We investigate generalized measurements, based on positive-operator-valued measures, and von Neumann measurements for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. In particular, we derive the conditions under which the failure probability of the measurement can reach its absolute lower bound, proportional to the fidelity of the states. The optimum measurement strategy yielding the fidelity bound of the failure probability is explicitly determined for a number of cases. One example involves two density operators of rank d that jointly span a 2d-dimensional Hilbert space and are related in a special way. We also present an application of the results to the problem of unambiguous quantum state comparison, generalizing the optimum strategy for arbitrary prior probabilities of the states.Comment: final versio

Abstract. The problem of discriminating among given nonorthogonal quantum states is underlying many of the schemes that have been suggested for quantum communication and quantum computing. However, quantum mechanics puts severe limitations on our ability to determine the state of a quantum system. In particular, nonorthogonal states cannot be discriminated perfectly, even if they are known, and various strategies for optimum discrimination with respect to some appropriately chosen criteria have been developed. In this article we review recent theoretical progress regarding the two most important optimum discrimination strategies. We also give a detailed introduction with emphasis on the relevant concepts of the quantum theory of measurement. After a brief introduction into the field, the second chapter deals with optimum unambiguous, i. e error-free, discrimination. Ambiguous discrimination with minimum error is the subject of the third chapter. The fourth chapter is devoted to an overview of the recently emerging subfield of discriminating multiparticle states. We conclude with a brief outlook where we attempt to outline directions of research for the immediate future. IntroductionIn quantum information and quantum computing the carrier of information is some quantum system and information is encoded in its state [1]. The state, however, is not an observable in quantum mechanics [2] and, thus, a fundamental problem arises: after processing the information -i.e. after the desired transformation is performed on the input state by the quantum processor -the information has to be read out or, in other words, the state of the system has to be determined. When the possible target states are orthogonal, this is a relatively simple task if the set of possible states is known. But when the possible target states are not orthogonal they cannot be discriminated perfectly, and optimum discrimination with respect to some appropriately chosen criteria is far from being trivial even if the set of the possible nonorthogonal states is known. Thus the problem of discriminating among nonorthogonal states is ubiquitous in quantum information and quantum computing, underlying many of the communication and computing schemes that have been suggested so far. It is the purpose of this article to review various theoretical schemes that have been developed for discriminating among nonorthogonal quantum states. The corresponding experimental

We describe a class of programmable devices that can discriminate between two quantum states. We consider two cases. In the first, both states are unknown. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. In the second case, we know one of the states while the other is unknown. One copy of the unknown state is fed into the single program register, and the data state which is guaranteed to be prepared in either the program state or the known state, is fed into the data register. The device will then tell us, again optimally, whether the data state matches the template or is the known state. We determine two types of optimal devices. The first performs discrimination with minimum error, the second performs optimum unambiguous discrimination. In all cases we first treat the simpler problem of only one copy of the data state and then generalize the treatment to n copies. In comparison to other works we find that providing n > 1 copies of the data state yields higher success probabilities than providing n > 1 copies of the program states.

