump et al. / Physics Letters A 279 (2001) 7–11 calculate the best teleportation fidelity. In the case of measurements in a partially entangled basis teleportation can be implemented probabilistically. Note that our approach uses a maximally entangled state shared between Alice and Bob, and is therefore different from the scheme considered in [10]. Let us first study the gain of information in a Bell measurement using a beamsplitter. Our notation is the same as in [8], and we also use their method to represent the Bell states in matrix form. We denote the photon modes with the creation operators a H,V and b H,V , where a and b stand for two different directions of propagation, and the indices H and V denote horizontal and vertical polarisation. Thus, the Bell states are ∣∣ψ±〉= 1 √ 2 ( a † Hb † V ± a V b H )|0〉, (2) ∣∣φ±〉= 1 √ 2 ( a † Hb † H ± a V b V )|0〉. The action of a realistic beamsplitter is given by [11]  a †