Symmetric States Research Articles

Fractional interpolation inequality and radially symmetric ground states

In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$ and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$ . We also prove an existence result of standing waves and prove their orbital stability.

Constructing an entangled Unruh Otto engine and its efficiency

Uniformly accelerated frame mimics a thermal bath whose temperature is proportional to the proper acceleration. Using this phenomenon we give a detailed construction of an Otto cycle between two energy eigenstates of a system, consists of two entangled qubits. In the isochoric stages the thermal bath is being provided via the vacuum fluctuations of the background field for a monopole interaction by accelerating them. We find that making of Otto cycle is possible when one qubit is accelerating in the right Rindler wedge and other one is moving in the left Rindler wedge; i.e. in anti-parallel motion, with the initial composite state is a non-maximally entangled one. However, the efficiency greater than that of the usual single qubit quantum Otto engine is not possible. We provide values of the available parameters which make Otto cycle possible. On the other hand, Otto cycle is not possible if one considers the non-maximally entangled state for parallel motion. Moreover, for both initial symmetric and anti-symmetric Bell states we do not find any possibility of the cycle for qubits’ parallel and anti-parallel motion.

Multipartite Entanglement Detection Via Projective Tensor Norms

We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $$S_1$$ to the Euclidean space $$\ell _2$$ , which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and Gühne by deriving systematic relations between the performance of these two criteria.

Symmetric states and dynamics of three quantum bits

The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

A tight lower bound for non-coherent index erasure

The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight $\Omega(\sqrt{n})$ lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle}of Hoyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

Theory of the Rydberg blockade with multiple intermediate-state excitations

We present a detailed theory of the Rydberg blockade, including contributions from multiple intermediate-state excitations. Two fields drive transitions between ground and Rydberg levels via an off-resonance intermediate state. Assuming a perfect blockade, we calculate the probability to excite fully symmetric collective states having either zero or one Rydberg excitation, but an arbitrary number of intermediate-state excitations. Both ``bare'' state and ``dressed'' state approaches are used for (1) constant amplitude driving fields and (2) adiabatic pulse driving fields. It is shown that a dressed state approach offers distinct advantages when multiple intermediate-state excitations occur. In the case of fixed amplitude fields, the multiple intermediate excitations can result in comblike modulated populations of individual states having one Rydberg excitation and $n\ensuremath{\ll}N$ intermediate-state excitations. However, when summed over all such state populations, most of the modulation disappears and the system is described to a good approximation by an effective two-level model. In the case of adiabatic, pulsed fields, there is no such modulation and an effective two-level model (in the dressed basis), corrected for light shifts, can be used to model the system. In addition to solving this problem using conventional methods, we show that similar results could be obtained using a form of the Holstein-Primakoff transformation.

Spontaneous chiral symmetry breaking in a random driven chemical system

Living systems have evolved to efficiently consume available energy sources using an elaborate circuitry of chemical reactions which, puzzlingly, bear a strict restriction to asymmetric chiral configurations. While autocatalysis is known to promote such chiral symmetry breaking, whether a similar phenomenon may also be induced in a more general class of configurable chemical systems—via energy exploitation—is a sensible yet underappreciated possibility. This work examines this question within a model of randomly generated complex chemical networks. We show that chiral symmetry breaking may occur spontaneously and generically by harnessing energy sources from external environmental drives. Key to this transition are intrinsic fluctuations of achiral-to-chiral reactions and tight matching of system configurations to the environmental drives, which together amplify and sustain diverged enantiomer distributions. These asymmetric states emerge through steep energetic transitions from the corresponding symmetric states and sharply cluster as highly-dissipating states. The results thus demonstrate a generic mechanism in which energetic drives may give rise to homochirality in an otherwise totally symmetrical environment, and from an early-life perspective, might emerge as a competitive, energy-harvesting advantage.

Symmetric ground state solutions for the Choquard Logarithmic equation with exponential growth

We investigate the existence of ground state solutions for the fractional Choquard Logarithmic equation (−Δ)1/2u+V(x)u+(ln|⋅|∗|u|2)u=f(u),x∈R, where V∈C(R,[0,∞)) and the f satisfies exponential critical growth. The present paper extends and complements the result of Böer and Miyagaki (0000). In particular, our paper has two typical features. Firstly, using a weaker assumption on f, we establish the energy inequality to exclude the vanishing case of the required Cerami sequence. Secondly, with the property of radial symmetry we shall use some new variational and analytic technique to establish our final result which is different to the arguments explored in Böer and Miyagaki (0000).

