Quantum correlations in the dimerized spin chain at zero and finite temperatures

By using the method of exact diagonalization, we investigate the quantum correlation measured by quantum discord of the dimerized spin chain at both zero and finite temperatures. The results disclose that the quantum discord is robust at any finite parameter α and temperature T, in contrast to entanglement which shows a sudden death when the parameter α or the temperature T reaches a critical point. At finite temperature, it is interesting to find that the quantum discord QD _{2i−1, 2i} can increase with temperature T no matter if the entanglement EoF _{2i−1, 2i} exists or not. The research on the relation between the quantum discord and the quantum phase transition in the dimerized spin chain indicates that the transition can be characterized by the first derivation of the quantum discord at zero and low temperatures.     

Quantum secret sharing based on quantum error-correcting codes

Quantum secret sharing(QSS) is a procedure of sharing classical information or quantum information by using quantum states.This paper presents how to use a [2k-1,1,k] quantum error-correcting code(QECC) to implement a quantum(k,2k 1) threshold scheme.It also takes advantage of classical enhancement of the [2k-1,1,k] QECC to establish a QSS scheme which can share classical information and quantum information simultaneously.Because information is encoded into QECC,these schemes can prevent intercept-resend attacks and be implemented on some noisy channels.     

A Scheme for Physical Implementation of a Ququadrit Quantum Computation with Cooled-Trapped Ions

The physics realization of a ququadrit quantum computation with cooled trapped ¹³⁸Ba⁺ ions in a Paul trap is investigated. The ground state level 6² S_1/2(m = −1/2) and three metastable levels: 5² D_3/2(m = −1/2), 5² D_5/2(m = −1/2), and 5² D_5/2(m = 1/2), of the fine-structure of the ¹³⁸Ba⁺ ion, are used to store the quantum information of ququadrits. The use of coherent manipulation of populations in single ququadrit, being a four-dimensional Hilbert space, produces a discrete Fourier transform and the manipulation of the first red band transitions with the introduction of an ancillary quantum channel between two ququadrits generates a conditional phase gate. The combination of the both above results in a universal two-ququadrit gate, called XOR⁽⁴⁾ gate corresponding to the controlled-NOT gate operation in qubit systems. The implementation of quantum Fourier transform for n ququadrits is performed by means of the conditional phase-shift gate. The feasibility of physical realization of ququadrit quantum computation with cooled-trapped ¹³⁸Ba⁺ ions is detailed analyzed and described, and the theoretical detection method of logical states is given. Higher entanglement between ququadrits than qutrits or qubits and more security of ququadrit quantum cryptography than qutrit's or qutrit's will lead to more extensive applications ququadrits in quantum information fields. In particular, it is pointed out that this scheme should be the highest dimensional quantum computation in cooled-trapped ions, the entanglement between ququadrits should be the highest dimensional entanglement in it, and the ququadrit quantum cryptography should be the most secure cryptography protocol in it.     

Quantum tricriticality in transverse Ising-like systems

The quantum tricriticality of d-dimensional transverse Ising-like systems is studied by means of a perturbative renormalization group approach focusing on static susceptibility. This allows us to obtain the phase diagram for 3 ≤ d < 4, with a clear location of the critical lines ending in the conventional quantum critical points and in the quantum tricritical one, and of the tricritical line for temperature T ≥ 0. We determine also the critical and the tricritical shift exponents close to the corresponding ground state instabilities. Remarkably, we find a tricritical shift exponent identical to that found in the conventional quantum criticality and, by approaching the quantum tricritical point increasing the non-thermal control parameter r, a crossover of the quantum critical shift exponents from the conventional value φ = 1/(d − 1) to the new one φ = 1/2(d − 1). Besides, the projection in the (r,T)-plane of the phase boundary ending in the quantum tricritical point and crossovers in the quantum tricritical region appear quite similar to those found close to an usual quantum critical point. Another feature of experimental interest is that the amplitude of the Wilsonian classical critical region around this peculiar critical line is sensibly smaller than that expected in the quantum critical scenario. This suggests that the quantum tricriticality is essentially governed by mean-field critical exponents, renormalized by the shift exponent φ = 1/2(d − 1) in the quantum tricritical region.     

Quantum mechanical version of classical Liouville theorem

In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of coherent state from |z〉to |sz-rz^*〉angle corresponds to the motion from a point z(q,p) to another point sz-rz^* with |s|²-|r|²=1. The evolution is governed by the so-called Fresnel operator U(s,r) recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation and obeys the group product rules. In another word, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.     

