Procedures are given below to construct symmetric and anti-symmetric quantum functions. If hidden in an oracle, such functions can be identified exactly, without iterative interrogation. This is another example of quantum search. The resulting positive (or negative) functions also serve to uniquely reorganize a superposition of states to give a basis state for testing purposes.
When applied to matter and antimatter states, the Pauli master equation (PME) may have two forms: time-symmetric, which is conventional, and time-antisymmetric, which is suggested in the present work. The symmetric and antisymmetric forms correspond to symmetric and antisymmetric extensions of thermodynamics from matter to antimatter --- this is demonstrated by proving the corresponding H-theorem. The two forms are based on the thermodynamic similarity of matter and antimatter and differ only in the directions of thermodynamic time for matter and antimatter (the same in the time-symmetric case…
We consider online strategies for discriminating between symmetric pure states with zero error when $n$ copies of the states are provided. Optimized online strategies involve local, possibly adaptive measurements on each copy and are optimal at each step, which makes them robust in front of particle losses or an abrupt termination of the discrimination process. We first review previous results on binary minimum and zero error discrimination with local measurements that achieve the maximum success probability set by optimizing over global measurements, highlighting their online features. We the…
We investigate a two-level spin system based anti-parity-time (anti-$\mathcal{PT}$)-symmetric qubit and study its decoherence as well as entanglement entropy properties. We compare our findings with that of the corresponding $\mathcal{PT}$-symmetric and Hermitian qubits. First we consider the time-dependent Dyson map to find the exact analytical dynamics for a general non-Hermitian qubit system coupled with a bath, then we specialize it to the case of the anti-$\mathcal{PT}$-symmetric qubit. We find that the decoherence function for the anti-$\mathcal{PT}$-symmetric qubit decays slower than th…
We investigate the properties of two anti-parity-time (anti-PT )-symmetric four-site scattering centers. The anti-PT -symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-PT symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-PT -symmetric scattering center possesses nonunitary scattering and the differenc…
Non-Hermitian parity-time ($\mathcal{PT}$) and anti-parity-time ($\mathcal{APT}$)-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in $\mathcal{PT}$- and $\mathcal{APT}$-symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in $\mathcal{PT}$- and $\mathcal{APT}$-symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe…
Two-photon Hong-Ou-Mandel (HOM) interference is a fundamental quantum effect with no classical counterpart. The exiting researches on two-photon interference were mainly limited in one degree of freedom (DoF), hence it is still a challenge to realize the quantum interference in multiple DoFs. Here we demonstrate the HOM interference between two hyper-entangled photons in two DoFs of polarization and orbital angular momentum (OAM) for all the sixteen hyper-entangled Bell states. We observe hyper-entangled two-photon interference with bunching effect for ten symmetric states (nine Boson-Boson st…
Symmetric logarithmic derivative (SLD) is a key quantity to obtain quantum Fisher information (QFI) and to construct the corresponding optimal measurements. Here we develop a method to calculate the SLD and QFI via anti-commutators. This method is originated from the Lyapunov representation and would be very useful for cases that the anti-commutators among the state and its partial derivative exhibits periodic properties. As an application, we discuss a class of states, whose squares linearly depend on the states themselves, and give the corresponding analytical expressions of SLD and QFI. A n…
Permutation-invariant, -equivariant, and -covariant functions and anti-symmetric functions are important in quantum physics, computer vision, and other disciplines. Applications often require most or all of the following properties: (a) a large class of such functions can be approximated, e.g. all continuous function, (b) only the (anti)symmetric functions can be represented, (c) a fast algorithm for computing the approximation, (d) the representation itself is continuous or differentiable, (e) the architecture is suitable for learning the function from data. (Anti)symmetric neural networks ha…
