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In Monras et al. (Phys. Rev. A, 2011, 84(1):012301 2011), the authors presented an entanglement measure for bipartite pure states based on local unitary operations. In this paper, motivated by this idea, we obtained an entanglement measure for multipartite quantum states with respect to k-partitions, which is called mirror entanglement measure for multipartite k-nonseparable states, it is simply denoted by k-MEM. We show that this measure is well-defined, i.e., it satisfies some necessary conditions of entanglement measure including vanishes iff the multipartite quantum states are k-separable, invariance under local unitary operation and monotonicity under local quantum operation and classical communication.

Although it is often assumed that spatial attention exists in the form of a unitary focus, the split-attention hypothesis proposes that attention can be simultaneously divided into two spatially noncontiguous positions and that the space in between can be ignored. However, whether split attention occurs directly based on the generation of attentional benefit or whether it requires a gradual divide from a unitary focus over time has not been clarified. In the present study, by using two spatial salient cues to direct the attention allocation of participants, we aimed to investigate whether attention requires time to divide from a unitary focus and whether the appearance time of split attention varies when the task difficulty level increases between experiments. The results showed that attention required time to divide from a unitary focus, and the position between the two cued positions was not excluded by attention when the stimulus-onset asynchrony (SOA) was 60 ms. However, as the task difficulty increased between experiments, the appearance time of split attention was earlier. These findings suggest that the appearance time of split attention has a certain flexibility and can be changed according to the task requirement, thus implying that split attention and unitary attention present some common attention mechanisms and that a split or unitary mode can be flexibly selected for an attention system.

Institutions—defined as strategies, norms and rules (Ostrom Understanding institutional diversity, Princeton University Press, Princeton, 2005)—are omnipresent in third sector contexts. In this paper, we present the Institutional Grammar (IG) as a theoretically informed approach to support institutional analysis in third sector research. More specifically, the IG coding syntax allows the researcher to systematically wade through rich text and (transcribed) spoken language to identify and dissect institutional statements into finer syntactical segments of interest to the researcher. It is a versatile method that can generate data for small- or large-N research projects and can be integrated with mixed-method research designs. After first introducing and describing the IG, we present a case study to illustrate how a IG-based syntactic analysis can be leveraged to inform third sector research. In the case, we ask: Do the rules embedded in regulatory text addressing the involuntary dissolution of charity organizations differ between bifurcated and unitary jurisdictions in the United States? Using IG’s ABDICO 2.0 syntax, we identify eleven “Activation Condition” (AC) categories that trigger action and assess variation among the 46 jurisdictions. We ultimately conclude that the rules do not differ between bifurcated and unitary jurisdictions, but that finding is not the primary concern. The case demonstrates IG as an important methodological advance that yields granular, structured analyses of rules, norms and strategies in third sector settings that may be difficult to identify with other methods. We then emphasize four areas of third sector research that could benefit from the addition of IG-based methods: analysis of (1) rule compliance, (2) inter-organizational collaboration, (3) comparative study of institutional design, and (4) the study of institutional change. We close the paper with some reflections on where IG-based analysis is headed.

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

In this paper, by using entanglement swapping of Bell states, an efficient quantum private comparison(QPC) protocol with a semi-honest party is proposed. The semi-honest third party (TP) is required to help two participants perform the comparison. She can record intermediate results and do some calculations in the whole process of the protocol execution, but she is not allowed to conspire with any participants. Moreover, TP cannot get two participants’ privacy information except the comparison results. The security analysis shows that the proposed protocol can resist both outsider attacks and insider attacks. Compared with other similar QPC protocols, the proposed one needs neither unitary operations nor the entanglement swapping of multiparticle quantum states, and it only needs the entanglement swapping of Bell states, which makes it more practical. Since three-bit classical information could be compared in each comparison, the proposed protocol has a good performance in its efficiency.

The unification problem in a propositional logic is to determine, given a formula φ, whether there exists a substitution σ such that σ(φ) is in that logic. In that case, σ is a unifier of φ. When a unifiable formula has minimal complete sets of unifiers, it is either infinitary, finitary, or unitary, depending on the cardinality of its minimal complete sets of unifiers. Otherwise, it is nullary. In this paper, we prove that in modal logic $\mathbf {K}+\square \square \bot $ , unifiable formulas are either finitary, or unitary.

