arXiv subjects

Elena R. Loubenets

Publications and source records attributed to Elena R. Loubenets.

At least 19 recordsLinked to original sources

arXiv2026

New bound on $S_{1}\times S_{2}$-setting Bell locality of a nonseparable Werner state

In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers $S_{1},S_{2}\geq1$ of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension $d\leq\min\{S_{1},S_{2}\}$ to satisfy all Bell inequalities under every $S_{1}\times S_{2}$-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous $-$ that is, to be $S_{1}\times S_{2}$-setting Bell local, for short. For a variety of $S_{1},S_{2}\geq1$ values, this new general locality condition is beyond Werner's and Barrett's locality conditions for a nonseparable Werner state. We also prove explicitly in the operator terms the optimization result by Terhal et. el. [Phys. Rev. Lett. \textbf{90,} 157903 (2003)] via semi-programming that every nonseparable Werner state with a dimension $d>\min\{S_{1},S_{2}\}$ is $S_{1}\times S_{2}$ -setting Bell local. The new results of the present article are important both for Bell nonlocality theory and for quantum applications based on Bell nonlocality.

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arXiv2026

Modelling optimal implementation of an arbitrary $N$-qubit quantum gate within the generalized Bloch vectors formalism

The optimal implementation of quantum gates for closed $N$-qubit systems is one of the key challenges for practical realization of many quantum information processing tasks. In the present article, based on the generalized Bloch vectors formalism [\emph{J. Phys. A: Math. Theor.} 54, 195301 (2021)] for a finite-dimensional quantum system, we develop a new general model for the optimal quantum gates implementation, which is formulated in terms of the Bloch vectors for the unitary evolution operator and the system Hamiltonians, drift and control, and has the unified form applicable for the implementation of an arbitrary $N$-qubit gate within any closed $N$-qubit system satisfying the controllability conditions. Within the developed optimal model, the cost functional has both the terminal part and also, the integral part with the special scaling, and this allows us to specify the quantum optimal control synthesis via solving the two-point boundary value problem (BVP) for the system of ordinary differential equations (ODEs), which can be explored numerically by any of the known BVP solvers for ODEs. The numerical experiments, conducted for the implementation within the developed optimal model of a variety of $N=1,2,3$ qubit gates, demonstrate the high accuracy of the model-based results.

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arXiv2025

High-spin measurements in an arbitrary two-qudit state

Violation of the CHSH inequality by a bipartite quantum state is now used in many quantum applications. However, the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements is still known only for $s=1/2$. In the present article, for an arbitrary state of two spin-$s$ qudits, each of dimension $d=2s+1\geq 2$, we introduce the notion of the spin-$s$ correlation matrix, which has dimension $3\times 3$ for all $s\geq \frac{1}{2}$; establish its relation to the general correlation $(d^{2}-1)\times (d^{2}-1)$ matrix of this state within the generalized Pauli representation and derive in terms of the spin-$s$ correlation matrix the explicit analytical expression for the maximal value of the CHSH expectation under local Alice and Bob spin-$s$ measurements in this state. Specifying this general expression for the two-qudit GHZ state, the nonlocal two-qudit Werner state, and some nonseparable pure two-qudit states, we find that, under local Alice and Bob high-spin ($s\geq1$) measurements in each of these nonseparable states, including the maximally entangled one, the CHSH inequality is not violated. Moreover, unlike the case of spin-$1/2$ measurements, where each pure nonseparable two-qubit state violates the CHSH inequality and the maximal value of its CHSH expectation increases monotonically with a growth of its entanglement, the situation under high-spin measurements is quite different -- for a pure two-qudit state with a higher degree of entanglement, the maximal value of the CHSH expectation turns out to be less than for a pure two-qudit state with lower entanglement and even for a separable one.

