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arXiv:2607.11759

Higher-order covariance matrices for non-Gaussian quantum states

Vojtěch Kala, Petr Marek, Nicolas J. Cerf

quant-phSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.

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9 pages, 1 figure

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Covariance matrices lie at the heart of the powerful and well-established symplectic framework for describing continuous-variable Gaussian quantum states. However, since this framework only relies on first- and second-order moments, it is not sufficient for the analysis of non-Gaussian states because their higher-order moments are essential to capture some of their key properties. Here, we define higher-order covariance matrices -- more precisely, covariance matrices built from higher-order quadrature monomials -- which provide a simple way to evaluate the effect of Gaussian transformations on non-Gaussian states and can be used, for example, to address nonlinear squeezing or non-Gaussian nullifiers. Higher-order covariance matrices can be estimated from homodyne measurement data using only a limited number of quadrature angles, which involves matrices of moderate dimension compared with a full simulation in the Fock basis. The dimension of a higher-order covariance matrix does not depend on the span of the quantum states in Fock basis and, furthermore, scales only polynomially with the number of modes.
The set of covariance matrices of a continuous-variable quantum system with a finite number of degrees of freedom is a strict subset of the set of real positive-definite matrices (PDMs) due to Heisenberg’s uncertainty principle. This has the implication that, in general, not every orthogonal transform of a quantum covariance matrix (CM) produces a PDM that obeys the uncertainty principle. A natural question thus arises, to find the set of quantum covariance matrices consistent with a given eigenspectrum. For the special class of pure Gaussian states the set of quantum covariance matrices with a given eigenspectrum consists of a single orbit of the action of the orthogonal symplectic group. The eigenspectrum of a CM of a state in this class is composed of pairs that each multiply to one. Our main contribution is finding a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by orthogonal symplectic transformations. We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra that identify the physically relevant thermal and squeezing parameters of a Gaussian state.
Entanglement is central to quantum physics, yet detecting and exploiting it in non-Gaussian systems remains a major challenge. In continuous variable platforms, standard inseparability criteria based on Gaussian statistics-such as the Duan-Simon criterion-fail when quantum correlations are encoded in higher moments of the field quadratures. Here we introduce a framework for detecting non-Gaussian entanglement using higher-order quadrature cumulants. In the Gaussian limit, the lowest order condition reduces to the Duan criterion, while higher-order violations reveal entanglement inaccessible to second-order methods. We experimentally demonstrate a tomography-free certification of inseparability in photon-subtracted squeezed states of light, where Gaussian witnesses fail despite the presence of entanglement. We further show that such higher-order inseparability enables enhanced quantum teleportation of Wigner negativity as compared to Gaussian protocols. These results identify higher-order cumulants as natural observables for non-Gaussian entanglement and open new routes to harnessing non-Gaussian resources in continuous variable quantum technologies.
Abstract Quantum photonic simulators are pivotal for designing and analyzing quantum photonic circuits that utilize both continuous-variable (CV) and discrete-variable (DV) encodings. This review critically examines the transition from Gaussian to non-Gaussian photonic models and the computational overhead that emerges from simulating systems with more than 20 modes or photon numbers exceeding 10 per mode. Gaussian states; such as coherent and squeezed states, are modeled efficiently using first-moment vectors and covariance matrices of dimension 2n×2n2n \times 2n2n×2n 2 n × 2 n , where n is the number of modes. Non-Gaussian elements like the cubic phase gate U=exp⁡(iγx^3)U = \exp (i \gamma \hat{x}^{3})U=exp(iγx^3) U = exp ( i γ x ˆ 3 ) and photon-subtracted states demand full Fock space representations and scale exponentially with photon number d2d^{2}d2 d 2 , where d is the Hilbert space dimension. We analyze simulators including , , and , benchmarking their support for tensor networks (bond dimension D >100), GPU acceleration, thus NVIDIA A100 with 1555 GB/s bandwidth, and SoC implementations such as Edge TPU with 8 TFLOPS. Special focus is given to noise models such as photon loss with transmissivity η , thermal noise modeled with nˉ=1.5\bar{n} = 1.5nˉ=1.5 n ¯ = 1.5 , and dark count Poisson distributions ( λ=0.1\lambda = 0.1λ=0.1 λ = 0.1 ). The review underscores scalable techniques and hybrid quantum-classical workflows vital for near-term photonic quantum computing.
Gaussian quantum states of bosonic systems are an important class of states. In particular, they play a key role in quantum optics as all processes generated by Hamiltonians up to second order in the field operators (i.e. linear optics and quadrature squeezing) preserve Gaussianity. A powerful approach to calculations and analysis of Gaussian states is using phase-space variables and symplectic transformations. The purpose of this note is to serve as a concise reference for performing phase-space calculations on Gaussian states. In particular, we list symplectic transformations for commonly used optical operations (displacements, beam splitters, squeezing), and formulae for tracing out modes, treating homodyne measurements, and computing fidelities.
The unavoidable interaction of quantum systems with their environment usually results in the loss of desired quantum resources. Suitably chosen system Hamiltonians, however, can, to some extent, counteract such detrimental decay, giving rise to the set of stabilizable states. Here we discuss the possibility to stabilize Gaussian states in continuous-variable systems. We identify necessary and sufficient conditions for such stabilizability and elaborate these on two benchmark examples, a single, damped mode and two locally damped modes. The obtained stabilizability conditions, which are formulated in terms of the states' covariance matrices, are, more generally, also applicable to non-Gaussian states, where they may similarly help to, e.g., discuss entanglement preservation and/or detection up to the second moments.
Non-Gaussian states are essential for achieving a quantum advantage in continuous-variable (CV) information processing. Among these, coherent superpositions of squeezed states are a foundational resource. While exact higher-order statistics are available in the undisplaced case, a complete and analytically tractable treatment with a common displacement has been missing. We introduce and solve the displaced Janus state---a coherent superposition of two squeezed coherent states that share the same displacement---and develop an analytical framework, based on a family of generalized squeezing polynomials, that yields closed-form expressions for arbitrary-order factorial moments and coherence functions g(k)(0){g}^{(k)}(0)g(k)(0), the full Wigner function, and the quantum Fisher information. The analysis shows how interference at a fixed mean, driven by a mismatch of the component covariances rather than by mean separation, can be precisely engineered to transform the extreme photon bunching of the constituents into strong sub-Poissonian and even perfect multiphoton suppression. We further provide a rigorous quantum Fisher information analysis, proving that parameters encoded by linear generators (for example, the number operator) are bounded by the standard quantum limit, whereas parameters encoded by quadratic generators (for example, squeezing transformations) achieve Heisenberg-limited scaling. Together, these results furnish a complete analytical toolkit for a versatile class of non-Gaussian states, establishing the displaced Janus state as a key primitive for hybrid quantum protocols, quantum metrology, and fault-tolerant continuous-variable computation.