We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems ͑''qubits''͒. Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction ͑in which the density matrix is linearly related to a set of measured quantities͒ and a maximum likelihood technique which requires numerical optimization ͑but has the advantage of producing density matrices that are always non-negative definite͒. In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.

Single-photons are key elements of many future quantum technologies, be it for the realisation of large-scale quantum communication networks 1 for quantum simulation of chemical and physical processes 2 or for connecting quantum memories in a quantum computer 3 . Scaling quantum technologies will thus require efficient, on-demand, sources of highly indistinguishable single-photons 4 . Semiconductor quantum dots inserted in photonic structures are ultrabright single photon sources [5][6][7] , but the photon indistinguishability is limited by charge noise induced by nearby surfaces 8 . The current state of the art for indistinguishability are parametric down conversion single-photon sources, but they intrinsically generate multiphoton events and hence must be operated at very low brightness to maintain high single photon purity 9,10 . To date, no technology has proven to be capable of providing a source that simultaneously generates near-unity indistinguishability and pure single-photons with high brightness. Here, we report on such devices made of quantum dots in electrically controlled cavity structures. We demonstrate on-demand, bright and ultra-pure single photon generation. Application of an electrical bias on deterministically fabricated devices 11,12 is shown to fully cancel charge noise effects. Under resonant excitation, an indistinguishability of 0.9956±0.0045 is evidenced with a g (2) (0)=0.0028±0.0012. The photon extraction of 65% and measured brightness of 0.154±0.015 make this source 20 times brighter than any source of equal quality. This new generation of sources open the way to a new level of complexity and scalability in optical quantum manipulation.

Using the process of spontaneous parametric downconversion in a novel two-crystal geometry, we have generated a source of polarization-entangled photon pairs which is more than ten times brighter, per unit of pump power, than previous sources, with another factor of 30 to 75 expected to be readily achievable. We have measured a high level of entanglement between photons emitted over a relatively large collection angle, and over a 10-nm bandwidth. As a demonstration of the source capabilities, we obtained a 242-σ violation of Bell's inequalities in less than three minutes, and observed near-perfect photon correlations when the collection efficiency was reduced. In addition, both the degree of entanglement and the state purity should be readily tunable. [5], and quantum computation [6]. At present, the most accessible and controllable source of entanglement arises from the process of spontaneous parametric down-conversion in a nonlinear optical crystal. Here we describe a proposal for, and experimental realization of, an ultrabright source of polarization-entangled photon pairs, using two such nonlinear crystals. Because nearly every pair of photons produced is polarization-entangled, the total flux of emitted polarization-entangled pairs should be hundreds of times greater than is achievable with the best previous source, for comparable pump powers. The new technique has the added advantage that the degree of entanglement and the purity of the state may be readily tunable, heretofore impossible.It is now well known that the photons produced via the down-conversion process share nonclassical correlations [7]. In particular, when a pump photon splits into two daughter photons, conservation of energy and momentum lead to entanglements in these two continuous degrees of freedom [8]. Yet conceptually, the simplest examples of entangled states of two photons are the polarizationentangled "Bell states":where H and V denote horizontal and vertical polarization, respectively, and for convenience we omit the normalization factor (1/ √ 2). For instance, HV − V H is the direct analog of the spin-singlet considered by Bell [2]. To date there have been only two methods for producing such polarization-entangled photon pairs, and each has fairly substantial limitations. The first was an atomic cascade -a two-photon decay process from one state of zero angular momentum to another. The resulting photons do display nonclassical correlations (they were used in the first tests of Bell's inequalities [9,10]), but the correlations decrease if the photons are not emitted backto-back, as is allowed by recoil of the parent atom.This problem was circumvented with parametric downconversion, since the emission directions of the photons are well-correlated. In several earlier experiments downconversion photon pairs of definite polarization were incident on a beamsplitter, and nonclassical correlations observed for those post-selected events in which photons traveled to different output ports [11]. However, the photons were actually created i...

Single photons are a fundamental element of most quantum optical technologies. The ideal single-photon source is an on-demand, deterministic, single-photon source delivering light pulses in a well-defined polarization and spatiotemporal mode, and containing exactly one photon. In addition, for many applications, there is a quantum advantage if the single photons are indistinguishable in all their degrees of freedom. Single-photon sources based on parametric down-conversion are currently used, and while excellent in many ways, scaling to large quantum optical systems remains challenging. In 2000, semiconductor quantum dots were shown to emit single photons, opening a path towards integrated single-photon sources. Here, we review the progress achieved in the past few years, and discuss remaining challenges. The latest quantum dot-based single-photon sources are edging closer to the ideal single-photon source, and have opened new possibilities for quantum technologies.