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No quantum circuit can turn a completely unknown unitary gate into its coherently controlled version. Yet, coherent control of unknown gates has been realised in experiments, making use of a different type of initial resources. Here, we formalise the task achieved by these experiments, extending it to the control of arbitrary noisy channels, and to more general types of control involving higher dimensional control systems. For the standard notion of coherent control, we identify the information-theoretic resource for controlling an arbitrary quantum channel on a $d$-dimensional system: specifically, the resource is an extended quantum channel acting as the original channel on a $d$-dimensional sector of a $(d+1)$-dimensional system. Using this resource, arbitrary controlled channels can be built with a universal circuit architecture. We then extend the standard notion of control to more general notions, including control of multiple channels with possibly different input and output systems. Finally, we develop a theoretical framework, called supermaps on routed channels, which provides a compact representation of coherent control as an operation performed on the extended channels, and highlights the way the operation acts on different sectors., 33 pages, 7 figures

In this article we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is two fold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of first Hamiltonian. In both the parts we investigate the dynamical trade off and balancing points for squashed entanglement and concurrence. During the study, we also found the results of entanglement sudden death (ESD) with Heisenberg Hamiltonian in Werner state under concurrence measure. In second part, we investigate the special result for bi-linear bi-quadratic Hamiltonian; it does not disturb squashed entanglement and concurrence in both the states and exhibit the robust character for both of the states., Comment: 12,5

The ensemble average of $| \sum_{j=1}^N e^{i k \lambda_j} |^2$ is of interest as a probe of quantum chaos, as is its connected part, the structure function. Plotting this average for model systems of chaotic spectra reveals what has been termed a dip-ramp-plateau shape. Generalising earlier work of Br\'ezin and Hikami for the Gaussian unitary ensemble, it is shown how the average in the case of the Laguerre unitary ensemble can be reduced to an expression involving the spectral density of the Jacobi unitary ensemble. This facilitates studying the large $N$ limit, and so quantifying the dip-ramp-plateau effect. When the parameter $a$ in the Laguerre weight $x^a e^{-x}$ scales with $N$, quantitative agreement is found with the characteristic features of this effect known for the Gaussian unitary ensemble. However, for the parameter $a$ fixed, the bulk scaled structure function is shown to have the simple functional form ${2 \over \pi} {\rm Arctan} \, k$, and so there is no ramp-plateau transition., Comment: 23 pages; v2 reference added, Introduction updated; v3 incorporates corrections and suggestions of the referees

We introduce and study two new classes of unital quantum channels. The first class describes a 2-parameter family of channels given by completely positive (CP) maps $M_3({\bf C}) \mapsto M_3({\bf C})$ which are both unital and trace-preserving. Almost every member of this family is factorizable and extreme in the set of CP maps which are both unital and trace-preserving, but is not extreme in either the set of unital CP maps or the set of trace-preserving CP maps. We also study a large class of maps which generalize the Werner-Holevo channel for $d = 3$ in the sense that they are defined in terms of partial isometries of rank $d-1$. Moreover, we extend this to maps whose Kraus operators have the form $t |e_j \rangle \langle e_j | \oplus V $ with $V \in M_{d-1} ({\bf C}) $ unitary and $t \in (-1,1)$. We show that almost every map in this class is extreme in both the set of unital CP maps and the set of trace-preserving CP maps. We analyze in detail a particularly interesting subclass which is extreme unless $t = -1/(d-1)$. For $d = 3$, this includes a pair of channels which have a dual factorization in the sense that they can be obtained by taking the partial trace over different subspaces after using the same unitary conjugation in $M_3({\bf C}) \otimes M_3({\bf C})$., Comment: Improved discussion of the question on factorizability when d = 4 and t = -1/3 in new Section 4.6.2

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing universal topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for loop-like excitations for 3D fermionic topological phases. Most surprisingly, we discovered new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with fermionic particles). The simplest example of such non-Abelian braiding statistics can be realized in interacting fermionic systems with a gauge group $\mathbb{Z}_2 \times \mathbb{Z}_8$ or $\mathbb{Z}_4 \times \mathbb{Z}_4$, and the physical origin of non-Abelian statistics can be viewed as attaching an open Majorana chain onto a pair of linked loops, which will naturally reduce to the well known Ising non-Abelian statistics via the standard dimension reduction scheme. Moreover, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also give rise to an alternative way to classify FSPT phases with unitary symmetries. We further compare the classification results for FSPT phases with arbitrary Abelian total symmetry $G^f$ and find systematical agreement with previous studies using other methods. We believe that the proposed framework of understanding three-loop braiding statistics (including both Abelian and non-Abelian cases) in interacting fermion systems applies for generic fermonic topological phases in 3D., 36 pages, 8 figures, all comments and suggestions are very welcome

