Optimal distinction between non-orthogonal quantum states

Given a finite set of linearly independent quantum states, an observer who examines a single quantum system may sometimes identify its state with certainty. However, unless these quantum states are orthogonal, there is a finite probability of failure. A complete solution is given to the problem of optimal distinction of three states, having arbitrary prior probabilities and arbitrary detection values. A generalization to more than three states is outlined. Electronic address: [email protected] Electronic address: [email protected] 1 1. Non-orthogonal quantum signals Quantum information theory is an emerging science, which combines two traditional disciplines: quantum mechanics and classical information theory. This subject has many fascinating potential applications for the transmission and processing of information, and yields results that cannot be achieved by classical means. A simple example is the use of quanta that have been prepared according to one of a finite set of states as signals for the transmission of information. The possibility of using non-orthogonal quantum states, which has no classical analogue, is especially interesting for its potential applications to cryptography (that is, for communication security) [1]. An observer, faced with such a set of signals whose prior probabilities are known, may follow various strategies. The approach favored by information theorists is to maximize the mutual information that can be acquired in the detection process [2]: each event is analyzed in a way from which it is possible to deduce definite posterior probabilities for the emission of the various signals, and the observer’s aim is to reduce as much as possible the Shannon entropy of the ensemble of signals. On the other hand, communication engineers attempt to guess what the signal actually was, and their aim is to miminize the number of errors [3]. Cryptographers, whose supply of signals is essentially unlimited but for whom security is paramount, do not want any error at all, but on the other hand they are ready to lose some fraction of the signals. The latter strategy is the one that will be investigated in this article. The case of just two non-orthogonal signals is quite simple and well known [4–6]. Recently, Chefles [7] investigated the case of N linearly independent signals, and obtained some partial results. In the following, we give a complete treatment of the case of three signals. Our method can readily be generalized to a larger number of signals (but explicit calculations become tedious). In the next section, we introduce a set of positive operator valued measures which describe generalized quantum measurements. (These are more general than the projection valued measures corresponding to the standard, von Neumann type of mesurement.) An explicit algorithm is developed, to ensure the positivity of the required matrices. Optimization (namely, how to maximize the information gain) is discussed in Sect. 3. We consider the possibility that the various signals may have different “values.” The information gain is defined as the expected average of the values of detected signals (this