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In this paper, we study the quantum channel on a von Neuamnn algebras $\mathcal{M}$ preserving a von Neumann subalgebra $\mathcal{N}$, namely $\mathcal{N}$-$\mathcal{N}$-bimodule unital completely positive map. By introducing the relative irreducibility of a bimodule quantum channel, we show that its eigenvalues with modulus 1 form a finite cyclic group, called its phase group. Moreover, the corresponding eigenspaces are invertible $\mathcal{N}$-$\mathcal{N}$-bimodules, which encode a categorification of the phase group. When $\mathcal{N}\subset \mathcal{M}$ is a finite-index irreducible subfactor of type II$_1$, we prove that any bimodule quantum channel is relative irreducible for the intermediate subfactor of its fixed points. In addition, we can reformulate and prove these results intrinsically in subfactor planar algebras without referring to the subfactor using the methods of quantum Fourier analysis., Comment: 27pages

In this paper, we leverage the insights from Quon, a picture language for quantum information, to develop a new class of quantum error-correcting codes termed Variational Graphical Quantum Error Correction~(VGQEC) codes. The VGQEC codes feature adjustable configuration parameters that play a pivotal role in determining the error-correcting capability of the codes. This key feature offers remarkable flexibility in customizing high-quality quantum error-correcting codes for various noise models. For instance, we will present a specific VGQEC code that exhibits a seamless transition of parameters, enabling the smooth transformation of the code from the five-qubit repetition code to the [[5,1,3]] code, and furthermore, the new VGQEC code has a better performance than the above two well-known codes under certain noise models. Meanwhile, we also propose a general physical scheme to implement and optimize VGQEC codes in realistic quantum devices. Lastly, we apply our approach to amplitude damping noise, and by numerical calculations, we discover an unexpected novel three-qubit code that can effectively mitigate the noise.

We introduce regular stratified piecewise linear manifolds to describe lattices and investigate the lattice model approach to topological quantum field theory in all dimensions. We introduce the unitary $n+1$ alterfold TQFT and construct it from a linear functional on an $n$-dimensional lattice model on an $n$-sphere satisfying three conditions: reflection positivity, homeomorphic invariance and complete finiteness. A unitary spherical $n$-category is mathematically defined and emerges as the local quantum symmetry of the lattice model. The alterfold construction unifies various constructions of $n+1$ TQFT from $n$-dimensional lattice models and $n$-categories. In particular, we construct a non-invertible unitary 3+1 alterfold TQFT from a linear functional and derive its local quantum symmetry as a unitary spherical 3-category of Ising type with explicit 20j-symbols, so that the scalar invariant of 2-knots in piecewise linear 4-manifolds could be computed explicitly., Comment: 65 pages, 24 figures, 5 tables

We explicitly construct a (unitary) $\mathbb{Z}/2\mathbb{Z}$ permutation gauging of a (unitary) modular category $\mathcal{C}$. In particular, the formula for the modular data of the gauged theory is provided in terms of modular data of $\mathcal{C}$, which provides positive evidence of the reconstruction program. Moreover as a direct consequence, the formula for the fusion rules is derived, verifying the conjectured formula of Edie-Michell-Jones-Plavnik. Our construction explicitly shows the genus-$0$ data of the gauged theory contains higher genus data of the original theory. As applications, we obtain an identity for the modular data that does not come from modular group relations, and we prove that representations of the symmetric mapping class group (associated to closed surfaces) coming from weakly group theoretical modular categories have finite images., Comment: Version 2, references added, typos fixed, readability improved

The properties of H-rich, type II-plateau supernova (SN II-P) progenitors remain uncertain, and this is primarily due to the complexities associated with red supergiant (RSG) wind-mass loss. Recent studies have suggested that the interaction of the ejecta with a standard RSG wind should produce unambiguous signatures in the optical (e.g., a broad, boxy H${\alpha}$ profile) and in the UV (especially Ly ${\alpha}$ and Mg ii ${\lambda}{\lambda}$ 2795, 2802) a few years following the explosion. Such features are expected to be generic in all SNe II-P and can be utilized to constrain RSG winds. Here, we investigate the possibility of detecting late-time (0.3-10 years since explosion) SNe II-P in the NUV with the China Space Station Telescope (CSST). Convolving the existing model spectra of ejecta-wind interactions in SNe II-P with the transmission functions of the CSST, we calculated the associated multiband light curves, in particular, the NUV (255 nm${\sim}$317 nm) band, as well as the $NUV-r$ color. We find that the CSST will be able to detect the NUV radiation associated with ejecta-wind interaction for hundreds SNe II-P out to a few hundred Mpc over its ten-year main sky survey. The CSST will therefore provide a sizable sample of SNe II-P with the NUV signatures of ejecta-wind interaction. This will be helpful for understanding the mass loss history of SN II-P progenitors and their origins., Comment: 10 pages, 7 figures, accepted for publication in Astronomy & Astrophysics

