D-dimensional System Research Articles

A novel quantum security multi-party extremum protocol in a d-dimensional quantum system

Secure multi-party extremum computation (SMEC) is a specific application scenario of secure multi-party computation, which allows multiple participants to compute the extremum of data without disclosing private information. The extremum includes maximum, minimum, sum of extremums, and difference of extremums. SMEC has wide applications in financial transactions, market analysis, sports events, healthcare, etc. Current protocol research mainly exists in the classical domain and cannot withstand quantum computing attacks. To address this issue, we propose a novel QSME protocol based on a d-dimensional quantum system, capable of computing the maximum and minimum values among multi-party data under unconditional security, and can compute the sum and difference of extremums without disclosing the maximum and minimum values, to adapt to complex application scenarios. The article proposes a coding method for a d-dimensional quantum system to further enhance security, provides correctness analysis, security analysis, robustness analysis, and comparative analysis, and proposes an experimental method for a d-dimensional quantum system to verify the effectiveness of the protocol, demonstrating strong practicality.

The iteration formula of (n, 2, d) full-correlated multi-component Bell function and its applications

No matter for a scientific or technological reason, constructing Bell inequalities for multi-partite and high-dimensional systems is a significant task. The Mermin-Ardehali-Belinskiĭ-Klyshko (MABK) inequality is expressed in the form of iteration formula and correlation functions, and is the generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multi-partite cases. In the sense of detecting the nonlocality of the noisy maximally entangled states with the most white noises for the corresponding quantum system, the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality, expressed in the joint probabilities form, is the high-dimensional analogue of the Clauser-Horne (CH) inequalities on joint probabilities. In the sense of detecting the nonlocality of the noisy Greenberger-Horne-Zeilinger(GHZ) states with the most white noises for the n-qudit system, it is a challenging task to construct the multi-partite analogue of the CGLMP inequality with iteration formula and correlation functions. In this paper, we generalize the multi-component correlation functions [Phys. Rev. A 71, 032 107 (2005)] for bipartite d-dimensional systems to n-partite d-dimensional ones, introduce the general full-correlated multi-component Bell function In,d , and construct the corresponding Bell inequality. By this way, we can reproduce both the CGLMP inequality and the Bell inequality in [Phys. Rev. A 71, 032 107 (2005)] for the case n = 2, and the MABK inequality for the case d = 2. Inspired by the iteration formula form of the MABK inequality, we prove that for prime d the general Bell function In,d can be reformulated by iterating two Bell functions In−1,d . As applications, for prime d, confined to the unbiased symmetric (d × 2)-port beam splitters and the noisy n-qudit GHZ states, we recover the most robust Bell inequalities for small n and d, such as for the (3, 2, 3), (4, 2, 3), (5, 2, 3), and (3, 2, 5) Bell scenarios, with the iteration formula and the most robust Bell inequalities for the (2, 2, d) scenario. This implies that the iteration formula is an efficient way of constructing the multi-partite analogues of the CGLMP inequality with correlation functions. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.

Thermodynamic Bounds on Generalized Transport: From Single-Molecule to Bulk Observables.

We prove that the transport of any differentiable scalar observable in d-dimensional nonequilibrium systems is bounded from above by the total entropy production scaled by the amount the observation "stretches" microscopic coordinates. The result-a time-integrated generalized speed limit-reflects the thermodynamic cost of transport of observables, and places underdamped and overdamped stochastic dynamics on equal footing with deterministic motion. Our work allows for stochastic thermodynamics to make contact with bulk experiments, and fills an important gap in thermodynamic inference, since microscopic dynamics is, at least for short times, underdamped. Requiring only averages but not sample-to-sample fluctuations, the proven transport bound is practical and applicable not only to single-molecule but also bulk experiments where only averages are observed, which we demonstrate by examples. Our results may facilitate thermodynamic inference on molecular machines without an obvious directionality from bulk observations of transients probed, e.g., in time-resolved x-ray scattering.

Nonlocality activation using local filtering operations based on CGLMP inequality

Entanglement is necessary but not sufficient to demonstrate nonlocality as there exist local entangled states which do not violate any Bell inequality. In recent years, the activation of nonlocality (known as hidden nonlocality) by using local filtering operations has gained considerable interest. In the original proposal of Popescu [Phys. Rev. Lett. 74, 2619 (1995)] the hidden nonlocality was demonstrated for the Werner class of states in d ≥ 5. In this paper, we demonstrate the hidden nonlocality for a class of mixed entangled states (convex mixture of a pure state and color noise) in an arbitrary d-dimensional system using suitable local filtering operations. For our demonstration, we consider the quantum violation of Collins-Linden-Gisin-Masser-Popescu (CGLMP) inequality which has hitherto not been considered for this purpose. We show that when the pure state in the aforementioned mixed entangled state is a maximally entangled state, the range of the mixing parameter for revealing hidden nonlocality increases with increasing the dimension of the system. Importantly, we find that for d ≥ 8, hidden non-locality can be revealed for the whole range of mixing parameter. Further, by considering another pure state, the maximally CGLMP-violating state, we demonstrate the activation of nonlocality by using the same local filtering operation. We have also shown the activation of quantum fidelity for mixed entangled states using the same local filtering operator.

