Traceless Operators Research Articles

Random quantum graphs

We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples(X1,⋯,Xd)(X_1,\cdots ,X_d)of traceless self-adjoint operators in then×nn\times nmatrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters:2≤d≤n2−32\le d\le n^2-3. Moreover, the automorphism group is generically abelian in the larger parameter range1≤d≤n2−21\le d\le n^2-2. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles ofXiX_i’s (mimicking the Erdős-RényiG(n,p)G(n,p)model) has trivial/abelian automorphism group almost surely.

Two-loop renormalization and mixing of gluon and quark energy-momentum tensor operators

In this paper, we present one- and two-loop results for the renormalization of the gluon and quark gauge-invariant operators which appear in the definition of the QCD energy-momentum tensor, in dimensional regularization. To this end, we consider a variety of Green's functions with different incoming momenta. We identify the set of twist-2 symmetric traceless and flavor singlet operators which mix among themselves and we calculate the corresponding mixing coefficients for the nondiagonal components. We also provide results for some appropriate regularization-independent (RI')-like schemes, which address this mixing, and we discuss their application to nonperturbative studies via lattice simulations. Finally, we extract the one- and two-loop expressions of the conversion factors between the proposed RI' and the MSbar schemes. From our results regarding the MSbar-renormalized Green's functions, one can easily derive conversion factors relating numerous variants of RI'-like schemes to MSbar. To make our results easily accessible, we also provide them as Supplemental Material, in the form of a Mathematica input file and, also, an equivalent text file.

Improved population operators for multi-state nonadiabatic dynamics with the mixed quantum-classical mapping approach.

The mapping approach addresses the mismatch between the continuous nuclear phase space and discrete electronic states by creating an extended, fully continuous phase space using a set of harmonic oscillators to encode the populations and coherences of the electronic states. Existing quasiclassical dynamics methods based on mapping, such as the linearised semiclassical initial value representation (LSC-IVR) and Poisson bracket mapping equation (PBME) approaches, have been shown to fail in predicting the correct relaxation of electronic-state populations following an initial excitation. Here we generalise our recently published modification to the standard quasiclassical approximation for simulating quantum correlation functions. We show that the electronic-state population operator in any system can be exactly rewritten as a sum of a traceless operator and the identity operator. We show that by treating the latter at a quantum level instead of using the mapping approach, the accuracy of traditional quasiclassical dynamics methods can be drastically improved, without changes to their underlying equations of motion. We demonstrate this approach for the seven-state Frenkel-exciton model of the Fenna-Matthews-Olson light harvesting complex, showing that our modification significantly improves the accuracy of traditional mapping approaches when compared to numerically exact quantum results.

Uncertainty relations in the presence of quantum memory for mutually unbiased measurements

In [Berta 2014 Entanglement], uncertainty relations in the presence of quantum memory was formulated for mutually unbiased bases using conditional collision entropy. In this paper, we generalize their results to the mutually unbiased measurements. Our primary result is an equality between the amount of uncertainty for a set of measurements and the amount of entanglement of the measured state, both of which are quantified by the conditional collision entropy. Implications of this equality relation are discussed. We further show that similar equality relation can be obtained for generalized symmetric informationally complete measurements. We also derive an interesting equality for arbitrary orthogonal basis of the space of Hermitian, traceless operators.

A d-dimensional stress tensor for Minkd+2 gravity

We consider the tree-level scattering of massless particles in (d+2)-dimensional asymptotically flat spacetimes. The mathcal{S} -matrix elements are recast as correlation functions of local operators living on a space-like cut ℳd of the null momentum cone. The Lorentz group SO(d + 1, 1) is nonlinearly realized as the Euclidean conformal group on ℳd. Operators of non-trivial spin arise from massless particles transforming in non-trivial representations of the little group SO(d), and distinguished operators arise from the soft-insertions of gauge bosons and gravitons. The leading soft-photon operator is the shadow transform of a conserved spin-one primary operator Ja, and the subleading soft-graviton operator is the shadow transform of a conserved spin-two symmetric traceless primary operator Tab. The universal form of the soft-limits ensures that Ja and Tab obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFTd, respectively.