We obtain an asymptotic formula for $$n\times n$$ n × n Toeplitz determinants as $$n\rightarrow \infty $$ n → ∞ , for non-negative symbols with any fixed number of Fisher–Hartwig singularities, which is uniform with respect to the location of the singularities. As an application, we prove a conjecture by Fyodorov and Keating (Philos Trans R Soc A 372: 20120503, 2014) regarding moments of averages of the characteristic polynomial of the Circular Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the momentum of impenetrable bosons in one dimension with periodic boundary conditions.

We study the orthogonal polynomials and the Hankel determinants associated with Gaussian weight with two jump discontinuities. When the degree $n$ is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected to the coupled Painlev\'e IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree $n$ grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlev\'{e} II system. As applications, we re-derive the recently found Tracy-Widom type expressions for the gap probability of there being no eigenvalues in a finite interval near the the extreme eigenvalue of large Gaussian unitary ensemble and the limiting conditional distribution of the largest eigenvalue in Gaussian unitary ensemble by considering a thinned process., Comment: 34 pages, 2 figures

Let ℌ be a complex Hilbert space with dim ℌ ≥ 3 and ℬ ( ℋ ) the algebra of all bounded linear operators on ℌ. The effect algebra E (ℌ) on ℌ is the set of all positive contractions in ℬ ( ℋ ) . We consider the automorphism of E(ℌ) with respect to convex sequential product ox on E(ℌ) for some λ ∈ [0, 1] defined by A ∘ λ B = λ A 1 2 B A 1 2 + ( 1 − λ ) B 1 2 A B 1 2 , ∀ A , B ∈ E ( ℋ ) . We show that an automorphism of E (ℌ) with respect to convex sequential product is just reduced by a unitary or an anti-unitary operator U on ℌ.

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator $L^1 \to L^\infty$, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles., 43 pages

The Levinson theorem relates the difference of the phase shifts at the threshold and at infinity to the number of bound states. The theorem is modified in view of the fact that Castillejo–Dalitz–Dyson (CDD) poles and Jaffe–Low primitives corresponding to zeros of the $$D$$ function on the unitary cut are present in the scattering amplitude. It is shown that, in general, the difference of the phase shifts at the threshold and at infinity is determined by the number of bound states, the number of CDD poles, and the number of primitives. Some consequences of this theorem that concern the properties of the nucleon–nucleon interaction are discussed.

The article analyzes the concept of a unitary state, considers decentralization in unitary states, and studies the views of various scientists on the processes of decentralization in unitary states. The most optimal form of government for Ukraine is determined to be a unitary republic with an expanded list of delegated powers to the regions. Key words: decentralization, deconcentration, devolution, delegation, unitarity, unitary state.

We prove Gaussian fluctuations for pair-counting statistics of the form $$\Sigma_{1\le i\ne j\le N}f(\theta_i-\theta_j)$$ for the circular unitary ensemble of random matrices in the large- $$N$$ limit under the condition that the variance increases slowly as $$N$$ increases.

In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $$ {\Phi}_t^{\otimes 2} $$ and Φt. A criterion of annihilation of entanglement by an arbitrary qubit mapping Φ⊗2 is found.

We propose a definition of partition quantum spaces. They are given by universal $C^*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first $d$ columns of easy quantum groups. However, we define them as universal $C^*$-algebras rather than as $C^*$-subalgebras of easy quantum groups. We also investigate the minimal number $d$ needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, $d$ takes the values one or two., Comment: 35 pages version 2: Notation changes

The symmetries of the generalized Calogero model and its discrete version are related to the well-known unitary symmetries of the isotropic oscillator. The correspondence is provided by the Dunkl-operator deformation of the standard quantum momentum.