Exact minimal effective amounts of three 1D continuous functions and their use in Anderson transitions

A conceptual quantity—the minimal effective amount of a quantum state ϕ(rj) in d-dimensional systems, defined by N∗=∑j=1Nmin{N|ϕ(rj)|2,1} , is newly proposed, where system sizes N=Ld . The effective dimension d IR can be calculated by N∗=h∗(L)LdIR , where h∗(L) does not change faster than any nonzero power. However, the nature of h∗(L) is unknown priori in any given model, but is at the same time very important for its numerical analysis. Hence, analytical results can provide insights on h∗(L) in more complex situations. In this paper, we get exact results of 1D continuous sine functions, exponential decay functions and power-law decay functions. They are used to distinguish extended and localized phases in the 1D uniform potential model, Anderson model and HMP (hopping rates modulated by a power-law function) model.

On unitary time evolution out of equilibrium

We consider d-dimensional quantum systems which for positive times evolve with a time-independent Hamiltonian in a nonequilibrium state that we keep generic in order to account for arbitrary evolution at negative times. We show how the one-point functions of local operators depend on the coefficients of the expansion of the nonequilibrium state on the basis of energy eigenstates. We express in this way the asymptotic offset and show under which conditions oscillations around this value stay undamped at large times. We also show how, in the case of small quenches, the structure of the general results simplifies and reproduces that known perturbatively.

High Schmidt Number Concentration in Quantum Bound Entangled States.

A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.

Finite-Time High-Probability Bounds for Polyak–Ruppert Averaged Iterates of Linear Stochastic Approximation

This paper provides a finite-time analysis of linear stochastic approximation (LSA) algorithms with fixed step size, a core method in statistics and machine learning. LSA is used to compute approximate solutions of a d-dimensional linear system [Formula: see text] for which [Formula: see text] can only be estimated by (asymptotically) unbiased observations [Formula: see text]. We consider here the case where [Formula: see text] is an a sequence of independent and identically distributed random variables sequence or a uniformly geometrically ergodic Markov chain. We derive pth moment and high-probability deviation bounds for the iterates defined by LSA and its Polyak–Ruppert-averaged version. Our finite-time instance-dependent bounds for the averaged LSA iterates are sharp in the sense that the leading term we obtain coincides with the local asymptotic minimax limit. Moreover, the remainder terms of our bounds admit a tight dependence on the mixing time [Formula: see text] of the underlying chain and the norm of the noise variables. We emphasize that our result requires the LSA step size to scale only with logarithm of the problem dimension d. Funding: The work of A. Durmus and E. Moulines was partly supported by [Grant ANR-19-CHIA-0002]. This project received funding from the European Research Council [ERC-SyG OCEAN Grant 101071601]. The research of A. Naumov and S. Samsonov was prepared within the framework of the HSE University Basic Research Program.

The breaking of Mermin-Wagner-Hohenberg's theorem for fractional systems

The Mermin-Wagner-Hohenberg theorem forbids the existence of long range order in two dimensional systems, since continuous symmetries cannot be spontaneously broken in d≤2. However, one of the premises for the validity of the theorem is that the energy-momentum dispersion of the quanta responsible for the order parameter fluctuations has a quadratic form, ε(k)=Ak2. In this paper we address the emergence of a fractional order energy-momentum dispersion in d-dimensional systems, leading to important consequences for the existence of long-range order in the case d=2. A first order renormalization group analysis in the ϵ-expansion is also performed in the fractional regime. It is shown that for fractional systems the universality classes are characterized by the number of spatial dimensions d and the number of components n of the order parameter, as in the case of integer order systems, but the labeling by the fractional order α is also required.

The complete new solutions to the blind millionaires’ problem in d-dimensional quantum system

The complete new solutions to the blind millionaires’ problem in d-dimensional quantum system

Characterizing an Uncertainty Diagram and Kirkwood-Dirac Nonclassicality Based on Discrete Fourier Transform.

