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 particle C as the intermediate particle that realises the global interaction between A and B. A similar scenario was also considered in a different context in [4, 5] . Initially, the total joint quantum state may or may not carry entanglement. In the following we will be only interested in bipartite entanglement, i.e. two out of the three particles A, B and C are grouped together. We quantify the initial entanglement between AC and B as E AC|B , and the final entanglement, after sending C to Bob, as E A|BC . As a quantifier of entanglement we will first use the relative entropy of entanglement, which is a well established and widely studied measure of entanglement for mixed states [6, 7] . It is defined as the minimal relative entropy S (ρ||σ) = Tr [ρ log ρ] − Tr [ρ log σ] between the given state ρ XY for two parties X and Y and the set of separable states S: Besides the fact that the relative entropy plays a crucial role in quantum information theory [8] , the significance of the relative entropy of entanglement is also provided by its close relation to the distillable entanglement [9] . In a naive approach to our original question, namely determining in a quantitative way the cost for sending a certain amount of entanglement, a natural conjecture
Related to teleportation cost, Streltsov et.al, REF posed the question of the cheapest way for distributing entanglement and provided the minimal quantum cost for sending an entangled composite state in a long distance.
Quantum cost for sending entanglement
Phys. Rev. Lett. 108, 250501 (2012)
Physics Medicine