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arXiv:2305.00834 · Communications in Mathematical Physics

Hilbert–Schmidt Estimates for Fermionic 2-Body Operators

Martin Ravn Christiansen

Published Jan 1, 2024• 4 citations
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Abstract

Abstract We prove that the 2-body operator $$\gamma _{2}^{\Psi }$$ γ 2 Ψ of a fermionic N-particle state $$\Psi $$ Ψ obeys $$\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{HS}}\le \sqrt{5}N$$ ‖ γ 2 Ψ ‖ HS ≤ 5 N , which complements the bound of Yang (Rev Mod Phys 34:694, 1962) that $$\Vert \gamma _{2}^{\Psi }\Vert _{\textrm{op}}\le N$$ ‖ γ 2 Ψ ‖ op ≤ N . This estimate furthermore resolves a conjecture of Carlen–Lieb–Reuvers (Commun Math Phys 344:655–671, 2016) concerning the entropy of the normalized 2-body operator. We also prove that the Hilbert–Schmidt norm of the truncated 2-body operator $$\gamma _{2}^{\Psi ,T}$$ γ 2 Ψ , T obeys the inequality $$\Vert \gamma _{2}^{\Psi ,T}\Vert _{\textrm{HS}}\le \sqrt{5N\,\textrm{tr}\,(\gamma _{1}^{\Psi }(1-\gamma _{1}^{\Psi }))}$$ ‖ γ 2 Ψ , T ‖ HS ≤ 5 N tr ( γ 1 Ψ ( 1 - γ 1 Ψ ) ) .

DOI
10.1007/s00220-023-04914-3

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Abstract We prove that the Hilbert–Schmidt norm of k -particle reduced density matrices of N -body fermionic states is bounded from above by CkNk/2C_kN^{k/2}Ck​Nk/2 C k N k / 2 - matching the scaling behaviour of Slater determinant states. This generalises a recent result of Christiansen [3] on 2-particle reduced density matrices to higher order density matrices. Moreover, our estimate directly yields a lower bound on the von Neumann entropy and the 2-Rényi entropy of reduced density matrices, thereby providing further insight into conjectures of Carlen–Lieb–Reuvers [2, 8].
We prove that the eigenvalues of a 2-body operator γ2Ψγ_{2}^Ψγ2Ψ​ associated to a fermionic NNN-particle state ΨΨΨ are highly constrained by the structure of the corresponding eigenvectors: If Φ=∑k=1∞λkuk∧vkΦ=\sum_{k=1}^{\infty}λ_{k}u_{k}\wedge v_{k}Φ=∑k=1∞​λk​uk​∧vk​ is the canonical form of an eigenvector ΦΦΦ with eigenvalue ΛΛΛ, then Λ≤(1+N−22∑k=1∞λk4)−1NΛ\leq(1+\frac{N-2}{2}\sum_{k=1}^{\infty}λ_{k}^{4})^{-1}NΛ≤(1+2N−2​∑k=1∞​λk4​)−1N. We also prove a lower bound on sup⁡∥Ψ∥=1⟨Φ,γ2ΨΦ⟩\sup_{\Vert Ψ\Vert =1}\langle Φ,γ_{2}^ΨΦ\ranglesup∥Ψ∥=1​⟨Φ,γ2Ψ​Φ⟩ for fixed ΦΦΦ, and state a conjecture motivated by these results.
Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.
We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of NNN indistinguishable fermions, based on states of M<NM<NM<N and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced MMM- and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of MMM-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized MMM-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of MMM and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of MMM-body DMs in some strongly correlated fermionic states is also provided.
The trace distance between two quantum states, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\ensuremath{\rho}$ or $\ensuremath{\sigma}$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.