Unitary Generators Research Articles

Resolvent Methods for Quantum Walks with an Application to a Thue–Morse Quantum Walk

In this expository note, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary generator into dynamical estimates for the corresponding quantum walk. To illustrate the general methods, we show how to apply them to a 1D coined quantum walk whose coins are distributed according to an element of the Thue--Morse subshift.

Noiseless Linear Amplification with General Local Unitary Operations**Supported by the National Natural Science Foundation of China under Grant Nos 11304013, 11204197, 11204379 and 11074244, the National Basic Research Program of China under Grant No 2011CBA00200, the Doctor Science Research Foundation of Ministry of Education of China under Grant No 20113402110059, and Civil Aerospace 2013669.

Noiseless linear amplification (NLA), first proposed by Ralpha et al., is a nondeterministic amplification process which gives gain to the Fock state |n〉 → gn|n〉, with g being the amplification gain. We here give a general framework for improving the NLA scheme with arbitrary general local unitary operations. We derive the improvement in the amplification gain in 0–1 photon subspace. In particular, we study if the local unitary is composed of single mode squeezing and coherent displacement operation. Finally, numerical simulations show that local unitary operation could give a further enhancement in the amplification gain as well as the success probability, making the NLA more feasible in future optic quantum communications.

Universal scaling of the logarithmic negativity in massive quantum field theory

We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r. We show that the former saturates to a finite value, and that the latter tends to zero, as . We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large-r analysis of the negativity allows for the detection of bound states.

Quantum Enhanced Estimation of a Multidimensional Field.

We present a framework for the quantum enhanced estimation of multiple parameters corresponding to noncommuting unitary generators. Our formalism provides a recipe for the simultaneous estimation of all three components of a magnetic field. We propose a probe state that surpasses the precision of estimating the three components individually, and we discuss measurements that come close to attaining the quantum limit. Our study also reveals that too much quantum entanglement may be detrimental to attaining the Heisenberg scaling in the estimation of unitarily generated parameters.

Quantum Eberlein compactifications and invariant means

We propose a definition of a � algebra, which is a weak form of a C � -bialgebra with a sort of unitary generator. Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stieltjes algebra B(G), has character space G E , which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space which contain a dense homomorphic image of G. We carry out a similar construction for locally compact quantum groups, leading to a maximal C � - Eberlein compactification. We show that C � -Eberlein algebras always admit invariant means, and we apply this to prove various splitting results, showing how the C � -Eberlein compactification splits as the quantum Bohr compactification and elements which are annihilated by the mean. This holds for matrix coefficients, but for Kac algebras, we show it also holds at the algebra level, generalising (in a semigroup-free way) results of Godement. 2010 Mathematics Subject Classification: Primary 43A07, 43A60, 46L89, 47D03; Secondary 22D25, 43A30, 43A65, 47L25

Fermion unification model based on the intrinsic SU(8) symmetry of a generalized Dirac equation

A natural generalization of the original Dirac spinor into a multi-component spinor is achieved, which corresponds to the single lepton and the three quarks of the first family of the standard model of elementary particle physics. Different fermions result from similarity transformations of the Dirac equation, but apparently there can be no more fermions according to the maximal multiplicity revealed in this study. Rotations in the fermion state space are achieved by the unitary generators of the U(1) and the SU(3) groups, corresponding to quantum electrodynamics (QED based on electric charge) and chromodynamics (QCD based on colour charge). In addition to hypercharge the dual degree of freedom of hyperspin emerges, which occurs due to the duplicity implied by the two related (Weyl and Dirac) representations of the Dirac equation. This yields the SU(2) symmetry of the weak interaction, which can be married to U(1) to generate the unified electroweak interaction as in the standard model. Therefore, the symmetry group encompassing all the three groups mentioned above is SU(8), which can accommodate and unify the observed eight basic stable fermions.