We investigate the scattering properties of an anti-parity-symmetric non-Hermitian system. The anti-parity-symmetric scattering center possesses imaginary nearest-neighbor hoppings and real onsite potentials, it has been experimentally realized through dissipative coupling and frequency detuning between atomic spin waves. We find that such anti-parity-symmetric system displays three salient features: Firstly, the reflection and transmission are both reciprocal. Secondly, the reflection and transmission probabilities satisfy R+T=1 or R-T=1, which depends on the parity of the scattering center s…
We propose an anti-parity-time (anti-PT ) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model, where the large non-Hermiticity constructively creates nontrivial topology and greatly expands the topological phase. In the anti-PT -symmetric SSH model, the gain and loss are alternatively arranged in pairs under the inversion symmetry. The appearance of degenerate point at the center of the Brillouin zone determines the topological phase transition, while the exceptional points unaffect the band topology. The large non-Hermiticity leads to unbalanced wavefunction distribution in the broken an…
The efficient sensing of weak environmental perturbations via special degeneracies called exceptional points in non-Hermitian systems has gained enormous traction in the last few decades. However, in contrast to the extensive literature on parity-time (PT) symmetric systems, the exotic hallmarks of anti-PT symmetric systems are only beginning to be realized now. Very recently, a characteristic resonance of vanishing linewidth in anti-PT symmetric systems was shown to exhibit tremendous sensitivity to intrinsic nonlinearities. Given the primacy of sensing in non-Hermitian systems, in general, a…
We reveal how the vector field links are untied under the influence of anti-parity-time-symmetric couplings in a dissipative sublattice-symmetric topological photonic crystal lattice. The topology of the quasi-one-dimensional two-band system is encoded in the geometric topology of the vector fields associated with the Bloch Hamiltonian. The linked vector fields reflect the topology of the nontrivial phase. The topological phase transition occurs concomitantly with the untying of the vector field link at the exceptional points. Counterintuitively, more dissipation constructively creates a nontr…
A numerical method to build an orthonormal basis of properly symmetrized hyperspherical harmonic functions is developed. As a part of it, refined algorithms for calculating the transformation coefficients between hyperspherical harmonics constructed from different sets of Jacobi vectors are derived and discussed. Moreover, an algorithm to directly determine the numbers of independent symmetric hyperspherical states (in case of bosonic systems) and antisymmetric hyperspherical-spinisospin states (in case of fermionic systems) entering the expansion of the A-body wave functions is presented. Num…
Non-Hermitian systems with anti-parity-time ($\mathcal{APT}$) symmetry have revealed rich physics beyond conventional systems. Here, we study optomechanics in an $\mathcal{APT}$-symmetric spinning resonator and show that, by tuning the rotating speed to approach the exceptional point (EP) or the non-Hermitian spectral degeneracy, nonreciprocal light transmission with a high isolation ratio can be realized. Accompanying this process, nonreciprocal group delay or advance is also identified in the vicinity of EP. Our work sheds new light on manipulating laser propagation with optomechanical EP de…
We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exp…
By engineering an anti-parity-time (anti-PT) symmetric cavity magnonics system with precise eigenspace controllability, we observe two different singularities in the same system. One type of singularity, the exceptional point (EP), is produced by tuning the magnon damping. Between two EPs, the maximal coherent superposition of photon and magnon states is robustly sustained by the preserved anti-PT symmetry. The other type of singularity, arising from the dissipative coupling of two anti-resonances, is an unconventional bound state in the continuum (BIC). At the settings of BICs, the coupled sy…
University of California, Berkeley（加州大学伯克利分校）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
A fundamental problem in quantum physics is to encode functions that are completely anti-symmetric under permutations of identical particles. The Barron space consists of high-dimensional functions that can be parameterized by infinite neural networks with one hidden layer. By explicitly encoding the anti-symmetric structure, we prove that the anti-symmetric functions which belong to the Barron space can be efficiently approximated with sums of determinants. This yields a factorial improvement in complexity compared to the standard representation in the Barron space and provides a theoretical …
