We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode $P$ functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two- and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.
We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode P functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two- and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.
One of the most relevant problems in the quantum theory is the question whether the appropriate state of the system can be described within a classical theory. The famous examples of nonclassical states are Schrodinger-cats states or entanglement states. In spite of many practical applications of entanglement as a special kind of nonclassicality (like quantum teleportation [1], dense coding [2], or implementation of super-fast [3] and fast algorithms [4]), we still have difficulties with describing and specifying it. One cannot find such an operational optimal method, which would allow us to maximally use the technological and scientific potential arising from the adaptation properties of quantum correlations. Thus, my thesis is focused on giving a proposal of a theoretical recipe for constructing experimentally achievable procedures for the study of quantum correlations. In my thesis I have focused on quantum correlations described in terms of nonclassicality (quantumness) and specific kind of it,i.e., entanglement. The nonclassicality definition, used by me in the thesis, is based on the Glauber-Sudarshan function (P -function). One can assume, that a state is nonclassical, if its P -function is negative or more singular than Dirac’s delta [5]. As follows, the state is nonclassical, when its P -function cannot be treated as a “real” probabilistic distribution. Based on specific properties of the P -function, Agarwal and Tara [6], as well as, Shchukin, Richter and Vogel [7, 8] proposed nonclassicality and entanglement [9] criteria based on matrices of moments of annihilation and creation operators. The operational procedures for analyzing nonclassicality and an efficient method for measuring such moments developed by Shchukin and Vogel [10] creates basis for my thesis. It can be seen that the nonclassicality criteria based on matrices of moments offer an effective way of deriving specific inequalities, which might be useful in the verification of nonclassicality of particular states generated in experiments. Therefore, criteria constructed based on the above definition of nonclassicality can be used to find practical and effective methods of generating and testing nonclassicality and, therefore, also quantum entanglement of optical fields. The results presented in my thesis can be divided into three main groups: 1. Finding operational and practical criteria to classify states in terms of nonclassicality and entanglement based on fundamental classical inequalities like Cauchy-Schwarz inequality. It was also shown how some known entanglement inequalities can be derived as nonclassicality inequalities or as sums of more than one inequality [Bartkowiak2010a]. 2. Describing properties and behaviour of quantum correlations for different optical fields (e.g. for multiand single-mode systems, for interacting and noninteracting modes). In particular, general occurrence of sudden vanishing of nonclassicality, which can be observed not only for twoor multimode but also for single-mode fields, was proven [Bartkowiak2011]. 3. Finding the methods of generating and testing of quantum correlations, which would be practical and easy to implement with available resources in both linear and nonlinear optics.
The study of nonclassical properties of two-mode quantum states is particularly useful in quantum information theory because of the possibilities of obtaining entanglement and other two-mode quantum correlations in these states. Here we have investigated the possibilities of the existence of nonclassicality in a two-mode New generalized binomial state (TMNGBS). Specifically two-mode antibunching, Quadrature squeezing, sum \& difference squeezing, and various entanglement criteria e.g Shchukin-Vogel entanglement criterion, the uncertainty relation of SU(1,1) Algebra and EPR entanglement criterion are explored in two mode particular example of quantum state named as New generalized binomial state. Earlier we studied nonclassicality in single-mode NGBS, here we are extending our study toward the two-mode version of a quantum state. Here we provide the general expressions of moments for a two-mode quantum state (Fock basis) and explore the quantification in a particular example NGBS. It is found that antibunching, squeezing, and SV entanglement are possible with different limits of depending parameters but the entanglement criteria (EPR, SU (1,1) algebra and Cauchy - Schwarz inequality based)for NGBS are not possible. This study opens up the possibility of exploring the two-mode nonclassicality in other states too.
Gaussian states with nonclassical properties such as squeezing and entanglement serve as crucial resources for quantum information processing. Accurately quantifying these properties within multimode Gaussian states has posed some challenges. To address this, we introduce a unified quantification: the "classical-nonclassical polarity," represented by P. For a single mode, a positive value of P captures the reduced minimum quadrature uncertainty below the vacuum noise, while a negative value represents an enlarged uncertainty due to classical mixtures. For multimode systems, a positive P indicates bipartite quantum entanglement. We show that the sum of the total classical-nonclassical polarity is conserved under arbitrary linear optical transformations for any two-mode and three-mode Gaussian states. For any pure multimode Gaussian state, the total classical-nonclassical polarity equals the sum of the mean photon number from single-mode squeezing and two-mode squeezing. Our results provide a new perspective on the quantitative relation between single-mode nonclassicality and entanglement, which may find applications in a unified resource theory of nonclassical features.
