Invariants for a Class of Nongeneric Three-Qubit States

We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations are presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them.     

Completely mixed state is a critical point for three-qubit entanglement

Pure three-qubit states have five algebraically independent and one algebraically dependent polynomial invariants under local unitary transformations and an arbitrary entanglement measure is a function of these six invariants. It is shown that if the reduced density operator of a some qubit is a multiple of the unit operator, than the geometric entanglement measure of the pure three-qubit state is absolutely independent of the polynomial invariants and is a constant for such tripartite states. Hence a one-particle completely mixed state is a critical point for the geometric measure of entanglement.     

Two Types of Maximally Entangled Bases and Their Mutually Unbiased Property in ℂd⊗ℂd′

We first construct a new maximally entangled basis in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\ (k\in Z^{+})\) which is diffrent from the one in Tao et al. (Quantum Inf. Process. 14, 2291 (2015)), then we generalize such maximally entangled basis into arbitrary bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}\). We also study the mutual unbiased property of the two types of maximally entangled bases in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\). In particular, explicit examples in \(\mathbb {C}^{2} \otimes \mathbb {C}^{4}\), \(\mathbb {C}^{2} \otimes \mathbb {C}^{8}\) and \(\mathbb {C}^{3} \otimes \mathbb {C}^{3}\) are presented.     

On PPT States in ${\cal C}^K \otimes {\cal C}^M\otimes {\cal C}^N$ Composite Quantum Systems

We study the general representations of positive partial transpose (PPT) states in ${\cal C}^K \otimes {\cal C}^M \otimes {\cal C}^N$. For the PPT states with rank-N a canonical form is obtained, from which a sufficient separability condition is presented.     

Genuine tripartite entanglement and quantum phase transition

A new simplified formula is presented to characterize genuine tripartite entanglement of （2 2 n）-dimensional quantum pure states. The formula turns out equivalent to that given in （Quant. Inf. Comp. 7（7） 584 （2007））, hence it also shows that the genuine tripartite entanglement can be described only on the basis of the local （2 2）-dimensional reduced density matrix. In particular, the two exactly solvable models of spin system studied by Yang （Phys. Rev. A 71 030302（R） （2005）） are reconsidered by employing the formula. The results show that a discontinuity in the first derivative of the formula or in the formula itself of the ground state just corresponds to the existence of quantum phase transition, which is obviously different from the concurrence.     

Mutually Unbiasedness between Maximally Entangled Bases and Unextendible Maximally Entangled Systems in $\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}}$

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system \(\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)\) are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in \(\mathbb {C}^{2}\otimes \mathbb {C}^{4}\) and \(\mathbb {C}^{2}\otimes \mathbb {C}^{8}\), respectively.     

Local Unitary Invariants for Multipartite Quantum Systems

We present an approach of constructing invariants under local unitary transformations for multipartite quantum systems. The invariants constructed in this way can be complement to that in [Science 340 (2013) 1205-1208]. Detailed examples are given to compute such invariant in detail. It is shown that these invariants can be used to detect the local unitary equivalence of degenerated quantum states.     

Mutually Unbiased Unextendible Maximally Entangled Bases in $\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}$

We study mutually unbiased unextendible maximally entangled bases (MUUMEBs) in bipartite stystem \(\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}\). By deriving the sufficient and necessary conditions that two MUUMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{4}\) need to satisfy, we first establish two pairs of MUUMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{4}\). Then we present the sufficient and necessary conditions that two MUUMEBs in bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}\) need to satisfy, thus generalize the main results of Halqem et al. (Int. J. Theor. Phys. 54(1), 326, 2015).     

Mutually Unbiased Maximally Entangled Bases for the Bipartite System ℂd⊗ℂdk

The construction of maximally entangled bases for the bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d}\) is discussed firstly, and some mutually unbiased bases with maximally entangled bases are given, where 2≤d≤5. Moreover, we study a systematic way of constructing mutually unbiased maximally entangled bases for the bipartite system \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{k}}\).     

The Refined BPS Index from Stable Pair Invariants

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi–Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a ${\mathbb{C}^{*}}$ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi–Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local ${\mathbb{P}^1}$ . We explicitly compute refined invariants in low degree for local ${\mathbb{P}^2}$ and local ${\mathbb{P}^1\,\times\,\mathbb{P}^1}$ and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov’s partition function and a refinement of Chern–Simons theory on a lens space. We also relate our product formula to wall crossing.     

Local Unitary Invariants for Multipartite States

We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full rank mixed states with non-degenerate eigenvalues, this set of invariants is also the set of the necessary and sufficient conditions for the local unitary equivalence of such two states.     

