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In \cite{Linde}, Kallosh and Linde discussed the SLOCC classification of black holes. However, the criteria for the SLOCC classification of black holes have not been given. In addition, the LU classification of black holes has not been studied in the past. In this paper we will consider both SLOCC and LU classification of the STU black holes with four integer electric charges $q_{i} $ and four integer magnetic charges $p^{i}$, $i=0,1,2,3$. Two STU black holes with eight charges are considered SLOCC (LU) equivalent if and only if their corresponding states of three qubits are SLOCC (LU) equivalent. Under this definition, we give criteria for the classification of the eight-charge STU black holes under SLOCC and under LU, respectively. We will study the classification of the black holes via the classification of SLOCC and LU entanglement of three qubits. We then identify a set of black holes corresponding to the state W of three qubits, which is of interest since it has the maximal average von Neumann entropy of entanglement. Via von Neumann entanglement entropy, we partition the STU black holes corresponding to pure states of GHZ SLOCC class into five families under LU.

In [M. Walter et al., Science 340, 1205, 7 June (2013)], via polytopes they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification. In this paper, we study entanglement classification of pure states of $n$ qubits via the basis state matrix (BSM) whose rows are the basis states. We propose a canonical form of BSM obtained by exchanging columns (i.e. permutation of qubits) and rows of BSM and then a necessary and sufficient condition for a genuinely entangled state of n qubits via a canonical form of BSM. Thus, for any $n$ qubits, the genuinely entangled states can be partitioned into two families. One family includes all states whose BSM cannot be transformed into the canonical form. The states with the BSM are always genuinely entangled no matter what the non-zero coefficients are. GHZ and W states belong to the family. The other includes all states whose BSM can be transformed into the canonical form, but for any canonical form of BSM, some two columns or rows of the corresponding coefficient matrix are not proportional. The cluster state belongs to the family.

Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of the ordered affine geometry. There are 22 axioms for the ordered affine geometry, of which the axiom I.7 is about the convergence of three lines (ignoring their directions). In this paper, we indicate that the axiom I.7 includes much redundancy, and demonstrate that the complicated axiom I.7 can be replaced equivalently with a simpler and more intuitive new axiom via a theorem prover., Comment: 23 pages, no figure. arXiv admin note: text overlap with arXiv:2205.05162

The black-hole--qubit correspondence has been proven to be ``useful for obtaining additional insight into one of the string black hole theory and quantum information theory by exploiting approaches of the other"[Phys. Rev. D 82, 026003 (2010)]. Though different classes of stringy black holes can be related to the well-known stochastic local operations and classical communication (SLOCC) entanglement classes of pure states, the string theory requires a more detailed classification than the SLOCC classification of three qubits. In this paper, we derive the entanglement family of three qubits under local unitary operations (LU), and use the black-hole--qubit correspondence to classify \textquotedblleft large\textquotedblright\ black holes into seven inequivalent families. In particular, we show that two black holes with 4 non-vanishing charges ($q_{0}$, $p^{1}$, $p^{2}$, and $p^{3}$) are LU equivalent if their difference is only in the signs of charges. Thus, the classification of black holes is independent of the signs of charges and is only related to the ratio of the absolute values of charges. This observation simplifies the classification task, as one would only need to consider either the classification of non-BPS black holes or the classification of BPS black holes, but not both. Moreover, through the LU classification, the physical basis for this black-hole--qubit correspondence can be observed, and a relation between the black-hole entropy and the von Neumann entanglement entropy is revealed. Therefore, the LU classification offers a more straightforward physical connection than the SLOCC classification. Based on the LU classification, we further study the properties of von Neumann entanglement entropy for each of the seven families, and find the black holes with the maximal von Neumann entanglement entropy.

How do dust grains in protoplanetary disks overcome rapid radial drift and grow from micron size particles to planets is not well understood. The key is to search for evidence of dust accumulation and growth as a function of radius in the disk. We investigate the radial profile of grain size in the DS Tau disk by fitting multi-band ALMA observations with self-consistent radiative transfer models. The best-fit grain sizes range from centimeters in the inner disk down to 30 micron in the outer regions. Such an inside-out decreasing tendency is consistent with theories of dust evolution. Based on the best-fit model, we find that dust of 2 Jupiter masses has been depleted within the gap. By taking the gas-to-dust mass ratio into account, the lost mass is enough to form the 3.5 Jupiter mass planet inferred by literature hydrodynamic simulations. Moreover, our modeling also indicates that at the interface region between the gap and the ring, the grain size profile shows a discontinuity, with its amplitude dependent on the dust model adopted in the radiative transfer analysis. Future multi-wavelength observations at higher angular resolutions are required to better constrain the grain size and its variation in the vicinity of disk substructures., Comment: 10 pages, 8 figures, accepted for publication in MNRAS