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arXiv2025

Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits

In the present paper, based on the general analytical expression [arXiv:2412.03470] for the maximum of the CHSH expectation under local Alice and Bob spin-$s$ measurements in a two-qudit state of dimension $d=2s+1$, $s\geq 1/2$, we analyze whether or not, under spin-$1$ measurements in an arbitrary two-qutrit state, the CHSH inequality is violated. We find analytically for a variety of pure nonseparable two-qutrit states and also, numerically for $1,000,000$ randomly generated pure nonseparable two-qutrit states, that, under local Alice and Bob spin-$1$ measurements in each of these nonseparable states, including maximally entangled, the CHSH inequality is not violated. These results together with the spectral decomposition of a mixed state lead us to the Conjecture that, under local Alice and Bob spin-$1$ measurements, every nonseparable two-qutrit state, pure or mixed, does not violate the CHSH inequality. For a variety of pure two-qutrit states, we further find the values of their concurrence and compare them with the values of their spin-$1$ CHSH parameter, which determines violation or nonviolation by a two-qutrit state of the CHSH inequality under spin-$1$ measurements. This comparison indicates that, in contrast to spin-$\frac{1}{2}$ measurements, where the spin-$\frac{1}{2}$ CHSH parameter of a pure two-qubit state is increasing monotonically with a growth of its entanglement, for a pure two-qutrit state, this is not the case. In particular, for the two-qutrit GHZ state, which is maximally entangled, the spin-$1$ CHSH parameter is equal to $\sqrt{\frac{8}{9}}$, while, for some separable pure two-qutrit states, this parameter can be equal to unity. Moreover, for the two-qutrit Horodecki state, the spin-$1$ CHSH parameter is equal to $4\sqrt{2}/21<1$ regardless of the entanglement type of this mixed state.

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arXiv2024

Geometric quantum discord of an arbitrary two-qudit state: the exact value and general upper bounds

The geometric quantum discord of a two-qudit state has been studied in many papers, however, its exact analytical value in the explicit form is known only for a general two-qubit state, a general qubit-qudit state and some special families of two-qudit states. Based on the general Bloch vectors formalism [J. Phys. A: Math. Theor. 54 195301 (2021)], we find the explicit exact analytical value of the geometric quantum discord for a general two-qudit state of an arbitrary dimension via the parameters of its correlation matrix and the Bloch vectors of its reduced states. This new general analytical result includes all the known exact results on the geometric quantum discord only as particular cases and proves rigorously that the lower bound on the geometric discord presented in [Phys. Rev. A 85, 024102 (2012)] constitutes its exact value for each two-qudit state. Moreover, our new general result allows us to find for an arbitrary two-qudit state, pure or mixed, the novel upper and lower bounds on its geometric quantum discord, expressed via the Hilbert space characteristics of this state.

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arXiv2024

The Bloch vectors formalism for a finite-dimensional quantum system

In the present article, we consistently develop the main issues of the Bloch vectors formalism for an arbitrary finite-dimensional quantum system. In the frame of this formalism, qudit states and their evolution in time, qudit observables and their expectations, entanglement and nonlocality, etc. are expressed in terms of the Bloch vectors -- the vectors in the Euclidean space $\mathbb{R}^{d^{2}-1}$ arising under decompositions of observables and states in different operator bases. Within this formalism, we specify for all $d\geq2$ the set of Bloch vectors of traceless qudit observables and describe its properties; also, find for the sets of the Bloch vectors of qudit states, pure and mixed, the new compact expressions in terms of the operator norms that explicitly reveal the general properties of these sets and have the unified form for all $d\geq2$. For the sets of the Bloch vectors of qudit states under the generalized Gell-Mann representation, these general properties cannot be analytically extracted from the known equivalent specifications of these sets via the system of algebraic equations. We derive the general equations describing the time evolution of the Bloch vector of a qudit state if a qudit system is isolated and if it is open and find for both cases the main properties of the Bloch vector evolution in time. For a pure bipartite state of a dimension $d_{1}\times d_{2}$, we quantify its entanglement in terms of the Bloch vectors for its reduced states. The introduced general formalism is important both for the theoretical analysis of quantum system properties and for quantum applications, in particular, for optimal quantum control, since, for systems where states are described by vectors in the Euclidean space, the methods of optimal control, analytical and numerical, are well developed.

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arXiv2023

Quantifying Bell nonlocality of a pure two-qudit state via its entanglement

For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in [J. Phys. A: Math. Theor. 55, 285301 (2022)] and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on the correlation properties of an arbitrary pure two-qubit state.

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arXiv2023

Conclusive discrimination by $N$ sequential receivers between $r\geq2$ arbitrary quantum states

In the present article, we develop a general framework for the description of discrimination between $r\geq2$ quantum states by $N\geq1$ sequential receivers in the case where each receiver obtains a conclusive result. This type of discrimination constitutes an $N$-sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number $r\geq2$ of arbitrary quantum states, pure or mixed, of an arbitrary dimension and any number $N\geq1$ of sequential receivers, is based on the notion of a quantum state instrument and this allows us to derive the new important general results. We, in particular, find a general condition on $r\geq2$ quantum states, under which, within the strategy where all types of receivers' quantum measurements are allowed, the optimal success probability is equal to that of the first receiver for any number $N\geq2$ of further sequential receivers. Furthermore, we extend our general framework to include an $N$-sequential conclusive discrimination between $r\geq2$ arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.

