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In tripartite discrete systems, two classes of genuine tripartite entanglement have been discovered, namely, the Greenberger-Horne-Zeilinger (GHZ) class and the W class. To date, much research effort has been concentrated on the polarization entangled three-photon GHZ and W states. Most studies of continuous variable multiparticle entanglement have been focused on Gaussian states. In this Brief Report, we examine two classes of three-photon entangled states in space and time. One class is a three-mode three-photon entangled state and the other is a two-mode triphoton state. These states show behavior similar to the GHZ and W states when one of the photons is not detected. The three-mode entangled state resembles a W state, while a two-mode three-photon state resembles a GHZ state when one of the photons is traced away. We characterize the distinction between these two states by comparing the second-order correlation functions $G^{(2)}$ with the third-order correlation function $G^{(3)}$., Comment: Published in Phys. Rev. A 79, 025802 (2009)

Computational ghost imaging has been demonstrated experimentally recently. In this comment we wish to clarify the difference between pseudothermal ghost imaging (PGI) and computational ghost imaging (CGI). In particular, to emphasize the physics that arises from the difference in the type of sources used. The experiment in [2] is not a PGI experiment but is a simulation of PGI. If we consider CGI at the single photon level, it becomes clear that CGI relies on two identical photons, one real and one simulated, to obtain an image., Comment: 4 pages with 1 figure

In this paper we study the resolution of images illuminated by sources composed of $N+1$ photons in which one non-degenerate photon is entangled with $N$ degenerate photons. The $N$ degenerate photons illuminate an object and are collected by an $N$ photon detector. The signal from the $N$ photon detector is measured in coincidence with the non-degenerate photon giving rise to a ghost image. We discuss the case of three photons in various configurations and generalize to $N+1$. Using the Rayleigh criterion, we find that the system may give an improvement in resolution by a factor of $N$ compared to using a classical source. For the case that the $N$-photon number detector is a point detector, a coherent image is obtained. If the $N$-photon detector is a bucket detector, the image is incoherent. The visibility of the image in both cases is 1. In the opposite case in which the non-degenerate photon is scattered by the object, then, using an $N$-photon point detector may reduce the Airy disk by a factor of $N$., Comment: 11 pages, 3 figures, has been submitted for publication. Comments are welcome

We propose a two-photon beating experiment based upon biphotons generated from a resonant pumping two-level system operating in a backward geometry. On the one hand, the linear optical-response leads biphotons produced from two sidebands in the Mollow triplet to propagate with tunable refractive indices, while the central-component propagates with unity refractive index. The relative phase difference due to different refractive indices is analogous to the pathway-length difference between long-long and short-short in the original Franson interferometer. By subtracting the linear Rayleigh scattering of the pump, the visibility in the center part of the two-photon beating interference can be ideally manipulated among [0, 100%] by varying the pump power, the material length, and the atomic density, which indicates a Bell-type inequality violation. On the other hand, the proposed experiment may be an interesting way of probing the quantum nature of the detection process. The interference will disappear when the separation of the Mollow peaks approaches the fundamental timescales for photon absorption in the detector., Comment: to appear in Phys. Rev. A (2008)

Generating nonclassical light offers a benchmark tool for the fundamental research and potential applications in quantum optics. Conventionally, it has become a standard technique to produce the nonclassical light through the nonlinear optical processes occurring in nonlinear crystals. In this review we describe using cold atomic-gas media to generate such nonclassical light, especially focusing on narrow-band biphoton generation. Compared with the standard procedure, the new biphoton source has such properties as long coherence time, long coherence length, high spectral brightness, and high conversion efficiency. In this paper we concentrate on the theoretical aspect of the entangled two-photon state produced from the four-wave mixing in a multilevel atomic ensemble. We show that both linear and nonlinear optical responses to the generated fields play an important role in determining the biphoton waveform and, consequently on the two-photon temporal correlation. There are two characteristic regimes determined by whether the linear or nonlinear coherence time is dominant. In addition, our model provides a clear physical picture that brings insight into understanding biphoton optics with this new source. We apply our model to recent work on generating narrow-band (and even subnatural linewidth) paired photons using the technique of electromagnetically induced transparency and slow-light effect in cold atoms, and find good agreements with experimental results., Comment: 12 pages, 4 figures, a review on narrow-band paired photons generated from an atomic-gase medium near resonance, submitted to JOSA B (2008)