The total amount of dust grains in protoplanetary disks is one of the key properties that characterize the potential for planet formation. With (sub-)millimeter flux measurements, literature studies usually derive the dust mass using an analytic form under the assumption of optically thin emission, which may lead to substantial underestimation. In this work, we conduct a parameter study with the goal of investigating the effects of disk structure and dust properties on the underestimation through self-consistent radiative transfer models. Different dust models, scattering modes and approaches for dust settling are considered and compared. The influences of disk substructures, such as rings and crescents, on the mass derivation are investigated as well. The results indicate that the traditional analytic method can underestimate the mass by a factor of a few to hundreds, depending on the optical depth along the line of sight set mainly by the true dust mass, disk size and inclination. As an application, we perform a detailed radiative transfer modeling of the spectral energy distribution of DoAr 33, one of the observed DSHARP disks. When the DSHARP dust opacities are adopted, the most probable dust mass returned from the Bayesian analysis is roughly 7 times higher than the value given by the analytic calculation. Our study demonstrates that estimating disk dust masses from radiative transfer modeling is one solution for alleviating the problem of insufficient mass for planet formation raised in the ALMA era., Comment: 16 pages, 9 figures, accepted for publication in A&A

Based on an ordering with directed lines and using constructions instead of existential axioms, von Plato proposed a constructive axiomatization of ordered affine geometry. There are 22 axioms for the ordered affine geometry, of which the axiom I.7 is about the convergence of three lines (ignoring their directions). In this paper, we indicate that the axiom I.7 includes much redundancy, and demonstrate that the complicated axiom I.7 can be replaced with a simpler and more intuitive new axiom (called ODO) which describes the properties of oppositely and equally directed lines. We also investigate a possibility to replace the axiom I.6 with ODO., Comment: 14 pages, no figure

In this paper, we derive a general formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann entropy of entanglement for Ac\'{\i}n et al.' Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Ac\'{\i}n et al.'s LU invariants. In particular, our result reveals a general relation among tangle, 3-tangle, and von Neumann entropy, together with a relation among their averages. The relations can help us find the entangled states satisfying distinct requirements for tangle, 3-tangle, and von Neumann entropy. Moreover, we obtain all the states of three qubits of which tangles, concurrence, 3-tangle and von Neumann entropy don't vanish and these states are endurable when one of three qubits is traced out. We indicate that for the three-qubit W state, its average von Neumann entropy is maximal only within the W SLOCC class, and that under ASD the three-qubit GHZ state is the unique state of which the reduced density operator obtained by tracing any two qubits has the maximal von Neumann entropy., Comment: 14 pages, no figure

In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class., Comment: 9 pages, no figures. Delete the motivation in previous version. Change abstract, introduction, main text, summary and references

We propose a criterion for the detection of genuine entanglement of pure multiqubit states. To this aim, we define an operator called the losing one qubit operator, which is different from the reduced density operator. The states obtained from a multiqubit state by applying the losing one qubit operator are referred to as its projected states. We show that all of the projected states of a pure product n-qubit state are pure product states provided that it cannot be written as a product of a single qubit state and a genuinely entangled (n-1)-qubit state. We also show that a pure n-qubit state is genuinely entangled provided that the state has at least two genuinely entangled (n-1)-qubit projected states. By repeating the losing process, we reduce the detection of entanglement of pure n-qubit states to the one of pure two-qubit states. Also we write a LISP program for the reduction process., Comment: 6 pages no figure

In [M. Walter et al., Science 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs., Comment: 15 pages, published version

In [Science 340, 1205, 7 June (2013)], via polytopes Michael Walter et al. proposed a sufficient condition detecting the genuinely entangled pure states. In this paper, we indicate that generally, the coefficient vector of a pure product state of $n$ qubits cannot be decomposed into a tensor product of two vectors, and show that a pure state of $n$ qubits is a product state if and only if there exists a permutation of qubits such that under the permutation, its coefficient vector arranged in ascending lexicographical order can be decomposed into a tensor product of two vectors. The contrapositive of this result reads that a pure state of $n$ qubits is genuinely entangled if and only if its coefficient vector cannot be decomposed into a tensor product of two vectors under any permutation of qubits. Further, by dividing a coefficient vector into $2^{i}$ equal-size block vectors, we show that the coefficient vector can be decomposed into a tensor product of two vectors if and only if any two non-zero block vectors of the coefficient vector are proportional. In terms of \textquotedblleft proportionality\textquotedblright , we can rephrase that a pure state of $n$ qubits is genuinely entangled if and only if there are two non-zero block vectors of the coefficient vector which are not proportional under any permutation of qubits. Thus, we avoid decomposing a coefficient vector into a tensor product of two vectors to detect the genuine entanglement. We also present the full decomposition theorem for product states of n qubits., Comment: 10 pages, no figure