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arXiv2022

Specifying nonlocality of a pure bipartite state and analytical relations between measures for bipartite nonlocality and entanglement

For a multipartite quantum state, the maximal violation of all Bell inequalities constitutes a measure of its nonlocality [Loubenets, J. Math. Phys. 53, 022201 (2012)]. In the present article, for the maximal violation of Bell inequalities by a pure bipartite state, possibly infinite-dimensional, we derive a new upper bound expressed in terms of the Schmidt coefficients of this state. This new upper bound allows us also to specify general analytical relations between the maximal violation of Bell inequalities by a bipartite quantum state, pure or mixed, and such entanglement measures for this state as "negativity" and "concurrence". To our knowledge, no any general analytical relations between measures for bipartite nonlocality and entanglement have been reported in the literature though, for a general bipartite state, specifically such relations are important for the entanglement certification and quantification scenarios. As an example, we apply our new results to finding upper bounds on nonlocality of bipartite coherent states intensively discussed last years in the literature in view of their experimental implementations.

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arXiv2021

Violation of general Bell inequalities by a pure bipartite quantum state

In the present article, based on the formalism introduced in [Loubenets, J. Math. Phys. 53, 022201 (2012)], we derive for a pure bipartite quantum state a new upper bound on its maximal violation of general Bell inequalities. This new bound indicates that, for an infinite dimensional pure bipartite state with a finite sum of its Schmidt coefficients, violation of any general Bell inequality is bounded from above by the value independent on a number of settings and a type of outcomes, continuous or discrete, specific to this Bell inequality. As an example, we apply our new general results to specifying upper bounds on the maximal violation of general Bell inequalities by infinite dimensional bipartite states having the Bell states like forms comprised of two binary coherent states $|α\rangle ,|-α\rangle$, with $α>0$. We show that, for each of these bipartite coherent states, the maximal violation of general Bell inequalities cannot exceed the value $3$ and analyse numerically the dependence of the derived analytical upper bounds on a parameter $α>0$.

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arXiv2021

New general lower and upper bounds under minimum-error quantum state discrimination

For the optimal success probability under minimum-error discrimination between $r\geq2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to $r>2$ the form of the Helstrom bound. For $r=2$, each of our new bounds, lower and upper, reduces to the Helstrom bound.

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arXiv2021

Two-sequential Conclusive Discrimination between Binary Coherent States via Indirect Measurements

A general scenario for an $N$-sequential conclusive state discrimination introduced recently in Loubenets and Namkung [arXiv:2102.04747] can provide a multipartite quantum communication realizable in the presence of a noise. In the present article, we propose a new experimental scheme for the implementation of a sequential conclusive discrimination between binary coherent states via indirect measurements within the Jaynes-Cummings interaction model. We find that if the mean photon number is less than 1.6, then, for our two-sequential state discrimination scheme, the optimal success probability is larger than the one presented in Fields, Varga, and Bergou [2020, IEEE Int. Conf. Quant. Eng. Comp.]. We also show that, if the mean photon number is almost equal to 1.2, then the optimal success probability nearly approaches the Helstrom bound.

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arXiv2020

Specifying the unitary evolution of a qudit for a general nonstationary Hamiltonian via the generalized Gell-Mann representation

Optimal realizations of quantum technology tasks lead to the necessity of a detailed analytical study of the behavior of a $d$-level quantum system (qudit) under a time-dependent Hamiltonian. In the present article, we introduce a new general formalism describing the unitary evolution of a qudit $(d\geq2)$ in terms of the Bloch-like vector space and specify how in a general case this formalism is related to finding time-dependent parameters in the exponential representation of the evolution operator under an arbitrary time-dependent Hamiltonian. Applying this new general formalism to a qubit case $(d=2)$, we specify the unitary evolution of a qubit via the evolution of a unit vector in $\mathbb{R}^{4}$ and this allows us to derive the precise analytical expression of the qubit unitary evolution operator for a wide class of nonstationary Hamiltonians. This new analytical expression includes the qubit solutions known in the literature only as particular cases.

