Unitary Basis Research Articles

Mixed Quantum-Classical Dynamics under Arbitrary Unitary Basis Transformations.

A common approach to minimizing the cost of quantum computations is by unitarily transforming a quantum system into a basis that can be optimally truncated. Here, we derive classical equations of motion subjected to similar unitary transformations and propose their integration into mixed quantum-classical dynamics, allowing this class of methods to be applied within arbitrary bases for both the quantum and classical coordinates. To this end, canonical positions and momenta of the classical degrees of freedom are combined into a set of complex-valued coordinates amenable to unitary transformations. We demonstrate the potential of the resulting approach by means of surface hopping calculations of an electronic carrier scattering onto a single impurity in the presence of phonons. Appropriate basis transformations, capturing both the localization of the impurity and the delocalization of higher-energy excitations, are shown to faithfully capture the dynamics within a fraction of the classical and quantum basis sets.

On bounded coordinates in bimodules

We provide a comprehensive study on uniformly left [Formula: see text]-bounded (respectively, [Formula: see text]-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are faithful normal states on [Formula: see text] and [Formula: see text], respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.

Orthogonal unitary bases and a subfactor conjecture

We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of M n ( C ) M_n(\mathbb {C}) admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta [Glasg. Math. J. 64 (2022), pp. 586–602], showing that any finite-index regular inclusion N ⊆ M N\subseteq M of I I 1 II_1 -factors admits an orthonormal unitary Pimsner-Popa basis.

Put crnogorske nacije od političke do etničke nacije

In the article we discuss the process of formation of the Montenegrin nation. Our aim is to prove that the key moment, which will decisively influence the subsequent formation of the Montenegrin ethnic nation, is the recognition of the state of Montenegro at the Berlin Congress in 1878. The act marks the formation of Montenegrin political nation, the nation with its own state. Hence, we have come to a conclusion that the formation of Montenegrin political nation had a decisive influence on the future formation of Montenegrin ethnic nation and the attempts to characterise the Montenegrin ethnic nation with elements that make up a nation. The Montenegrins who express their allegiance to the Montenegrin ethnic nation also cherish the state of Montenegro. Nonetheless, the citizens of Montenegro who declare themselves as belonging to the Serbian ethnic nation also regard the state of Montenegro as sacred. Thus, they strongly defend their allegiance to the Montenegrin political nation but contrary to ethnic Montenegrins, they claim that Montenegro is a Serbian state. In that sense, the Serbs from Montenegro envision (possible) new unification with Serbia on a federal basis and not on a unitary basis. We have already witnessed that such a state, which existed from 1992 until 2006 under the name of the Federal Republic of Yugoslavia and, subsequently, Serbia and Montenegro, has no future. By not accepting (possible) unification on a unitary but on a federal basis,, Serbs in Montenegro do not realize that due to such an attitude they are at risk of becoming members of Montenegrin ethnic nation rather than pleading allegiance to the Montenegrin political nation.

Unextendible product operator basis

Quantum nonlocality is associated with the local indistinguishability of orthogonal states. Unextendible product basis (UPB), a widely used tool in quantum information, exhibits nonlocality, which is the powerful resource for quantum information processing. In this work, we extend the definitions of nonlocality and genuine nonlocality from states to operators. We also extend UPB to the notions of unextendible product operator basis, unextendible product unitary operator basis (UPUOB), and strongly UPUOB. We construct their examples and show the nonlocality of some strongly UPUOBs under local operations and classical communications. We study the phenomenon of these operators acting on quantum states. As an application, we distinguish the two-dimensional strongly UPUOB, which only consumes three ebits of entanglement. Our results imply that such UPUOBs exhibit nonlocality as UPBs, and the distinguishability of them requires entanglement resources.

Quantum Cuntz-Krieger algebras

Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^\ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ C^\ast -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to K K KK -equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C ∗ C^\ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K KK -theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.

Schrieffer-Wolff transformations for experiments: Dynamically suppressing virtual doublon-hole excitations in a Fermi-Hubbard simulator

