We study the maximum-confidence (MC) measurement strategy for discriminating among nonorthogonal symmetric qudit states. Restricting to linearly dependent and equally likely pure states, we find the optimal positive operator valued measure (POVM) that maximizes our confidence in identifying each state in the set and minimizes the probability of obtaining inconclusive results. The physical realization of this POVM is completely determined and it is shown that after an inconclusive outcome, the input states may be mapped into a new set of equiprobable symmetric states, restricted, however, to a subspace of the original qudit Hilbert space. By applying the MC measurement again onto this new set, we can still gain some information about the input states, although with less confidence than before. This leads us to introduce the concept of "sequential maximum-confidence" (SMC) measurements, where the optimized MC strategy is iterated in as many stages as allowed by the input set, until no further information can be extracted from an inconclusive result. Within each stage of this measurement our confidence in identifying the input states is the highest possible, although it decreases from one stage to the next. In addition, the more stages we accomplish within the maximum allowed, the higher will be the probability of correct identification. We will discuss an explicit example of the optimal SMC measurement applied in the discrimination among four symmetric qutrit states and propose an optical network to implement it.Comment: 14 pages, 4 figures. Published versio

Quantum mechanics forbids perfect discrimination among nonorthogonal states through a single shot measurement. To optimize this task, many strategies were devised that later became fundamental tools for quantum information processing. Here, we address the pioneering minimum-error (ME) measurement and give the first experimental demonstration of its application for discriminating nonorthogonal states in high dimensions. Our scheme is designed to distinguish symmetric pure states encoded in the transverse spatial modes of an optical field; the optimal measurement is performed by a projection onto the Fourier transform basis of these modes. For dimensions ranging from D = 2 to D = 21 and nearly 14000 states tested, the deviations of the experimental results from the theoretical values range from 0.3% to 3.6% (getting below 2% for the vast majority), thus showing the excellent performance of our scheme. This ME measurement is a building block for high-dimensional implementations of many quantum communication protocols, including probabilistic state discrimination, dense coding with nonmaximal entanglement, and cryptographic schemes.Quantum mechanics establishes fundamental bounds to our capability of distinguishing among states with nonvanishing overlap: if one is given at random one of two or more nonorthogonal states and asked to identify it from a single shot measurement, it will be impossible to accomplish the task deterministically and with full confidence. This constraint has deep implications both foundational, underlying the debate about the epistemic and ontic nature of quantum states [1][2][3][4], and practical, warranting secrecy in quantum key distribution [5,6]. Beyond that, the problem of discriminating nonorthogonal quantum states plays an important role in quantum information and quantum communications [7].A wide variety of measurement strategies have been devised in order to optimize the state discrimination process according to a predefined figure of merit [7]. The pioneering one was the minimum-error (ME) measurement [8][9][10] where each outcome identifies one of the possible states and the overall error probability is minimized. Other fundamental strategies conceived later [11][12][13][14][15][16][17] employ the ME discrimination in the step next to a transformation taking the input states to more distinguishable ones [18], which enables us to identify them with any desired confidence level (within the allowed bounds) and a maximum success probability. Nowadays, the ME measurement is central to a range of applications, including quantum imaging [19], quantum reading [20], image discrimination [21], error correcting codes [22], and quantum repeaters [23], thus stressing its importance.Closed-form solutions for ME measurements are known only for a few sets of states. One of these is the set of symmetric pure states (defined below) prepared with equal * msolisp@udec.cl † lneves@fisica.ufmg.br prior probabilities [24]. Discriminating among them with minimum error sets the bounds on the eavesdropping...

We study the possibility of performing quantum state tomography via equidistant states. This class of states allows us to propose a non-symmetric informationally complete POVM based tomographic scheme. The scheme is defined for odd dimensions and involves an inversion which can be analytically carried out by Fourier transform.

We propose an optimal discrimination scheme for a case of four linearly independent nonorthogonal symmetric quantum states, based on linear optics only. The probability of discrimination is in agreement with the optimal probability for unambiguous discrimination among N symmetric states [Phys. Lett. A 250, 223 (1998)]. The experimental setup can be extended for the case of discrimination among 2 M nonorthogonal symmetric quantum states.