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L. A. Markovich

Publications and source records attributed to L. A. Markovich.

At least 19 recordsLinked to original sources
arXiv2024

Not all Probability Density Functions are Tomograms

This paper delves into the significance of the tomographic probability density function (pdf) representation of quantum states, shedding light on the special classes of pdfs that can be tomograms. Instead of using wave functions or density operators on Hilbert spaces, tomograms, which are the true pdfs, are used to completely describe the states of quantum systems. Unlike quasi-pdfs, like the Wigner function, tomograms can be analysed using all the tools of classical probability theory for pdf estimation, which can allow a better quality of state reconstruction. This is particularly useful when dealing with non-Gaussian states where the pdfs are multi-mode. The knowledge of the family of distributions plays an important role in the application of both parametric and non-parametric density estimation methods. We show that not all pdfs can play the role of tomograms of quantum states and introduce the conditions that must be fulfilled by pdfs to be "quantum".

quant-ph↗
arXiv2023

Phase shift rule with the optimal parameter selection

The phase shift rules enable the estimation of the derivative of a quantum state with respect to phase parameters, providing valuable insights into the behavior and dynamics of quantum systems. This capability is essential in quantum simulation tasks where understanding the behavior of complex quantum systems is of interest, such as simulating chemical reactions or condensed matter systems. However, parameter shift rules are typically designed for Hamiltonian systems with equidistant eigenvalues. For systems with closely spaced eigenvalues, effective rules have not been established. We provide insights about the optimal design of a parameter shift rule, tailored to various sorts of spectral information that may be available. The proposed method lets derivatives be calculated for any system, regardless of how close the eigenvalues are to each other. It also optimizes the number of phase shifts, which reduces the amount of gate resources needed.

quant-ph↗
arXiv2023

High-Precision Observable Estimation with Single Qubit Quantum Memory

The estimation of multi-qubit observables is a key task in quantum information science. The standard approach is to decompose a multi-qubit observable into a weighted sum of Pauli strings. The observable can then be estimated from projective single qubit measurements according to the Pauli strings followed by a classical summation. As the number of Pauli strings in the decomposition increases, shot-noise drastically builds up, and the accuracy of such estimation can be considerably compromised. Access to a single qubit quantum memory, where measurement data may be stored and accumulated can circumvent the build-up of shot noise. Here, we describe a many-qubit observable estimation approach to achieve this with a much lower number of interactions between the multi-qubit device and the single qubit memory compared to previous approaches. Our algorithm offers a reduction in the required number of measurements for a given target variance that scales $N^{\frac{2}{3}}$ with the number of Pauli strings $N$ in the observable decomposition. The low number of interactions between the multi-qubit device and the memory is desirable for noisy intermediate-scale quantum devices.

quant-ph↗
arXiv2023

Hermitian and unitary almost-companion matrices of polynomials on demand

We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and Unitary ACMs starting from appropriate third degree polynomials, with implications for their use in physical-mathematical problems such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degree.

quant-ph↗
arXiv2019

Highly parallel algorithm for the Ising ground state searching problem

Finding an energy minimum in the Ising model is an exemplar objective, associated with many combinatorial optimization problems, that is computationally hard in general, but occurs in all areas of modern science. There are several numerical methods, providing solution for the medium size Ising spin systems. However, they are either computationally slow and badly parallelized, or do not give sufficiently good results for the large systems. In this paper, we present a highly parallel algorithm, called Mean-field Annealing from a Random State (MARS), incorporating the best features of the classical simulated annealing (SA) and Mean-Field Annealing (MFA) methods. The algorithm is based on the mean-field descent from a randomly selected configuration and temperature. Since a single run requires little computational effort, the effectiveness can be achieved by massive parallelisation. MARS shows excellent performance both on the large Ising spin systems and on the set of exemplary maximum cut benchmark instances in terms of both solution quality and computational time.

quant-ph↗
arXiv2018

Optimal nonlinear filtering of quantum state

We extend the optimal filtering equation known from the Stratonovich filtering theory on the quantum process case. The used observation model is based on an indirect measurement method, where the measurement is performed on an ancilla system that is interacted with an unknown one. Observation model for single qudit system is proposed.

quant-ph↗
arXiv2017

Gamma kernel estimation of multivariate density and its derivative on the nonnegative semi-axis by dependent data

In this paper, we consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on $[0,\infty)$. To this end we use the class of kernel estimators with asymmetric gamma kernel functions. The gamma kernels are nonnegative. They change their shape depending on the position on the semi-axis and are robust to the boundary bias problem. We investigate the mean integrated squared error (MISE) assuming dependent data with strong mixing and find the optimal bandwidth of the kernel as a minimum of the MISE. We derive the bias, the variance and the covariance of the density and of its partial derivative.

math.PR↗
arXiv2017

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent SU(2) problems. The general integral of exemplary differential Riccati equations, not previously considered in the specialized literature, is explicitly determined to illustrate both mathematical usefulness and easiness of applicability of our proposed treatment. The possibility of exploiting the general integral of a given differential Riccati equation to solve an SU(2) quantum dynamical problem, is succinctly pointed out.