In a recent paper [quant-ph/0607008] Lapaire and Sipe argue that one can discuss interference experiments using entangled photons in terms of single photon wave functions. Furthermore, they argue that contrary to the claim of the authors of the postponed compensation experiment [2], the single photon wave functions overlap on the beam splitter when interference is observed. In this comment, we show that the claim in [2] is correct and we argue that the idea of single photon wave functions in entangled states is misleading.

We theoretically discuss the preservation of squeezing and continuous variable entanglement of two mode squeezed light when the two modes are subjected to unequal transmission. One of the modes is transmitted through a slow light medium while the other is sent through an optical fiber of unit transmission. Balanced homodyne detection is used to check the presence of squeezing. It is found that loss of squeezing occurs when the mismatch in the transmission of the two modes is greater than 40% while near ideal squeezing is preserved when the transmissions are equal. We also discuss the effect of this loss on continuous variable entanglement using strong and weak EPR criteria and possible applications for this experimental scheme., Comment: 7 pages, 4 figures

A discussion of discrete Wigner functions in phase space related to mutually unbiased bases is presented. This approach requires mathematical assumptions which limits it to systems with density matrices defined on complex Hilbert spaces of dimension p^n where p is a prime number. With this limitation it is possible to define a phase space and Wigner functions in close analogy to the continuous case. That is, we use a phase space that is a direct sum of n two-dimensional vector spaces each containing p^2 points. This is in contrast to the more usual choice of a two-dimensional phase space containing p^(2n) points. A useful aspect of this approach is that we can relate complete separability of density matrices and their Wigner functions in a natural way. We discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd. Special attention is required for two qubits (p=2) and our technique fails to establish the separability property for more than two qubits., Comment: Some misprints have been corrected and a proof of the separability of the A matrices has been added

The 'Ghost' interference experiment is analyzed when the source of entangled photons is a multimode Optical Parametric Amplifier(OPA) whose weak limit is the two-photon Spontaneous Parametric Downconversion(SPDC) beam. The visibility of the double-slit pattern is calculated, taking the finite coincidence time window of the photon counting detectors into account. It is found that the coincidence window and the bandwidth of light reaching the detectors play a crucial role in the loss of visibility on coincidence detection, not only in the 'Ghost' interference experiment but in all experiments involving coincidence detection. The differences between the loss of visibility with two-mode and multimode OPA sources is also discussed. PACS: 42.65.Yj, 42.50.Dv, 42.65.Lm

We examine the possibility of storing and retrieving a single photon using electromagnetically induced transparency (EIT). We consider the theory of a proof of principle two-photon interference experiment, in which an atomic vapor cell is placed in one arm of a two-photon interferometer. Since the two-photon state is entangled, we can examine the degree to which entanglement survives. We show that while the experiment might be difficult, it should be possible to perform. We also show that the two-photon interference pattern has oscillatory behavior. pacs 42.50.Ct, 42.50.Dv, 42.50.Gy, 42.50.St

A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: || ^{2}=1/d. The MUB problem is to prove or disprove the the existence of a maximal set of d+1 bases. It has been shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381, (1989)] that such a collection exists if d is a power of a prime number p. We revisit this problem and use dX d generalizations of the Pauli spin matrices to give a constructive proof of this result. Specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the MUB problem. Additionally we give formulas from which the orthogonal bases can be readily computed. We show how the techniques developed here provide a natural way to analyze the separability of the bases. The techniques used require properties of algebraic field extensions, and the relevant part of that theory is included in an Appendix.

