We investigate (k_{1},k_{2}) extendibility of fermionic Gaussian states, a property central to quantum correlations and approximations of separability. We show that these states are (k_{1},k_{2})-extendible if and only if they admit a fermionic Gaussian extension, yielding a complete covariance-matrix characterization and a simple semidefinite program (SDP) whose size scales linearly with the number of modes. This provides necessary conditions for arbitrary fermionic states and is sufficient within the Gaussian setting. Our main result is a finite de Finetti-type theorem: we derive trace-norm bounds between (k_{1},k_{2})-extendible fermionic Gaussian states and separable states, improving previous exponential scaling to linear in the number of modes, with complementary relative entropy and squashed entanglement bounds. For two modes, upper and lower bounds match at order 1/sqrt[k_{1}k_{2}]. Extendibility also provides operational support for one of the different notions of separability in fermionic systems. Finally, for fermionic Gaussian channels, we provide a simple semidefinite program criterion for antidegradability and show that entanglement-breaking channels coincide with replacement channels, implying no nontrivial entanglement-breaking fermionic Gaussian channels exist.
We investigate (k_{1},k_{2}) extendibility of fermionic Gaussian states, a property central to quantum correlations and approximations of separability. We show that these states are (k_{1},k_{2})-extendible if and only if they admit a fermionic Gaussian extension, yielding a complete covariance-matrix characterization and a simple semidefinite program (SDP) whose size scales linearly with the number of modes. This provides necessary conditions for arbitrary fermionic states and is sufficient within the Gaussian setting. Our main result is a finite de Finetti-type theorem: we derive trace-norm bounds between (k_{1},k_{2})-extendible fermionic Gaussian states and separable states, improving previous exponential scaling to linear in the number of modes, with complementary relative entropy and squashed entanglement bounds. For two modes, upper and lower bounds match at order 1/sqrt[k_{1}k_{2}]. Extendibility also provides operational support for one of the different notions of separability in fermionic systems. Finally, for fermionic Gaussian channels, we provide a simple semidefinite program criterion for antidegradability and show that entanglement-breaking channels coincide with replacement channels, implying no nontrivial entanglement-breaking fermionic Gaussian channels exist.
A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order D−1/2 for D-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a k-copy Gaussian state that is n-exchangeable under graded fermionic copy permutations is O(k/n)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as O(r−1), and hence, by recoverability, the state admits an O(r−1/2) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.
Extendibility of bosonic Gaussian states is a key issue in continuous-variable quantum information. We show that a bosonic Gaussian state is k-extendible if and only if it has a Gaussian k-extension, and we derive a simple semidefinite program, whose size scales linearly with the number of local modes, to efficiently decide k-extendibility of any given bosonic Gaussian state. When the system to be extended comprises one mode only, we provide a closed-form solution. Implications of these results for the steerability of quantum states and for the extendibility of bosonic Gaussian channels are discussed. We then derive upper bounds on the distance of a k-extendible bosonic Gaussian state to the set of all separable states, in terms of trace norm and Rényi relative entropies. These bounds, which can be seen as "Gaussian de Finetti theorems," exhibit a universal scaling in the total number of modes, independently of the mean energy of the state. Finally, we establish an upper bound on the entanglement of formation of Gaussian k-extendible states, which has no analogue in the finite-dimensional setting.
We prove a separability rigidity theorem for finite-mode Gaussian states. Let ρG be Gaussian and let I be a fixed partition of the modes. If ρG dominates a nonzero positive operator T that is separable across I, 0=T≼ρG, then ρG itself is separable across I. The dominated operator is arbitrary and need not be Gaussian. The theorem has three applications. First, any Gaussian channel admitting a nonzero entanglement-breaking completely positive subchannel is itself entanglement breaking, even when that subchannel is non-Gaussian. Second, if a resource state dominates a positive multiple of a faithful separable state, no finite-copy, postselected separability-preserving protocol can prepare an entangled Gaussian state exactly with nonzero success probability. Third, a connected graph of entangled Gaussian two-mode marginals certifies genuine multipartite entanglement of every compatible global state, even if the global state is non-Gaussian; the certification is robust to sufficiently small trace-norm errors in the marginals. As a structural corollary, mixing states separable across different partitions cannot produce new Gaussian states, so full inseparability and genuine multipartite entanglement coincide for finite-mode Gaussian states and identical copies of a biseparable Gaussian state cannot activate genuine multipartite entanglement.
Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.
Fermionic Gaussian states form a central class of classically tractable quantum states, while fermionic non-Gaussianity provides the resource required to go beyond free-fermion dynamics. A key challenge is to quantify this resource through monotones that are both mathematically rigorous and experimentally accessible. Here, we show that the fermionic entropy, defined through the squared Frobenius norm of the correlation matrix, is a strong pure-state Gaussian monotone. Its simple closed-form expression also makes it directly measurable: we show that the associated fermionic purity can be unbiasedly estimated up to additive error ε using O(ε−2) two-copy measurements, independently of the system size. Moreover, we prove that the fermionic entropy obeys asymptotic continuity and, as a direct consequence, establish its operational meaning as the upper bound to the asymptotic rate of non-Gaussianity distillation. We further derive a linear sample complexity bound for tolerant testing of fermionic Gaussian states, providing a quadratic improvement over the state of the art. As a further application of our results, we study unitary designs generated by Matchgate circuits supplemented with Majorana-local non-Gaussian gates. We prove that a linear number of such gates is necessary even to achieve an approximate state 2-design with error below 0.4%. Combined with known nearly linear upper bounds for relative-error designs, this determines the optimal doping level, up to logarithmic factors, across all relevant design notions and reveals the extensive non-Gaussianity cost required to generate Haar-like quantum dynamics in this architecture.
We prove an exponential variant of the Gaussian de Finetti theorem: the subsystems of permutation-invariant, free-fermionic Gaussian states are well approximated by convex combinations of almost-i.i.d. states that are Gaussian on subsets of their parts. Our result provides an error bound between the original state and its approximants that decays exponentially in the number of unconstrained parts, becoming super-exponential when the subsystem under consideration is small. The dependence of our bound on the local Hilbert space dimension is polylogarithmic, giving an exponential improvement over the standard de Finetti theorem of [Nat. Phys. 3, 645-649]. Previous works considered Gaussian-symmetric states, which are supported on the trivial irrep of the tensor matchgate representation. We extend these results to the broader class of Gaussian-invariant states, which contains, for example, i.i.d. copies of single-replica mixed Gaussian states. We show that Gaussian-invariant states are precisely the partial traces of Gaussian-symmetric states on locally enlarged replicas and always admit a purification into a larger Gaussian-symmetric state. As an application, we derive explicit convergence bounds for a hierarchy of convex relaxations of polynomial optimisation over fermionic Gaussian states. For variational energy minimisation, the hierarchy gives lower bounds converging to the global Gaussian minimum. Combined with upper bounds from Gaussian trial states, these bounds quantify how far a variational calculation can be from the best energy attainable within the Gaussian family.