Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
A method is introduced for the verification of nonclassicality in terms of moments of nonclassicality quasiprobability distributions. The latter are easily obtained from experimental data and will be denoted as nonclassicality moments. Their relation to normally-ordered moments is derived, which enables us to verify nonclassicality by using well established criteria. Alternatively, nonclassicality criteria are directly formulated in terms of nonclassicality moments. The latter converge in proper limits to the usually used criteria, as is illustrated for squeezing and sub-Poissonian photon stat…
We derive simple practical procedures revealing the quantum behavior of angular momentum variables by the violation of classical upper bounds on the statistics. Data analysis is minimum and definite conclusions are obtained without evaluation of moments, or any other more sophisticated procedures. These nonclassical tests are very general and independent of other typical quantum signatures of nonclassical behavior such as sub-Poissonian statistics, squeezing, or oscillatory statistics, being insensitive to the nonclassical behavior displayed by other variables.
Nonclassicality criteria based on intensity moments and derived from the usual matrix approach are compared to those provided by the majorization theory. The majorization theory is shown to give a greater number of more suitable nonclassicality criteria. Fifteen experimentally useful criteria of the majorization theory containing the intensity moments up to the fifth order are identified. Their performance is experimentally demonstrated on the set of eleven potentially nonclassical states generated from a twin beam by postselection based on detecting a given number of photocounts in one arm by…
We present an experimental method for complete identification of the nonclassicality of Gaussian states in the whole phase space. Our method relies on nonclassicality witnesses written in terms of measured integrated intensity moments up to the third order, provided that appropriate local coherent displacements are applied to the state under consideration. The introduced approach, thus, only requires linear detectors for measuring intensities of optical fields, that is very convenient and powerful from the experimental point of view. Additionally, we demonstrate that the proposed technique not…
In the spirit of Quantum Non-Demolition Measurements, we show that exploiting suitable vibronic couplings and repeatedly measuring the atomic population of a confined ion, it is possible to distill center of mass vibrational states with well defined square of angular momentum or, alternatively, angular momentum projection Schrödinger cat states.
According to Born's rule quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. This means that both contribute to any observed nonclassical effect that is usually attributed just to the observed light state. This is relevant since typical measurement are highly nonclassical by themselves, such as number states and quadrature eigenstates. We show that nonclassical effects only arise provided that the measurement is itself nonclassical. Otherwise there is a classical-like model accounting for the observed statistics.
Characteristic functions are shown to be useful for highly sensitive measurements. Redistributions of motional Fock states of a trapped atom can be directly monitored via the most fragile nonclassical part of the characteristic function. The method can also be used for decoherence measurements in optical quantum-information systems.
Nonclassicality cannot be a single-observable property since the statistics of any quantum observable is compatible with classical physics. We develop a general procedure to reveal nonclassical behavior from the joint measurement of multiple observables. This requires enlarged spaces and couplings with auxiliary degrees of freedom so it must be followed by an inversion procedure to infer system properties from the observed statistics. In particular this discloses nonclassical properties of standard examples of classical-like behavior, such as SU(2) and Glauber coherent states. Moreover, when c…
We show that one single experiment can test simultaneously and independently both the nonclassicality of states and measurements by the violation or fulfillment of classical bounds on the statistics. Nonideal measurements affected by imperfections can be characterized by two bounds depending on whether we test the ideal measurement or the real one.
Correlations in twin beams composed of many photon pairs are studied using an intensified CCD camera. Joint signal-idler photon-number distribution and quantum phase-space quasi-distributions determined from experimental data have nonclassical features.
The effects of von Neumann postselected measurement on nonclassicality of single-photon-added coherent state (SPACS) are studied. Explicit expressions and analytical results for various field properties of SPACS such as the photon number distribution, the Mandel Q_{m} factor and the squeezing parameter of field quadrature after postselected von Neumann measurement are investigated. The results showed that the nonclassicality of SPACS after measurement changed dramatically than initial state. The measurement let SPACS possess more strong sub-Poissonian photon statistics in some definite couplin…
We propose to experimentally test the nonclassicality of quantum states through homodyne tomography. For single-mode states we check violations of inequalities involving the photon-number probability. For two-mode states we test the nonclassicality by reconstructing some suitable number-operator functions. The test can be performed with available quantum efficiency of homodyne detection, by measuring the pertaining quantities on the corresponding noisy states.