Quantum sets and clifford algebras

The mathematical language presently used for quantum physics is a high-level language. As a lowest-level or basic language I construct a quantum set theory in three stages: (1) Classical set theory, formulated as a Clifford algebra of “S numbers” generated by a single monadic operation, “bracing,” Br = {…}. (2) Indefinite set theory, a modification of set theory dealing with the modal logical concept of possibility. (3) Quantum set theory. The quantum set is constructed from the null set by the familiar quantum techniques of tensor product and antisymmetrization. There are both a Clifford and a Grassmann algebra with sets as basis elements. Rank and cardinality operators are analogous to Schroedinger coordinates of the theory, in that they are multiplication or “Q-type” operators. “P-type” operators analogous to Schroedinger momenta, in that they transform theQ-type quantities, are bracing (Br), Clifford multiplication by a setX, and the creator ofX, represented by Grassmann multiplicationc(X) by the setX. Br and its adjoint Br* form a Bose-Einstein canonical pair, andc(X) and its adjointc(X)* form a Fermi-Dirac or anticanonical pair. Many coefficient number systems can be employed in this quantization. I use the integers for a discrete quantum theory, with the usual complex quantum theory as limit. Quantum set theory may be applied to a quantum time space and a quantum automaton. This material is based upon work supported in part by NSF Grant No. PHY8007921.     

Polarization correlation in the two-photon decay of atomic hydrogen: nonlocality versus entanglement

From the work by Perrie et al. [Phys. Rev. Lett. 54, 1790 (1985)], photon pairs from the 2s _1/2 → 1s _1/2 (two-photon) decay of atomic hydrogen are known to be quantum mechanically correlated. In these experiments, the polarization states of the photons emitted in back-to-back geometry were shown to violate the Bell inequality as a qualitative sign of nonlocality and entanglement. In the present contribution, we analyze how these nonlocal quantum correlations, as given by the violation of the Bell inequality, differ from the concurrence as a true entanglement measure. Results are shown for both quantifiers in dependence of the decay geometry and the initial polarization of the atoms for the 2s _1/2 → 1s _1/2 and 3d _5/2 → 1s _1/2 two-photon decay of atomic hydrogen. These results display the difference between nonlocality and entanglement and, hence, may stimulate further experiments on nonlocal quantum correlations in atomic systems.     

A Meinardus Theorem with Multiple Singularities

We solve the quantum version of the A ₁ T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A ₁ Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.     

Quantum Computational Logics and Possible Applications

In quantum computational logics meanings of formulas are identified with quantum information quantities: systems of qubits or, more generally, mixtures of systems of qubits. We consider two kinds of quantum computational semantics: (1) a compositional semantics, where the meaning of a compound formula is determined by the meanings of its parts; (2) a holistic semantics, which makes essential use of the characteristic “holistic” features of the quantum-theoretic formalism. The compositional and the holistic semantics turn out to characterize the same logic. In this framework, one can introduce the notion of quantum-classical truth table, which corresponds to the most natural way for a quantum computer to calculate classical tautologies. Quantum computational logics can be applied to investigate different kinds of semantic phenomena where holistic, contextual and gestaltic patterns play an essential role (from natural languages to musical compositions).     

On the Spectroscopic Structure of Two Interacting Electrons in a Quantum Dot

The shifted 1/N expansion technique used by El-Said [Phys. Rev. B 61 (2000) 13026], to study the relative Hamiltonian of two interacting electrons confined in a quantum dot, is investigated. El-Said's results from shifted large-N (or 1/N) expansion technique are revised and results from an alternative method are also reported. The distinctive role of the central spike term, (m ²-1/4)/q ², in determining spectral properties of the above problem is shown, moreover.     

Systematics of perturbative semiclassical quantum defect expansions probed by RKR-QDTand a Fisher-information-based criterion

The systematics of perturbative semiclassical quantum defect expansions corresponding to a hydrogenic potential plus a perturbing term of the form -A/2r^κ, k\geqslant 2\kappa \geqslant 2, are studied as a function of expansion order N. Towards this task the expansions μ _Nare first used as input for constructing associated N-dependent atomic RKR-QDT potential curves. Subsequently the coordinate Fisher information for the energy levels supported by those curves as well as its rate ε with respect to N is semiclassically computed. Then, the plot of relative quantum defect error between successive orders, δμ _N+1,N, with respect to ε serves as convergence indicator for both approximate potentials and quantum defects. For a given κ and when the quantum defect expansion proves to be of limited accuracy the plot reveals an A- and N-dependent scatter of points and “saturation” (the relative error remains almost constant with respect to ε). More importantly, when ε is equal to or lower than the value of ε (N=1) for which πμ ₁\leqslant 1/2_{1}\leqslant 1/2 the relative error exhibits a κ-, A- and N-independent power-law dependence, δμ _N+1,N∝ ε ^m, clearly distinguishing the N=1 order (m=1/2) from all other N>1 orders (m=1). These power-laws may be employed for setting-up confidence level bounds on perturbative expansions.     