In this paper, we investigate an uncertainty diagram and Kirkwood-Dirac (KD) nonclassicality based on discrete Fourier transform (DFT) in a d-dimensional system. We first consider the uncertainty diagram of the DFT matrix, which is a transition matrix from basis A to basis B. Here, the bases A, B are not necessarily completely incompatible. We show that for the uncertainty diagram of the DFT matrix, there is no "hole" in the region of the (nA,nB) plane above and on the line nA+nB=d+1. Then, we present where the holes are in the region strictly below the line and above the hyperbola nAnB=d. Finally, we provide an alternative proof of the conjecture about KD nonclassicality based on DFT.

Measurement incompatibility and quantum advantage in communication

Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems assisted with shared randomness is the same as the set obtained from d-dimensional quantum strategies restricted to compatible measurements with shared randomness in any communication scenario. Thus, measurement incompatibility is necessary for quantum advantage in communication, and any quantum advantage (with or without shared randomness) in communication acts as a witness to the incompatibility of the measurements at the receiver's end in a semi-device-independent way. We introduce a class of communication tasks - a general version of random access codes - to witness incompatibility of an arbitrary number of quantum measurements with arbitrary outcomes acting on d-dimensional systems, and provide generic upper bounds on the success metric of these tasks for compatible measurements. We identify all sets of three incompatible rank-one projective qubit measurements that random access codes can witness. Finally, we present the generic relationship between different sets of probability distributions - classical, quantum with or without shared randomness, and quantum restricted to compatible measurements with or without shared randomness - produced in communication scenarios.

Universal construction of decoders from encoding black boxes

Isometry operations encode the quantum information of the input system to a larger output system, while the corresponding decoding operation would be an inverse operation of the encoding isometry operation. Given an encoding operation as a black box from a d-dimensional system to a D-dimensional system, we propose a universal protocol for isometry inversion that constructs a decoder from multiple calls of the encoding operation. This is a probabilistic but exact protocol whose success probability is independent of D. For a qubit (d=2) encoded in n qubits, our protocol achieves an exponential improvement over any tomography-based or unitary-embedding method, which cannot avoid D-dependence. We present a quantum operation that converts multiple parallel calls of any given isometry operation to random parallelized unitary operations, each of dimension d. Applied to our setup, it universally compresses the encoded quantum information to a D-independent space, while keeping the initial quantum information intact. This compressing operation is combined with a unitary inversion protocol to complete the isometry inversion. We also discover a fundamental difference between our isometry inversion protocol and the known unitary inversion protocols by analyzing isometry complex conjugation and isometry transposition. General protocols including indefinite causal order are searched using semidefinite programming for any improvement in the success probability over the parallel protocols. We find a sequential "success-or-draw" protocol of universal isometry inversion for d=2 and D=3, thus whose success probability exponentially improves over parallel protocols in the number of calls of the input isometry operation for the said case.

Purifying teleportation

Coupling to the environment typically suppresses quantum properties of physical systems via decoherence mechanisms. This is one of the main obstacles in practical implementations of quantum protocols. In this work we show how decoherence effects can be reversed/suppressed during quantum teleportation in a network scenario. Treating the environment quantumly, we show that under a general pure dephasing coupling, performing a second teleportation step can probabilistically reverse the decoherence effects if certain commutativity conditions hold. This effect is purely quantum and most pronounced for qubit systems, where in 25 % of instances the decoherence can be reversed completely. As an example, we show the effect in a physical model of a qubit register coupled to a bosonic bath. We also analyze general d-dimensional systems, identifying all instances of decoherence suppression. Our results are proof-of-concept but we believe will be relevant for the emerging field of quantum networks as teleportation is the key building block of network protocols.

Cramér–Rao, Fisher–Shannon and LMC–Rényi Complexity-like Measures of Multidimensional Hydrogenic Systems with Application to Rydberg States

Statistical measures of complexity hold significant potential for applications in D-dimensional finite fermion systems, spanning from the quantification of the internal disorder of atoms and molecules to the information–theoretical analysis of chemical reactions. This potential will be shown in hydrogenic systems by means of the monotone complexity measures of Cramér–Rao, Fisher–Shannon and LMC(Lopez-Ruiz, Mancini, Calbet)–Rényi types. These quantities are shown to be analytically determined from first principles, i.e., explicitly in terms of the space dimensionality D, the nuclear charge and the hyperquantum numbers, which characterize the system’ states. Then, they are applied to several relevant classes of particular states with emphasis on the quasi-spherical and the highly excited Rydberg states, obtaining compact and physically transparent expressions. This is possible because of the use of powerful techniques of approximation theory and orthogonal polynomials, asymptotics and generalized hypergeometric functions.