BPS Invariants of Semi-Stable Sheaves on Rational Surfaces

Bogomolnyi–Prasad–Sommerfield (BPS) invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface ${\Sigma_\ell}$ takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder–Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank ≤  3 sheaves on ${\Sigma_\ell}$ and the projective plane ${\mathbb{P}^2}$ . The applied techniques can be applied iteratively to compute invariants for higher rank.     

Entanglement dynamics and decoherence of an atom coupled to a dissipative cavity field

In this paper, we investigate the entanglement dynamics anddecoherence in the interacting system of a strongly driventwo-level atom and a single mode vacuum field in the presence ofdissipation for the cavity field. Starting with an initial productstate with the atom in a general pure state and the field in avacuum state, we show that the final density matrix is supportedon \({\mathbb C}^2\otimes{\mathbb C}^2\) space, and therefore, theconcurrence can be used as a measure of entanglement between theatom and the field. The influences of the cavity decay on thequantum entanglement of the system are also discussed. We alsoexamine the Bell-CHSH violation between the atom and the field andshow that there are entangled states for which the Bell-BCSHinequality is not violated. Using the above system as a quantumchannel, we also investigate the quantum teleportation of ageneric qubit state and also a two-qubit entangled state, and showthat in both cases the atom-field entangled state can be useful toteleport an unknown state with fidelity better than any classicalchannel.     

Study of the FCNH Coupling with Boosted Higgs at LHC*

We present a study about the flavor changing coupling of the top quark with the Higgs boson through the channel $pp\to H t/\bar{t}$ with $H\to b\bar{b}$ at LHC. The final states considered for the such process are $l^\pm+\mathbb{E}_{T}+3b$. We focus on the boosted region in the phase space of the Higgs boson. The backgrounds and events are simulated and analyzed. The sensitivities for the FCNH couplings are estimated. It is found that it is more sensitive for $y_{\rm tu}$ than $y_{\rm tq}$ at LHC. The upper limits of the FCNH couplings can be set at LHC with 3000 ${\rm fb}^{-1}$ integrated luminosity as $\vert y_{\rm tu}\vert^2=1.1\times10^{-3}$ and $\vert y_{\rm tc}\vert^2=7.2\times 10^{-3}$ at 95% C.L.     

A Categorification of the Boson–Fermion Correspondence via Representation Theory of sl(∞)

In recent years different aspects of categorification of the boson–fermion correspondence have been studied. In this paper we propose a categorification of the boson–fermion correspondence based on the category of tensor modules of the Lie algebra sl(∞) of finitary infinite matrices. By \({\mathbb{T}^{+}}\) we denote the category of “polynomial” tensor sl(∞)-modules. There is a natural “creation” functor \({{\mathcal{T}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\), \({M \mapsto N \otimes M, \quad M,N \in \mathbb{T}^{+}}\). The key idea of the paper is to employ the entire category \({\mathbb{T}}\) of tensor sl(∞)-modules in order to define the “annihilation” functor \({{\mathcal{D}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\) corresponding to \({{\mathcal{T}_{N}}}\). We show that the relations allowing one to express fermions via bosons arise from relations in the cohomology of complexes of linear endofunctors on \({{\mathbb{T}^{+}}}\).     

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3} orbifold point of local \mathbb P²{{\mathbb P}^2}, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3}, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.     

Representations of Spacetime as Unitary Operation Classes; or Against the Monoculture of Particle Fields

Spacetime is modeled as a homogeneous manifoldgiven by the classes of unitary U(2) operations in thegeneral complex operations GL( ). The residual representations of thisnoncompact symmetric space of rank two are characterized by two continuousreal invariants, one invariant interpreted as a particlemass for a positive unitary subgroup and the second onefor an indefinite unitary subgroup related to nonparticle interpretable interaction ranges.Fields represent nonlinear spacetimeGL( )/U(2) by theirquantization and include necessarily nonparticlecontributions in the timelike part of their flat-space Feynman propagator.     

Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented.The concepts of Lie symmetry and higher order adiabatic invariants of Poincar\'{e} equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.     

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A{{\mathcal A}} of von Neumann algebras on \mathbb R{\mathbb R} . In this first part, we focus on the completely rational net A{{\mathcal A}} . Our main result here states that, if A{{\mathcal{A}}} is completely rational, there exists exactly one locally normal KMS state j{\varphi} . Moreover, j{\varphi} is canonically constructed by a geometric procedure. A crucial r?le is played by the analysis of the “thermal completion net” associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.