Quantum Associative Neural Network with Nonlinear Search Algorithm

Based on analysis on properties of quantum linear superposition, to overcome the complexity of existing quantum associative memory which was proposed by Ventura, a new storage method for multiply patterns is proposed in this paper by constructing the quantum array with the binary decision diagrams. Also, the adoption of the nonlinear search algorithm increases the pattern recalling speed of this model which has multiply patterns to O( log₂^{2n -t} ) = O( n - t )O( {\log_{2}}^{2^{n -t}} ) = O( n - t ) time complexity, where n is the number of quantum bit and t is the quantum information of the t quantum bit. Results of case analysis show that the associative neural network model proposed in this paper based on quantum learning is much better and optimized than other researchers’ counterparts both in terms of avoiding the additional qubits or extraordinary initial operators, storing pattern and improving the recalling speed.     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

The quantum poisson-Lie T-duality and mirror symmetry

Poisson-Lie T-duality in quantum N=2 superconformal Wess-Zumino-Novikov-Witten models is considered. The Poisson-Lie T-duality transformation rules of the super-Kac-Moody algebra currents are found from the conjecture that, as in the classical case, the quantum Poisson-Lie T-duality transformation is given by an automorphism which interchanges the isotropic subalgebras of the underlying Manin triple in one of the chirality sectors of the model. It is shown that quantum Poisson-Lie T-duality acts on the N=2 super-Virasoro algebra generators of the quantum models as a mirror symmetry acts: in one of the chirality sectors it is a trivial transformation while in another chirality sector it changes the sign of the U(1) current and interchanges the spin-3/2 currents. A generalization of Poisson-Lie T-duality for the quantum Kazama-Suzuki models is proposed. It is shown that quantum Poisson-Lie T-duality acts in these models as a mirror symmetry also. Zh. éksp. Teor. Fiz. 116, 11–25 (July 1999) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Non-commutative Low-dimension Spaces and Superspaces Associated with Contracted Quantum Groups and Supergroups

Quantum planes, which correspond to all one-parameter solutions of Quantum Yang-Baxter Equation (QYBE) for the two-dimensional case of GL-groups, are summarized and their geometrical interpretations are given. It is shown that the quantum dual plane is associated with an exotic solution of QYBE and the well-known quantum h-plane may be regarded as the quantum analog of the flag (or fiber) plane. Contractions of the quantum supergroup G L _q (12) and corresponding quantum superspace C _q (12) are considered in Cartesian basis. The contracted quantum superspace C _h (12);) is interpreted as the non-commutative analog of the superspace with the fiber odd part.     

Lower Bound of Minimal Time Evolution in Quantum Mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S ², must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2×2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector P of the evolving quantum state with the vector of the 2×2 hermitian Hamiltonians and it is shown that the Hamiltonian H can be parameterized by two independent parameters and Θ.     

Low frequency asymptotics for the spin-weighted spheroidal equation in the case of s=1/2

The spin-weighted spheroidal equation in the case of s=1/2 is thoroughly studied by using the perturbation method from the supersymmetric quantum mechanics.The first-five terms of the superpotential in the series of parameter β are given.The general form for the n-th term of the superpotential is also obtained,which could also be derived from the previous terms W k,k < n.From these results,it is easy to obtain the ground eigenfunction of the equation.Furthermore,the shape-invariance property in the series of parameter β is investigated and is proven to be kept.This nice property guarantees that the excited eigenfunctions in the series form can be obtained from the ground eigenfunction by using the method from the supersymmetric quantum mechanics.We show the perturbation method in supersymmetric quantum mechanics could completely solve the spin-weight spheroidal wave equations in the series form of the small parameter β.     

Transport through artificial single-molecule magnets: Spin-pair state sequential tunneling and Kondo effects

The transport properties of an artificial single-molecule magnet based on a CdTe quantum dot doped with a single Mn+2 ion(S=5/2) are investigated by the non-equilibrium Green function method.We consider a minimal model where the Mn-hole exchange coupling is strongly anisotropic so that spin-flip is suppressed and the impurity spin S and a hole spin s entering the quantum dot are coupled into spin pair states with(2S+1) sublevels.In the sequential tunneling regime,the differential conductance exhibits(2S+1) possible peaks,corresponding to resonance tunneling via(2S+1) sublevels.At low temperature,Kondo physics dominates transport and(2S+1) Kondo peaks occur in the local density of states and conductance.These peaks originate from the spin-singlet state formed by the holes in the leads and on the dot via higher-order processes and are related to the parallel and antiparallel spin pair states.