We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large $N$ results are obtained by using Szeg\"o theorem with a Fisher-Hartwig singularity. In the large $N$ (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space., Comment: 22 pages

We associate to each Temperley-Lieb-Jones C*-tensor category $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ with parameter $\delta$ in the discrete range $\{2\cos(\pi/(k+2))\,:\,k=1,2,\ldots\}\cup\{2\}$ a certain C*-algebra $\mathcal{B}$ of compact operators. We use the unitary braiding on $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ to equip the category $\mathrm{Mod}_{\mathcal{B}}$ of (right) Hilbert $\mathcal{B}$-modules with the structure of a braided C*-tensor category. We show that $\mathcal{T}\!\mathcal{L}\mathcal{J}(\delta)$ is equivalent, as a braided C*-tensor category, to the full subcategory $\mathrm{Mod}_{\mathcal{B}}^f$ of $\mathrm{Mod}_{\mathcal{B}}$ whose objects are those modules which admit a finite orthonormal basis. Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories., Comment: In the latest version, we corrected a couple of typos and reformatted the bibliography. The paper will appear in essentially this form in Communications in Mathematical Physics (2020)

We present an application of the Schur–Weyl duality in the one-dimensional Hubbard model in the case of a half-filled system of any number of atoms. We replace the actions of the dual symmetric and unitary groups in the whole 4N-dimensional Hilbert space by the actions of the dual groups in the spin and pseudo-spin spaces. The calculations significantly reduce the dimension of the eigenproblem of the one-dimensional Hubbard model.

In order to avoid the risk of information leakage during the information mutual transmission between two authorized participants, i.e., Alice and Bob, a quantum dialogue protocol based on the entanglement swapping between any two Bell states and the shared secret Bell state is proposed. The proposed protocol integrates the ideas of block transmission, two-step transmission and unitary operation encoding together using the Bell states as the information carriers. Besides the entanglement swapping between any two Bell states, a shared secret Bell state is also used to overcome the information leakage problem, which not only makes Bob aware of the prepared initial state but also is used for Bob's encoding and entanglement swapping. Security analysis shows that the proposed protocol can resist the general active attacks from an outside eavesdropper Eve. Moreover, the relation between the maximal amount of information Eve can gain and the detection probability is derived., Comment: 8 pages, 1 figure, 1 table

In near-term quantum computers, the computations are realized via unitary operators. The optimization problem fed into the quantum computer sets an objective function that is to be estimated via several measurement rounds. Here, we define a procedure for objective function approximation in gate-model quantum computers. The proposed solution optimizes the process of objective function estimation for optimization problems in gate-model quantum computers and quantum devices.

We obtain eigenstates of interacting disorder Hamiltonians using unitary displacement transformations that rotate the state of the system. The method generates excited states if the strength of these transformations is chosen to optimize the energy, while decreasing the energy variance. We apply the method to analyse the localization properties of one-dimensional spinless fermions with short range interactions, reaching relatively large system sizes. We quantify the degree of localization through the evolution of the inverse participation ratio as a function of the energy variance.

We realize the second Hopf fibration for a two-qubit system without using the quaternionic language. In this respect, we explore the geometrical features emerging from this Hopf fibration. Further, we investigate the metric tensor and the SO(4) non-abelian gauge field defined on the $$S^4$$ -base in terms of the entanglement quantified by the Wootters concurrence on the associated Hopf bundle. Finally, by transforming an entangled two-qubit state in the Schmidt form, we examine the different quantum phases acquired by this state under $$U(2) \times U(2)$$ local unitary operations in relation to the entanglement as well as the geometry of the corresponding state manifold.

We study the compatibility between the conformal symmetry together with the unitarity and the continuous higher-form symmetries. We show that the d-dimensional unitary conformal field theories are not consistent with continuous p-form symmetries for certain (d,p), assuming that the corresponding conserved current is a conformal primary operator. We further discuss several dynamical applications of this constraint., Comment: 14 pages, 5 tables

We study a 3d/2d dimensional degression which is a Kaluza-Klein type mechanism in AdS3 space foliated into AdS2 hypersurfaces. It is shown that an AdS3 massless particle of spin s = 1, 2, …, ∞ degresses into a couple of AdS2 particles of equal energies E = s. Note that the Kaluza-Klein spectra in higher dimensions are always infinite. To formulate the AdS3/AdS2 degression we consider branching rules for AdS3 isometry algebra o(2,2) representations decomposed with respect to AdS2 isometry algebra o(1,2). We find that a given o(2,2) higher-spin representation lying on the unitary bound (i.e. massless) decomposes into two equal o(1,2) modules. In the field-theoretical terms, this phenomenon is demonstrated for spin-2 and spin-3 free massless fields. The truncation to a finite spectrum can be seen by using particular mode expansions, (partial) diagonalizations, and identities specific to two dimensions.

