We study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.
We suggest a formalism to illustrate the entanglement of identical particles in the first quantization language (1QL). Our 1QL formalism enables one to exploit all the well-established quantum information tools to understand the indistinguishable ones, including the reduced density matrix and familiar entanglement measures. The rigorous quantitative relation between the amount of entanglement and the spatial coherence of particles is possible in this formalism. Our entanglement detection process is a generalization of the entanglement extraction protocol for identical particles with mode split…
A suitable deformation of the Hopf algebra of the creation and annihilation operators for a complex scalar field, initially quantized in Minkowski space--time, induces the canonical quantization of the same field in a generic gravitational background. The deformation parameter q turns out to be related to the gravitational field. The entanglement of the quantum vacuum appears to be robust against interaction with the environment.
Second quantization is an essential topic in senior undergraduate and postgraduate level Quantum Mechanics course. However, it seems that there is a lack of transparent and natural derivation of this formalism from the first-quantization one in most existing textbooks. Without introducing the concept of a simple harmonic oscillator and taking the case of an assembly of identical fermions as an example, we provide a less ad hoc derivation of the second quantization formalism based on the equivalence of the fully antisymmetric many-fermion states and the Fock states.
The theories defined by Lagrangians containing second time derivative are considered. It is shown that if the second derivatives enter only the terms multiplied by coupling constant one can consistently define the perturbative sector via Dirac procedure. The possibility of introducing standard canonical variables is analysed in detail. The ambiguities in quantization procedure are pointed out.
We define a multi-partite entanglement measure for stabilizer states, which can be computed efficiently from a set of generators of the stabilizer group. Our measure applies to qubits, qudits and continuous variables.
The event formalism is a non-linear extension of quantum field theory designed to be compatible with the closed time-like curves that appear in general relativity. Whilst reducing to standard quantum field theory in flat space-time the formalism leads to testably different predictions for entanglement distribution in curved space. In this paper we introduce a more general version of the formalism and use it to analyse the practicality of an experimental test of its predictions in the earth's gravitational well.
We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed.
Using the fact that the nonintegrable phase factor can reformulate the gauge theory in terms of path dependent vector potentials, the quantization condition for the nonintegrable phase is investigated. It is shown that the path-dependent formalism can provide compact description of the flux quantization and the charge quantization at the existence of a magnetic monopole. Moreover, the path-dependent formalism gives suggestions for searching of the quantized flux in different configurations and for other possible reasons of the charge quantization. As an example, the developed formalism is empl…
A method is developed for realizing entanglement and general second quantized fermionic and bosonic fields in the framework of the fermionic projector.
It is a general belief that all fundamental interactions need to be quantized. However, all attempts to develop a quantum theory of gravity presented various problems, leading to a recent active debate about how to probe its quantum nature. In the present work we provide a proof for the necessity of quantizing fundamental interactions demonstrating that a quantum version is needed for any non trivial conservative interaction whose strength is a function of the relative distance between two objects. Our proof is based on a consistency argument that in the presence of a classical field two inter…
There is a debate in course about the quantification of entanglement of indistinguishable particles and recently a new method due to Lo Franco and Compagno (LFC) [Sci. Rep. \textbf{6}, 20603 (2016)] appeared with the aim of settling the controversies of this debate. In this work we show that the LFC approach can be recovered using the second quantization formalism and a local single-particle basis to perform the trace over one particle state. The use of the second quantization formalism has the advantage of being well known in many-body physics. We also compare the LFC and second quantization …
We review and generalize the recently introduced framework of entropy vectors for detecting and quantifying genuine multipartite entanglement in high dimensional multicomponent quantum systems. We show that these ideas can be extended to discriminate among other forms of multipartite entanglement. In particular, we develop methods to test whether density matrices are: decomposable, i.\ e.\ separable with respect to certain given partitions of the subsystems; $k$-separable, i.\ e.\ separable with respect to $k$ partitions of the subsystems; $k$-partite entangled, i.e. there is entanglement amon…
