This paper proposes a new quantum control method which controls the Shannon entropy of quantum systems. For both discrete and continuous entropies, controller design methods are proposed based on probability density function control, which can drive the quantum state to any target state. To drive the entropy to any target at any prespecified time, another discretization method is proposed for the discrete entropy case, and the conditions under which the entropy can be increased or decreased are discussed. Simulations are done on both two- and three-dimensional quantum systems, where division and prediction are used to achieve more accurate tracking.
This paper proposes a new quantum control method which controls the Shannon entropy of quantum systems. For both discrete and continuous entropies, controller design methods are proposed based on probability density function control, which can drive the quantum state to any target state. To drive the entropy to any target at any prespecified time, another discretization method is proposed for the discrete entropy case, and the conditions under which the entropy can be increased or decreased are discussed. Simulations are done on both two- and three-dimensional quantum systems, where division and prediction are used to achieve more accurate tracking.
This paper explores a novel approach to quantum control, termed information entropy gradient-driven quantum control. The central premise is that the rate of change of information entropy within a quantum system provides a fundamentally different and potentially more robust control signal compared to traditional methods relying on precise target state definitions. We propose a methodology where the gradient of the information entropy is continuously measured and utilized to drive the system towards desired quantum states. This approach aims to mitigate the inherent challenges associated with achieving high-fidelity control in quantum systems, particularly those stemming from measurement uncertainty and the difficulty of specifying precise target states. The theoretical framework presented outlines the key equations and concepts underlying this method, highlighting its potential advantages for applications in quantum computing, quantum communication, and quantum sensing. The core claim is that leveraging quantum system information entropy changes as control signals can enable efficient state manipulation, avoiding the precision issues encountered in traditional control methods. The core mechanism involves precise measurement of the quantum system's information entropy and feedback to the system for fine-tuning and state transfer. We demonstrate, through mathematical modeling, the feasibility and potential benefits of this approach.
We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy S(t) associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in 1+1 dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law ∼t−α, two distinct asymptotic regimes emerge: S(t)≃21lnt for α>1/2 and S(t)≃αlnt for 0<α<1/2. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case α=1/2 and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.
It is shown that Shannon entropy, Fisher information, and quantum-mechanical kinetic energy may all be viewed as measures of correlation. We begin with a fundamental correlation-destroying map: a joint probability density ρ(x,y) is replaced by ρx(x)ρy(y), the product of its marginal distributions. In the discrete case, Shannon's entropy is nondecreasing under this map. For continuous distributions, this fundamental map leads to a well-behaved, coordinate-invariant correlation measure, analogous to the discrete Shannon entropy: I[ρ]=h[ρx]+h[ρy]−h[ρ]. In the continuous case, however, another measure appears: Fisher information. The relative Fisher information J(ρ∥ρxρy) behaves similarly under the fundamental map. For a normalized real quantum wavefunction, this decrease is exactly proportional to the decrease in mean kinetic energy, ⟨T⟩ψ−⟨T⟩Φ=8mℏ2J(ρ∥ρxρy),Φ=ρxρy. In Jaynes's language, the result supports a principle of minimum justified correlation: given physical constraints and a set of possible distributions or related amplitudes satisfying those constraints, select from that set those with the least correlation.
This is Part I of a two-part paper. Here we develop the entropy, Fisher-information, and kinetic-energy identities above, and state the principle they support. Part II addresses questions which Part I raises but leaves unanswered: multiple solutions, time dependence, the role of spin, a route to the Schrödinger equation itself, and a proposed experimental test. AI has been used.