We introduce the 3-alterfold topological quantum field theory (TQFT) by extending the quantum invariant of 3-alterfolds. The bases of the TQFT are explicitly characterized and the Levin-Wen model is naturally interpreted in 3-alterfold TQFT bases. By naturally considering the RT TQFT and TV TQFT as sub-TQFTs within the 3-alterfold TQFT, we establish their equivalence. The 3-alterfold TQFT is unitary when the input fusion category is unitary. Additionally, we extend the 3-alterfold TQFT to the Morita context and demonstrate that Morita equivalent fusion categories yield equivalent TV TQFTs. We also provide a simple pictorial proof of complete positivity criteria for unitary categorization when the 3-alterfold TQFT is unitary. Expanding our scope to high-genus surfaces by replacing the torus, we introduce the high genus topological indicators and proving the equivariance under the mapping class group actions., Comment: 35 pages

Binary stars are as common as single stars. Binary stars are of immense importance to astrophysicists because that they allow us to determine the masses of the stars independent of their distances. They are the cornerstone of the understanding of stellar evolutionary theory and play an essential role in cosmic distance measurement, galactic evolution, nucleosynthesis and the formation of important objects such as cataclysmic variable stars, X-ray binaries, Type Ia supernovae, and gravitational wave-producing double compact objects. In this article, we review the significant theoretical and observational progresses in addressing binary stars in the new millennium. Increasing large survey projects have led to the discovery of enormous numbers of binary stars, which enables us to conduct statistical studies of binary populations, and therefore provide unprecedented insight into the stellar and binary evolution physics. Meanwhile, the rapid development of theoretical concepts and numerical approaches for binary evolution have made a substantial progress on the alleviation of some long-standing binary-related problems such as the stability of mass transfer and common envelope evolution. Nevertheless, it remains a challenge to have a full understanding of fundamental problems of stellar and binary astrophysics. The upcoming massive survey projects and increasingly sophisticated computational methods will lead to future progress., Comment: An invited review published in Progress in Particle and Nuclear Physics; An open access to the published version, see https://doi.org/10.1016/j.ppnp.2023.104083

Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Nonetheless, non-physical maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. To date, there exists no effective method for implementing these non-physical maps with quantum devices. In this work, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively realize the action of an arbitrary Hermitian-preserving map by encoding its output into a quantum process. We analyze the performances of this algorithm, including its sample complexity and robustness, and prove its optimality in certain cases. When combined with algorithms such as the Hadamard test and quantum phase estimation, it allows for the extraction of information and generation of states from outputs of Hermitian-preserving maps, enabling various applications. Utilizing positive but not completely positive maps, this algorithm provides exponential advantages in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the recovery of noiseless quantum states from multiple copies of noisy states by implementing the inverse map of the corresponding noise channel, offering an intriguing approach to handling quantum errors. Our findings present a pathway for systematically and efficiently implementing non-physical actions with quantum devices, thereby boosting the exploration of potential quantum advantages across a wide range of information processing tasks., Comment: 34 pages, 10 figures, comments are welcome

In this paper, we introduce the concept of 3-alterfolds with embedded separating surfaces. When the separating surface is decorated by a spherical fusion category, we obtain quantum invariants of 3-alterfold, which is consistent with many topological moves. These moves provide evaluation algorithms for various presentations of 3-alterfold, e.g. Heegaard splittings, triangulations, link surgeries. In particular, we obtain quantum invariants of 3-manifolds containing surfaces, generalizing those of 3-manifolds containing framed links. Moreover, in this framework, we topologize fundamental algebraic concepts. For instance, we implement the Drinfeld center by tube diagrams as a blow up of framed links. The topologized center leads to a quick proof of the equality between the Reshetikhin-Turaev invariants and the Turaev-Viro invariants for spherical fusion categories. In addition, we topologize half-braiding, $S$-matrix and the generalized Frobenius-Schur indicators, etc. In particular, the latter leads to a quick topological proof of the equivariance of the generalized Frobenius-Schur indicators., Comment: 53 pages, 168 figures

Large-scale variational quantum algorithms are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To gain further clarity on this matter, we present a novel polynomial-scale method based on the path integral of observable's back-propagation on Pauli paths (OBPPP). This method efficiently approximates expectation values of operators in variational quantum algorithms with bounded truncation error in the presence of single-qubit Pauli noise. Theoretically, we rigorously prove: 1) For a constant minimal non-zero noise rate $\gamma$, OBPPP's time and space complexity exhibit a polynomial relationship with the number of qubits $n$, the circuit depth $L$. 2) For variable $\gamma$, in scenarios where more than two non-zero noise factors exist, the complexity remains $\mathrm{Poly}\left(n,L\right)$ if $\gamma$ exceeds $1/\log{L}$, but grows exponential with $L$ when $\gamma$ falls below $1/L$. Numerically, we conduct classical simulations of IBM's zero-noise extrapolated experimental results on the 127-qubit Eagle processor [Nature \textbf{618}, 500 (2023)]. Our method attains higher accuracy and faster runtime compared to the quantum device. Furthermore, our approach allows us to simulate noisy outcomes, enabling accurate reproduction of IBM's unmitigated results that directly correspond to raw experimental observations. Our research reveals the vital role of noise in classical simulations and the derived method is general in computing the expected value for a broad class of quantum circuits and can be applied in the verification of quantum computers.

