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.

Abstract. We present a theoretical analysis of the process of spontaneous parametric down conversion (SPDC) in a non-linear crystal characterized by a linearly-chirped χ (2) grating along the direction of propagation. Our analysis leads to an expression for the joint spectral amplitude, based on which we can derive various spectral-temporal properties of the photon pairs and of the heralded single photons obtained from the photon pairs, including: the singlephoton spectrum, the chronocyclic Wigner function and the Schmidt number. The simulations that we present are for the specific case of a collinear SPDC source based on a PPLN crystal with the signal and idler photons emitted close to the telecom window. We discuss the mechanism for spectral broadening due to the presence of a linearly chirped χ (2) grating, showing that not only the width but also to some extent the shape of the SPDC spectrum maybe controlled. Also, we discuss how the fact that the different spectral components are emitted on different planes in the crystal leads to single-photon chirp.

We study the chronocyclic character, i.e. the joint temporal and spectral properties, of the singlephoton constituents of photon pairs generated by spontaneous parametric down conversion. In particular we study how single photon properties, including purity and single-photon chirp, depend on photon pair properties, including the type of signal-idler spectral and correlations and the level of pump chirp.

Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reveals the topological phase as fringes shifts when local operations are performed. Dimensions d = 2, 3 and 4 were tested, showing the expected theoretical values.Geometrical phases were introduced long ago in a seminal work on polarization transformations in classical optics [1,2]. Later, geometrical and topological features were demonstrated to play a fundamental role in the realm of quantum theory [3,4]. Tomita and Chiao verified experimentally the existence of Berry's phase and its topological properties at the classical level in helically wound optical fiber [5]. Geometric phases were measured by using coincidence detection of photon pairs produced in parametric down-conversion in conjunction with a Michelson interferometer [6,7], by changing adiabatically the polarization state of the photon pairs [8], with single photons in a mixed state of polarization by using a Mach-Zenhder interferometer [9] or in a polarization pure state by using a polarimetric technique [10]. More recently, they have been proposed as a robust tool for implementing unitary gates for quantum computation [11,12]. From this perspective, the geometric phase on entangled bipartite systems and the role of entanglement in its topological nature have been discussed both theoretically [13][14][15] and experimentally [16][17][18] for twoqubit systems. Later, they were generalized to pairs of qudits of any dimension [19,20] and to multiple qubits [21], showing that the dimension of the Hilbert space plays a crucial role in determining fractional topological phases in both cases. A recent review on this rich subject can be found in Ref. [22]. Although experimental schemes for demonstrating these fractional values have been proposed [23, 24], they have not been implemented so far. In a broader context, fractional phases may be revealed in quantum Hall systems, related to different homotopy classes in the configuration space of anyons. This nontrivial topology has been conjectured to be a possible resource for fault tolerant quantum computation [25].In this work we present experimental results of fractional topological phase measurements on entangled qu-dits. A qudit is a quantum state that belongs to a d-dimensional Hilbert space (d > 2). A quantum system in a general qudit state can be written in terms of d states that form a basis in the d-dimensional Hilbert space. Photonic qudits with dimensions d = 3 and 4 and qubits (d = 2) were encoded on the transverse ...