Toroidal orbifolds of [formula omitted] and [formula omitted] symmetries of noncommutative tori

The Hexic transform ρ of the noncommutative 2-torus Aθ is the canonical order 6 automorphism defined by ρ(U)=V, ρ(V)=e−πiθU−1V, where U, V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUV. The Cubic transform is κ=ρ2. These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C⁎-algebras Aθρ, Aθκ are noncommutative Z6, Z3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ, a projection e of trace q2θ−pq is constructed in Aθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθe contains a Hexic invariant q×q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C∞-projections of the continuous field {At}0<t<1 of noncommutative tori C⁎-algebras such that ρ(E(t))=E(t). It turns out that the projection E(t) is the support projection of a canonical C∞-positive element that has the appearance of a noncommutative 2-dimensional Theta function. The topological invariants (or ‘quantum’ numbers) of E(t), e, and related projections are computed by a new and quicker method than in previous works. (They would also give topological invariants for finitely generated projective modules over noncommutative orbifolds associated to Z6 and Z3 symmetries of noncommutative tori.) We remark that these results have some bearing on research work related to noncommutative orbifolds used in string theory.

Noncoherent unitary space-time codes: Constellation design and decoding algorithm

Two novel schemes of unitary space-time constellations generation based on zero vectors adding are proposed for the multiple-antenna communication system. In the first scheme, T 2 zero row vectors are added into conventional unitary matrices directly, and the number of new unitary matrices obtained by different positions of the added zero vectors in T symbol duration is ⌊T / T 2 ⌋ times larger than that of conventional unitary matrices. In the second scheme, one part of the required constellations is created by the first scheme and the other part is obtained by the conventional design. This means that more information bits can be transmitted by the new constellations. According to their special construction, two corresponding decoding algorithms are proposed with low complexity in flat fading channel, respectively. At the same time, the probability of miss detection is deduced for the decoding algorithms. Performance analysis and simulation results show that the proposed constellations outperform the conventional constellations and the proposed decoding algorithms are efficient and simple.

Requirement of Dissonance in Assisted Optimal State Discrimination

A fundamental problem in quantum information is to explore what kind of quantum correlations is responsible for successful completion of a quantum information procedure. Here we study the roles of entanglement, discord, and dissonance needed for optimal quantum state discrimination when the latter is assisted with an auxiliary system. In such process, we present a more general joint unitary transformation than the existing results. The quantum entanglement between a principal qubit and an ancilla is found to be completely unnecessary, as it can be set to zero in the arbitrary case by adjusting the parameters in the general unitary without affecting the success probability. This result also shows that it is quantum dissonance that plays as a key role in assisted optimal state discrimination and not quantum entanglement. A necessary criterion for the necessity of quantum dissonance based on the linear entropy is also presented. PACS numbers: 03.65.Ta, 03.67.Mn, 42.50.Dv.

Realistic Approach of the Relations of Uncertainty of Heisenberg

Due to the requirements of the principle of causality in the theory of relativity, one cannot make a device for the simultaneous measuring of the canonical conjugate variables in the conjugate Fourier spaces. Instead of admitting that a particle’s position and its conjugate momentum cannot be accurately measured at the same time, we consider the only probabilities which can be determined when working at subatomic level to be valid. On the other hand, based on Schwinger's action principle and using the quadridimensional form of the unitary transformation generator function of the quantum operators in the paper, the general form of the evolution equation for these operators is established. In the nonrelativistic case one obtains the Heisenberg's type evolution equations which can be particularized to derive Heisenberg's uncertainty relations. The analysis of the uncertainty relations as implicit evolution equations allows us to put into evidence the intrinsic nature of the correlation expressed by these equations in straight relations with the measuring process. The independence of the quantisation postulate from the causal evolution postulate of quantum mechanics is also put into discussion.

Trap-free manipulation in the Landau-Zener system

The analysis of traps, i.e., locally but not globally optimal controls, for quantum control systems has attracted a great interest in recent years. The central problem that has been remained open is to demonstrate for a given system either existence or absence of traps. We prove the absence of traps and hence completely solve this problem for the important tasks of unconstrained manipulation of the transition probability and unitary gate generation in the Landau-Zener system---a system with a wide range of applications across physics, chemistry and biochemistry. This finding provides the first example of a controlled quantum system which is completely free of traps. We also discuss the impact of laboratory constraints due to decoherence, noise in the control pulse, and restrictions on the available controls which when being sufficiently severe can produce traps.