We introduce $\mathfrak{F}$-positive elements in planar algebras. We establish the Perron-Frobenius theorem for $\mathfrak{F}$-positive elements. We study the existence and uniqueness of the Perron-Frobenius eigenspace. When it is not one-dimensional, we characterize its multiplicative structure. Moreover, we consider the Perron-Frobenius eigenspace as the space of logical qubits for mixed states in quantum information. We describe the relation between these mathematical results and quantum error correction, especially to the Knill-Laflamme theorem., Comment: Some revisions have been made to Section 4

In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when $n\geq3$. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifusion rings with sparse known data. We give numerous examples to illustrate the efficiency and the power of these criteria., Comment: 34 pages

Common envelope evolution (CEE) physics plays a fundamental role in the formation of binary systems, such as mergering stellar gravitational wave sources, pulsar binaries and type Ia supernovae. A precisely constrained CEE has become more important in the age of large surveys and gravitational wave detectors. We use an adiabatic mass loss model to explore how the total energy of the donor changes as a function of the remnant mass. This provides a more self-consistent way to calculate the binding energy of the donor. For comparison, we also calculate the binding energy through integrating the total energy from the core to the surface. The outcome of CEE is constrained by total energy conservation at the point at which both component's radii shrink back within their Roche lobes. We apply our results to 142 hot subdwarf binaries. For shorter orbital period sdBs, the binding energy is highly consistent. For longer orbital period sdBs in our samples, the binding energy can differ by up to a factor of 2. The CE efficiency parameter $\beta_\mathrm{CE}$ becomes smaller than $\alpha_\mathrm{CE}$ for the final orbital period $\log_{10} P_{\mathrm{orb}}/\mathrm{d} > -0.5$. We also find the mass ratios $\log_{10} q$ and CE efficiency parameters $\log_{10} \alpha_{\mathrm{CE}}$ and $\log_{10} \beta_{\mathrm{CE}}$ linearly correlate in sdBs, similarly to De Marco et al. (2010) for post-AGB binaries., Comment: 18 pages, 17 figures, 3 tables, Accepted for publication in ApJ, minor updates for the final proof

In this paper, we introduce Frobenius von Neumann algebras and study quantum convolution inequalities. In this framework, we unify quantum Young's inequality on quantum symmetries such as subfactors, and fusion bi-algebras studied in quantum Fourier analysis. Moreover, we prove quantum entropic convolution inequalities and characterize the extremizers in the subfactor case. We also prove quantum smooth entropic convolution inequalities. We obtain the positivity of comultiplications of subfactor planar algebras, which is stronger than the quantum Schur product theorem. All these inequalities provide analytic obstructions of unitary categorification of fusion rings stronger than Schur product criterion., Comment: 25 pages

We present the triangular prism equations for fusion categories, demonstrating their equivalence to the pentagon equations in the spherical case, modulo a change of basis. These equations offer valuable insights for managing complexity through localization. Additionally, we establish an enhanced version of Z. Wang's conjecture regarding the second Frobenius-Schur indicator in pivotal fusion categories. As applications, we introduce new categorification criteria and complete the classification of unitary 1-Frobenius simple integral fusion categories up to rank 8 and up to FPdim 20000., Comment: 34 pages. Improved English; merged sections; enhanced bounds; addressed gaps concerning positive characteristic. Feedback is welcome!

In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson., Comment: 20pages, 1 figure

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,q)), q prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings R_q interpolates to all integers q>=2, providing (when q is not prime-power) the first example of infinite family of non group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring R_q (with q non prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed., Comment: 29 pages. Minor updates. Accepted in International Journal of Mathematics

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation., Comment: 23 pages; published version, containing the latest updates

The first author constructed a $q$-parameterized spherical category $\sC$ over $\mathbb{C}(q)$ in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables., Comment: 25 pages

The mathematical picture language project that we began in 2016 has already yielded interesting results. We also point out areas of mathematics and physics where we hope that it will prove useful in the future., Comment: 19 pages

{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform $\FS$, as a map between suitably defined $L^{p}$ spaces, leading to a new uncertainty principle for relative entropy. We cite several applications of the quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a new topological inequality, and we outline several open problems.

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite., Comment: 13 pages, 21 figures

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras., Comment: 39 pages; 8 figures; Accepted in Advances in Mathematics. Acknowledgement and affiliations are updated

We use one photon to simulate an n-qubit quantum system for the first time. We propose a new scheme to realize universal quantum computation in polynomial time O(n^5). A generating set of gates can be realized with high accuracy in the lab. We conclude that photonic quantum computation is one of the promising approaches to universal quantum computation.

In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters $0

The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin-Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems., Comment: 16 pages