The basic formulas for the unitary principal series representations of the group SL(2, ℂ) are given, and the decomposition of a tensor product of two representations into irreducible representations is considered. A simple proof of the completeness of the 3j-symbols is given.

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

In this paper, an improved controlled bidirectional quantum teleportation protocol of the special three-qubit state is proposed. In a little bit more detail, under the control of the third supervisor Charlie, Alice wants to send one special three-qubit entangled state to Bob, and at the meantime, Bob also wants to transmit another special three-qubit entangled state to Alice. In other words, both Alice and Bob can be the sender and receiver simultaneously. To achieve this aim, a specific eleven-qubit entangled state is shared among Alice, Bob and Charlie in advance acting as the quantum channel. Then, Alice and Bob first implement the GHZ-state measurement and Bell-state measurement respectively, and following Charlie’s single-qubit measurement. Finally, upon the foregoing measurement results, Alice and Bob can respectively implement the specific unitary operators on their local particles to recover the initial state transmitted by the other.

Why did Bernhard Riemann (1826-1866), arguably the most original mathematician of his generation, spend the last year of life investigating the mechanism of hearing? Fighting tuberculosis and the hostility of eminent scientists such as Hermann Helmholtz, he appeared to forsake mathematics to prosecute a case close to his heart. Only sketchy pages from his last paper remain, but here we assemble some significant clues and triangulate from them to build a broad picture of what he might have been driving at. Our interpretation is that Riemann was a committed idealist and from this philosophical standpoint saw that the scientific enterprise was lame without the "poetry of hypothesis". He believed that human thought was fundamentally the dynamics of "mind-masses" and that the human mind interpenetrated, and became part of, the microscopic physical domain of the cochlea. Therefore, a full description of hearing must necessarily include the perceptual dimensions of what he saw as a single manifold. The manifold contains all the psychophysical aspects of hearing, including the logarithmic transformations that arise from Fechner's law, faithfully preserving all the subtle perceptual qualities of sound. For Riemann, hearing was a unitary physical and mental event, and parallels with modern ideas about consciousness and quantum biology are made. A unifying quantum mechanical model for an atom of consciousness-drawing on Riemann's mind-masses and the similar "psychons" proposed by Eccles-is put forward., Foundations of Science, 27 (3)

Let $$(\pi ,\,H_\pi )$$ be a unitary representation of a locally compact group G. The characterization of the minimality of the topological center induced by $$\pi $$ is already accessed. Also, some conditions which guarantee that the center is maximal are known. Here, we set our goal to characterizing the maximality of the center by concentrating on weakly almost G-periodic operators. In fact, we present a new point of view on such operators that allows us to get the desired result. Moreover, we consider some topological centers that are always maximal.

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups [3,2]. In this article, we prove similar result on isolating the cuspidal spectrum in [3] for the function field case., The paper has been accepted for publication in SCIENCE CHINA Mathematics. A gap in the previous version is fixed

In this paper, we theoretically propose and demonstrate a non-unitary beam-splitter (BS) by introducing coupling losses at the interface of the plasmonic waveguide and multimode dielectric waveguide (DW). The coupling losses enable us to modify the reflection and transmission factors, which can result in arbitrary shift of interference curves of two outputs. Specially, the lossy non-unitary BS can tune the amplitudes and phases of two outputs, even making them change synchronously, regardless of input phase differences. After a $$\pi /2$$ phase delay in one arm, these two outputs are fed into another multimode DW. This DW is a normal unitary BS, working like the Michelson interferometer, where anti-synchronous interference can take place. At last, the whole device outputs an invariant zero energy state in one port, exhibiting a phase-insensitive performance. Our study provides a versatile design platform to realize non-unitary/unitary BS and construct more multi-functional devices.