In strongly interacting systems with a separation of energy scales, low-energy effective Hamiltonians help provide insights into the relevant physics at low temperatures. The emergent interactions in the effective model are mediated by virtual excitations of high-energy states: For example, virtual doublon-hole excitations in the Fermi-Hubbard model mediate antiferromagnetic spin-exchange interactions in the derived effective model, known as the $t-J-3s$ model. Formally this procedure is described by performing a unitary Schrieffer-Wolff basis transformation. In the context of quantum simulation, it can be advantageous to consider the effective model to interpret experimental results. However, virtual excitations such as doublon-hole pairs can obfuscate the measurement of physical observables. Here we show that quantum simulators allow one to access the effective model even more directly by performing measurements in a rotated basis. We propose a protocol to perform a Schrieffer-Wolff transformation on Fermi-Hubbard low-energy eigenstates (or thermal states) to dynamically prepare approximate $t-J-3s$ model states using fermionic atoms in an optical lattice. Our protocol involves performing a linear ramp of the optical lattice depth, which is slow enough to eliminate the virtual doublon-hole fluctuations but fast enough to freeze out the dynamics in the effective model. We perform a numerical study using exact diagonalization and find an optimal ramp speed for which the state after the lattice ramp has maximal overlap with the $t-J-3s$ model state. We compare our numerics to experimental data from our Lithium-6 fermionic quantum gas microscope and show a proof-of-principle demonstration of this protocol. More generally, this protocol can be beneficial to studies of effective models by enabling the suppression of virtual excitations in a wide range of quantum simulation experiments.

On-manifold probabilistic Iterative Closest Point: Application to underwater karst exploration

This paper proposes MpIC, an on-manifold derivation of the probabilistic Iterative Correspondence (pIC) algorithm, which is a stochastic version of the original Iterative Closest Point. It is developed in the context of autonomous underwater karst exploration based on acoustic sonars. First, a derivation of pIC based on the Lie group structure of is developed. The closed-form expression of the covariance modeling the estimated rigid transformation is also provided. In a second part, its application to 3D scan matching between acoustic sonar measurements is proposed. It is a prolongation of previous work on elevation angle estimation from wide-beam acoustic sonar. While the pIC approach proposed is intended to be a key component in a Simultaneous Localization and Mapping framework, this paper focuses on assessing its viability on a unitary basis. As ground truth data in karst aquifer are difficult to obtain, quantitative experiments are carried out on a simulated karst environment and show improvement compared to previous state-of-the-art approach. The algorithm is also evaluated on a real underwater cave dataset demonstrating its practical applicability.

A few remarks on Pimsner–Popa bases and regular subfactors of depth 2

AbstractWe prove that a finite index regular inclusion of$II_1$-factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of$II_1$-factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

Quantum space, quantum time, and relativistic quantum mechanics

We treat space and time as bona fide quantum degrees of freedom on an equal footing in Hilbert space. Motivated by considerations in quantum gravity, we focus on a paradigm dealing with linear, first-order Hamiltonian and momentum constraints that lead to emergent features of temporal and spatial translations. Unlike the conventional treatment, we show that Klein-Gordon and Dirac equations in relativistic quantum mechanics can be unified in our paradigm by applying relativistic dispersion relations to eigenvalues rather than treating them as operator-valued equations. With time and space being treated on an equal footing in Hilbert space, we show symmetry transformations to be implemented by unitary basis changes in Hilbert space, giving them a stronger quantum mechanical footing. Global symmetries, such as Lorentz transformations, modify the decomposition of Hilbert space; and local symmetries, such as $U(1)$ gauge symmetry are diagonal in coordinate basis and do not alter the decomposition of Hilbert space. We briefly discuss extensions of this paradigm to quantum field theory and quantum gravity.

Optimal quantum tomography with constrained measurements arising from unitary bases

The purpose of this paper is to introduce techniques of obtaining optimal ways to determine a [Formula: see text]-level quantum state or distinguish such states. It entails designing constrained elementary measurements extracted from maximal abelian subsets of a unitary basis [Formula: see text] for the operator algebra [Formula: see text] of a Hilbert space [Formula: see text] of finite dimension [Formula: see text] or, after choosing an orthonormal basis for [Formula: see text], for the ⋆-algebra [Formula: see text] of complex matrices of order [Formula: see text]. Illustrations are given for the techniques. It is shown that the Schwinger basis [Formula: see text] of unitary operators can give for [Formula: see text], a product of primes [Formula: see text] and [Formula: see text], the ideal number [Formula: see text] of rank one projectors that have a few quantum mechanical overlaps (or, for that matter, a few angles between the corresponding unit vectors). Finally, we give a combination of the tensor product and constrained elementary measurement techniques to deal with all [Formula: see text], though with more overlaps or angles depending on the factorization of [Formula: see text] as a product of primes or their powers like [Formula: see text] with [Formula: see text], all primes, [Formula: see text] for [Formula: see text], or other types. A comparison is drawn for different forms of unitary bases for the Hilbert space factors of the tensor product like [Formula: see text] or [Formula: see text], where [Formula: see text] is the Galois field of size [Formula: see text] and [Formula: see text] is the ring of integers modulo [Formula: see text]. Even though as Hilbert spaces they are isomorphic, but quantum mechanical system-wise, these tensor products are different.In the process, we also study the equivalence relation on unitary bases defined by R. F. Werner [J. Phys. A: Math. Gen. 34 (2001) 7081–7094], connect it to local operations on maximally entangled vectors bases, find an invariant for equivalence classes in terms of certain commuting systems, called fan representations, and, relate it to mutually unbiased bases and Hadamard matrices. Illustrations are given in the context of Latin squares and projective representations as well.

