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arXiv:1912.12313 · Entropy

Symmetric Logarithmic Derivative of Fermionic Gaussian States

Angelo C. M. Carollo, Bernardo Spagnolo, Davide Valenti

Published Jun 22, 2018• 66 citations
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Abstract

In this article, we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium steady states with Fermionic many-body systems.

DOI
10.3390/e20070485

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In this article, we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium steady states with Fermionic many-body systems.
We derive explicit expressions for the quantum Fisher information and the symmetric logarithmic derivative (SLD) of a quantum state in the exponential form $\ensuremath{\rho}=exp(G)$; the SLD is expressed in terms of the generator GGG. Applications include quantum-metrology problems with Gaussian states and general thermal states. Specifically, we give the SLD for a Gaussian state in two forms, in terms of its generator and its moments; the Fisher information is also calculated for both forms. Special cases are discussed, including pure, degenerate, and very noisy Gaussian states.
Entanglement and magic are fundamental resources that capture the complexity of quantum many-body systems. Non-local magic isolates the irreducible nonstabilizerness intrinsically tied to entanglement. However, evaluating this quantity generally requires a prohibitive minimization over the full Hilbert space, making it computationally inaccessible beyond a few qubits. Here, we overcome this bottleneck by establishing a closed-form expression for the non-local stabilizer entropies of fermionic Gaussian states over local Gaussian unitaries, which we prove at Rényi index α=2α=2α=2 for arbitrary subsystem size, and which can be evaluated in polynomial time directly from the eigenvalues of the reduced Majorana covariance matrix. We apply this framework to characterize fermionic non-local magic across diverse physical regimes: we derive an exact Page-like curve for typical random states, reveal logarithmic scaling at the quantum critical point of the XY model, and establish a quasiparticle picture for magic generation during out-of-equilibrium quantum quenches. Crucially, because our result relies solely on two-point correlation functions, it provides a scalable route for the experimental estimation of fermionic non-local magic in large-scale quantum processors via fermionic shadow tomography.
We formulate, with full generality, the asymptotic estimation theory for Gaussian states in terms of their first and second moments. By expressing the quantum Fisher information (QFI) and the elusive symmetric logarithmic derivative (SLD) in terms of the state's moments (and their derivatives) we are able to obtain the noncommutative extension of the well known expression for the Fisher information of a Gaussian probability distribution. Focusing on models with fixed first moments and identical Williamson 'diagonal' states --which include pure state models--, we obtain their SLD and QFI, and elucidate what features of the Wigner function are fundamentally accessible, and at what rates. In addition, we find the optimal homodyne detection scheme for all such models, and show that for pure state models they attain the fundamental limit.
Bosonic Gaussian thermal states form a fundamental class of states in quantum information science. This paper explores the information geometry of these states, focusing on characterizing the distance between two nearby states and the geometry induced by a parameterization in terms of their mean vectors and Hamiltonian matrices. In particular, for the family of bosonic Gaussian thermal states, we derive expressions for their Fisher-Bures, Kubo-Mori, and ααα-zzz information matrices with respect to their mean vectors and Hamiltonian matrices. An important application of our formulas consists of fundamental limits on how well one can estimate these parameters. We additionally establish formulas for the derivatives and the symmetric logarithmic derivatives of bosonic Gaussian thermal states. The former could have applications in gradient descent algorithms for quantum machine learning when using bosonic Gaussian thermal states as an ansatz, and the latter in formulating optimal strategies for single parameter estimation of bosonic Gaussian thermal states. Finally, the expressions for the aforementioned information matrices could have additional applications in natural gradient descent algorithms when using bosonic Gaussian thermal states as an ansatz.
We investigate the quantum Cramer-Rao bounds on the joint multiple-parameter estimation with the Gaussian state as a probe. We derive the explicit right logarithmic derivative and symmetric logarithmic derivative operators in such a situation. We compute the corresponding quantum Fisher information matrices, and find that they can be fully expressed in terms of the mean displacement and covariance matrix of the Gaussian state. Finally, we give some examples to show the utility of our analytical results.