math-ph↗
arXiv2017

Separability and entanglement in the Hilbert space reference frames related through the generic unitary transform for four level system

Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, $X$-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.

quant-ph↗
arXiv2017

Breakdown of separability due to confinement, submitted to Report on mathematical physics

A simple system of two particles in a bidimensional configurational space $S$ is studied. The possibility of breaking in $S$ the time independent Schrödinger equation of the system into two separated one-dimensional one-body Schrödinger equations is assumed. In this paper, we focus on how the latter property is countered by imposing such boundary conditions as confinement in a limited region of $S$ and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate (an impenetrability condition). Our investigation demonstrates the reducibility of the problem under scrutiny into that of a single particle living in a limited domain of its bidimensional configurational space. These general ideas are illustrated introducing the coordinates $X_c$ and $x$ of the center of mass of two particles and of the associated relative motion, respectively. The effects of the confinement and the impenetrability are then analyzed by studying with the help of an appropriate Green's function and the time evolution of the covariance of $X_c$ and $x$. Moreover, to calculate the state of the single particle constrained within a square, a rhombus, a triangle and a rectangle the Green's function expression in terms of Jacobi $θ_3$-function is applied. All the results are illustrated by examples.

quant-ph↗
arXiv2016

Entopy-energy inequality for superconducting qutrit

We compare the entropy-energy inequality and the von Neumann entropic inequality for three level atom implemented on superconducting circuits with Josephson junction. The positivity of entropy and energy relations for the qutrit system are used for verification of state tomography of qudit systems. The results obtained are valid for generic quantum states (qudits) and are illustrated on the example of the temperature density matrix of the single qutrit state.

quant-ph↗
arXiv2016

Photon distributions and information nonnegativity versus quadrature uncertainty relation

Using entropic inequalities for Shannon entropies new inequalities for some classical polynomials are obtained. To this end, photon distribution functions for one-, two- and multi-mode squeezed states in terms of Hermite, Laguerre, Legendre polynomials and Gauss' hypergeometric functions are used. The dependence between the violation of the quadrature uncertainty relation, the sign and the existence of the distribution function of such states is considered.

quant-ph↗
arXiv2015

Steering in spin tomographic probability representation

The steering property known for two qubit-state in terms of specific inequalities for correlation function is translated for the state of qudit with the spin $j=3/2$. Since most steering detection inequalities are based on the correlation functions we introduce analogs of such functions for the single qudit systems. The tomographic probability representation for the qudit states is applied. The connection between the correlation function in two qubit system and the single qudit is presented in an integral form with a intertwining kernel calculated explicitly in tomographic probability terms.

quant-ph↗
arXiv2015

Steering and correlations for the single qudit state on the example of $j=3/2$

The phenomenon of quantum steering and probabilistic meaning of the correlations are discussed for the state of the single qudit. The method of qubit portrait of the qudit states is used to extend the known steering detection inequality to the system without subsystems. The example of the $X$-state with $j=3/2$ is studied in detail.

quant-ph↗
arXiv2015

Inequalities for purity parameters for multipartite and single qudit states

We analyze a recently found inequality for eigenvalues of the density matrix and purity parameter describing either a bipartite system state or a single qudit state. The Minkowski type trace inequality for the density matrices of the qudit states is rewritten in terms of the purities. The properties of the obtained inequality are discussed. The $X$-states of the two qubits and the single qudit are considered in detail. A study of the relation of the obtained purity inequalities with the entanglement is presented.

quant-ph↗
arXiv2015

Gamma kernel estimation of the density derivative on the positive semi-axis by dependent data

We estimate the derivative of a probability density function defined on $[0,\infty)$. For this purpose, we choose the class of kernel estimators with asymmetric gamma kernel functions. The use of gamma kernels is fruitful due to the fact that they are nonnegative, change their shape depending on the position on the semi-axis and possess good boundary properties for a wide class of densities. We find an optimal bandwidth of the kernel as a minimum of the mean integrated squared error by dependent data with strong mixing. This bandwidth differs from that proposed for the gamma kernel density estimation. To this end, we derive the covariance of derivatives of the density and deduce its upper bound. Finally, the obtained results are applied to the case of a first-order autoregressive process with strong mixing. The accuracy of the estimates is checked by a simulation study. The comparison of the proposed estimates based on independent and dependent data is provided.

math.ST↗
arXiv2014

New Minkowski type inequalities and entropic inequalities for quantum states of qudits

The two-parameter Minkowski like inequality written for composite quantum system state is obtained for arbitrary Hermitian nonnegative matrix with trace equal to unity. The inequality can be used as entropic and information inequality for density matrix of noncomposite finite quantum system, e.g., for a single qudit state. The analogs of strong subadditivity condition for the single qudit is discussed in context of obtained Minkowski like inequality.

quant-ph↗