We analyze the R + R2 model of quantum gravity where terms quadratic in the curvature tensor are added to the General Relativity action. This model was recently proved to be a self-consistent quantum theory of gravitation, being both renormalizable and unitary. The model can be made practically indistinguishable from General Relativity at astrophysical and cosmological scales by the proper choice of parameters., Comment: 12 pages

We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph., 17 pages, second version (added an appendix)

The lightcone singularity at the origin is resolved by blowing up the singular point to $CP^1$. The Lorentz group acts on the resolved lightcone and has $CP^1$ as a special orbit. Using Wigner's method of associating unitary irreducible representations of the Poincar\'e group to particle states, we find that the special orbit gives rise to new vacuum states. These vacuum states are labelled by the principal series representations of $SL(2,C)$. Some remarks are included on the applications of these results to gauge theories and asymptotically flat spacetimes., Comment: 8 pages, new material added

We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator $J^{(n)}$ of modular weight $n$ over a spacelike surface passing through $x = 0$. For $\vert n \vert \geq 2$ the modular Hamiltonian associated with a division of space at $x = 0$ picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at $x = 0$. The endpoint contribution is a sum of light-ray moments of the perturbing operator $J^{(n)}$ and its descendants. For perturbations on null planes only moments of $J^{(n)}$ itself contribute., LaTeX, 35 pages. v2: references added, version to appear in JHEP

We consider the construction of exactly solvable models of twisted carbon nanotubes by applying Darboux transformation to the square of the Dirac-like Hamiltonian, describing the low-energy dynamics of charge carriers. It is shown that obtained two pseudo-Schrodinger expressions are intertwined, demonstrating supersymmetry structure. It is proved, that for the considered class of pseudo-scalar external perturbations, the unitary disentanglement of negative/ positive states as within Foldy-Wouthuysen (FW) approach, so by Eriksen or SU(2) methods, can be carried out exactly, thus demonstrating the stability of the Dirac sea. The necessity of introduction of the probability density operator into the theory, while considering the problem in FW representation, is underlined.

We investigated the determination of the pole mass for the broken symmetry phase of the Abelian Higgs model in the $R_{\xi }$, Landau and the unitary gauges. In the case of the $R_{\xi }$ and Landau gauges, we obtained the same finite renormalized two-point functions, which were scale independent. Especially in the Landau gauge, the basic one-loop integral with zero mass $B(0,0)$\ appearing in the renormalized two-point function which was originated from the $\xi $ dependent masses, canceled out in the resulting pole mass. In the case of the unitary gauge, the problem of renormalization occurs in the two-point function, and if we take only the finite part of the $% p^{4}$ term in the two point function, then we obtain the same result for the pole mass as in case of the $R_{\xi }$ and Landau gauges.

We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their coadjoint orbits, using geometric quantization. The sufficient condition for the contractibility of a representation is expressed via cocycles on coadjoint orbits. This condition is verified explicitly for the contraction of SU$$_2$$2 into $$\mathbb {H}$$H. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.

An algebraic approach based on unitary algebras to calculate Franck–Condon factors (FCF’s) is presented. The method is based on the unitary group approach, which consists in adding a scalar boson to the $$\nu $$-D harmonic oscillator space, taking advantage of the transformation brackets connecting the energy, coordinate and momentum representations. In this scheme the solutions are given in terms of an expansion of harmonic oscillator functions. In this way the overlaps are expressed in terms of simple scalar product of the eigenvectors. As a benchmark to illustrate our approach the case of two 1D-Morse potentials is presented. Discussions concerned with both non-Condon contributions and the harmonic limit are included. FCF’s involving the 1D-Morse and asymmetric double Morse potentials are also considered. As an application, the FCF’s involving the S–S stretching in the $$\hbox {S}_2\hbox {O}$$ molecule are described in terms of two Morse potentials.

