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.

The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.

Quantum computers are unnecessary for exponentially-efficient computation or simulation if the Extended Church-Turing thesis---a foundational tenet of computer science---is correct. The thesis would be directly contradicted by a physical device that efficiently performs a task believed to be intractable for classical computers. Such a task is BosonSampling: obtaining a distribution of n bosons scattered by some linear-optical unitary process. Here we test the central premise of BosonSampling, experimentally verifying that the amplitudes of 3-photon scattering processes are given by the permanents of submatrices generated from a unitary describing a 6-mode integrated optical circuit. We find the protocol to be robust, working even with the unavoidable effects of photon loss, non-ideal sources, and imperfect detection. Strong evidence against the Extended Church-Turing thesis will come from scaling to large numbers of photons, which is a much simpler task than building a universal quantum computer.Comment: See also Crespi et al., arXiv:1212.2783; Spring et al., arXiv:1212.2622; and Tillmann et al., arXiv:1212.224

The fundamental problem faced in quantum chemistry is the calculation of molecular properties, which are of practical importance in fields ranging from materials science to biochemistry. Within chemical precision, the total energy of a molecule as well as most other properties, can be calculated by solving the Schrödinger equation. However, the computational resources required to obtain exact solutions on a conventional computer generally increase exponentially with the number of atoms involved 1,2 . This renders such calculations intractable for all but the smallest of systems. Recently, an efficient algorithm has been proposed enabling a quantum computer to overcome this problem by achieving only a polynomial resource scaling with system size 2,3,4 . Such a tool would therefore provide an extremely powerful tool for new science and technology. Here we present a photonic implementation for the smallest problem: obtaining the energies of H 2 , the hydrogen molecule in a minimal basis. We perform a key algorithmic step-the iterative phase estimation algorithm 5,6,7,8 -in full, achieving a high level of precision and robustness to error. We implement other algorithmic steps with assistance from a classical computer and explain how this non-scalable approach could be avoided. Finally, we provide new theoretical results which lay the foundations for the next generation of simulation experiments using quantum computers. We have made early experimental progress towards the long-term goal of exploiting quantum information to speed up quantum chemistry calculations.Experimentalists are just beginning to command the level of control over quantum systems required to explore their information processing capabilities. An important long-term application is to simulate and calculate properties of other many-body quantum systems. Pioneering experiments were first performed using nuclear-magnetic-resonance-based systems to simulate quantum oscillators 9 , leading up to recent simulations of a pairing Hamiltonian 7,10 . Very recently the phase transitions of a two-spin quantum magnet were simulated 11 using an ion-trap system. Here we simulate a quantum chemical system and calculate its energy spectrum, using a photonic system. Molecular energies are represented as the eigenvalues of an associated time-independent HamiltonianĤ and can be efficiently obtained to fixed accuracy, using a quantum algorithm with three distinct steps 6 : encoding a molecular wavefunction into qubits; simulating its time evolution using quantum logic gates; and extracting the approximate energy using the phase estimation algorithm 3,12 . The latter is a general-purpose quantum algorithm for evaluating the eigenvalues of arbitrary Hermitian or unitary operators. The algorithm estimates the phase, φ, accumulated by a molecular eigenstate, |Ψ , under the action of the time-evolution operator,Û =e −iĤt/ , i.e.,where E is the energy eigenvalue of |Ψ . Therefore, estimating the phase for each eigenstate amounts to estimating the eigenvalues of the Hamiltonia...

Entanglement is widely believed to lie at the heart of the advantages offered by a quantum computer. This belief is supported by the discovery that a noiseless (pure) state quantum computer must generate a large amount of entanglement in order to offer any speed up over a classical computer. However, deterministic quantum computation with one pure qubit (DQC1), which employs noisy (mixed) states, is an efficient model that generates at most a marginal amount of entanglement. Although this model cannot implement any arbitrary algorithm it can efficiently solve a range of problems of significant importance to the scientific community. Here we experimentally implement a first-order case of a key DQC1 algorithm and explicitly characterise the non-classical correlations generated. Our results show that while there is no entanglement the algorithm does give rise to other non-classical correlations, which we quantify using the quantum discord-a stronger measure of nonclassical correlations that includes entanglement as a subset. Our results suggest that discord could replace entanglement as a necessary resource for a quantum computational speed-up. Furthermore, DQC1 is far less resource intensive than universal quantum computing and our implementation in a scalable architecture highlights the model as a practical short-term goal.In contrast to the highly pure multi-qubit states required for the conventional models of quantum computing [1, 2], DQC1 employs only a single qubit in a pure state, alongside a register of qubits in the fully mixed state [3]. While this model is strictly less powerful than a universal quantum computer (where one can implement any arbitrary algorithm) it can still efficiently solve important problems that are thought to be classically intractable. The application originally identified was the efficient simulation of some quantum systems [3]. Since then exponential speed-ups have been identified in estimating: the average fidelity decay under quantum maps [4]; quadratically signed weight enumerators [5]; and the Jones Polynomial in knot theory [6]. Recently it has been shown that DQC1 also affords efficient parameter estimation at the quantum metrology limit [7]. Furthermore, attempts to find an efficient way of classically simulating DQC1 have failed [8]. These results provide strong evidence that a device capable of implenting scalable DQC1 algorithms would be an extremely useful tool.Besides the practical applications, DQC1 is also fascinating from a fundamental perspective. For example, it is straightforward to show that a model employing only fully mixed qubits offers no advantage over a classical computer. It is therefore surprising that the addition of only a single pure qubit offers such a dramatic increase in computational power. Furthermore, an important quantum information result is that a pure state quantum computer can only offer an advantage over a classical approach if it generates an amount of entanglement that grows with the size of the problem being tackled [9,10]. This support...