We discuss the problem of when a set of measurements made on an entangled source can be simulated with a classically correlated source. This is discussed in general and some examples are given. The question of which aspects of quantum imaging can be simulated by classically correlated sources is examined.

Let $H^{[ N]}=H^{[ d_{1}]}\otimes ... \otimes H^{[ d_{n}]}$ be a tensor product of Hilbert spaces and let $\tau_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state $\rho_{0}$. Let $\tilde{\tau}_{0}$ denote the closest separable state to $\rho_{0}$ along the line segment from $I/N$ to $\rho_{0}$ where $I$ is the identity matrix. Following [pitrubmat] a witness $W_{0}$ detecting the entanglement of $\rho_{0}$ can be constructed in terms of $I, \tau_{0}$ and $\tilde{\tau}_{0}$. If representations of $\tau_{0}$ and $\tilde{\tau}_{0}$ as convex combinations of separable projections are known, then the entanglement of $\rho_{0}$ can be detected by local measurements. G\"{u}hne \textit{et. al.} in [bruss1] obtain the minimum number of measurement settings required for a class of two qubit states. We use our geometric approach to generalize their result to the corresponding two qudit case when $d$ is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, $\tau_{0}=\tilde{\tau}_{0}$. We illustrate our general approach with a two parameter family of three qubit bound entangled states for which $\tau_{0} \neq \tilde{\tau}_{0}$ and we show our approach works for $n$ qubits. In [pitt] we elaborated on the role of a ``far face'' of the separable states relative to a bound entangled state $\rho_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times $I$ and a separable density $\mu_{0}$ on the far face from $\rho_{0}$. Up to a normalization this coincides with the witness obtained in [bruss1] for the particular example analyzed there.

We present theoretical and experimental study of preparing maximally entangled two-photon polarization states, or Bell states, using femtosecond pulse pumped spontaneous parametric down-conversion (SPDC). First, we show how the inherent distinguishability in femtosecond pulse pumped type-II SPDC can be removed by using an interferometric technique without spectral and amplitude post-selection. We then analyze the recently introduced Bell state preparation scheme using type-I SPDC. Theoretically, both methods offer the same results, however, type-I SPDC provides experimentally superior methods of preparing Bell states in femtosecond pulse pumped SPDC. Such a pulsed source of highly entangled photon pairs is useful in quantum communications, quantum cryptography, quantum teleportation, etc., Comment: 11 pages, two-column format, to appear in PRA

A finite dimensional quantum mechanical system is modeled by a density rho, a trace one, positive semi-definite matrix on a suitable tensor product space H[N] . For the system to demonstrate experimentally certain non-classical behavior, rho cannot be in S, a closed convex set of densities whose extreme points have a specificed tensor product form. Two mathematical problems in the quantum computing literature arise from this context: (1) the determination whether a given rho is in S and (2) a measure of the ``entanglement'' of such a rho in terms of its distance from S. In this paper we describe these two problems in detail for a linear algebra audience, discuss some recent results from the quantum computing literature, and prove some new results.We emphasize the roles of densities rho as both operators on the Hilbert space H[N] and also as points in a real Hilbert space M. We are able to compute the nearest separable densities tau0 to rho0 in particular classes of inseparable densities and we use the Euclidean distance between the two in M to quantify the entanglement of rho0. We also show the role of tau0 in the construction of separating hyperplanes, so-called entanglement witnesses in the quantum computing literature.

We show in this Comment that the interpretation of experimental data as well as the theory presented in Atat\"ure et al. [Phys. Rev. Lett. 84, 618 (2000)] are incorrect and discuss why such a scheme cannot be used to "recover" high-visibility quantum interference., Comment: Comment on Atat\"ure et al. [Phys. Rev. Lett. 84, 618 (2000)], 2nd revision, To appear in Phys. Rev. Lett. April, (2001)

Great progress has been made recently in establishing conditions for separability of a particular class of Werner densities on the tensor product space of $n$ $d$--level systems (qudits). In this brief note we complete the process of establishing necessary and sufficient conditions for separability of these Werner densities by proving the sufficient condition for general n and d.