The unitary Birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices, such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity. If the dimension n of the unitary matrix equals a power of a prime p, i.e. if n = p w , then the Birkhoff decomposition does not need all n! possible permutation matrices, as the epicirculant permutation matrices suffice. This group of permutation matrices is isomorphic to the general affine group GA( w , p ) of order only p w ( p w − 1 ) ( p w − p ) . . . ( p w − p w − 1 ) ≪ ( p w ) ! .

We study the limiting behavior of the k-th eigenvalue $$x_k$$ of unitary invariant ensembles with Freud-type and uniform convex potentials. As both k and $$n-k$$ tend to infinity, we obtain Gaussian fluctuations for $$x_k$$ in the bulk and soft edge cases, respectively. Multi-dimensional central limit theorems, as well as moderate deviations, are also proved. This work generalizes earlier results in the GUE and unitary invariant ensembles with monomial potentials of even degree. In particular, we obtain the precise asymptotics of corresponding Christoffel–Darboux kernels as well.

We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of "quantum majorization" from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels, and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving., Comment: 25 pages, 7 Figures. To appear in IEEE Transactions on Information Theory

The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A.Wassermann for type $A$ affine VOAs, and later by T.Loke for $c, 61 pages, final revision

Having a suitable measure to quantify the coherence of quantum states, a natural task is to evaluate the power of quantum channels for creating or destroying the coherence of input quantum states. In the present paper, by introducing the maximal coherent Gaussian states based on the relative entropy measure of coherence, we propose the (generalized) cohering power and the (generalized) decohering power of Gaussian unitary operations for continuous-variable systems. Some basic properties are obtained and the cohering power and decohering power of two special kinds of Gaussian unitary operations are calculated.

We derive positivity bounds on low energy effective field theories which admit gapped, analytic, unitary, Lorentz invariant, and possibly non-local UV completions, by considering 2 to 2 scatterings of Jaffe fields whose Lehmann-K\"{a}ll\'{e}n spectral density can grow exponentially. Several properties of S-matrix, such as analyticity properties, are assumed in our derivation. Interestingly, we find that some of the positivity bounds obtained in the literature, such as sub-leading order forward-limit bounds, must be satisfied even when UV completions fall into non-localizable theories in Jaffe's language, unless momentum space Wightman functions grow too rapidly at high energy. Under this restriction on the growth rate, such bounds may provide IR obstructions to analytic, unitary, and Lorentz invariant UV completions., Comment: 24 pages, 1 figure. v3: matches published version

We revisit the one-dimensional Hubbard model in the case of strong correlations, i.e. for on-site repulsion constant strongly greater than hopping constant u ≫ t, and assume that each atom has only a single relevant orbital. We present in detail the example of half-filled system with four atoms. We reconfirm that when the sites are almost independent atoms, i.e. in the atomic limit, the spin-spin coupling term appears, every site carries exactly one electron and hopping is suppressed. We compare the results with one-dimensional Heisenberg Hamiltonian using the translational and unitary symmetries, and separation of the spin and pseudo-spin degrees of freedom.

We study the unitary representations of the non-compact real forms of the complex Lie superalgebra $${\mathfrak{sl}({\rm n}|{\rm m})}$$ . Among them, only the real form $${\mathfrak{su}({\rm p}, {\rm q}|{\rm m})}$$ with $${({\rm p} + {\rm q}= {\rm n)}}$$ admits nontrivial unitary representations, and all such representations are of the highest-weight type (or the lowest-weight type). We extend the standard oscillator construction of the unitary representations of non-compact Lie superalgebras over standard Fock spaces to generalised Fock spaces which allows us to define the action of oscillator determinants raised to non-integer powers. We prove that the proposed construction yields all the unitary representations including those with continuous labels. The unitary representations can be diagrammatically represented by non-compact Young diagrams. We apply our general results to the physically important case of four-dimensional conformal superalgebra $${\mathfrak{su}(2,2|4)}$$ and show how it yields readily its unitary representations including those corresponding to supermultiplets of conformal fields with continuous (anomalous) scaling dimensions.

In this paper, we illustrate a new concept regarding unitary elements defined on Lorentz cone, and establish some basic properties under the so-called unitary transformation associated with Lorentz cone. As an application of unitary transformation, we achieve a weaker version of the triangle inequality and several (weak) majorizations defined on Lorentz cone.