A teleportation protocol for certain class of ten-qubit state by utilizing an eight-qubit entangled state as a quantum channel has been proposed. In this paper, we present an optimal scheme for the teleportation of a ten-qubit state by using a stochastic local operation and classical communication(SLOCC) equivalent to four-qubit χ state as entanglement channel. Only von Neumann type measurement, controlled-not (CNOT) operations and appropriate unitary operations are needed in this scheme. Receiver Bob can reconstruct the initial state by introducing the appropriate unitary transformation and auxiliary particles.

By using a Wick symbol transform, a characterization theorem is proven for the white noise operators in terms of a new spaces of (p, q)-entire functions with certain growth rates determined by Young functions and a suitable (p, q)-exponential map. Next, we investigate the main aim of this paper which is to give unitary equivalent representations of the (p, q)-square oscillator white noise algebra in the basis of noncommutative white noise theory.

We characterize twisted convolutions associated with the Pedersen transform for unitary irreducible representations of nilpotent Lie groups. For $$1\le p

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

We give an integral formulation for the length frequencies in classical Coxeter groups. From this we derive a precise asymptotic for the peak of length frequency taken among elements in a Coxeter group. An interpretation of the peak of frequency in terms of an integral of characters afforded by tensor products of unitary representations.

While the creation and still outstanding implementation of the Unified Patent Jurisdiction system continue to cause contentious debates among academics and practitioners, the system introduced in the early 1990s for adjudication of conflicts concerning unitary trade marks (and, later on, unitary design rights) had the charm of simplicity. Building on the national court systems instead of providing for a genuine EU judiciary, the scheme could easily be fitted into existing structures. On the other hand, the imperfect, limping character of the system creates issues inter alia concerning international jurisdiction and applicable law. Furthermore, attributing jurisdiction in infringement litigation to separate national court hierarchies may jeopardize the coherence of the system. While the prospects for reform may be dim insofar as the system in its entirety is concerned, the detrimental effects of fragmentation could be overcome at least to some extent by creating a unitary catalogue of sanctions.

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

Quantum secret sharing (QSS) provides an efficient way for managing keys securely. Generally, the dishonest participant is a stronger attacker than others in QSS, and then the participant attack should be particularly analyzed. In this paper, the security of two QSS protocols based on GHZ states and local measurements has been further reviewed. Using the participant attack model, Hillery et al. (Phys. Rev. A 59, 1829, 1999) and general HBB-like protocols are analyzed, and they are both insecure: the dishonest participant can restore all of the key without being detected just using local unitary operation and local measurements. Even inserting some nonorthogonal states in the phase of distributing particles, the dishonest participant can also restore all of the key without being detected just using local unitary operation and local measurements. Two key factors of insecurity are deduced: the dishonest participant and the sender have the same ability to control GHZ states, and there are no monitoring strategies in the secret recovery phase. Therefore, how to design multi-party key distribution protocols fixing the two security vulnerabilities is meaningful.

Quantum secret sharing (QSS) protocols are designed to allow a secret message to be divided into several shadows, and the secret can be reconstructed cooperatively. Motivated by the eight quantum entangled (EQE) state, a tripartite verifiable QSS scheme is proposed. It involves three agents, in four phases, i.e., shadow distribution phase, eavesdropping detection phase, measurement and secret sharing phase, verification and secret recover phase. The protocol can not only share and recover the secret correctly, but also verify whether there are dishonest participants in communication. In addition, the protocol is extended to a multiparty quantum secret sharing scheme by utilizing the unitary operations. Security analysis shows that the proposed schemes achieve the fundamental security requirements against the external and internal attackers with the aid of the quantum decoy states. It is shown that the efficiency in classical and quantum transmission is higher than the other protocols.

Sefer Ḥasidim (The Book of the Pious) has long served as a crucial source for medieval Jewish historiography. Yet the dual question of who composed the anonymous text and how its varying recensions came into existence has been a contentious one among scholars of medieval Ashkenaz. In particular, opinions have been split on the issue of the book’s authorship. Ever since the 1538 publication of the editio princeps, Judah he-Ḥasid (“the Pious,” d. 1217) has been credited as the work’s singular “author,” but in the intervening years numerous theories of composite authorship have been proposed as well. The present article reassesses notions of “authorship” in medieval Ashkenaz and does so in dialogue with Ivan Marcus’s recent Sefer Ḥasidim and the Ashkenazic Book in Medieval Europe (2018), a work that seeks to deconstruct the reductive category of unitary “books” in medieval Ashkenaz, but which simultaneously reifies Judah’s self-conscious “authorial identity.” In contrast, I argue on methodological and conceptual grounds that “authorship” is a problematic category in medieval Ashkenazic culture and suggest that in the case of Sefer Ḥasidim there are textual reasons to doubt that a single individual (Judah he-Ḥasid or anyone else) was solely responsible for “authoring” the text in its entirety.

