ar X iv : 1 20 3 . 12 64 v 2 [ qu an t - ph ] 7 M ar 2 01 2 Quantum cost for sending entanglement

Establishing quantum entanglement between two distant parties is an essential step of many protocols in quantum information processing [1, 2]. One possibility for providing long-distance entanglement is to create an entangled composite state within a lab and then physically send one subsystem to a distant lab. However, is this the “cheapest” way? Here, we investigate the minimal “cost” that is necessary for establishing a certain amount of entanglement between two distant parties. We prove that this cost is intrinsically quantum, and is specified by quantum correlations. Our results provide an optimal protocol for entanglement distribution and show that quantum correlations are the essential resource for this task. Imagine that one wants to send a letter in the oldfashioned way. The postage cost that the sender has to invest depends on the amount of the transmitted substance, quantified by the weight of the letter. If the receiver had already provided some pre-paid envelope, the sender may have to add an appropriate stamp if he/she wants to send a heavier letter. Naturally, the allowed weight of the letter is smaller or equal to a limit which is linked to the total postage. Now, imagine that a sender wants to send quantum entanglement to a receiver. How does the cost that the sender has to invest depend on the amount of entanglement sent, quantified by some entanglement measure? Is this cost reduced when sender and receiver already shared some pre-established entanglement? And what is the nature of this cost can one pay in classical quantities, or does one have to invest a quantum cost? One might be tempted to consider these questions and their answers as obvious matters. However, quantum mechanics has often surprised us with puzzling features: Counterintuitively, as shown in [3], separable states (i.e. states without entanglement) can be used to distribute entanglement. What is then the resource that makes this process possible and enables entanglement distribution without actually sending an entangled state? In order to address this question in a well-defined and quantitative way we will consider the following setting, see Fig. 1: the sender is called Alice (A), and the distant receiver Bob (B). Each of them has a quantum particle in his/her possession. In addition, they have a third quantum particle or ancilla (C) available, which is at the beginning located in Alice’s lab, and then sent (via a noiseless quantum channel) to Bob’s lab. This is a general model for any interaction: One can consider the b A b B b C Initial setup