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...

Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors “scans” overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d / 2, we performed numerical simulations for single systems in an economic (n = 4) and a costly (n = d) scenario, as well as for multiqubit systems (n = 6logd). This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10−2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data—key for its success—are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.

The quantum analogue of ptychography, a powerful coherent diffractive imaging technique, is a simple method for reconstructing d -dimensional pure states. It relies on measuring partially overlapping parts of the input state in a single orthonormal basis and feeding the outcomes to an iterative phase retrieval algorithm for postprocessing. We provide a proof of concept demonstration of this method by determining pure states given by superpositions of d transverse spatial modes of an optical field. A set of n rank- r projectors, diagonal in the spatial mode basis, is used to generate n partially overlapping parts of the input, and each part is projectively measured in the Fourier transformed basis. For d up to 32, we successfully reconstructed hundreds of random states using n = 5 and n = d rank- ⌈ d / 2 ⌉ projectors. The extension of quantum ptychography for other types of photonic spatial modes is outlined.