Heavy-hole bilayer trions of transition metal dichalcogenides by analytical treatment to model He-isoelectronic ions upto dipole factor of Green’s function expansion of Coulomb interaction

Electronic structures of heavy-hole trions and He-isoelectronic ions show dependence on transitions among two-electron bound states constituted of hydrogenic orbitals for which Coulomb (exchange) interaction causes nontrivial secular divergence to Schrödinger equation (SE). Coulomb interaction between a pair of electrons triggers nonadiabatic transitions due conical intersections and larger vibrational amplitudes of nuclei. Therefore, existing state-of-art theories ubiquitously urge analytical integrals for multipoles of electrostatic Green’s function expansion. Born–Oppenheimer (BO) approximation separates the hamiltonian into electronic and nuclear coordinates at adiabatic limit. Employing associated Laguerre polynomial and Whittaker-M function basis sets of hydrogenic orbitals for the pair of electrons (electronic coordinate) furnishes analytical, terminable, simple and finitely summed integrals in terms of Lauricella functions. The exact integrals also remedy the difficulty arisen from different scaling factors of bound states for higher order perturbation calculations. Analytical perturbation calculations for monopole and dipole factors using singly and doubly excited states (|n2sn4s〉, |n2pn4p〉) with spherical and dumbbell symmetries show sufficient convergence of ground-state energy correction of He-isoelectronic series within 0.20–6.89% of reported results. Calculation of current-density with ground-state upto first order perturbation correction clearly shows that singly and doubly excited spherically symmetric states do not contribute to local current of He-isoelectronic series for monopole factor. Binding energies of heavy-hole trions of transition metal dichalcogenides (TMDCs) are found to be in good agreement with experimentally observed values.

Spatial transfer of entangled states in the correlated quantum dots system

We propose a protocol for spatial ‘stretching’ of an initially prepared entangled state along chains of quantum dots (QDs) or quantum wires arranged in perpendicular directions. We theoretically analyze such stretching for a triplet superposition of symmetric and antisymmetric single-electron states with opposite spins formed in a diamond-like arrangement of four QDs. Under specific choice of parameters of the system, the probability amplitude to find the system in an entangled state localized at the end dots of the chains can reach unity at particular instants of time.

Domains of the adsorbed benzene monolayer on graphene

We consider the adsorbed benzene monolayer on graphene and show that the monolayer’s structure can be very complicated even for benzene, the simplest representative of the cyclic hydrocarbons. We base this on previous the first principal investigations, which found different energy states of the adsorbed benzene molecule. We find the existence of the six domains (star domains) for the stable molecular monolayer. The representatives of these domains (grains) at the graphene hexagonal layer are star vectors that describe transitions from symmetric high energy state into the stable low symmetric position. In the hexagon lattice of any benzene close pack domain, we introduce the primitive unit vectors, which are twice longer than the graphene primitive unit ones. The symmetry elements of the adsorbed benzene monolayer on graphene are found, left and right domains are introduced. Besides, half-translation of the benzene domains on the graphene primitive unit vectors leads to a further increase in the variety of the domain types. We introduce the selection procedure to decrease the number of the possible domains, find zero-block cells, and the complete number of the benzene domains — eight. We find all types of the structural domains and show their relation with the graphene unit cell construction.

State transformations within entanglement classes containing permutation-symmetric states

The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit-states cannot be obtained from nor transformed to any state, which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that this situation persists for generic n-qubit symmetric states, we consider particular families for which we can determine that on the contrary they are endorsed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. The only classes found for which every state can be converted to a more weakly entangled state are the n-qubit GHZ and W classes. In the course of the study of LOCC transformations we also characterize the local symmetries of symmetric states.

Measure of Tripartite Quantum Correlation in Multiqubit Symmetric Pure State

We propose a direct measure of tripartite quantum correlation in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third-order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third-order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only the tripartite quantum correlations among the N atoms and propose a measure and quantification of these correlations. We also propose the necessary and sufficient condition for the presence of tripartite quantum correlation in such multi-atomic systems. We conjecture the way of determining the third-order moments of the pseudo-spin operators of the two-level atoms in the assembly, experimentally.

Cryo-EM structure of transmembrane AAA+ protease FtsH in the ADP state

AAA+ proteases regulate numerous physiological and cellular processes through tightly regulated proteolytic cleavage of protein substrates driven by ATP hydrolysis. FtsH is the only known family of membrane-anchored AAA+ proteases essential for membrane protein quality control. Although a spiral staircase rotation mechanism for substrate translocation across the FtsH pore has been proposed, the detailed conformational changes among various states have not been clear due to absence of FtsH structures in these states. We report here the cryo-EM structure for Thermotoga maritima FtsH (TmFtsH) in a fully ADP-bound symmetric state. Comparisons of the ADP-state structure with its apo-state and a substrate-engaged yeast YME1 structure show conformational changes in the ATPase domains, rather than the protease domains. A reconstruction of the full-length TmFtsH provides structural insights for the dynamic transmembrane and the periplasmic domains. Our structural analyses expand the understanding of conformational switches between different nucleotide states in ATP hydrolysis by FtsH.