Measurements serve as the intermediate communication layer between the quantum world and our classical perception. So, the question which measurements efficiently extract information from quantum systems is of central interest. Using quantum steering as a nonclassical phenomenon, we show that there are instances, where the results of all two-outcome measurements can be explained in a classical manner, while the results of some three-outcome measurements cannot. This points at the important role of the number of outcomes in revealing the nonclassicality hidden in a quantum system. Moreover, our…
We theoretically generate nonclassical states from coherent state heralded by Knill-Laflamme-Milburn (KLM)-type SU(3) interference. Injecting a coherent state in signal mode and two single-photon sources in other two auxiliary modes of SU(3) interferometry, a broad class of useful nonclassical states are obtained in the output signal port after making two single-photon-counting measurements in the two output auxiliary modes. The nonclassical properties, in terms of anti-bunching effect and squeezing effect as well as the negativity of the Wigner function, are studied in detail by adjusting the…
Based on the measurement of quantum correlation functions, the quantum statistical properties of spectral measurements are studied for broadband radiation fields. The spectral filtering of light before its detection is compared with the direct detection followed by the spectral analysis of the recorded photocurrents. As an example, the squeezing spectra of the atomic resonance fluorescence are studied for both types of filtering procedures. The conditions for which the detection of the nonclassical signatures of the radiation is possible are analyzed. For the considered example, photocurrent f…
Derivation and experimental violation of Bell-like inequalities involve the measurement of in-compatible observables. Simple complementarity forbids the existence of such joint probabilitydistribution. Moreover, the measurement of incompatible observables requires different experimen-tal procedures, which no necessarily must share a common joint statistics. In this work, we avoidthese difficulties by proposing a joint simultaneous measurement experiment. We can obtain theexact individual statistics of all the observables involved in the Bell inequalities after a suitable datainversion. A lack …
We describe the creation of nonclassical states of microwave radiation via ideal dichotomic single photon detection, i.e., a detector that only indicates presence or absence of photons. Ideally, such a detector has a back action in the form of the subtraction operator. Using the non-linearity of this back action, it is possible to create a large family of nonclassical states of microwave radiation, including squeezed and multi-component cat states, starting from a coherent state. We discuss the applicability of this protocol to current experimental designs of Josephson Photomultipliers (JPMs).
We derive exceedingly simple practical procedures revealing the quantum nature of states and measurements by the violation of classical upper bounds on the statistics of arbitrary measurements. Data analysis is minimum and definite conclusions are obtained without evaluation of moments, or any other more sophisticated procedures. These nonclassical tests are independent of other typical quantum signatures such as sub-Poissonian statistics, quadrature squeezing, or oscillatory statistics. This approach can be equally well applied to very diverse situations such as single- and two-mode fields, o…
We extend the definition of quantum discord as a quantifier of nonClassical correlations in a quantum state to the case where weak measurements are performed on subsystem $A$ of a bipartite system $AB$. The properties of weak discord are explored for several families of quantum states. We find that in many cases weak quantum discord is identical to normal discord and in general the values of the two are very close to each other. Weak quantum discord reduces to discord in the appropriate limits as well. We also discuss the implications of these observations on the interpretations of quantum dis…
The measurement of joint signal-idler photon-number distribution of a field obtained from spontaneous parametric downconversion using an intensified CCD camera is presented. It is shown that a classicality criterion is violated directly by the measured data. Characteristic dimensions of the area of correlation are determined in the same experimental setup.
We report the experimental reconstruction of the statistical properties of an ultrafast pulsed type-II parametric down conversion source in a periodically poled KTP waveguide at telecom wavelengths, with almost perfect photon-number correlations. We used a photon-number-resolving time-multiplexed detector based on a fiber-optical setup and a pair of avalanche photodiodes. By resorting to a germane data-pattern tomography, we assess the properties of the nonclassical light states states with unprecedented precision.
Joint measurements of two observables reveal that every state is nonclassical, with the only trivial exception of the state with density matrix proportional to the identity. This naturally includes states considered universally as classical-like, such as SU(2) and Glauber coherent states. We show that this holds because we can always find a joint measurement whose statistics is not separable.