We construct $\ell $-spin-flipping matrices from the coefficient matrices of pure states of $n$ qubits and show that the $\ell $-spin-flipping matrices are congruent and unitary congruent whenever two pure states of $n$ qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the $\ell $-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the $\ell $-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the $\ell $-spin-flipping matrices under LU and then permits a reduction of LU classification of n qubits to calculation of singular values of the $\ell $-spin-flipping matrices. Furthermore, we show that the invariance of singular values of the $\ell $-spin-flipping matrices $\Omega _{1}^{(n)}$ implies the invariance of the concurrence for even $n$ qubits and the invariance of the n-tangle for odd $n$ qubits. Thus, the concurrence and the n-tangle can be used for LU classification and computing the concurrence and the n-tangle only performs additions and multiplications of coefficients of states., Comment: 7 pages, no figures

We investigate the proportional relationships for spectrums and for SJNFs (Standard Jordan Normal Forms) of the matrices constructed from coefficient matrices of two SLOCC (stochastic local operations and classical communication) equivalent states of $n$ qubits. Invoking the proportional relationships for spectrums and for SJNFs, pure states of $n$ ($\geq 4$) qubits are partitioned into 12 groups and 34 families under SLOCC, respectively. Specially, it is true for four qubits., Comment: 5 pages,no figure, submit to PRL

Rotating wave approximation (RWA) plays a key rule in quantum optics to solve some Schr\"{o}dinger equation approximately. For example, it is well known that RWA has been used to calculate the transition probability. However, so far no one shows the validity of RWA. Recently Larson [J. Larson, Phys. Rev. Lett. 108, 033601 (2012)] gave the first example showing that the application of RWA in the Jaynes-Cummings model may lead to false conclusions. In this paper, we analytically and numerically show that the application of RWA leads to false conclusions about the transition probability for the near or exact resonance., Comment: 14 pages, 2 figures This paper has been withdrawn by the authors because Eq. (22) holds only for some case and the figures ploted from the data obtained with 15*10^{6} iterations are not precice. The figures will be ploted with 15*10^{8}iterations to solve the ode by using workstation

The quantitative adiabatic condition (QAC), or quantitative condition, is a convenient (a priori) tool for estimating the adiabaticity of quantum evolutions. However, the range of the applicability of QAC is not well understood. It has been shown that QAC can become insufficient for guaranteeing the validity of the adiabatic approximation, but under what conditions the QAC would become necessary has become controversial. Furthermore, it is believed that the inability for the QAC to reveal quantum adiabaticity is due to induced resonant transitions. However, it is not clear how to quantify these transitions in general. Here we present a progress to this problem by finding an exact relation that can reveal how transition amplitudes are related to QAC directly. As a posteriori condition for quantum adiabaticity, our result is universally applicable to any (nondegenerate) quantum system and gives a clear picture on how QAC could become insufficient or unnecessary for the adiabatic approximation, which is a problem that has gained considerable interest in the literature in recent years., Comment: 17 pages, 1 figure; accepted by New Journal of Physics

We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.

We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |l,n> with l=1, ..., [n/2] are inequivalent under SLOCC.

We study the entanglement classification under stochastic local operations and classical communication (SLOCC) for odd n-qubit pure states. For this purpose, we introduce the rank with respect to qubit i for an odd n-qubit state. The ranks with respect to qubits 1,2,...,n give rise to the classification of the space of odd n qubits into 3^n families., Comment: Submitted to QIC on November 12, 2010

In this paper, we study SLOCC determinant invariants of order 2^{n/2} for any even n qubits which satisfy the SLOCC determinant equations. The determinant invariants can be constructed by a simple method and the set of all these determinant invariants is complete with respect to permutations of qubits. SLOCC entanglement classification can be achieved via the vanishing or not of the determinant invariants. We exemplify the method for several even number of qubits, with an emphasis on six qubits., Comment: J. Phys. A: Math. Theor. 45 (2012) 075308

In this paper we derived the precise formula in a sine function form of the norm of the amplitude in the desired state, and by means of he precise formula we presented the necessary and sufficient phase condition for any quantum algorithm with arbitrary phase rotations. We also showed that the phase condition: identical rotation angles, is a sufficient but not a necessary phase condition., Comment: 16 pages. Modified some English sentences and some proofs. Removed a table. Corrected the formula for kol on page 10. No figures

This paper presented two general quantum search algorithms. We derived the iterated formulas and the simpler approximate formulas and the precise formula for the amplitude in the desired state. A mathematical proof of Grover's algorithm being optimal among the algorithms with arbitrary phase rotations was given in this paper. This first reported the non-symmetric effects of different rotating angles, and gave the first-order approximate phase condition when rotating angles are different., Comment: 13 pages, misusing tex formatting commands in title, shorted the titles, corrected typos, added the justifications to the section 2