The physical model of non-interacting fermion quantum channel is proposed by fermionic linear optics.Based on covariance matrix property of even Gaussian state and majorization theory,a formula of fermion quantum channel minimum output entropy for even Gaussian input state is deduced.A output entropy correspondence between Gaussian state and even Gaussian state is established by adding one extra fermionic mode to n-qubit fermion system.A formula of tensor product state capacity for non-interacting fermion channel is deduced according this output entropy correspondence and under assumption that minimum output entropy is achieved on a Gaussian input state.For fermionic channel correct of proposed minimum output entropy formula is confirmed by the iterative minimization algorithm of output entropy.The numerical results is that for non-interacting fermion noisy quantum channel the proposed formula agrees with the numerical results with a precision about 10e-9.
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.
Fermionic non-Gaussianity, or fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet tractable and operationally meaningful ways to quantify it remain limited. We address this challenge by developing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable quantities for pure fermionic states, both derived from the Williamson normal form of the covariance matrix. The first family, occupation number entropies, is defined as the Tsallis-
α entropy of the occupation numbers. We prove that one member of this family is monotonic under Gaussian protocols, establishing it as a computable convex resource monotone. It consequently lower bounds the number of non-Gaussian gates needed for state preparation. The second family, natural-orbital participation entropies, is given by the Rényi-
α entropy of the squared amplitudes of the state in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. They quantify state compressibility in this basis and thus upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure. We further study representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, highlighting the role of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry, opening new directions for studying the complexity of quantum many-body systems and providing practical tools to assess the classical simulability of fermionic states relevant for quantum advantage.
We define fermionic convolution and demonstrate its utility in characterizing fermionic non-Gaussian components, which are essential to the computational advantage of fermionic systems. Using fermionic convolution, we propose an efficient protocol that tests the fermionic Gaussianity of pure states using three copies of the input state. We also introduce "Non-Gaussian Entropy," an experimentally accessible resource measure that quantifies fermionic non-Gaussianity. These results provide new insights into the study of fermionic quantum computation.