Suppose in a quantum network, there are $n$ qubits hold by Alice, Bob and Charlie, denoted by systems $A$, $B$ and $C$, respectively. We require the qubits to be described by a stabilizer state and assume the system $A$ is entangled with the combined system $BC$. An interesting question to ask is when it is possible to transfer all the entanglement to system $A$ and $B$ by local operation on $C$ and classical communication to $AB$, namely \textit{entanglement trimming}. We find a necessary and sufficient condition and prove constructively for this entanglement trimming, which we name it as "th…
We investigate how stabilizer theory can be used for constructing sufficient conditions for entanglement. First, we show how entanglement witnesses can be derived for a given state, provided some stabilizing operators of the state are known. These witnesses require only a small effort for an experimental implementation and are robust against noise. Second, we demonstrate that also nonlinear criteria based on uncertainty relations can be derived from stabilizing operators. These criteria can sometimes improve the witnesses by adding nonlinear correction terms. All our criteria detect states clo…
Experiments witnessing the entanglement between two particles interacting only via the gravitational field have been proposed as a test whether gravity must be quantized. In the language of quantum information, a non-quantum gravitational force would be modeled by local operations with classical communication (LOCC), which cannot generate entanglement in an initially unentangled state. This idea is criticized as too constraining on possible alternatives to quantum gravity. We present a parametrized model for the gravitational interaction of quantum matter on a classical spacetime, inspired by …
In this dissertation, I present a general method for studying quantum error correction codes (QECCs). This method not only provides us an intuitive way of understanding QECCs, but also leads to several extensions of standard QECCs, including the operator quantum error correction (OQECC), the entanglement-assisted quantum error correction (EAQECC). Furthermore, we can combine both OQECC and EAQECC into a unified formalism, the entanglement-assisted operator formalism. This provides great flexibility of designing QECCs for different applications. Finally, I show that the performance of quantum l…
We find an interesting relationship between multipartite bound entangled states and the stabilizer formalism. We prove that if a set of commuting operators from the generalized Pauli group on $n$ qudits satisfy certain constraints, then the maximally mixed state over the subspace stabilized by them is an unlockable bound entangled state. Moreover, the properties of this state, such as symmetry under permutations of parties, undistillability and unlockability, can be easily explained from the stabilizer formalism without tedious calculation. In particular, the four-qubit Smolin state and its re…
Observable signatures of the quantum nature of gravity at low energies have recently emerged as a promising new research field. One prominent avenue is to test for gravitationally induced entanglement between two mesoscopic masses prepared in spatial superposition. Here we analyze such proposals and what one can infer from them about the quantum nature of gravity, as well as the electromagnetic analogues of such tests. We show that it is not possible to draw conclusions about mediators: even within relativistic physics, entanglement generation can equally be described in terms of mediators or …
Quantum Space-Time and Phase Space with fuzzy geometric structure are studied as possible formalism for quantization of massive particles and fields. In this approach the state of nonrelativistic particle m described by the fuzzy point of fuzzy (weakly) ordered Manifold X. Due to m partial ordering in X m coordinate x principal uncertainty is generic in this ansatz. It's shown that in 1+1 dimension such Fuzzy Mechanics (FM) is equivalent to Path Integral formalism of Quantum Mechanics (QM).
Polymer quantization is as a useful toy model for the mathematical aspects of loop quantum gravity and is interesting in its own right. Analyzing entropies of physically equivalent states in the standard Hilbert space and the polymer Hilbert space we show that they converge in the limit of vanishing polymer scale. We derive a general bound that relates entropies of physically equivalent states in unitarily inequivalent representations.
We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the Hamiltonian, we quantize canonically the classical particle in a viscous media. We study the free quantum particle in a viscous media and the dissipative quantum tunnelling. It is found that the dissipation influences tunnelling probability by a factor that closely resembles the result of Caldeira and Leggett.
In this tenth paper of the series we aim at showing that our formalism, using the Wigner-Moyal Infinitesimal Transformation together with classical mechanics, endows us with the ways to quantize a system in any coordinate representation we wish. This result is necessary if one even think about making general relativistic extensions of the quantum formalism. Besides, physics shall not be dependent on the specific representation we use and this result is necessary to make quantum theory consistent and complete.
We apply the general formalism of nilpotent polynomials [Mandilara et al, Phys. Rev. A 74, 022331 (2006)] to the problem of pure-state multipartite entanglement classification in four qubits. In addition to establishing contact with existing results, we explicitly show how the nilpotent formalism naturally suggests constructions of entanglement measures invariant under the required unitary or invertible class of local operations. A candidate measure of fourpartite entanglement is also suggested, and its behavior numerically tested on random pure states.
