For finite quantum many-particle systems modeled with say $m$ fermions in $N$ single particle states and interacting with $k$-body interactions ($k \leq m$), the wavefunction structure is studied using random matrix theory. Hamiltonian for the system is chosen to be $H=H_0(t) + λV(k)$ with the unperturbed $H_0(t)$ Hamiltonian being a $t$-body operator and $V(k)$ a $k$-body operator with interaction strength $λ$. Representing $H_0(t)$ and $V(k)$ by independent Gaussian orthogonal ensembles (GOE) of random matrices in $t$ and $k$ fermion spaces respectively, first four moments, in $m$-fermion spaces, of the strength functions $F_κ(E)$ are derived; strength functions contain all the information about wavefunction structure. With $E$ denoting the $H$ energies or eigenvalues and $κ$ denoting unperturbed basis states with energy $E_κ$, the $F_κ(E)$ give the spreading of the $κ$ states over the eigenstates $E$. It is shown that the first four moments of $F_κ(E)$ are essentially same as that of the conditional $q$-normal distribution given in: P.J. Szabowski, Electronic Journal of Probability {\bf 15}, 1296 (2010). This naturally gives asymmetry in $F_κ(E)$ with respect to $E$ as $E_κ$ increases and also the peak value changes with $E_κ$. Thus, the wavefunction structure in quantum many-fermion systems with $k$-body interactions follows in general the conditional $q$-normal distribution.
We demonstrate that a conditional wavefunction theory enables a unified and efficient treatment of the equilibrium structure and nonadiabatic dynamics of correlated electron-ion systems. The conditional decomposition of the many-body wavefunction formally recasts the full interacting wavefunction of a closed system as a set of lower dimensional (conditional) coupled `slices'. We formulate a variational wavefunction ansatz based on a set of conditional wavefunction slices, and demonstrate its accuracy by determining the structural and time-dependent response properties of the hydrogen molecule.…
In the same way as the quantum no-cloning theorem and quantum key distribution in two preceding papers, entanglement-assisted quantum teleportation and Grover's search algorithm are generalized by transferring them to an abstract setting, including usual quantum mechanics as a special case. This again shows that a much more general and abstract access to these quantum mechanical features is possible than commonly thought. A non-classical extension of conditional probability and, particularly, a very special type of state-independent conditional probability are used instead of Hilbert spaces an…
A well-known feature of quantum mechanics is the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over any classical communication channel. It is demonstrated that this quantum key distribution allows a much more general and abstract access than commonly thought. The results include some generalizations for the Hilbert space version of quantum key distribution,but base upon a general non-classical extension of conditional probability. A special state-independent conditional probability is identified as origin of the superior security of quant…
Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, $π$-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of co…
This paper introduces the FermiFab toolbox for many-particle quantum systems. It is mainly concerned with the representation of (symbolic) fermionic wavefunctions and the calculation of corresponding reduced density matrices (RDMs). The toolbox transparently handles the inherent antisymmetrization of wavefunctions and incorporates the creation/annihilation formalism. Thus, it aims at providing a solid base for a broad audience to use fermionic wavefunctions with the same ease as matrices in Matlab, say. Leveraging symbolic computation, the toolbox can greatly simply tedious pen-and-paper calcu…
The many-anyons wavefunction is constructed via the superposition of all the permutations on the direct product of single anyon states and its interchange properties are examined. The phase of permutation is not a representation but the word metric of the permutation group . Amazingly the interchange phase yields a finite capacity of one quantum state interpolating between Fermion and Boson and the mutual exchange phase has no explicit effect on statistics. Finite capacity of quantum state is defined as Gentile statistics and it is different from the fractional exclusion statistics. Some discu…
Massachusetts Institute of Technology（麻省理工学院）；The NSF AI Institute for Artificial Intelligence and Fundamental Interactions（美国国家科学基金会人工智能与基础交互研究所）；Harvard University（哈佛大学）；University of Michigan（密歇根大学）；University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We present a neural flow wavefunction, Gauge-Fermion FlowNet, and use it to simulate 2+1D lattice compact quantum electrodynamics with finite density dynamical fermions. The gauge field is represented by a neural network which parameterizes a discretized flow-based transformation of the amplitude while the fermionic sign structure is represented by a neural net backflow. This approach directly represents the $U(1)$ degree of freedom without any truncation, obeys Guass's law by construction, samples autoregressively avoiding any equilibration time, and variationally simulates Gauge-Fermion syst…
We present the Schmidt decomposition for arbitrary wavefunctions of two indistinguishable bosons, extending the recent studies of entanglement or quantum correlations for two fermion systems [J. Schliemann et al., Phys. Rev. B {\bf 63}, 085311 (2001) and quant-ph/0012094]. We point out that the von Neumann entropy of the reduced single particle density matrix remains to be a good entanglement measure for two identical particles.