Development and applications of a unitary group adapted state specific multi-reference coupled cluster theory with internally contracted treatment of inactive double excitations

Any multi-reference coupled cluster (MRCC) development based on the Jeziorski-Monkhorst (JM) multi-exponential ansatz for the wave-operator Ω suffers from spin-contamination problem for non-singlet states. We have very recently proposed a spin-free unitary group adapted (UGA) analogue of the JM ansatz, where the cluster operators are defined in terms of spin-free unitary generators and a normal ordered, rather than ordinary, exponential parametrization of Ω is used. A consequence of the latter choice is the emergence of the "direct term" of the MRCC equations that terminates at exactly the quartic power of the cluster amplitudes. Our UGA-MRCC ansatz has been utilized to generate both the spin-free state specific (SS) and the state universal MRCC formalisms. It is well-known that the SSMRCC theory requires suitable sufficiency conditions to resolve the redundancy of the cluster amplitudes. In this paper, we propose an alternative variant of the UGA-SSMRCC theory, where the sufficiency conditions are used for all cluster operators containing active orbitals and the single excitations with inactive orbitals, while the inactive double excitations are assumed to be independent of the model functions they act upon. The working equations for the inactive double excitations are thus derived in an internally contracted (IC) manner in the sense that the matrix elements entering the MRCC equations involve excitations from an entire combination of the model functions. We call this theory as UGA-ICID-MRCC, where ICID is the acronym for "Internally Contracted treatment of Inactive Double excitations." Since the number of such excitations are the most numerous, choosing them to be independent of the model functions will lead to very significant reduction in the number of cluster amplitudes for large active spaces, and is worth exploring. Moreover, unlike for the excitations involving active orbitals, where there is inadequate coupling between the model and the virtual functions in the SSMRCC equations generated from sufficiency conditions, our internally contracted treatment of inactive double excitations involves much more complete couplings. Numerical implementation of our formalism amply demonstrates the efficacy of the formalism.

Formulation and implementation of a unitary group adapted state universal multi-reference coupled cluster (UGA-SUMRCC) theory: Excited and ionized state energies

The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.

Unitary group adapted state-specific multi-reference coupled cluster theory: Formulation and pilot numerical applications

We present the formulation and the implementation of a spin-free state-specific multi-reference coupled cluster (SSMRCC) theory, realized via the unitary group adapted (UGA) approach, using a multi-exponential type of cluster expansion of the wave-operator Ω. The cluster operators are defined in terms of spin-free unitary generators, and normal ordered exponential parametrization is utilized for cluster expansion instead of pure exponentials. Our Ansatz for Ω is a natural spin-free extension of the spinorbital based Jeziorski-Monkhorst (JM) Ansatz. The normal ordered cluster Ansatz for Ω results in a terminating series of the direct term of the MRCC equations, and it uses ordinary Wick algebra to generate the working equations in a straightforward manner. We call our formulation as UGA-SSMRCC theory. Just as in the case of the spinorbital based SSMRCC theory, there are redundancies in the cluster operators, which are exploited to ensure size-extensivity and avoidance of intruders via suitable sufficiency conditions. Although there already exists in the literature a spin-free JM-like Ansatz, introduced by Datta and Mukherjee, its structure is considerably more complex than ours. The UGA-SSMRCC offers an easier access to spin-free MRCC formulation as compared to the Datta-Mukherjee Ansatz, which at the same time provides with quite accurate description of electron correlation. We will demonstrate the efficacy of the UGA-SSMRCC formulation with a set of numerical results. For non-singlet cases, there is pronounced M(s) dependence of the energy for the spinorbital based SSMRCC results. Although M(s) = 1 results are closer to full configuration interaction (FCI), the extent of spin-contamination is more. In most of the cases, our UGA-SSMRCC results are closer to FCI than the spinorbital M(s) = 0 results.

