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Magnetochemistry

On the Use of Classical and Quantum Fisher Information in Molecular Magnetism

Filippo Troiani

Published Sep 8, 2016• 2 citations
Publisher

Abstract

The present paper discusses the use of two information-theoretical quantities—namely, the classical and quantum Fisher information—in the context of molecular magnetism. These functions quantify the suitability of a given observable to the estimation of a physical parameter and provide the highest precision allowed by quantum mechanics in such an estimation process. The quantum Fisher information also quantifies the degree of macroscopicity of a quantum state. As illustrative examples of such applications, we compute the classical and quantum Fisher information of the Fe 4 molecular nanomagnet, used as a probe of an applied magnetic field or as a platform for generating Schrödinger cat states.

DOI
10.3390/magnetochemistry2030033

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International Journal of Quantum Chemistry•Jan 1, 2008•105 citations
The present paper discusses the use of two information-theoretical quantities—namely, the classical and quantum Fisher information—in the context of molecular magnetism. These functions quantify the suitability of a given observable to the estimation of a physical parameter and provide the highest precision allowed by quantum mechanics in such an estimation process. The quantum Fisher information also quantifies the degree of macroscopicity of a quantum state. As illustrative examples of such applications, we compute the classical and quantum Fisher information of the Fe 4 molecular nanomagnet, used as a probe of an applied magnetic field or as a platform for generating Schrödinger cat states.
We address the characterization of molecular nanomagnets at the quantum level and analyze the performance of local measurements in estimating the physical parameters in their spin Hamiltonians. To this aim, we compute key quantities in quantum estimation theory, such as the classical and the quantum Fisher information, in the prototypical case of a heterometallic antiferromagnetic ring. We show that local measurements, performed only on a portion of the molecule, allow a precise estimate of the parameters related to both magnetic defects and avoided level crossings.
Molecular nanomagnets are compounds characterized by a high-spin magnetic core that is protected by organic ligands. They have recently gained attention as potential quantum information carriers in solid-state quantum computing platforms, simultaneously exhibiting classical macroscopic properties and quantum features in light of their complex nature and configuration. Addressing the condition when they manifest unquestionable quantum behavior is key to guarantee their effectiveness as resources for quantum information processing. We address the quantumness of molecular nanomagnets using a recently formulated criterion [cf. Krisnanda et al., Phys. Rev. Lett. 119, 120402 (2017)] demonstrating that these systems exhibit an intrinsic quantum nature, as evidenced by their ability to generate and enhance quantum correlations between two non-interacting probes. Our analysis, which is performed addressing various dynamical regimes, paves the way to the design of experimentally viable tests of non-classicality in multipartite registers consisting of ensembles of molecular nanomagnets.
Molecular nanomagnets are compounds characterized by a high-spin magnetic core that is protected by organic ligands. They have recently gained attention as potential quantum information carriers in solid-state quantum computing platforms, simultaneously exhibiting classical macroscopic properties and quantum features in light of their complex nature and configuration. Addressing the condition when they manifest unquestionable quantum behavior is key to guarantee their effectiveness as resources for quantum information processing. We address the quantumness of molecular nanomagnets using a recently formulated criterion [T. Krisnanda , ] demonstrating that these systems exhibit an intrinsic quantum nature, as evidenced by their ability to generate and enhance quantum correlations between two noninteracting probes. Our analysis, which is performed addressing various dynamical regimes, paves the way to the design of experimentally viable tests of nonclassicality in multipartite registers consisting of ensembles of molecular nanomagnets.
In its simplest form, decoherence occurs when a quantum state is entangled with a second state, but the results of measurements made on the second state are not accessible. As the second state has effectively ``measured'' the first, in this paper we argue that the quantum Fisher information is the relevant metric for predicting and quantifying this kind of decoherence. The quantum Fisher information is usually used to determine an upper bound on how precisely measurements on a state can be used to estimate a classical parameter, and as such it is an important resource. Quantum-enhanced metrology aims to create nonclassical states with large quantum Fisher information and utilize them in precision measurements. In the process of doing this it is possible for states to undergo decoherence; for instance atom-light interactions used to create coherent superpositions of atomic states may result in atom-light entanglement. Highly nonclassical states, such as spin-cat states (Schr\"odinger cat states constructed from superpositions of collective spins) are shown to be highly susceptible to this kind of decoherence. We also investigate the required field occupation of the second state, such that this decoherence is negligible.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
Abstract The classical Fisher information concept is generalized to cover the complex probability amplitudes (wave functions) of quantum mechanics. Its contributions due to the probability and probability‐current densities are identified. General properties of this local information measure are examined and the Schrödinger functional for the kinetic energy is interpreted as the average Fisher information. The information current and source densities are introduced in terms of which the relevant continuity equation is formulated, expressing the local balance of the information content. The superposition principles for the particle probabilities, probability currents, and the Fisher information densities are examined. The interference (nonadditive) contributions to these quantities due to the quantum mechanical mixing of individual states are identified. An illustrative application of using the nonadditive Fisher information of the atomic orbital (AO) resolution in indexing the chemical bond within the 2‐AO model is presented. This AO‐phase sensitive index is capable of distinguishing the bonding, nonbonding, and antibonding electronic states, as do the familiar bond‐orders of quantum chemistry. It is interpreted as a measure of an extra delocalization of electrons due to the bond formation. This is in contrast to the so called quadratic bond‐multiplicity measures and the nonprojected entropy/information concepts of the communication theory of the chemical bond, which both loose the memory about the relative AO‐phases in molecular orbitals, thus failing to distinguish between the bonding and antibonding electron configurations. Finally, the Schrödinger variational principle of quantum mechanics and the related Kohn‐Sham principle of Density‐Functional theory are interpreted as constrained variational principles of the Fisher information contained in the electron probability distributions. They are examples of the Extreme Physical Information principle related to the measurement process. The Fisher‐information principle for the adiabatic (Born‐Oppenheimer) approximation of the molecular quantum mechanics is also examined. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008