We find and classify the simplest $$ \mathcal{N} $$ N = 2 SUSY multiplets on AdS4 which contain partially massless fields. We do this by studying representations of the $$ \mathcal{N} $$ N = 2, d = 3 superconformal algebra of the boundary, including new shortening conditions that arise in the non-unitary regime. Unlike the $$ \mathcal{N} $$ N = 1 case, the simplest $$ \mathcal{N} $$ N = 2 multiplet containing a partially massless spin-2 is short, containing several exotic fields. More generally, we argue that $$ \mathcal{N} $$ N = 2 supersymmetry allows for short multiplets that contain partially massless spin-s particles of depth t = s − 2.

The topological order of a (2+1)D topological phase of matter is characterized by its chiral central charge and a unitary modular tensor category that describes the universal fusion and braiding properties of its anyonic quasiparticles. I discuss the topologically invariant quantities associated with these and identify ones that are useful for determining the topological order. I propose a variety of physical experiments that probe these quantities and detail the relation of the measured data to the topological invariants., 32 pages, 10 figures; v2: minor corrections and updates

Quantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks -- which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels -- we develop fundamental restrictions on the error incurred in such transformations and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity., Comment: 15+25 pages, 4 figures. v3: close to published version (changes in presentation, title modified; main results unaffected). See also related work by Fang and Liu at arXiv:2010.11822

Certification and quantification of correlations for multipartite states of quantum systems appear to be a central task in quantum information theory. We give here a unitary quantum-mechanical perspective of both entanglement and Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode states. This originates in the Heisenberg uncertainty relations for the canonical quadrature operators of the modes. Correlations of two-party $(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)$-mode states are examined by using the variances of a pair of suitable EPR-like observables. It turns out that the uncertainty sum of these nonlocal variables is bounded from below by local uncertainties and is strengthened differently for separable states and for each one-way unsteerable state. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of $(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)$-mode states in both possible ways of steering. When the states and the performed measurements are Gaussian, then these conditions are precisely the previously known criteria of separability and one-way unsteerability.

The development of a quadratic unitary coupled-cluster singles and doubles (qUCCSD) based self-consistent polarization propagator method is reported. We present a simple strategy for truncating the commutator expansion of the unitary version of coupled-cluster transformed Hamiltonian H̄. The qUCCSD method for the electronic ground state includes up to double commutators for the amplitude equations and up to cubic commutators for the energy expression. The qUCCSD excited-state eigenvalue equations include up to double commutators for the singles-singles block of H̄, single commutators for the singles-doubles and doubles-singles blocks, and the bare Hamiltonian for the doubles-doubles block. Benchmark qUCCSD calculations of the ground-state properties and excitation energies for representative molecules demonstrate significant improvement of the accuracy and robustness over the previous UCC3 scheme derived using Møller-Plesset perturbation theory.

Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the $\mathfrak{su}(1,1)$ Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras as generators of the SU$(1,1)$ and SU$(2)$ unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states.

In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles $$\begin{aligned} w(z,t)=A\theta (z-t)e^{-z^2+tz}, \end{aligned}$$ here $$\theta (x)$$ is the Heaviside function, where $$A> 0$$ , $$t>0$$ , and $$z\in [0,\infty )$$ . We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities $$R_n(t)$$ and $$r_n(t)$$ satisfy the coupled Riccati equations, from which we also prove that $$R_n(t)$$ satisfies a particular Painleve IV equation. Even more, $$\sigma _n(t)$$ , allied to $$R_n(t)$$ , satisfies both the discrete and continuous Jimbo–Miwa–Okamoto $$\sigma $$ -form of the Painleve IV equation. Finally, we consider the situation when n gets large, the second order linear differential equation satisfied by the polynomials $$P_n(x)$$ orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.

We analytically study quantum dissipative dynamics described by the Caldirola-Kanai model with inter-particle interactions. We have found that the dissipative quantum dynamics of the Caldirola-Kanai model can be exactly mapped to a dissipationless quantum dynamics under a negative external harmonic potential, even when the particles are strongly interacting. In particular, we show that the mapping is valid for the unitary Fermi gas, which is relevant for cold atoms and nuclear matters., 12 pages