Finite resolution quantum nondemolition (QND) measurements allow a determination of light field properties while preserving some of the original quantum coherence of the input state. It is thus possible to measure correlations between the photon number and a quadrature component of the same light field mode. Nonclassical features emerge as photon number quantization is resolved. In particular, a strong anti-correlation of quantization and coherence is observable in QND measurements of photon number, and a correlation between measurement induced quantum jumps and quadrature component measuremen…
Quantifiers and witnesses for the nonclassicality of measurements and of states
Institute for Quantum Computing, University of Waterloo（量子计算研究所，滑铁卢大学）；Department of Physics & Astronomy, University of Waterloo（物理与天文学系，滑铁卢大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）
A quantum decaying system can reveal its nonclassical behavior by being noninvasively measured. Correlations of weak measurements in the noninvasive limit violate the classical bound for a universal class of systems. The violation is related to the incompatibility between exponential decay and unitary evolution. The phenomenon can be experimentally observed by continuous weak measurements and a large class of observables. The nonclassical nature of such a system allows us to treat it as a potential quantum resource.
Criteria defining higher-order sub-Poissonian-like fields are given using five different quantities: moments of (I) integrated intensity, (II) photon number, (III) integrated-intensity fluctuation, (IV) photon-number fluctuation, and (V) elements of photocount and photon-number distributions. Relations among the moment criteria are revealed. Performance of the criteria is experimentally investigated using a set of potentially sub-Poissonian fields obtained by post-selection from a twin beam. The criteria based on moments of integrated intensity and photon number and those using the elements of…
We study a novel optical setup which is able to select a specific Fock component from a generic input state. The device allows to synthesize number states and superpositions of few number states, and to measure the photon distribution and the density matrix of a generic input signal.
Interpretation of the nonclassical total probability formula arising in some quantum experiments is provided based on stochastic models described by means of a sequence of random vectors changing in the measurement procedures.
Simple joint measurements of pairs of observables reveal that states considered universally as classical-like, such as SU(2) spin coherent states, Glauber coherent states, and thermal states are actually nonclassical. We show that this holds because we can find a joint measurement the statistics of which is not separable. Eventually this may be extended to all states different from the maximally mixed state.
We exploit results from the classical Stieltjes moment problem to bring out the totality of all the information regarding phase insensitive nonclassicality of a state as captured by the photon number distribution p_n. Central to our approach is the realization that n !p_n constitutes the sequence of moments of a (quasi) probability distribution, notwithstanding the fact that p_n can by itself be regarded as a probability distribution. This leads to classicality restrictions on p_n that are local in n involving p_n's for only a small number of consecutive n's, enabling a critical examination of…
Singularity or negativity of Glauber P-function is a widespread notion of nonclassicality, with important implications in quantum optics and with the character of an irreducible resource. Here we explore how P-nonclassicality may be generated by conditional Gaussian measurements on bipartite Gaussian states. This nonclassical steering may occur in a weak form, which does not imply entanglement, and in a strong form that implies EPR-steerability and thus entanglement. We show that field quadratures are the best measurements to remotely generate nonclassicality, and exploit this result to derive…
We have observed the three-photon correlation in nonclassical light sources by using an indirect measurement scheme based on the dead time effect of photon-counting detectors. We first developed a general theory which enables us to extract the three-photon correlation from the two-photon correlation of an arbitrary light source measured with detectors with finite dead times. We then confirmed the validity of our measurement scheme in experiments done with the cavity-QED microlaser operating with a large intra-cavity mean photon number exhibiting both sub- and super-poissoniaq photon statistics…
Numerous inequalities involving moments of integrated intensities and revealing nonclassicality and entanglement in bipartite optical fields are derived using the majorization theory, non-negative polynomials, the matrix approach, as well as the Cauchy-Schwarz inequality. Different approaches for deriving these inequalities are compared. Using the experimental photocount histogram generated by a weak noisy twin beam monitored by a photon-number-resolving iCCD camera the performance of the derived inequalities is compared. A basic set of ten inequalities suitable for monitoring the entanglement…
Quantifying nonclassicality in qubit systems via positive operator-valued measures
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院斐兹克量子中心）；Department of Physics, Isfahan University of Technology（伊斯法罕理工大学物理系）
The quantum teleportation process is composed of a joint measurement performed upon two subsystems A and B (uncorrelated), followed by a unitary transformation (parameters of which depend on the outcome of the measurement) performed upon a third subsystem C (EPR correlated with system B). The information about the outcome of the measurement is transferred by classical means. The measurement performed upon the systems A and B collapses their joint wavefunction into one of the four {\it entangled} Bell states. It is shown here that this measurement process plus a possible measurement on the thir…