Numerous inequalities involving moments of integrated intensities and revealing nonclassicality and entanglement in bipartite optical fields are derived using the majorization theory, nonnegative polynomials, the matrix approach, and 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 intensified CCD camera, the performance of the derived inequalities is compared. A basic set of 10 inequalities suitable for monitoring entanglement of a twin beam is suggested. Inequalities involving moments of photocounts (photon numbers) as well as some containing directly the elements of photocount (photon-number) distributions are also discussed as a tool for revealing nonclassicality.
Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.
Abstract These two accompanying papers are concerned with entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. The main focus is on two mode entanglement, but multi-mode entanglement is also considered. The bosons may be atoms or molecules as in cold quantum gases. The previous paper I dealt with the general features of quantum entanglement and its specific definition in the case of systems of identical bosons. Entanglement is a property shared between two (or more) quantum sub-systems. In defining entanglement for systems of identical massive particles, it was concluded that the single particle states or modes are the most appropriate choice for sub-systems that are distinguishable, that the general quantum states must comply both with the symmetrization principle and the super-selection rules (SSR) that forbid quantum superpositions of states with differing total particle number (global SSR compliance). Further, it was concluded that (in the separable states) quantum superpositions of sub-system states with differing sub-system particle number (local SSR compliance) also do not occur. The present paper II determines possible tests for entanglement based on the treatment of entanglement set out in paper I. Several inequalities involving variances and mean values of operators have been previously proposed as tests for entanglement between two sub-systems. These inequalities generally involve mode annihilation and creation operators and include the inequalities that define spin squeezing. In this paper, spin squeezing criteria for two mode systems are examined, and spin squeezing is also considered for principle spin operator components where the covariance matrix is diagonal. The proof, which is based on our SSR compliant approach shows that the presence of spin squeezing in any one of the spin components requires entanglement of the relevant pair of modes. A simple Bloch vector test for entanglement is also derived. Thus we show that spin squeezing becomes a rigorous test for entanglement in a system of massive bosons, when viewed as a test for entanglement between two modes. In addition, other previously proposed tests for entanglement involving spin operators are considered, including those based on the sum of the variances for two spin components. All of the tests are still valid when the present concept of entanglement based on the symmetrization and SSR criteria is applied. These tests also apply in cases of multi-mode entanglement, though with restrictions in the case of sub-systems each consisting of pairs of modes. Tests involving quantum correlation functions are also considered and for global SSR compliant states these are shown to be equivalent to tests involving spin operators. A new weak correlation test is derived for entanglement based on local SSR compliance for separable states, complementing the stronger correlation test obtained previously when this is ignored. The Bloch vector test is equivalent to one case of this weak correlation test. Quadrature squeezing for single modes is also examined but not found to yield a useful entanglement test, whereas two mode quadrature squeezing proves to be a valid entanglement test, though not as useful as the Bloch vector test. The various entanglement tests are considered for well-known entangled states, such as binomial states, relative phase eigenstates and NOON states—sometimes the new tests are satisfied while than those obtained in other papers are not. The present paper II then outlines the theory for a simple two mode interferometer showing that such an interferometer can be used to measure the mean values and covariance matrix for the spin operators involved in entanglement tests for the two mode bosonic system. The treatment is also generalized to cover multi-mode interferometry. The interferometer involves a pulsed classical field characterized by a phase variable and an area variable defined by the time integral of the field amplitude, and leads to a coupling between the two modes. For simplicity the center frequency was chosen to be resonant with the inter-mode transition frequency. Measuring the mean and variance of the population difference between the two modes for the output state of the interferometer for various choices of interferometer variables is shown to enable the mean values and covariance matrix for the spin operators for the input quantum state of the two mode system to be determined. The paper concludes with a discussion of several key experimental papers on spin squeezing.