Effects of noise in continuous variables quantum communication and measurement

Introduction The foundation of the classical theory of information were laid in 1948 by C. Shannon in the pi- oneering paper A Mathematical Theory of Communication [1]. This theory is mainly concerned with encoding and sending classical bits of information, such as alphabet letters, over communi- cation channels that are governed by the laws of classical physics. When we allow the carriers of the information and the channels to be quantum ones, the properties of the whole apparatus change accordingly, so that we are left with a completely new physical object to analyze. This is the field of quantum communication theory. An example of the remarkable differences between the classical and quantum settings, is the problem concerning the difficulty to correctly identity the received signal: in a classical contest, though it could be hard to read a message if the communi- cation channel is noisy, it is always possible to recover part or all of the information sent through the channel, at least in principle, distinguishing between the different states used to encode the information. Conversely, in a quantum setting, this is not always true, not even in principle for a noiseless channel: there is never a downright separation between two states that are too “close”, unless they are orthogonal. This implies that a considerable effort must be directed in the choice of the optimal measuring strategy, a task that gave rise to a new branch of study on itself (see, for example, reference [2]), keeping in mind that even the “closeness” of two quantum states could have different (though equivalent) definitions. Nevertheless, quantum communication theory deserves our attention because its possibilities are much greater than the classical ones. Dense coding, quantum teleportation, quantum distillation, quantum cloning (see [3] for a comprehensive review and detailed explanation) are all powerful tools out of the reach of classical communication. A typical quantum communication setting implies the presence of two parties, a sender and a receiver, that need to exchange information. We are going to call the sender Alice and the receiver Bob, as it is customary. After choosing a set of quantum states with which encoding the information (the alphabet), Alice sends these states through a quantum channel; upon receiving the encoded information, Bob tries to correctly identify the states, in order to assign a (previously agreed) meaning to each. All the steps in the process can be realized in many different ways, but an element that can not be avoided is the spoiling of the quantum states along their travel, between Alice and Bob. This spoiling process is due to the presence of noise and dissipation. The later leads to a progressive destruction of the signal, reducing its amplitude due to the presence of imperfection in the media convoying the information, but without affecting significantly its quantum entanglement; the former actually modifies the states, soiling them and producing effects v