By using a systematic optimization approach we determine quantum states of light with definite photon number leading to the best possible precision in optical two mode interferometry. Our treatment takes into account the experimentally relevant situation of photon losses. Our results thus reveal the benchmark for precision in optical interferometry. Although this boundary is generally worse than the Heisenberg limit, we show that the obtained precision beats the standard quantum limit thus leading to a significant improvement compared to classical interferometers. We furthermore discuss alternative states and strategies to the optimized states which are easier to generate at the cost of only slightly lower precision.

We demonstrate that the Wigner function of the Einstein-Podolsky-Rosen state, though positive definite, provides a direct evidence of the nonlocal character of this state. The proof is based on an observation that the Wigner function describes correlations in the joint measurement of the phase space displaced parity operator.

We propose an experiment demonstrating the nonlocality of a quantum singlet-like state generated from a single photon incident on a beam splitter. Each of the two spatially separated apparatuses in the setup performs a strongly unbalanced homodyning, employing a single photon counting detector. We show that the correlation functions violating the Bell inequalities in the proposed experiment are given by the joint two-mode Q-function and the Wigner function of the optical singlet-like state. This establishes a direct relationship between two intriguing aspects of quantum mechanics: the nonlocality of entangled states and the noncommutativity of quantum observables, which underlies the nonclassical structure of phase space quasidistribution functions. PACS Number(s): 03.65.Bz, 42.50.Dv A fundamental step providing a bridge between classical and quantum physics has been given by Wigner in form of a quantum mechanical phase space distribution: the Wigner function [1]. From the pioneering work of Weyl, Wigner and Moyal, it follows that the noncommutativity of quantum observables leads to a real abundance of different in form quantum mechanical phase space quasidistributions. A description of quantum phenomena in terms of the Wigner or the positive-Q quasidistributions, provided a milestone step towards a c-number formulation of quantum effects in phase space [2]. Due to Einstein, Podolsky, and Rosen (EPR) [3], followed by the seminal contribution of Bell [4], the meaning of quantum reality and quantum nonlocality has become a central issue of the modern interpretation and understanding of quantum phenomena [5]. Such concepts like entanglement and quantum nonlocality have generated a real flood of theoretical work devoted to various connections of the quantum description with different views or representations of the quantum formalism.Despite all these theoretical works a direct link between various phase space distributions and the nonlocality of quantum mechanics has been missing. In some works [6] the quantum phase space has been treated as a model for a hidden variable theory, and the incompatibility of quantum mechanics with local theories has been attributed to the nonpositive character of the Wigner function. In this context it has been argued that the original EPR wave function cannot violate the positionmomentum Bell inequality, because the corresponding Wigner function is positive everywhere.It is the purpose of this Letter to propose an experimental demonstration of nonlocal effects in phase space exhibited by a quantum optical singlet-like state generated from a single photon. The entanglement will be represented by a correlated state of light, which refers to two spatially separated modes of the electromagnetic field. We show that the proposed experiment establishes a direct relationship between quantum nonlocality and the positive phase space Q-function, as well as the nonpositive Wigner function. We demonstrate that for a certain class of experiments these two quasiprobability distributions are nonlocal ...

We present a universal technique for quantum state estimation based on the maximum-likelihood method. This approach provides a positive definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive definite Hermitian matrices.Comment: 4 pages, 3 figures (5 eps files). Submitted to Phys. Rev. A as a Rapid Communicatio

We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e. the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.Comment: 12 pages, 5 figure

We report the development of a photon-number resolving detector based on a fiber-optical setup and a pair of standard avalanche photodiodes. The detector is capable of resolving individual photon numbers, and operates on the well-known principle by which a single mode input state is split into a large number (eight) of output modes. We reconstruct the photon statistics of weak coherent input light from experimental data, and show that there is a high probability of inferring the input photon number from a measurement of the number of detection events on a single run.Many quantum information strategies require the preparation of nonclassical states. For example, the method of linear optical quantum computing proposed by Knill, Laflamme, and Milburn [1] demands the preparation of single photon states as well as maximally entangled photon multiplets. A number of schemes have been proposed for the preparation of such states, including single photon emitters [2,3,4] and conditionally prepared photon pairs from parametric downconversion [5,6,7]. Conditional state preparation requires the ability to distinguish states of different photon number, which is not possible using conventional photodetectors. Photon number resolution is also desirable to enhance the security of quantum cryptographic schemes [8,9]. In this case it is important to measure the photon statistics of the source at the sending and receiving stations. For implementations that use weak coherent states this means distinguishing between the detection of, say, one or two photons.According to the quantum theory of photodetection, the signal obtained from an ideal noise-free detector has a discrete form corresponding to the absorption of an integer number of quanta from the incident radiation. In practice, however, the granularity of the output signal is concealed by the noise of the detection mechanism. When very low light levels are detected using devices with single-photon sensitivity such as photomultipliers or Geiger-mode operated avalanche photodiodes (APDs), the electronic signal can be reliably converted into a binary message telling us with high efficiency whether an absorption event has occurred or not. However, the intrinsic noise of the gain mechanism necessary to bring the initial energy of absorbed radiation to the macroscopic level completely masks the information on exactly how many photons have triggered that event.There are several methods for constructing photon number resolving detectors. Among those demonstrated to date are the segmented photomultiplier [10], the superconducting bolometer [11], and the superconducting transimpedance amplifier [12]. These detectors operate at cryogenic temperatures and have single photon quantum efficiencies ranging from about 20% for the superconducting devices to approximately 70% in the case of the segmented photomultiplier. On the other hand, conventional room temperature APDs have intrinsic quantum efficiencies up to 80%, though they respond only to the presence or absence of radiation. The ease of...