We show how losses in photodetection and in quantum-state measurements can be numerically compensated after the measurements have been performed. When the overall efficiency exceeds -, ', our recipe works for all quantum states. For smaller e%ciencies, however, the convergence of the compensation procedure depends on the quantum state under investigation.PACS number(s): 03.65.8z, 42.50.0v Detector inefficiencies and losses are present in every real experiment in quantum optics. Apart from attenuating the signal they create extra noise as a consequence of the fluctuation-dissipation theorem. This noise causes quantum decoherence and diminishes our ability to observe subtle quantum phenomena such as interference in phase space [1,2]. The effect of losses is especially important in the recent measurements of the quantum state of light [3,4]. How can we compensate detection losses'? It can be done physically by preamplification [5] or numerically by deconvolution of the recorded data. In a recent proposal [6] for the tomographic reconstruction of the density matrix, the decon volution is woven into the reconstruction algorithm.There the compensation of losses is possible, but only when the detection efficiency g exceeds the critical value -, '. Is this an artifact of the particular algorithm or is g = -, ' the general bound?In this Brief Report we separate the detection from the compensation procedure. We assume a photon-number distribution or, more generally, a density matrix as given. We show how the compensation of losses can be achieved. Again, only when the efficiency is larger than the critical value -, is this possible for every density matrix. In this respect, g= -, ' is a bound also for our method. On the other hand, we show that in certain cases the critical q can be less than -, '.The detection efficiency and other losses (e.g. , those due to mode mismatch) can be effectively taken into account with a simple beam-splitter model [7], where a fictitious semitransparent mirror is placed in front of the ideal photodetector. The same simple picture can be applied to a homodyne detection scheme [8]. Here the measuring apparatus can be considered as an ideal homodyne detector with a single beam splitter placed in front of it that accounts for all the losses (Fig. 1). Mode 1 is the signal being in the state p".g. It is attenuated by the beam splitter, while mode 2 is the channel of the losses 'Permanent address: where a vacuum input p""=~0)(0~is formally introduced. This vacuum mode models the extra quantum noise involved in inefficient detection. An elegant way [9,10] of treating the beam splitter is to apply the Jordan-Sch winger formalism, originally developed in the theory of angular momenta. Setting the phase parameters to zero for simplicity, we find for the unitary transformation of the beam splitter -i 2 arccos~gE2 S(g)=e whereHere & & and d2 denote the annihilation operators for the signal and the noise mode, respectively. The transmittance q of the beam splitter is identified with the overall detectio...

In quantum state filtering one wants to determine whether an unknown quantum state, which is chosen from a known set of states, {|ψ1 , . . . |ψN }, is either a specific state, say |ψ1 , or one of the remaining states, {|ψ2 , . . . |ψN }. We present the optimal solution to this problem, in terms of generalized measurements, for the case that the filtering is required to be unambiguous. As an application, we propose an efficient, probabilistic quantum algorithm for distinguishing between sets of Boolean functions, which is a generalization of the Deutsch-Jozsa algorithm.PACS numbers: 03. 03.65.Ta, Optimal discrimination among quantum states plays a central role in quantum information theory. Interest in this problem was prompted by the suggestion to use nonorthogonal quantum states for communication in certain secure quantum cryptographic protocols, most notably in the one based on the two-state procedure as developed by Bennett [1]. The reason why until recently the area has shown relatively slow progress within the rapidly evolving field of quantum information is that it poses quite formidable mathematical challenges. Except for a handful of very special cases, no general exact solution has been available involving more than two arbitrary states. In this paper we present an exact solution to an optimum measurement problem involving an arbitrary number of quantum states, with no restriction on the states. The resulting method has the potential for widespread applications in quantum information processing. In particular, it lends itself quite naturally to a quantum generalization of probabilistic classical algorithms. Whenever it is possible to find a one-to-one mapping of classical alternatives onto quantum states, our method can discriminate among these quantum alternatives in a single step with optimum success probability.We illustrate the strength of the method on the example of a probabilistic quantum algorithm to discriminate between sets of Boolean functions. A Boolean function on n bits is one that returns either 0 or 1 as output for every possible value of the input x, where 0 ≤ x ≤ 2 n − 1. The function is uniform (or constant) if it returns the same output on all of its arguments, i.e. either all 0's or all 1's; it is balanced (or even) if it returns 0's on half of its arguments and 1's on the other half; and it is biased if it returns 0's on m 0 of its arguments and 1's on the remaining m 1 = 2 n − m 0 arguments (m 0 = m 1 = 0 or 2 n − 1). Classically, if one is given an unknown function and told that it is either balanced or uniform, one needs 2 (n−1) + 1 measurements to decide which. Deutsch and Jozsa [2] developed a quantum algorithm that can accomplish this task in one step. To discriminate a biased Boolean function from an unknown balanced one, 2 (n−1) + m 1 + 1 measurements are needed classically, where, without loss of generality, we have assumed that m 1 < m 0 . Here we propose a probabilistic quantum algorithm that can unambiguously discriminate a known biased Boolean function from a given se...