The MLS conjecture states that every finite simple group has a minimal logarithmic signature. The aim of this paper is proving the existence of a minimal logarithmic signature for some simple unitary groups PSUn(q). We report a gap in the proof of the main result of Hong et al. (Des. Codes Cryptogr. 77: 179–191, 2015) and present a new proof in some special cases of this result. As a consequence, the MLS conjecture is still open.

We compare two different cohomology groups on unitary Shimura curves over totally real number fields; these groups are used in the construction of p-adic families of modular forms in this context. More precisely, we describe the space of finite slope overconvergent modular forms of any weight in terms of a convenient overconvergent cohomology.

We present a protocol for controlled cyclic remote preparation of an arbitrary single-qudit state via a seven-qudit cluster state. In the protocol, Alice can help the remote agent Bob prepare an arbitrary single-qudit state, Bob can help the agent Charlie prepare an arbitrary single-qudit state and at the same time Charlie can help Alice prepare an arbitrary single-qudit state under the controller David’s control. Alice, Bob and Charlie first perform positive operator-valued measurement (POVM) on their entangled particles according to the information of the prepared state, then perform generalized X-basis measurement. The controller performs generalized X-basis measurement on his entangled particle. The arbitrary single-qudit states can be cyclic remote prepared under the controller’s control. The protocol is more convenient in application since it only requires single-particle measurement and single-particle unitary operations for controlled cyclic remote preparation of the single-qudit states.

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

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 document a nonparametric test for weak separability between time use and consumption goods applied to unique time-use data on Belgian singles and households. The test outcomes reject the hypothesis of weak separability, indicating that household consumption decisions are not made independent from time-use allocations. Second, we conduct a nonparametric evaluation of unitary and collective rationality tests under a comprehensive setting studying choices on both private and public consumption in combination with time spent on labour supply and domestic work. This makes the analysis less vulnerable to separability issues and obtains a more realistic description of household trade-offs between consumption and time use (i.e. leisure and domestic work) than does the existing work. Moreover, our analysis uses nonparametric revealed preference tools and abstains from imposing functional form assumptions on the individual utilities and the household decision process. The outcomes confirm previous findings validating the use of collective models to analyse multi-member decision making. Lastly, we show the robustness of our outcomes to settings typically considered in existing work (missing detailed time-use information).

The unitary group U(n) is a symmetric space and has the point-symmetry for every point $$x\in U(n)$$ . A great antipodal set on U(n) is a “good” finite subset of U(n) related to the point-symmetries. On the other hand, a great antipodal set on U(n) is an analogue of a pair of antipodal points on spheres. It is known that a finite subset of a sphere is a tight spherical 1-design if and only if it is a pair of antipodal points. In this paper, we investigate a relation between great antipodal sets on U(n) and design theory on U(n). Moreover, we give a relation between a great antipodal set on U(n) and a Hamming cube graph $${\mathcal {Q}}_n$$ .

BB84-state is the non-orthogonal single-photon state which has the advantage of easy implementation compared with the quantum multi-photon entanglement states. In this paper, based on BB84-state, by introducing a trusted third-party voting center, a quantum voting scheme is proposed. In this scheme, by performing corresponding unitary operation on BB84-state, all voters send their voting information to the tallyman Charlie, then Charlie counts all votes under the supervision of voting management center Bob, which ensures that the protocol can resist inside attacks. Moreover, by utilizing the decoy particles, our scheme can efficiently prevent outside attacks. Compared with other related quantum voting protocols, our protocol has higher qubit efficiency and fewer interactive times.