I derive a tight bound between the quality of estimating the state of a single copy of a d-level system, and the degree the initial state has to be altered in course of this procedure. This result provides a complete analytical description of the quantum mechanical trade-off between the information gain and the quantum state disturbance expressed in terms of mean fidelities. I also discuss consequences of this bound for quantum teleportation using nonmaximally entangled states.PACS numbers: 03.67.-a, 03.65.BzAs a general rule, the more information is obtained from an operation on a quantum system, the more its state has to be altered. This heuristic statement was first exemplified by the Heisenberg microscope gedankenexperiment [1], where the spatial resolution of the apparatus was shown to scale inversely with the uncertainty of the momentum transfered during the observation. Presently, the disturbance caused by the information gain has become an issue of practical significance, as it underlies the security of quantum key distribution [2].The balance between the information gain and the state disturbance attracts currently a lot of interest, particularly in the context of quantum cryptography [3]. Information theory provides a selection of concepts to quantify both the information gain and the state disturbance. The choice of measures for these two effects is usually dictated by the relevance to a specific application. In most cases, however, derivation of the actual balance represents a highly nontrivial task, especially if one is tempted to resign from numerical means. The purpose of this Letter is to present a formulation of the information gain versus state disturbance trade-off which is completely solvable using elementary analytical techniques. This formulation is motivated by recent works on quantum state estimation [4], where the information obtained from the operation is converted into an estimate for the initial state of the system.The problem considered in this Letter can be formulated as follows. Suppose we are given a single d-level particle in a completely unknown pure state |ψ . We want to make a guess about the quantum state of this particle, but at the same time we would like to alter the state as little as possible. One can associate two fidelities with such a procedure. The first one, which we will denote by F , describes how much the state after the operation resembles the original one. The second fidelity, denoted by G, characterizes the average quality of our guess. It is natural to expect that these two quantities cannot take simultaneously too large values. What is the actual quantitative bound between them?Two extreme cases are well known: if nothing is done to the particle we have F = 1, but then our guess about the state of the particle has to be random, which yields G = 1/d. On the other hand, the optimal estimation strategy for a single copy [5] yields G = 2/(d + 1), but then the particle after the operation cannot provide any more information on the initial state; thus also F = 2/(...

We introduce a novel measure to quantify the non-Gaussian character of a quantum state: the quantum relative entropy between the state under examination and a reference Gaussian state. We analyze in details the properties of our measure and illustrate its relationships with relevant quantities in quantum information as the Holevo bound and the conditional entropy; in particular a necessary condition for the Gaussian character of a quantum channel is also derived. The evolution of non-Gaussianity (nonG) is analyzed for quantum states undergoing conditional Gaussification towards twin-beam and de-Gaussification driven by Kerr interaction. Our analysis allows to assess nonG as a resource for quantum information and, in turn, to evaluate the performances of Gaussification and de-Gaussification protocols. PACS numbers: 03.67.-a, 03.65.Bz, 42.50.DvIntroduction-The use of Gaussian states and operations allows the implementation of relevant quantum information protocols including teleportation, dense coding and quantum cloning [1]. Indeed, the Gaussian sector of the Hilbert space plays a crucial role in quantum information processing with continuous variables (CV), especially for what concerns quantum optical implementations [2]. On the other hand, quantum information protocols required for long distance communication, as for example entanglement distillation and entanglement swapping, require nonG operations [3]. Besides, it has been demonstrated that using nonG states and operations teleportation [4,5,6] and cloning [7] of quantum states may be improved. Indeed, de-Gaussification protocols for singlemode and two-mode states have been proposed [4,5,6,8,9] and realized [10]. From a more theoretical point of view, it should be noticed that any strongly superadditive and continuous functional is minimized, at fixed covariance matrix (CM), by Gaussian states. This is crucial to prove extremality of Gaussian states and Gaussian operations [11,12] for various quantities such as channel capacities [13], multipartite entanglement measures [14] and distillable secret key in quantum key distribution protocols. Overall, nonG appears to be a resource for CV quantum information and a question naturally arises on whether a convenient measure to quantify the nonG character of a quantum state may be introduced. Notice that the notion of nonG already appeared in classical statistics in the framework of independent component analysis [15].