Local sum uncertainty relations for angular momentum operators of bipartite permutation symmetric systems

We show that violation of the variance based local sum uncertainty relation (LSUR) for angular momentum operators of a bipartite system, proposed by Hofmann and Takeuchi [Phys. Rev. A 68 032103 (2003)], reflects entanglement in the equal bipartitions of an N-qubit symmetric state with even qubits. We establish the one-to-one connection with the violation of LSUR with negativity of covariance matrix [Phys. Lett. A 364 203 (2007)] of the two-qubit reduced system of a permutation symmetric N-qubit state.

Iron(III) Complexes of a Hexadentate Thioether-Appended 2-Aminophenol Ligand: Redox-Driven Spin State Switchover.

A green complex [Fe(L3)] (1), supported by the deprotonated form of a hexadentate noninnocent redox-active thioether-appended 2-aminophenolate ligand (H4L3 = N,N'-bis(2-hydroxy-3,5-di-tert-butylphenyl)-2,2'-diamino(diphenyldithio)ethane), has been synthesized and structurally characterized at 100(2) K and 298(2) K. In CH2Cl2, 1 displays two oxidative and a reductive one-electron redox processes at E1/2 values of -0.52 and 0.20 V, and -0.85 V versus the Fc+/Fc redox couple, respectively. The one-electron oxidized 1+ and one-electron reduced 1- forms, isolated as a blackish-blue solid 1(PF6)·CH2Cl2 (2) and a gray solid [Co(η5-C5H5)2]1·DMF (3), have been structurally characterized at 100(2) K. Structural parameters at 100 K of the ligand backbone and metrical oxidation state values unambiguously establish the electronic states as [FeIII{(LAPO,N)2-}{(LISQO,N)•-}{(LS,S)0}] (1) (two tridentate halves are electronically asymmetric-ligand mixed-valency), [FeIII{(LISQO,N)•-}{(LISQO,N)•-}{(LS,S)0}]+ (1+), and [FeIII{(LAPO,N)2-}{(LAPO,N)2-}{(LS,S)0}]- (1-) [dianionic 2-amidophenolate(2-) (LAPO,N)2- and monoanionic 2-iminobenzosemiquinonate(1-) π-radical (Srad = 1/2) (LISQ)•- redox level]. Mössbauer spectral data of 1 at 295, 200, and 80 K reveal that it has a major low-spin (ls)-Fe(III) and a minor ls-Fe(II) component (redox isomers), and at 7 K, the major component exists exclusively. Thus, in 1, the occurrence of a thermally driven valence-tautomeric (VT) equilibrium (asymmetric) [FeIII{(LAPO,N)2-}{(LISQO,N)•-}{(LS,S)0}] ⇌ (symmetric) [FeII{(LISQO,N)•-}{(LISQO,N)•-}{(LS,S)0}] (80-295 K) is implicated. Mössbauer spectral parameters unequivocally establish that 1+ is a ls-Fe(III) complex. In contrast, the monoanion 1- contains a high-spin (hs)-Fe(III) center (SFe = 5/2), as is deduced from its Mössbauer and EPR spectra. Complexes 1-3 possess total spin ground states St = 0, 1/2, and 5/2, respectively, based on 1H NMR and EPR spectra, the variable-temperature (2-300 K) magnetic behavior of 2, and the μeff value of 3 at 300 K. Broken-symmetry density functional theory (DFT) calculations at the B3LYP-level of theory reveal that the unpaired electron in 1+/2 is due to the (LISQ)•- redox level [ls-Fe(III) (SFe = 1/2) is strongly antiferromagnetically coupled to one of the (LISQ)•- radicals (Srad = 1/2)], and 1-/3 is a hs-Fe(III) complex, supported by (L3)4- with two-halves in the (LAP)2- redox level. Complex 1 can have either a symmetric or asymmetric electronic state. As per DFT calculation, the former state is stabilized by -3.9 kcal/mol over the latter (DFT usually stabilizes electronically symmetric structure). Time-dependent (TD)-DFT calculations shed light on the origin of observed UV-vis-NIR spectral absorptions for 1-3 and corroborate the results of spectroelectrochemical experiments (300-1100 nm) on 1 (CH2Cl2; 298 K). Variable-temperature (218-298 K; CH2Cl2) absorption spectral (400-1000 nm) studies on 1 justify the presence of VT equilibrium in the solution-state.