The special affine Fourier transform (SAFT) generalizes a number of well known unitary transformations, signal processing transformations and optics related mathematical operations. For example, Fourier transform, fractional Fourier transform, linear canonical transform, etc. The short-time special affine Fourier transform (STSAFT) is a novel time-frequency analysis tool, which solve the limitation of SAFT in time-frequency signal description. In this paper, we generalize some different uncertainty principles (UPs) for the SAFT and STSAFT of complex signals. The UPs for the STSAFT of complex signals are obtained in both local and global cases. Then the uncertainty principle for two STSAFT time-frequency domains is discussed. Finally, we provide a class of signals to illustrate our results.

An answer is obtained to a question posed in 2008 by V. Lyubashevsky about an efficient algorithm for calculating the parameter θ(f) that characterizes the value of the sup-norm of the product of elements of the ring of truncated polynomials modulo a given unitary polynomial f (x) with real coefficients. The results are used to estimate the decryption failure probability in NTRU-like cryptosystems.

We initiate a study of involutions in the setting of complex quasi Jordan algebras and discuss the notions of self-adjoint and unitary elements; besides other results, we also obtain a Russo-Dye type theorem for unital involutive split quasi Jordan Banach algebras.

Security, efficiency and universality are the major concerns in distributed computation for how to communicate securely as there are a large number of nodes that require interactive authentication. These become more challenging especially for the multi-party quantum environment as it is highly complicated and heterogeneous in nature. Hence, based on bell states and unitary operation, we propose a multi-party quantum group key distribution protocol. The initial two participants are generated by election algorithm to ensure the fairness among the participants in our protocol. Next, mutual authentication is provided by two Bell states of four orthogonal maximal states, and the group session key is obtained by combining unitary operation. The main innovations of this scheme are that the authenticated information with the group key is transmitted among the ordered participants, and the efficiency of our protocol is acceptable which the time complexity of QKD is $$ O\left(\left\lceil {\mathit{\log}}_2^N\right\rceil \right) $$ with the N-party. Through security analysis and efficiency analysis, the protocol can effectively resist internal attacks and external attacks, and has feasible efficiency.

A novel practical quantum secure direct communication (QSDC) protocol is constructed by utilizing the evolution law of atom via cavity QED. It transmits the atoms from one communicant to the other communicant in a two-step manner. It adopts two-atom product states as the initial quantum resource and the decoy atom technique to guarantee the security against the outside attack. It only needs single-atom measurements, and neither the quantum entanglement swapping nor the unitary operations.

Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect nontrivially. We prove that the diameter of $\Delta_G$ has a tight upper bound of 5, thereby resolving a question posed by Shen (2010). Furthermore, a diameter of 5 is achieved only by the baby monster group and certain unitary groups of odd prime dimension., Comment: 8 pages. Corrected minor non-mathematical typos

In this note we point out an error in a recent paper on flag-transitive block designs with gcd $$(r,\lambda )=1$$ , and we prove a result correcting the error.

The history of the development of today’s very dependable three-piece inflatable penile prostheses is fascinating. In its infancy, the three piece was plagued with frequent revisions and a relatively complex insertion and consequently unitary and two-piece prostheses flourished with the implanting urologists. While the surgery was less difficult because these devices did not require the dreaded reservoir insertion, they often proved unsatisfactory to patients in flaccidity, erection and longevity. By the turn of this century all of the unitary and two-piece inflatables had been withdrawn from the market except the Ambicor two-piece inflatable penile prosthesis. This paper covers the history of the various unitary and 2-piece inflatable models before examining a critical question: is the Ambicor 2-piece still a needed implant today?

We give a brief overview of the black hole information problem and describe qualitatively the recent research that shows that the von Neumann entropy of Hawking radiation follows the Page curve, and hence is consistent with unitary, information preserving time evolution.