To quantify single mode nonclassicality, we start from an operational approach. A positive semi-definite observable is introduced to describe a measurement setup. The quantification is based on the negativity of the normally ordered version of this observable. Perfect operational quantumness corresponds to the quantum-noise-free measurement of the chosen observable. Surprisingly, even moderately squeezed states may exhibit perfect quantumness for a properly designed measurement. The quantification is also considered from an axiomatic viewpoint, based on the algebraic structure of the quantum s…
We show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states. Specifically, spatially-separated parties who perform local measurements on a maximally-entangled state using randomly chosen measurement bases can, with significant probability, generate nonclassical correlations that violate a Bell inequality. For n parties using a Greenberger-Horne-Zeilinger state, this probability of violation rapidly tends to unity as the number of parties increases. We also show that, even with both a randomly chosen two-qubit p…
We experimentally examine the nonclassical character of a class of non-Gaussian states known as phase-diffused squeezed states. These states may show no squeezing effect at all, and therefore provide an interesting example to test nonclassicality criteria. The characteristic function of the Glauber-Sudarshan representation (P function) proves to be a powerful tool to detect nonclassicality. Using this criterion we find that phase-diffused squeezed states are always nonclassical, even if the squeezing effect vanishes. Testing other criteria of nonclassicality based on higher-order squeezing and…
A framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit qua…
We consider the classical correlations that two observers can extract by measurements on a bipartite quantum state, and we discuss how they are related to the quantum mutual information of the state. We show with several examples how complementarity gives rise to a gap between the quantum and the classical correlations, and we relate our quantitative finding to the so-called classical correlation locked in a quantum state. We derive upper bounds for the sum of classical correlation obtained by measurements in different mutually unbiased bases and we show that the complementarity gap is also pr…
It is often asserted that quantum effects can be observed in coincidence detection rates or other correlations, but never in the rate of single-photon detection. We observe nonclassical interference in a singles rate, thanks to the intrinsic nonlinearity of photon counters. This is due to a dependence of the effective detection efficiency on the quantum statistics of the light beam. Such measurements of detector response to photon pairs promise to shed light on the microscopic aspects of silicon photodetectors, and on general issues of quantum measurement and decoherence.
Rotated quadratures carry the phase-dependent information of the electromagnetic field, so they are somehow conjugate to the photon number. We analyze this noncanonical pair, finding an exact uncertatinty relation, as well as a couple of weaker inequalities obtained by relaxing some restrictions of the problem. We also find the intelligent states saturating that relation and complete their characterization by considering extra constraints on the second-order moments of the variables involved. Using these moments, we construct performance measures tailored to diagnose photon-added and Schröding…
It is demonstrated that a weak measurement of the squared quadrature observable may yield negative values for coherent states. This result cannot be reproduced by a classical theory where quadratures are stochastic $c$-numbers. The real part of the weak value is a conditional moment of the Margenau-Hill distribution. The nonclassicality of coherent states can be associated with negative values of the Margenau-Hill distribution. A more general type of weak measurement is considered, where the pointer can be in an arbitrary state, pure or mixed.
Contextuality is a signature of operational nonclassicality in the outcome statistics of an experiment. This notion of nonclassicality applies to a breadth of physical phenomena. Here, we establish its relation to two fundamental nonclassical entities in quantum theory; measurement incompatibility and steering. We show that each is necessary and sufficient the failure of operational contextuality. We exploit the established connection to contextuality to provide a novel approach to problems in measurement incompatibility and steering.
We consider macroscopic correlations in a bipartite system consisting of 2N particles described by a generalised probabilistic theory. In particular, we discuss a case of N PR-boxes shared between two parties. We characterise macroscopic measurements as collective measurements of the same property on all the boxes in the same region. Such measurements are assumed to reveal only the average value of the measured collective property. We show that for two measurements per observer and N greater or equal 2 there always exist a joint probability distribution explaining all observable data and there…
While quantum mechanics exquisitely describes the behavior of microscopic systems, one ongoing challenge is to explore its applicability to systems of larger size and mass. Unfortunately, quantum states of increasingly macroscopic objects are more easily corrupted by unintentional measurements from the classical environment. Additionally, even the intentional measurements from the observer can further perturb the system. In optomechanics, coherent light fields serve as the intermediary between the fragile mechanical states and our inherently classical world by exerting radiation pressure force…
