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Symmetry

Controlling Problem within a Class of Two-Level Positive Maps

Farrukh Maksutovich Mukhamedov, Izzat Qaralleh

Published Oct 31, 2022• 1 citation
Publisher

Abstract

This paper aims to define the set of unital positive maps on M2(C) by means of quantum Lotka–Volterra operators which are quantum analogues of the classical Lotka–Volterra operators. Furthermore, a quantum control problem within the class of quantum Lotka–Volterra operators are studied. The proposed approach will lead to the understanding of the behavior of the classical Lotka–Volterra systems within a quantum framework.

DOI
10.3390/sym14112280

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This paper aims to define the set of unital positive maps on M2(C) by means of quantum Lotka–Volterra operators which are quantum analogues of the classical Lotka–Volterra operators. Furthermore, a quantum control problem within the class of quantum Lotka–Volterra operators are studied. The proposed approach will lead to the understanding of the behavior of the classical Lotka–Volterra systems within a quantum framework.
The present paper deals with a connection between quantum quadratic operators (QQOs) and quasi QQOs on 𝕄2(ℂ). We show that QQOs and quasi QQOs on 𝕄2(ℂ) coincide in the class of Volterra type of operators. To establish this result, we first describe these two kind of operators on the commutative part of 𝕄2(ℂ). Furthermore, in the last section, we introduce a quantum analogue of Volterra operators and provide concrete examples of such kind of operators. It is established that the considered examples also satisfy quasiness condition as well. The obtained results will allow to produce (with explicit conditions) a class of unital, but not trace-preserving positive maps of 𝕄2(ℂ).
We propose a general theoretical framework that is suitable to study a wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we propose definitions of invariant and attractive quantum subsystem, characterize Markovian invariance properties, and provide sufficient conditions for attraction. The general framework and results are illustrated by addressing the potential of output-feedback Markovian control strategies for quantum pure state-stabilization. In particular, constructive results for the synthesis of stabilizing semigroups in arbitrary finite-dimensional Markovian systems are established. I. BACKGROUND AND MOTIVATIONS Stabilization problems are of central relevance for many quantum control applications, ranging from state preparation of quantum-optical and nano-mechanical systems to generation of noise-protected realizations of quantum information in realistic devices [1]. Dynamical systems undergoing Markovian evolution [2], [3] are both widely relevant from a physical standpoint and present distinctive control challenges – preventing, in particular, open-loop quantum-engineering and stabilization methods based on dynamical decoupling to be viable [4], [5]. However, we show here how a wide class of stabilization problems can be effectively treated in a general framework, provided by attractive quantum subsystems. After introducing the main ideas and definitions along with some general results, we shall explore their application to pure-state preparation problems for Markovian output-feedback control. We refer to the forthcoming journal version of the present paper [6] for detailed proofs we shall omit or merely sketch in the following sections. Consider a separable Hilbert space H over the complex field C. Let B(H) represent the set of linear bounded operators on H, H(H) denoting the real subspace of Hermitian operators, with I, O being the identity and the zero operator, respectively. In the standard statistical formulation of quantum mechanics [7], [8], the dimension of the Hilbert space H associated with the quantum system of interest, Q, is determined by the physics of the problem. In what follows, we consider finite-dimensional systems, i.e. dim(H) 0 being described by a TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], [1]. A differential equation for the density operator of I may be derived provided that a forward composition law holds: Definition 1 (QDS): A quantum dynamical semigroup is a one-parameter family of TPCP maps {Tt(·), t ≥ 0} that satisfies: (i) T0 = I, (ii) Tt ◦ Ts = Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) is a continuous function of t, ∀ρ ∈ D(HI), ∀X ∈ B(HI). Due to the trace and positivity preserving assumptions, a QDS is a semigroup of contractions. It has been proved [10], [13] that the Hille-Yoshida generator for a QDS exists and can be cast in the following canonical form:
The Caputo fractional order Lotka-Volterra system is time-consuming in practical applications, since its starting point is fixed. To tackle this problem, a short memory fractional order Lotka-Volterra system (SMFrLVS) is proposed, where the chaotic attractor of the short memory fractional order Lotka-Volterra system is achieved by the predictor-corrector method. Then, a multilayer fractional order Lotka-Volterra system with short memory (MSMFrLVS) is introduced, whose chaotic behaviors are explored via Poincare sections and frequency power spectra. A quantum image encryption algorithm is proposed by combining MSMFrLVS with quantum dual-scale triangular map. A quantum circuit of the dual-scale triangular map is designed with ADDER-MOD 2n . At the permutation stage, the plaintext image is transformed into quantum form with the generalized quantum image representation model. The resulting quantum image is divided into sub-blocks and scrambled by the quantum dual-scale triangular map. Subsequently, the intra and the inter permutation operations on bit-planes are realized by sorting pseudo-random sequence and by quantum Gray code, respectively. At the diffusion stage, the initial values of the MSMFrLVS are generated with a plaintext correlation mechanism. The ciphertext image can be acquired by carrying out three-level diffusion operations. It is demonstrated that the proposed quantum image encryption algorithm performs better than some typical image encryption algorithm in terms of security, robustness, computational complexity and encryption speed.