Within the context of quantum teleportation, a proposed intuitive model to explain bipartite entanglement describes the scheme as being the same qubit of information evolving along and against the flow of time of an external observer. We investigate the physicality of such a model by applying the time-reversal of the Schrodinger equation in the teleportation context. To do so, we first lay down the theory of time-reversal applied to the circuit model and then show that the outcome of a teleportation-like circuit is consistent with the usual tensor product treatment, thus independent of the phy…
The relativistic 20-component wave equation, describing particles with spin 1/2 and two mass states, is analyzed. The projection operators extracting states with definite energy and spin projections, and density matrix are obtained. The canonical quantization of the field with two mass states in the formalism of the first order is performed and the chronological pairing of the 20-component operators was found.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
We present an extension of Nelson's stochastic quantum mechanics to finite temperature. Utilizing the formulation of Thermo Field Dynamics (TFD), we can show that Ito's stochastic equations for tilde and non-tilde particle positions reproduce the TFD-type Schrödinger equation which is equivalent to the Liouville-von Neumann equation. In our formalism, the drift terms in the Ito's stochastic equation have the temperature dependence and the thermal fluctuation is induced through the correlation of the non-tilde and tilde particles. We show that our formalism satisfy the position-momentum uncerta…
The paper investigates the non-local property of quantum mechanics in the quantum hydrodynamic analogy (QHA) given by Madelung. The role of the quantum potential in generating the non-local dynamics of quantum mechanics is analyzed. The work shows how in presence of noise the non-local properties as well as the quantization of the action are perturbed. The resulting stochastic QHA dynamics much depend by the strength of the interaction: Strongly bounded systems (such as linear ones) lead to quantum entangled stochastic behavior, while weakly bounded ones may be not able to maintain the quantum…
We propose a general formalism for analytical description of multiatomic ensembles interacting with a single mode quantized cavity field under the assumption that most atoms remain un-excited on average. By combining the obtained formalism with the nilpotent technique for the description of multipartite entanglement we are able to overview in a unified fashion different probabilistic control scenarios of entanglement among atoms or examine atomic ensembles. We then apply the proposed control schemes to the creation of multiatom states useful for quantum information.
Signal analysis is built upon various resolutions of the identity in signal vector spaces, e.g. Fourier, Gabor, wavelets, etc. Similar resolutions are used as quantizers of functions or distributions, paving the way to a time-frequency or time-scale quantum formalism and revealing interesting or unexpected features. Extensions to classical electromagnetism viewed as a quantum theory for waves and not for photons are mentioned.
History state formalism for time series with application to finance
Instituto de Física La Plata, CONICET（拉普拉塔物理研究所，阿根廷国家科学研究委员会）；Departamento de Física, Universidad Nacional de La Plata（拉普拉塔国立大学物理系）；Comisión de Investigaciones Científicas (CIC)（科学研究委员会）
We introduce a formalism that connections entanglement witnesses and the distillation and activation properties of a state. We apply this formalism to two cases: First, we rederive the results presented in quant-ph/0104095 by Eggeling et al., namely that on copy of any bipartite state with non--positive partial transpose (NPPT) is either distillable, or activable. Second, we show that there exist three--partite NPPT states, with the property that two copies can neither be distilled, nor activated.
Quantum entanglement plays an important role in quantum computation and communication. It is necessary for many protocols and computations, but causes unexpected disturbance of computational states. Hence, static analysis of quantum entanglement in quantum programs is necessary. Several papers studied the problem. They decided qubits were entangled if multiple qubits unitary gates are applied to them, and some refined this reasoning using information about the state of each separated qubit. However, they do not care about the fact that unitary gate undoes entanglement and that measurement may …
In this paper, the master equation for the coupled lossy waveguides is solved using the thermofield dynamics(TFD) formalism. This formalism allows the use of the underlying symmetry algebras SU(2) and SU(1,1), associated with the Hamiltonian of the coupled lossy waveguides,to compute entanglement and decoherence as a function of time for various input states such as NOON states and thermal states.
75 years after the term "entanglement" was coined to a peculiar feature inherent to quantum systems, the connection between quantum and classical mechanics remains an open problem. Drawing on recent results obtained in semiclassical systems, we discuss here the fate of entanglement in a closed system as Planck's constant becomes vanishingly small. In that case the generation of entanglement in a quantum system is perfectly reproduced by properly defined correlations of the corresponding classical system. We speculate on what these results could imply regarding the status of entanglement and of…
The Berezin--Simon (BS) quantization is a rigorous version of the ``operator formalism'' of quantization procedure. The goal of the paper is to present a rigorous real-time (not imaginary-time) path-integral formalism corresponding to the BS operator formalism of quantization; Here we consider the classical systems whose phase space $M$ is a (possibly non-compact) Kähler manifold which satisfies some conditions, with a Hamiltonian $H:M\rightarrow\mathbb{R}$. For technical reasons, we consider only the cases where $H$ is smooth and bounded. We use Güneysu's extended version of the Feynman--Kac …
General two-particle system is considered within the formalism of Fokker-type action integrals. It is assumed that the system is invariant with respect to the Aristotle group which is a common subgroup of the Galileo and Poincaré groups. It is shown that equations of motion of such system admit circular orbit solutions. The dynamics of perturbations of these solutions is studied. It is described by means of a linear homogeneous set of time-nonlocal equations and is analyzed in terms of eigenfrequencies and eigenmodes. The Hamiltonian description of the system is built in the almost circular or…
We investigate two aspects of the elementary example of POVMs on the Euclidean plane, namely their status as quantum observables and their role as quantizers in the integral quantization procedure. The compatibility of POVMs in the ensuing quantum formalism is discussed, and a Naimark dilation is found for the quantum operators. The relation with Toeplitz quantization is explained. A physical situation is discussed, where we describe the linear polarization of the light with the use of Stokes parameters. In particular, the case of sequential measurements in a real bidimensional Hilbert space i…
The tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.
Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of quantum geometric formalism. In such formalism the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty dx. It's shown that m evolution with minimal number of additional assumptions obeys to schroedinger and dirac formalisms in norelativistic and relativistic cases correspondingly. It's argued that particle's interactions on such fuzzy manifold should be gauge invariant.
We present a method to apply the well-known matrix product state (MPS) formalism to partially separable states in solid state systems. The computational effort of our method is equal to the effort of the standard density matrix renormalisation group (DMRG) algorithm. Consequently, it is applicable to all usually considered condensed matter systems where the DMRG algorithm is successful. We also show in exemplary cases, that polymerisation properties of ground states are closely connected to properties of partial separability, even if the ground state itself is not partially separable.
The quantum space-time and the phase space with fuzzy structure is investigated as the possible quantization formalism. In this theory the state of nonrelativistic particle corresponds to the element of fuzzy ordered set (Foset) - fuzzy point. Due to Foset partial (weak) ordering, particle's space coordinate x acquires principal uncertainty dx. It's shown that Shroedinger formalism of Quantum Mechanics can be completely derived from consideration of particle evolution in fuzzy phase space with minimal number of axioms.
We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function $p(x)$, the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactl…
The quantization rules recently proposed by M. Navarro (and independently I.V. Kanatchikov) for a finite-dimensional formulation of quantum field theory are applied to the Klein-Gordon and the Dirac fields to obtain the quantum equations of motion of both fields. In doing so several problems arise. Solving these difficulties leads us to propose a new classical canonical formalism, which, in turn, leads us to new, improved rules of quantization. We show that the new classical equations of motion and rules of quantization overcome several known unsatisfactory features of the previous formalism. …
Rarita-Schwinger (RS) quantum free field is reexamined in the context of deformation quantization. It is found out that the subsidiary condition does not introduce any change either in the Wigner function or in other aspects of the deformation quantization formalism, in relation to the Dirac field case. This happens because the vector structure of the RS field imposes constraints on the space of wave function solutions and not on the operator structure. The RS propagator was also calculated within this formalism.
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two Bosons or Fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated …
A twistor model of a free massless spinning particle in 4-dimensional Minkowski space is studied in terms of spacetime and spinor variables. This model is specified by a simple action, referred to here as the gauged Shirafuji action, that consists of twistor variables and gauge fields on the 1-dimensional parameter space. We consider the canonical formalism of the model by following the Dirac formulation for constrained Hamiltonian systems. In the subsequent quantization procedure, we obtain a plane-wave solution with momentum spinors. From this solution and coefficient functions, we construct…
This article investigates small oscillations of a vortex ring with zero thickness that evolves under the Local Induction Equation (LIE). We deduce the differential equation that describes the dynamics of these oscillations. We suggest the new approach to the Hamiltonian description of this dynamic system. This approach is based on the extension of the set of dynamical variables by adding the circulation $Γ$ as a dynamical variable. The constructed theory is invariant under the transformations of the Galilei group. The appearance of this group allows for a new viewpoint on the energy of a vorte…
The basic elements of the geometric approach to a consistent quantization formalism are summarized, with reference to the methods of the old quantum mechanics and the induced representations theory of Lie groups. A possible relationship between quantization and discretization of the configuration space is briefly discussed.
For low dimension systems admitting a moment equation representation (MER), the development of an effective eigenenergy bounding theory applicable to all discrete states had remained elusive, until now. Whereas Handy et al (1988 Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it Moment Problem} based, Eigenvalue Moment Method (EMM), for generating arbitrarily tight bounds to the multidimensional, positive, bosonic ground state, its extension to arbitrary excited states seemed intractable. We have discovered a new, moment representation based, quantization formalism that achiev…
An extended Wigner function formalism is introduced for describing the quantum dynamics of particles with internal degrees of freedom in the presence of spatially inhomogeneous fields. The approach is used for quantitative simulations of molecular beam experiments involving space-spin entanglement, such as the Stern-Gerlach and the Rabi experiment. The formalism allows a graphical visualization of entanglement and decoherence processes.