Spectra of operators in large N tensor models

We study the operators in the large $N$ tensor models, focusing mostly on the fermionic quantum mechanics with $O(N)^3$ symmetry which may be either global or gauged. In the model with global symmetry we study the spectra of bilinear operators, which are in either the symmetric traceless or the antisymmetric representation of one of the $O(N)$ groups. In the symmetric traceless case, the spectrum of scaling dimensions is the same as in the SYK model with real fermions; it includes the $h=2$ zero-mode. For the operators anti-symmetric in the two indices, the scaling dimensions are the same as in the additional sector found in the complex tensor and SYK models; the lowest $h=0$ eigenvalue corresponds to the conserved $O(N)$ charges. A class of singlet operators may be constructed from contracted combinations of $m$ symmetric traceless or antisymmetric two-particle operators. Their two-point functions receive contributions from $m$ melonic ladders. Such multiple ladders are a new phenomenon in the tensor model, which does not seem to be present in the SYK model. The more typical $2k$-particle operators do not receive any ladder corrections and have quantized large $N$ scaling dimensions $k/2$. We construct pictorial representations of various singlet operators with low $k$. For larger $k$ we use available techniques to count the operators and show that their number grows as $2^k k!$. As a consequence, the theory has a Hagedorn phase transition at the temperature which approaches zero in the large $N$ limit. We also study the large $N$ spectrum of low-lying operators in the Gurau-Witten model, which has $O(N)^6$ symmetry. We argue that it corresponds to one of the generalized SYK models constructed by Gross and Rosenhaus. Our paper also includes studies of the invariants in large $N$ tensor integrals with various symmetries.

Spinning geodesic Witten diagrams

We present an expression for the four-point conformal blocks of symmetric traceless operators of arbitrary spin as an integral over a pair of geodesics in Anti-de Sitter space, generalizing the geodesic Witten diagram formalism of Hijano et al. [1] to arbitrary spin. As an intermediate step in the derivation, we identify a convenient basis of bulk threepoint interaction vertices which give rise to all possible boundary three point structures. We highlight a direct connection between the representation of the conformal block as geodesic Witten diagram and the shadow operator formalism.

Additive Bounds of Minimum Output Entropies for Unital Channels and an Exact Qubit Formula

We investigate minimum output (R\'enyi) entropy of qubit channels and unital quantum channels. We obtain an exact formula for the minimum output entropy of qubit channels, and bounds for unital quantum channels. Interestingly, our bounds depend only on the operator norm of the matrix representation of the channels on the space of trace-less Hermitian operators. Moreover, since these bounds respect tensor products, we get bounds for the capacity of unital quantum channels, which is saturated by the Werner-Holevo channel. Furthermore, we construct an orthonormal basis, besides the Gell-Mann basis, for the space of trace-less Hermitian operators by using discrete Weyl operators. We apply our bounds to discrete Weyl covariant channels with this basis, and find new examples in which the minimum output R\'enyi $2$-entropy is additive.

Isometries of the spaces of self-adjoint traceless operators

In this paper we describe the structure of all isometries of the space of self-adjoint traceless operators on a finite dimensional Hilbert space under the metrics coming from the operator norm or the Schatten norms. We prove that up to a translation and multiplication by −1, they are unitary or antiunitary similarity transformations.

Holographic two-point functions in conformal gravity

In this paper we compute the holographic two-point functions of four dimensional conformal gravity. Precisely we calculate the two-point functions for Energy- Momentum (EM) and Partially Massless Response (PMR) operators that have been identified as two response functions for two independent sources in the dual CFT. The correlation function of EM with PMR tensors turns out to be zero which is expected according to the conformal symmetry. The two-point function of EM is that of a transverse and traceless tensor, and the two-point function of PMR which is a traceless operator contains two distinct parts, one for a transverse-traceless tensor operator and another one for a vector field, both of which fulfill criteria of a CFT. We also discuss about the unitarity of the theory.

Spinning conformal correlators

We develop the embedding formalism for conformal field theories, aimed at doing computations with symmetric traceless operators of arbitrary spin. We use an indexfree notation where tensors are encoded by polynomials in auxiliary polarization vectors. The efficiency of the formalism is demonstrated by computing the tensor structures allowed in n-point conformal correlation functions of tensors operators. Constraints due to tensor conservation also take a simple form in this formalism. Finally, we obtain a perfect match between the number of independent tensor structures of conformal correlators in d dimensions and the number of independent structures in scattering amplitudes of spinning particles in (d + 1)-dimensional Minkowski space.

Mutually unbiased bases and trinary operator sets forNqutrits

A complete orthonormal basis of N-qutrit unitary operators drawn from the Pauli Group consists of the identity and 9^N-1 traceless operators. The traceless ones partition into 3^N+1 maximally commuting subsets (MCS's) of 3^N-1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3^N are isomorphic to all MCS's, and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS's. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like). For three qutrits, 728 operators partition into 28 MCS's with less rigid structure allowing for the coexistence of separable, partially-entangled, and totally entangled (GHZ-like) bases. However, a minimum of 16 GHZ-like bases must occur. Every basis state is described by an N-digit trinary number consisting of the eigenvalues of N observables constructed from the corresponding MCS.