High-dimensional entangled states of light provide novel possibilities for quantum information, from fundamental tests of quantum mechanics to enhanced computation and communication protocols. In this context, the frequency degree of freedom combines the assets of robustness to propagation and easy handling with standard telecommunication components. Here we use an integrated semiconductor chip to engineer the wavefunction and exchange statistics of frequency-entangled photon pairs directly at the generation stage, without post-manipulation. Tuning the spatial properties of the pump beam allow…
A prominent tool to study the dynamics of open quantum systems is the reduced density matrix. Yet, approaching open quantum systems by means of state vectors has well known computational advantages. In this respect, the physical meaning of the so-called conditional states in Markovian and non-Markovian scenarios has been a topic of recent debate in the construction of stochastic Schrödinger equations. We shed light on this discussion by acknowledging the Bohmian conditional wavefunction as the proper mathematical object to represent, in terms of state vectors, an arbitrary subset of degrees of…
A logarithmic Schrödinger equation with time-dependent coupling to the non-linearity is presented as a model of collisional decoherence of the wavefunction of a quantum particle in position-space. The particular mathematical form of the logarithmic Schrödinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schrödinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the non-linear equation of motion for the wavefunction numerically and compare it it to the solut…
In this letter we analyze the effect of the spin dimensionality of a physical system in two mathematical formulations of the uncertainty principle: a generalized Heisenberg uncertainty relation valid for all antisymmetric N-fermion wavefunctions, and the Fisher-information- based uncertainty relation valid for all antisymmetric N-fermion wavefunctions of central potentials. The accuracy of these spin-modified uncertainty relations is examined for all atoms from Hydrogen to Lawrencium in a self-consistent framework.
Using a separable many-body variational wavefunction, we formulate a self-consistent effective Hamiltonian theory for fermionic many-body system. The theory is applied to the two-dimensional Hubbard model as an example to demonstrate its capability and computational effectiveness. Most remarkably for the Hubbard model in 2-d, a highly unconventional quadruple-fermion non-Cooper-pair order parameter is discovered.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M>N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1(2m+1). Within this model the complimentary wavefunction for v = 1-1/(2m + 1) is found explicitly and extremely simple ground state wavefunctions for arbitra…
We investigate pair correlations in trapped fermion and boson gases as a means to probe the quantum states producing the density fluctuations. We point out that "opposite sign correlations" (meaning pair correlations that are positive for fermions and negative for bosons) unambiguously indicate that the quantum many-particle state cannot be "free." In particular, a system of fermions that exhibits positive pair correlations cannot be described by any Slater determinant wavefunction. This insight may help one to interpret results of current experiments on ultracold atomic gases.
Weak measurements of photon position can be used to obtain direct experimental evidence of the wavefunction of a photon between generation and ultimate detection. Significantly, these measurement results can also be understood as complex valued conditional probabilities of intermediate photon positions. It is therefore possible to interpret the quantum state as a complex valued probability distribution from which measurement probabilities can be derived according to Bayesian rules. The conventional measurement probabilities derived from squares of the wavefunction then describes the effects of…
We present two 2-body Hamiltonians that approximate the exact PH-Pfaffian wavefunction with their ground states for all the system sizes where this wavefunction has been numerically constructed to date. The approximate wavefunctions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimisation procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a non-interacting model. They belong to a finite regio…
Wavefunction structure is analyzed for dense interacting many-boson systems using Hamiltonian $H$, which is a sum of one-body $h(1)$ and an embedded GOE of $k$-body interaction $V(k)$ with strength $λ$. In the first analysis, a complete analytical description of the variance of the strength function as a function of $λ$ and $k$ is derived and the marker $λ_t$ defining thermalization region is obtained. In the strong coupling limit ($λ> λ_t$), the conditional $q$-normal density describes Gaussian to semi-circle transition in strength functions as body rank $k$ of the interaction increases. In t…
By introducing concepts of beam shaping into quantum mechanics, we show how interference effects of the quantum wavefunction describing multiple electrons can exactly balance the repulsion among the electrons. With proper shaping of the fermionic wavefunction, we propose non-diffracting quantum wavepackets of multiple electrons that can also carry orbital angular momentum, in the form of multi-electron non-diffracting vortex beams. The wavefunction is designed to compensate for both the repulsion between electrons and for the diffraction-broadening. This wavefunction shaping facilitates the us…
In this paper, we investigate a five-dimensional Dirac fermion on a quantum graph that consists of a single vertex and $N$ loops. We find that the model possesses a rich structure of boundary conditions for wavefunctions on the quantum graph and they can be classified into $(2N+1)$ distinct categories. It is then shown that there appear degenerate four-dimensional chiral massless fermions in the four-dimensional mass spectrum. We briefly discuss how our model could naturally solve the problems of the fermion generation, the fermion mass hierarchy and the origin of the $CP$-violating phase.
Wave-function microscopy: Derivation and anatomy of exact algebraic spinful wave functions and full Wigner-molecular spectra of a few highly correlated rapidly rotating ultracold fermionic atoms
We introduce a procedure to systematically search for a local unitary transformation that maps a wavefunction with a non-trivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one and two qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign …
Entanglement is typically created via systematic intervention in the time evolution of an initially unentangled state, which can be achieved by coherent control, carefully tailored non-demolition measurements, or dissipation in the presence of properly engineered reservoirs. In this paper we show that two-component Fermi gases at ~μK temperatures naturally evolve, in the presence of reactive two-body collisions, into states with highly entangled (Dicke-type) spin wavefunctions. The entanglement is a steady-state property that emerges---without any intervention---from uncorrelated initial state…