The Role of Redundant Bases and Shrinkage Functions in Image Denoising.

Wavelet denoising is a classical and effective approach for reducing noise in images and signals. Suggested in 1994, this approach is carried out by rectifying the coefficients of a noisy image, in the transform domain, using a set of shrinkage functions (SFs). A plethora of papers deals with the optimal shape of the SFs and the transform used. For example, it is widely known that applying SFs in a redundant basis improves the results. However, it is barely known that the shape of the SFs should be changed when the transform used is redundant. In this paper, we introduce a complete picture of the interrelations between the transform used, the optimal shrinkage functions, and the domains in which they are optimized. We suggest three schemes for optimizing the SFs and provide bounds of the remaining noise, in each scheme, with respect to the other alternatives. In particular, we show that for subband optimization, where each SF is optimized independently for a particular band, optimizing the SFs in the spatial domain is always better than or equal to optimizing the SFs in the transform domain. Furthermore, for redundant bases, we provide the expected denoising gain that can be achieved, relative to the unitary basis, as a function of the redundancy rate.

The Role of the Medial Prefontal Cortex in Self-Agency in Schizophrenia.

Schizophrenia is a disorder of the self. In particular, patients show cardinal deficits in self-agency (i.e., the experience and awareness of being the agent of one’s own thoughts and actions) that directly contribute to positive psychotic symptoms of hallucinations and delusions and distort reality monitoring (defined as distinguishing self-generated information from externally-derived information). Predictive coding models suggest that the experience of self-agency results from a minimal prediction error between the predicted sensory consequence of a self-generated action and the actual outcome. In other words, the experience of self-agency is thought to be driven by making reliable predictions about the expected outcomes of one’s own actions. Most of the agency literature has focused on the motor system; here we present a novel viewpoint that examines agency from a different lens using distinct tasks of reality monitoring and speech monitoring. The self-prediction mechanism that leads to self-agency is necessary for reality monitoring in that self-predictions represent a critical precursor for the successful encoding and memory retrieval of one’s own thoughts and actions during reality monitoring to enable accurate self-agency judgments (i.e., accurate identification of self-generated information). This self-prediction mechanism is also critical for speech monitoring where we continually compare auditory feedback (i.e., what we hear ourselves say) with what we expect to hear. Prior research has shown that the medial prefrontal cortex (mPFC) may represent one potential neural substrate of this self-prediction mechanism. Unfortunately, patients with schizophrenia (SZ) show mPFC hypoactivity associated with self-agency impairments on reality and speech monitoring tasks, as well as aberrant mPFC functional connectivity during intrinsic measures of agency during resting states that predicted worsening psychotic symptoms. Causal neurostimulation and neurofeedback techniques can move the frontiers of schizophrenia research into a new era where we implement techniques to manipulate excitability in key neural regions, such as the mPFC, to modulate patients’ reliance on self-prediction mechanisms on distinct tasks of reality and speech monitoring. We hypothesize these findings will show that mPFC provides a unitary basis for self-agency, driven by reliance on self-prediction mechanisms, which will facilitate the development of new targeted treatments in patients with schizophrenia.

Rigidity and a common framework for mutually unbiased bases and k‐nets

AbstractMany deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called ‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a ‐net over a ‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) ‐nets, while the choice of leads to collections of MUBs. In this framework, we derive a rigidity property which hence automatically applies to both objects. For ‐nets that can be completed to affine planes, this was already known by a completely different, combinatorial argument. For ‐nets that cannot be completed and for MUBs, this result is new, and it implies that the only vectors unbiased to all but bases of a complete collection of MUBs in are the elements of the remaining bases (up to phase factors). Further, we show that this bound is tight with counterexamples for in every prime‐square dimension. Applying our rigidity result, we prove that if a large enough collection of MUBs constructed from a certain unitary error basis (like, the generalized Pauli operators) can be extended to a complete system, then every basis of the completion must come from the same error basis. In turn, we use this to show that certain large systems of MUBs cannot be completed.