A sum rule for nonlinear optical susceptibilities

It is explicitly shown, for optical processes arbitrarily comprising two-, three- or four-photon interactions, that the sum over all matter states of any optical susceptibility is exactly zero. The result remains true even in frequency regions where damping is prominent. Using a quantum electrodynamical framework to render the photonic nature of the fundamental interactions, the result emerges in the form of a traceless operator in Hilbert space. The generality of the sum rule and its significance as a thermodynamic limit are discussed, and the applicability to real systems is assessed.

Virial coefficients of non-abelian anyons

We study a system of non-Abelian anyons in the lowest Landau level of a strong magnetic field. Using diagrammatic techniques, we prove that the virial coefficients do not depend on the statistics parameter. This is true for all representations of all non-Abelian groups for the statistics of the particles and relies solely on the fact that the effective statistical interaction is a traceless operator.

Homotopic groups of photon propagators in isotropic, bi-isotropic and Faraday materials

With the use of an evolutionary approach the branches of solutions of the tensor dispersion photon equations in isotropic, bi-isotropic and Faraday media are investigated. It is shown that the homotopic groups of photon propagators with the generators being the refractive index operators correspond to these solutions. The latter operators are divided into two groups. These are operators with non-zero trace and traceless operators. The traceless operators correspond to meeting photons and are represented by infinite sets of isometries and involutions of the three-dimensional space. The analysis of polarization states is carried out for photons meeting in non-absorbing and absorbing isotropic media and curvatures, torsions and Darboux-Lagrange vectors of the spiral lines related to such states are found.

Complex Maxwell groups in the description of evanescent photons

For inhomogeneous electromagnetic waves in isotropic media the operator evolution solutions of Maxwell equations are obtained. These solutions are functions of the complex spatial variable. In the case of homogeneous waves an evolution operator is associated with a set of right-handed and left-handed generalized helices. In this set the helices of straight circular cylinders are geodetic lines. It is shown that one of the branches of the evolution operators is generated by traceless operators and that it correspond to standing evanescent waves. These evolution operators are elements of the group . For this Maxwell group a new parametrization is introduced and appropriate composition laws are derived. The introduced parametrization is compared with the known Fedorov parametrization.

The oscillator basis for octupole collective motion in nuclei

A complete classification and a method of construction of the wavefunctions of the octupole oscillator, according to the physical group chain U(7) contains/implies R(7) contains/implies R(3), are given. The method is based on the concepts of elementary factors (elementary permissible diagrams) for the reduction process in the chain U(7) contains/implies R(3) and of traceless operators for the extraction of irreducible representations of R(7) from those of U(7). The seven quantum numbers used for labelling the wavefunctions are the number of phonons, the seniority, the spin and its projection, the numbers of quartets and sextets of phonons coupled to spin zero, and a label for a 'residual coupling'. A connection between the present problem and the theory of covariants of algebraic forms is mentioned. The construction of the oscillator basis is a first step towards a description of the octupole motions, in their full variety, in nuclei.

Field Theory Describing Arbitrary Two-Body Scattering Amplitude as a Born Term

We propose in this paper the local relativistic quantum field-theoretic formulation of the most general isobar model, with unstable two-particle isobars having arbitrary mass and spin spectrum. Such infinite family of isobars is described by infinite-component field operator with its components given by symmetric, traceless and divergenceless field operators ϕu1…uI (x;s), where I describes spin (I = 0, 1, 2…) and the continuous parameter S the spectrum of asymptotic masses. The asymptotic free states defined by generalized LSZ limits of ϕu1…uI (x;s), can be described by means of stable multiparticle states. In such a field theory one can choose the parameters in such a way that already a Born term describes an arbitrarily chosen two-body scattering amplitude. As an example we present the generalization of Van Hove model of Reggeons to the case of complex Regge trajectory, describing Regge family of resonances on the second sheet. Finally the correspondence with conventional theory is discussed. It is shown that the Feynman diagrams in local field-theoretic isobar model can be interpreted in the framework of non local theory describing in some approximation an equivalent interaction between stable decay products.

Absence of Ordering in Certain Isotropic Systems

The method of Mermin and Wagner [Phys. Rev. Lett. 17, 1133 (1966)] is used to show that one- and two-dimensional spin systems interacting with a general isotropic interaction H=12 ∑ ijn Iij(n) (Si·Sj)n,where the exchange interactionsIij(n) are of finite range, cannot order in the sense that 〈Oi〉=0for all traceless operators Oi defined at a single site i. Mermin and Wagner have proved the above for the case n = 1 with Oi = Si, i.e., for the Heisenberg Hamiltonian. The proof allows us to rule out the possibility that a small isotropic biquadratic exchange (Si·Sj)2 could induce ferromagnetism or antiferromagnetism in a two-dimensional Heisenberg system.