Pauli graphs, Riemann hypothesis, and Goldbach pairs

Let consider the Pauli group $\mathcal{P}_q=<X,Z>$ with unitary quantum generators $X$ (shift) and $Z$ (clock) acting on the vectors of the $q$-dimensional Hilbert space via $X|s> =|s+1>$ and $Z|s> =\omega^s |s>$, with $\omega=\exp(2i\pi/q)$. It has been found that the number of maximal mutually commuting sets within $\mathcal{P}_q$ is controlled by the Dedekind psi function $\psi(q)=q \prod_{p|q}(1+\frac{1}{p})$ (with $p$ a prime) \cite{Planat2011} and that there exists a specific inequality $\frac{\psi (q)}{q}>e^{\gamma}\log \log q$, involving the Euler constant $\gamma \sim 0.577$, that is only satisfied at specific low dimensions $q \in \mathcal {A}=\{2,3,4,5,6,8,10,12,18,30\}$. The set $\mathcal{A}$ is closely related to the set $\mathcal{A} \cup \{1,24\}$ of integers that are totally Goldbach, i.e. that consist of all primes $p2$) is equivalent to Riemann hypothesis. Introducing the Hardy-Littlewood function $R(q)=2 C_2 \prod_{p|n}\frac{p-1}{p-2}$ (with $C_2 \sim 0.660$ the twin prime constant), that is used for estimating the number $g(q) \sim R(q) \frac{q}{\ln^2 q}$ of Goldbach pairs, one shows that the new inequality $\frac{R(N_r)}{\log \log N_r} \gtrapprox e^{\gamma}$ is also equivalent to Riemann hypothesis. In this paper, these number theoretical properties are discusssed in the context of the qudit commutation structure.

Extended rotation algebras: Adjoining spectral projections to rotation algebras

Denote by Aθ the rotation algebra corresponding to the rotation 2πθ. The C*-algebra 𝔹θ generated by Aθ together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra 𝔹θ is shown to have a unique tracial state and to be nuclear provided that θ is irrational. Moreover, we study the ideal structure of the C*-algebra 𝔹θ. In particular, it is shown that 𝔹θ is simple if neither the commutative sub-C*-algebra generated by the spectral projections of u in question (assumed to be a set invariant under Ad v) nor the corresponding commutative sub-C*-algebra associated to v contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra 𝔹θ generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to θ. (The spectral projections for each half-open interval from nθ to (n + 1)θ are then included for each integer n.) Using simplicity of 𝔹θ for θ irrational, the natural field of C*-algebras on the unit circle with fibres 𝔹θ is shown to be continuous at irrational points. This field is lower semicontinuous on the whole circle. Much more useful is an upper semicontinuous field which is obtained by desingularizing this field at rational points on the circle. The fibres of the desingularized field at rational points are certain (computable) type I C*-algebras. Using this new field, we are able to show that 𝔹θ is an AF algebra with K0(𝔹θ) ≅ ℤ + θℤ for generic θ, in the sense of Baire category, with the class of the unit being 1 ∈ ℤ.