Emergent Dark States from Superradiant Dynamics in Multilevel Atoms in a Cavity

We investigate the collective decay dynamics of atoms with a generic multilevel structure (angular momenta $F\leftrightarrow F'$) coupled to two light modes of different polarization inside a cavity. In contrast to two-level atoms, we find that multilevel atoms can harbour eigenstates that are perfectly dark to cavity decay even within the subspace of permutationally symmetric states (collective Dicke manifold). The dark states arise from destructive interference between different internal transitions and are shown to be entangled. Remarkably, the superradiant decay of multilevel atoms can end up stuck in one of these dark states, where a macroscopic fraction of the atoms remains excited. This opens the door to the preparation of entangled dark states of matter through collective dissipation useful for quantum sensing and quantum simulation. Our predictions should be readily observable in current optical cavity experiments with alkaline-earth atoms or Raman-dressed transitions.

Optimum performance-based design of unsymmetrical 2D steel moment frame

The most commonly used analysis method in performance-based design (PBD) is nonlinear static analysis (NSA). In unsymmetrical 2D frames, unlike for the symmetrical state, NSA should be performed in two lateral loading directions. This complicates the process of achieving a feasible optimal design in addition to increasing the volume of calculations. In this study, to overcoming this issue, a two-step approach is proposed for the optimal PBD of unsymmetrical 2D steel moment-resisting frames (SMRFs). In this approach, in two independent steps, the structure is analyzed with a lateral loading pattern based on the first mode shape in the positive and negative directions. The implementation of the second step is conditional on the satisfactory completion of the first step. The objective function takes into account the differences between successful and unsuccessful steps. The constraints considered are based on the acceptance criteria for SMRF according to FEMA-356 at each performance level. The effectiveness of the proposed approach has been investigated by employing four meta-heuristic optimization algorithms to determine the optimum design for case studies of SMRF structures having three and nine stories.

Tcc+ and its partners

Inspired by the ${T}_{cc}^{+}$ signal discovered by the LHCb Collaboration, we systematically investigate the doubly heavy tetraquark states with the molecule configuration $[{Q}_{1}{\overline{q}}_{2}{]}_{{\mathbf{1}}_{c}}[{Q}_{3}{\overline{q}}_{4}{]}_{{\mathbf{1}}_{c}}$ ($Q=c$ and $b$, $q=u$, $d$, and $s$) in a nonrelativistic quark model. The model involves a color screening confinement potential, meson-exchange interactions, and one-gluon-exchange interactions. The state ${T}_{cc}^{+}$ with $I{J}^{P}=0{1}^{+}$ is a very loosely bound deuteronlike state with a binding energy around 0.34 MeV and a huge size of 4.32 fm. Both the meson exchange force and the coupled channel effect play a pivotal role. Without the meson exchange force, there does not exist the ${T}_{cc}^{+}$ molecular state. In strong contrast, the QCD valence bond forms clearly in the ${T}_{bb}^{\ensuremath{-}}$ system when we turn off the meson-exchange force, which is very similar to the hydrogen molecule in QED. Moreover, the ${T}_{bb}^{\ensuremath{-}}$ becomes a heliumlike QCD-atom if we increase the bottom quark mass by a factor of three. Especially, the states ${T}_{bb}^{\ensuremath{-}}$ with ${01}^{+}$, ${T}_{bc}^{0}$ with ${00}^{+}$ and ${01}^{+}$, and the $V$-spin antisymmetric states ${T}_{bbs}^{\ensuremath{-}}$ with $\frac{1}{2}{1}^{+}$, ${T}_{bcs}^{0}$ with $\frac{1}{2}{0}^{+}$ and $\frac{1}{2}{1}^{+}$ can form a compact, hydrogen moleculelike or deuteronlike bound state with different binding dynamics. The high-spin states ${T}_{bc}^{0}$ with ${02}^{+}$ and ${T}_{bcs}^{0}$ with $\frac{1}{2}{2}^{+}$ can decay into $D$-wave $\overline{B}D$ and ${\overline{B}}_{s}D$ although they are below the thresholds ${\overline{B}}^{*}{D}^{*}$ and ${\overline{B}}_{s}^{*}{D}^{*}$, respectively. The isospin and $V$-spin symmetric states are unbound. We also calculate their magnetic moments and axial charges.

Symmetric and asymmetric receptor conformation continuum induced by a new insulin.

Cone snail venoms contain a wide variety of bioactive peptides, including insulin-like molecules with distinct structural features, binding modes, and biochemical properties. Here, we report a fully active humanized cone snail venom insulin with an elongated A chain and a truncated B chain, and use cryo-electron microscopy and protein engineering to elucidate its interactions with the human insulin receptor ectodomain. We reveal how an extended A chain can compensate for deletion of B-chain residues, which are essential for activity of human insulin but also compromise therapeutic utility by delaying dissolution from the site of subcutaneous injection. This finding suggests approaches to developing improved therapeutic insulins. Curiously, the receptor displays a continuum of conformations from the symmetric state to a highly asymmetric low-abundance structure that displays novel coordination of a single humanized venom insulin using elements from both of the previously characterized site-1 and site-2 interactions.