Proper Orthogonal Decomposition of High-Speed Particle Image Velocimetry in an Open Cavity

Two-dimensional velocity data of flow over an open cavity acquired by pulse-burst particle image velocimetry at 37.5 kHz for freestream Mach numbers of 0.6, 0.8, and 0.94 have been analyzed using proper orthogonal decomposition (POD). The snapshot application of the POD resulted in modes consistent with previous studies on cavity flows under similar conditions. The time-resolved nature of velocity datasets also allowed for the application of the classical POD in both space and time which, for a stationary flow, is solved in the frequency domain. Eigenvalue spectra and convergence plots revealed that these POD modes are organized by capturing the maximum energy associated with dominant frequencies. The extracted eigenvectors revealed a strong dependency on frequency, with the most dominant modes corresponding to frequencies associated with Rossiter modes known to dominate open cavities. Mach number comparisons from the classical POD application reveal that their modes efficiently capture energy of dominant frequencies associated with the flow dynamics, regardless of the freestream conditions. Using a similarity parameter between spatial unitary basis functions, it was shown that the modes at different Mach numbers were quantitatively similar at the same Strouhal and POD mode numbers.

Gas flow measuring system using signal processing on the basis of entropy estimations

Purpose. To increase the accuracy of gas flow measurement in tachometric transducers based on the improvement of structural, hardware and algorithmic support of information and measuring systems. Methodology. The gas consumption value is determined by the parameters of information and measurement signals. Sensor signals interacting with the environment are traditionally processed on the basis of amplitude and frequency methods. The research methodology is based on the information theory, methods of statistical and spectral analysis, digital signal processing, the theory of gas dynamics, based on mathematical modeling in a computational experiment, as well as the theory of errors and measurement results uncertainty. The statistical characteristics of the measuring signals of the converter presented in the unitary basis are studied. Findings. The conducted research resulted in development of an information-measuring system to control the sensitivity threshold of the transducers of the primary volume and the volume of gas consumption based on the developed primary transducer, which allows providing relative standard uncertainty of cost determination within 0.5%. A special processor has been developed to calculate the entropy estimates of signal information. Originality. For the first time, a method for the formation and processing of information-measuring signals, which is based on the use of pressure pulsations due to the movement of the measuring element of the converter in the toroidal measuring cell, is proposed. Implementation of the measuring element of a spherical converter, whose density is almost commensurable with the density of the controlled medium is offered. Practical value. The proposed method allows providing a lower sensitivity threshold compared to the industrial implementation of tachometric type transducers.

Quantum teleportation with infinite reference-frame uncertainty

We present two new schemes for quantum teleportation between parties whose local reference frames are misaligned by the action of a compact Lie group G. These schemes require no prior alignment of reference frames and are unaffected by arbitrary changes in reference frame alignment during execution, suiting them to situations of rapid reference frame drift. Our tight scheme yields improved purity compared to standard teleportation, in some cases substantially --- this includes the case of qubit teleportation under arbitrary SU(2) reference frame uncertainty--- while communicating no information about either party's reference frame alignment at any time. Our perfect scheme performs perfect teleportation, but does communicate some reference frame information. The mathematical foundation of these schemes is a unitary error basis permuted up to a phase by the conjugation action of a finite subgroup of G.

Notice 2019-30, 2019-2020 Priority Guidance Plan – Sourcing of Cloud Services Income

Proposed Reg §1.861-19 was recently issued to define cloud services income and its most typical classification as a service. Subsequently, it was reported that Treasury was looking for tax community input on the need for income sourcing rules for cloud services income. This submission to Treasury and the IRS establishes the need and importance of clear sourcing rules for cloud services income. Further, it provides suggestions for how such sourcing rules might be structured. In particular, it suggests that such rules be applied on a unitary basis rather than the traditional separate-entity basis on which most tax rules are applied. It also suggests that the location of DEMPE functions (development, enhancement, maintenance, protection, and exploitation) must be an important factor in the determination of income source. The ongoing and quickly developing international effort orchestrated by the OECD and the Inclusive Framework to revamp the application of national taxing rights make new sourcing rules a matter that is critical for Treasury and the IRS to now consider.

An integrated model of materno-foetal cardiac dysfunction in severe pre-eclampsia.

Maternal cardiovascular deterioration in severe pre-eclampsia is due to a combination of factors in the setting of severe trophoblastic ischaemia and the outpouring of maternal cathecolamines, leading to increased left ventricular afterload and increasing ventricular volumes, resulting in increased left ventricular stroke work and demand myocardial ischaemia. This is the substrate for ventricular arrhythmias. Foetal cardiac dysfunction is most likely on the basis of the increased afterload, consequent upon widespread vasoconstriction, due to angiogenic imbalances. In this integrated model, chronic trophoblastic ischaemia is the central role player by releasing vasoactive substances that induce haemodynamic alterations in the maternofoetal complex, augmented and modified by 'latent' maternal cardiovascular dysfunction and increased maternal cathecolamine secretion on the one hand, and altered foetal signalling mechanisms on the other, all three components of the materno-placental-foetal complex being in constant interaction with each other. This unified hypothesis may explain the development of both maternal and foetal morbidity and/or mortality on a unitary basis in severe, complicated preeclampsia.

English

We present an infinite number of construction schemes involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary connections, algebraic objects which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.