Limit cycles in piecewise polynomial Hamiltonian systems allowing nonlinear switching boundaries

This paper aims to study the limit cycles of planar piecewise polynomial Hamiltonian systems of degree n with the switching boundary y=xm, where m and n are positive integers. We answer a version of the 16th Hilbert problem for such systems, providing an upper bound for the maximum number of limit cycles in the function of m and n. We also are devoted to giving a lower bound via perturbing piecewise linear Hamiltonian systems having at the origin a global center. For this, a complete classification of the center conditions is established. In pursuit of a better lower bound, we require that this global center is nonlinear induced by the piecewise linearity, instead of the normal linear differential center considered in most of the existing articles. This renders that the traveling time of the unperturbed periodic orbits in each smooth zone is not calculable explicitly. Thus it is difficult to use the known Melnikov functions and averaged functions to study the bifurcating limit cycles. To overcome this difficulty, we develop an arbitrary order Melnikov-like function, which does not depend on the traveling time, for general d-dimensional piecewise smooth integrable systems allowing nonlinear switching boundaries. Finally, employing the new Melnikov-like function to the considered perturbation problem, we obtain a lower bound and an upper bound for the maximum number of limit cycles bifurcating from the unperturbed periodic orbits up to any order.

Generalized Solution of Inverse Problem for Ising Connection Matrix on d-Dimensional Hypercubic Lattice

We analyze a connection matrix of a -dimensional Ising system and solve the inverse problem, restoring the constants of interaction between spins, based on the known spectrum of its eigenvalues. When the boundary conditions are periodic, we can account for interactions between spins that are arbitrarily far. In the case of the free boundary conditions, we have to restrict ourselves with interactions between the given spin and the spins of the first coordination spheres.

Exponential and Weibull models for spherical and spherical-shell diffusion-controlled release systems with semi-absorbing boundaries

We consider the classical problem of particle diffusion in d-dimensional radially-symmetric systems with absorbing boundaries. A key quantity to characterise such diffusive transport is the evolution of the proportion of particles remaining in the system over time, which we denote by P(t). Rather than work with analytical expressions for P(t) obtained from solution of the corresponding continuum model, which when available take the form of an infinite series of exponential terms, single-term low-parameter models are commonly proposed to approximate P(t) to ease the process of fitting, characterising and interpreting experimental release data. Previous models of this form have mainly been developed for circular and spherical systems with an absorbing boundary. In this work, we consider circular, spherical, annular and spherical-shell systems with absorbing, reflecting and/or semi-absorbing boundaries. By proposing a moment matching approach, we develop several simple one and two parameter exponential and Weibull models for P(t), each involving parameters that depend explicitly on the system dimension, diffusivity, geometry and boundary conditions. The developed models, despite their simplicity, agree very well with values of P(t) obtained from stochastic model simulations and continuum model solutions.

Dephasing effects on the low-energy dynamics of ϕ4-model

In this paper, we investigate the Lindblad dynamics arising due to single particle loss and local dephasing in a d-dimensional nonrelativistic quantum system. Using Schwinger–Keldysh functional techniques, we derive the static mean field equation for the symmetry broken phase. Fluctuations over mean field are studied within the Bogoliubov approximation, and the momentum distribution and effective temperature are calculated. An effective temperature arises which depends on the dephasing rate [Formula: see text] and mean field particle density. Furthermore, a perturbative renormalization group (RG) upto one loop order is used to study the underlying dynamics of the model. The conditions for the system to be perturbative are obtained.

Implementation of quantum measurements using classical resources and only a single ancillary qubit

We propose a scheme to implement general quantum measurements, also known as Positive Operator Valued Measures (POVMs) in dimension d using only classical resources and a single ancillary qubit. Our method is based on probabilistic implementation of d-outcome measurements which is followed by postselection of some of the received outcomes. We conjecture that success probability of our scheme is larger than a constant independent of d for all POVMs in dimension d. Crucially, this conjecture implies the possibility of realizing arbitrary nonadaptive quantum measurement protocol on d-dimensional system using a single auxiliary qubit with only a constant overhead in sampling complexity. We show that the conjecture holds for typical rank-one Haar-random POVMs in arbitrary dimensions. Furthermore, we carry out extensive numerical computations showing success probability above a constant for a variety of extremal POVMs, including SIC-POVMs in dimension up to 1299. Finally, we argue that our scheme can be favorable for experimental realization of POVMs, as noise compounding in circuits required by our scheme is typically substantially lower than in the standard scheme that directly uses Naimark’s dilation theorem.