Prototype-based models like the Generalized Learning Vector Quantization (GLVQ) belong to the class of interpretable classifiers. Moreover, quantum-inspired methods get more and more into focus in machine learning due to its potential efficient computing. Further, its interesting mathematical perspectives offer new ideas for alternative learning scenarios. This paper proposes a quantum computing-inspired variant of the prototype-based GLVQ for classification learning. We start considering kernelized GLVQ with real- and complex-valued kernels and their respective feature mapping. Thereafter, we explain how quantum space ideas could be integrated into a GLVQ using quantum bit vector space in the quantum state space$${\mathcal {H}}^{n}$$Hnand show the relations to kernelized GLVQ. In particular, we explain the related feature mapping of data into the quantum state space$${\mathcal {H}}^{n}$$Hn. A key feature for this approach is that$${\mathcal {H}}^{n}$$Hnis an Hilbert space with particular inner product properties, which finally restrict the prototype adaptations to be unitary transformations. The resulting approach is denoted as Qu-GLVQ. We provide the mathematical framework and give exemplary numerical results.

We consider continuous unitary quantum walks of a single particle and of closed systems and discuss more realistic approaches to modeling quantum walks.

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.

To detect frauds from some internal participants or external attackers, some verifiable threshold quantum secret sharing schemes have been proposed. In this paper, we present a new verifiable threshold structure based on a single qudit using bivariate polynomial. First, Alice chooses an asymmetric bivariate polynomial and sends a pair of values from this polynomial to each participant. Then Alice and participants implement in sequence unitary transformation on the d-dimensional quantum state based on unbiased bases, where those unitary transformations are contacted by this polynomial. Finally, security analysis shows that the proposed scheme can detect the fraud from external and internal attacks compared with the exiting schemes and is comparable to the recent schemes.

Recently, various control problems for quantum systems, such as individual atoms, molecules, and electron states at quantum points, have been actively researched. In this paper, we briefly discuss methods of constructing controls for quantum system by means of gradient algorithms, genetic algorithms, and the speed gradient. The violation of the asymptotic stabilizability condition for the problem of generation of unitary operations is proved by the speed gradient method in two-level quantum systems.

Consider the free orthogonal quantum groups $O_N^+(F)$ and free unitary quantum groups $U_N^+(F)$ with $N \geq 3$. In the case $F = {\rm id}_N$ it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $L_\infty(O_N^+)$ is strongly solid. Moreover, Isono obtains strong solidity also for $L_\infty(U_N^+)$ . In this paper we prove for general $F \in GL_N(\mathbb{C})$ that the von Neumann algebras $L_\infty(O_N^+(F))$ and $L_\infty(U_N^+(F))$ are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani--Sauvageot., Comment: Accepted for Mathematische Annalen

It is almost universally believed that in quantum theory the two following statements hold: 1) all transformations are achieved by a unitary interaction followed by a von Neumann measurement; 2) all mixed states are marginals of pure entangled states. I name this doctrine the dogma of purification ontology. The source of the dogma is the original von Neumann axiomatisation of the theory, which largely relies on the Schroedinger equation as a postulate, which holds in a nonrelativistic context, and whose operator version holds only in free quantum field theory, but no longer in the interacting theory. In the present paper I prove that both ontologies of unitarity and state purity are unfalsifiable, even in principle, and therefore axiomatically spurious. I propose instead a minimal four-postulate axiomatisation: 1) associate a Hilbert space HA to each system A; 2) compose two systems by the tensor product rule HAB = HAxHB; 3) associate a transformation from system A to B to a quantum operation, i.e. to a completely positive trace-non-increasing map between the trace-class operators of A and B; 4) (Born rule) evaluate all joint probabilities through that of a special type of quantum operation: the state preparation. I then conclude that quantum paradoxes-such as the Schroedinger-cat's, and, most relevantly, the information paradox-are originated only by the dogma of purification ontology, and they are no longer paradoxes of the theory in the minimal formulation. For the same reason, most interpretations of the theory (e.g. many-world, relational, Darwinism, transactional, von Neumann-Wigner, time-symmetric, ...) interpret the same dogma, not the strict theory stripped of the spurious postulates., Comment: To be published on Foundations of Physics

The skew information has important applications in quantum theory. In this paper, a measure for correlations in terms of the skew information is introduced and discussed. Its fundamental properties are obtained, which include the measure is locally unitary invariant and convex on the set of states with the fixed marginal state, moreover, it decreases under local random unitary operation. Lastly, we propose an entanglement criterion in terms of the quantum skew information and illustrate the significance of this criterion by some examples.