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arXiv2019

The generalized Gell-Mann representation and violation of the CHSH inequality by a general two-qudit state

We formulate and prove the main properties of the generalized Gell-Mann representation for traceless qudit observables with eigenvalues in $[-1,1]$ and analyze via this representation violation of the CHSH inequality by a general two-qudit state. For the maximal value of the CHSH expectation in a two-qudit state with an arbitrary qudit dimension $d\geq2$, this allows us to find two new bounds, lower and upper, expressed via the spectral properties of the correlation matrix for a two-qudit state. We have not yet been able to specify if the new upper bound improves the Tsirelson upper bound for each two-qudit state. However, this is the case for all two-qubit states, where the new lower bound and the new upper bound coincide and reduce to the precise two-qubit CHSH\ result of Horodeckis, and also, for the Greenberger-Horne-Zeilinger (GHZ) state with an odd $d\geq2,$ where the new upper bound is less than the upper bound of Tsirelson. Moreover, we explicitly find the correlation matrix for the two-qudit GHZ state and prove that, for this state, the new upper bound is attained for each dimension $d\geq2$ and this specifies the following new result: for the two-qudit GHZ state, the maximum of the CHSH expectation over traceless qudit observables with eigenvalues in $[-1,1]$ is equal to $2\sqrt{2}$ if $d\geq2$ is even and to $\frac{2(d-1)}{d}\sqrt{2}$ if $d>2$ is odd.

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arXiv2019

Quantum analog of the original Bell inequality for two-qudit states with perfect correlations/anticorrelations

For an even qudit dimension $d\geq 2,$ we introduce a class of two-qudit states exhibiting perfect correlations/anticorrelations and prove via the generalized Gell-Mann representation that, for each two-qudit state from this class, the maximal violation of the original Bell inequality is bounded from above by the value $3/2$ - the upper bound attained on some two-qubit states. We show that the two-qudit Greenberger-Horne-Zeilinger (GHZ) state with an arbitrary even $d\geq 2$ exhibits perfect correlations/anticorrelations and belongs to the introduced two-qudit state class. These new results are important steps towards proving in general the $\frac{3}{2}$ upper bound on quantum violation of the original Bell inequality. The latter would imply that similarly as the Tsirelson upper bound $2\sqrt{2}$ specifies the quantum analog of the CHSH inequality for all bipartite quantum states, the upper bound $\frac{3}{2}$ specifies the quantum analog of the original Bell inequality for all bipartite quantum states with perfect correlations/ anticorrelations. Possible consequences for the experimental tests on violation of the original Bell inequality are briefly discussed.

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arXiv2019

Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations

We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3/2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3/2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.

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arXiv2018

New concise upper bounds on quantum violation of general multipartite Bell inequalities

Last years, bounds on the maximal quantum violation of general Bell inequalities were intensively discussed in the literature via different mathematical tools. In the present paper, we analyze quantum violation of general Bell inequalities via the LqHV (local quasi hidden variable) modelling framework, correctly reproducing the probabilistic description of every quantum correlation scenario. The LqHV mathematical framework allows us to derive for all d and N a new upper bound (2d-1)^{N-1} on the maximal violation by an N-qudit state of all general Bell inequalities, also, new upper bounds on the maximal violation by an N-qudit state of general Bell inequalities for S settings per site. These new upper bounds essentially improve all the known precise upper bounds on quantum violation of general multipartite Bell inequalities. For some S, d and N, the new upper bounds are attainable.

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arXiv2018

Bell's nonlocality in a general nonsignaling case: quantitatively and conceptually

Quantum violation of Bell inequalities is now used in many quantum information applications and it is important to analyze it both quantitatively and conceptually. In the present paper, we analyze violation of multipartite Bell inequalities via the local probability model - the LqHV (local quasi hidden variable) model [Loubenets, J. Math. Phys. 53, 022201 (2012)], incorporating the LHV model only as a particular case and correctly reproducing the probabilistic description of every quantum correlation scenario, more generally, every nonsignaling scenario. The LqHV probability framework allows us to construct nonsignaling analogs of Bell inequalities and to specify parameters quantifying violation of Bell inequalities - Bell's nonlocality - in a general nonsignaling case. For quantum correlation scenarios on an N-qudit state, we evaluate these nonlocality parameters analytically in terms of dilation characteristics of an N-qudit state and also, numerically - in d and N. In view of our rigorous mathematical description of Bell's nonlocality in a general nonsignaling case via the local probability model, we argue that violation of Bell inequalities in a quantum case is not due to violation of the Einstein-Podolsky-Rosen (EPR) locality conjectured by Bell but due to the improper HV modelling of "quantum realism".

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