Flexibility in the Patterning and Control of Axial Locomotor Networks in Lamprey

In lower vertebrates, locomotor burst generators for axial muscles generally produce unitary bursts that alternate between the two sides of the body. In lamprey, a lower vertebrate, locomotor activity in the axial ventral roots of the isolated spinal cord can exhibit flexibility in the timings of bursts to dorsally-located myotomal muscle fibers versus ventrally-located myotomal muscle fibers. These episodes of decreased synchrony can occur spontaneously, especially in the rostral spinal cord where the propagating body waves of swimming originate. Application of serotonin, an endogenous spinal neurotransmitter known to presynaptically inhibit excitatory synapses in lamprey, can promote decreased synchrony of dorsal-ventral bursting. These observations suggest the possible existence of dorsal and ventral locomotor networks with modifiable coupling strength between them. Intracellular recordings of motoneurons during locomotor activity provide some support for this model. Pairs of motoneurons innervating myotomal muscle fibers of similar ipsilateral dorsoventral location tend to have higher correlations of fast synaptic activity during fictive locomotion than do pairs of motoneurons innervating myotomes of different ipsilateral dorsoventral locations, suggesting their control by different populations of premotor interneurons. Further, these different motoneuron pools receive different patterns of excitatory and inhibitory inputs from individual reticulospinal neurons, conveyed in part by different sets of premotor interneurons. Perhaps, then, the locomotor network of the lamprey is not simply a unitary burst generator on each side of the spinal cord that activates all ipsilateral body muscles simultaneously. Instead, the burst generator on each side may comprise at least two coupled burst generators, one controlling motoneurons innervating dorsal body muscles and one controlling motoneurons innervating ventral body muscles. The coupling strength between these two ipsilateral burst generators may be modifiable and weakening when greater swimming maneuverability is required. Variable coupling of intrasegmental burst generators in the lamprey may be a precursor to the variable coupling of burst generators observed in the control of locomotion in the joints of limbed vertebrates.

Dissipative effects on a generation scheme of a W state in an array of coupled Josephson junctions

The dynamics of an open quantum system, consisting of three superconducting qubits interacting with independent reservoirs, is investigated to elucidate the effects of the environment on a unitary generation scheme of W states (Migliore R et al 2006 Phys. Rev. B 74 104503). To this end a microscopic master equation is constructed and its exact resolution predicts the generation of a Werner-like state instead of the W state. A comparison between our model and a more intuitive phenomenological model is also considered, in order to find the limits of the latter approach in the case of structured reservoirs.

The spin-free analogue of Mukherjee's state-specific multireference coupled cluster theory

In this paper, we develop a rigorously spin-adapted version of Mukherjee's state-specific multireference coupled cluster theory (SS-MRCC, also known as Mk-MRCC) [U. S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171 (1999)] for reference spaces comprising open-shell configurations. The principal features of our approach are as follows: (1) The wave operator Ω is written as Ω = ∑(μ)Ω(μ)|φ(μ)>c(μ), where {φ(μ)} is the set of configuration state functions spanning a complete active space. (2) In contrast to the Jeziorski-Monkhorst Ansatz in spin-orbital basis, we write Ω(μ) as a power series expansion of cluster operators R(μ) defined in terms of spin-free unitary generators. (3) The operators R(μ) are either closed-shell-like n hole-n particle excitations (denoted as T(μ)) or they involve valence (active) destruction operators (denoted as S(μ)); these latter type of operators can have active-active scatterings, which can also carry the same active orbital labels (such S(μ)'s are called to have spectator excitations). (4) To simulate multiple excitations involving powers of cluster operators, we allow the S(μ)'s carrying the same active orbital labels to contract among themselves. (5) We exclude S(μ)'s with direct spectator scatterings. (6) Most crucially, the factors associated with contracted composites are chosen as the inverse of the number of ways the S(μ)'s can be joined among one another leading to the same excitation. The factors introduced in (6) have been called the automorphic factors by us. One principal thrust of this paper is to show that the use of the automorphic factors imparts a remarkable simplicity to the final amplitude equations: the equations consist of terms that are at most quartic in cluster amplitudes, barring only a few. In close analogy to the Mk-MRCC theory, the inherent linear dependence of the cluster amplitudes leading to redundancy is resolved by invoking sufficiency conditions, which are exact spin-free analogues of the spin-orbital based Mk-MRCC theory. This leads to manifest size-extensivity and an intruder-free formulation. Our formalism provides a relaxed description of the nondynamical correlation in presence of dynamical correlation. Pilot numerical applications to doublet systems, e.g., potential energy surfaces for the first two excited (2)A' states of asymmetric H(2)S(+) ion and the ground (2)Σ(+)state of BeH radical are presented to assess the viability of our formalism over an wide range of nuclear geometries and the manifest avoidance of intruder state problem.

Algebraic reformulation of connes embedding problem and the free group algebra

We give a modification of the I. Klep and M. Schweighofer algebraic reformulation of Connes’ embedding problem by considering *-algebra of the countably generated free group. This allows one to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.