Using the finite Fourier transform, we introduce a generalization of Pauli-spin matrices for $d$-dimensional spaces, and the resulting set of unitary matrices $S(d) $ is a basis for $d\times d$ matrices. If $N=d_{1}\times d_{2}\times...\times d_{b}$ and $H^{[ N]}=\bigotimes H^{% [ d_{k}]}$, we give a sufficient condition for separability of a density matrix $\rho $ relative to the $H^{[ d_{k}]}$ in terms of the $L_{1}$ norm of the spin coefficients of $\rho >.$ Since the spin representation depends on the form of the tensor product, the theory applies to both full and partial separability on a given space $H^{[ N]}$% . It follows from this result that for a prescribed form of separability, there is always a neighborhood of the normalized identity in which every density matrix is separable. We also show that for every prime $p$ and $n>1$ the generalized Werner density matrix $W^{[ p^{n}]}(s) $ is fully separable if and only if $s\leq (1+p^{n-1}) ^{-1}$.

Necessary conditions for separability are most easily expressed in the computational basis, while sufficient conditions are most conveniently expressed in the spin basis. We use the Hadamard matrix to define the relationship between these two bases and to emphasize its interpretation as a Fourier transform. We then prove a general sufficient condition for complete separability in terms of the spin coefficients and give necessary and sufficient conditions for the complete separability of a class of generalized Werner densities. As a further application of the theory, we give necessary and sufficient conditions for full separability for a particular set of $n$-qubit states whose densities all satisfy the Peres condition.

We study first-order interference in spontaneous parametric down-conversion generated by two pump pulses that do not overlap in time. The observed modulation in the angular distribution of the signal detector counting rate can only be explained in terms of a quantum mechanical description based on biphoton states. The condition for observing interference in the signal channel is shown to depend on the parameters of the idler radiation., Comment: 5 pages, two-column, submitted to PRA

We report a two-photon interference effect, in which the entangled photon pairs are generated from two laser pulses well-separated in time. In a single pump pulse case, interference effects did not occur in our experimental scheme. However, by introducing a second pump pulse delayed in time, quantum interference was then observed. The visibility of the interference fringes shows dependence on the delay time between two laser pulses. The results are explained in terms of indistinguishability of biphoton amplitudes which originated from two temporally separated laser pulses., Comment: two-column, 4pages, submitted to PRA, minor changes

When a subset of particles in an entangled state is measured, the state of the subset of unmeasured particles is determined by the outcome of the measurement. This first measurement may be thought of as a state preparation for the remaining particles. In this paper, we examine how the duration of the first measurement effects the state of the unmeasured subsystem. The state of the unmeasured subsytem will be a pure or mixed state depending on the nature of the measurement. In the case of quantum teleportation we show that there is an eigenvalue equation which must be satisfied for accurate teleportation. This equation provides a limitation to the states that can be accurately teleported., Comment: 24 pages, 3 figures, submitted to Phys. Rev. A

We review the formalism for describing the two photon state produced in spontaneous parametric down conversion.

The purpose of these workshops is to bring together an international selection of scientists to discuss the latest developments in Squeezed States in various branches of physics, and in the understanding of the foundations of quantum mechanics. At the third workshop, special attention was given to the influence that quantum optics is having on our understanding of quantum measurement theory. The fourth meeting in this series will be held in the People's Republic of China.

The generation of entangled states of two systems from product states is discussed for the case in which the paths of the two systems do not overlap. A particular method of measuring allows one to project out the nonlocal entangled state. An application to the production of four photon entangled states is outlined.

Two types of Einstein-Podolsky-Rosen experiments were demonstrated recently in our laboratory. It is interesting to see that in an interference experiment (wave-like experiment) the photon exhibits its particle property, and in a beam-splitting experiment (particle-like experiment) the photon exhibits its wave property. The two-photon states are produced from Type 1 and Type 2 optical spontaneous parametric down conversion, respectively.