Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.
Non-Hermitian degeneracies are usually discussed through spectral coalescence, whereas entanglement is a property of eigenvectors and need not be fixed by the eigenvalues alone. We formulate a compact bipartite model that separates these two notions. A Hermitian operator with a degenerate eigenspace is transformed by an invertible non-unitary similarity map. The resulting Hamiltonian is non-Hermitian and retains a non-defective degeneracy at every finite value of the non-Hermiticity parameter. For an exactly solvable two-qubit realization, the right eigenstates evolve continuously from product states to maximally entangled states although the spectrum is unchanged. We distinguish the positive right-state reduced density matrix from the generally non-positive biorthogonal reduction, for which entropy may become complex. The same two-qubit solution gives a closed partial-transpose negativity and a Schmidt-gauged non-local magic. Entanglement grows monotonically with the non-Hermiticity parameter, whereas the non-local magic vanishes for both the product and maximally entangled limits and is largest at an intermediate coupling. In larger bipartite spaces, the Page entropy and Haar-averaged purity provide reference values for eigenstate typicality. These diagnostics separate non-defective degeneracy, exceptional-point sensitivity, Haar-typical entanglement, and non-stabilizer correlations without relying on a proliferation of basis-dependent spectral quantities.
Abstract Starting from a Hermitian operator with two distinct eigenvalues, we construct a non-Hermitian bipartite system in Gaussian orthogonal ensemble according to random matrix theory, where we introduce the off-diagonal fluctuations through random eigenkets and realizing the bipartite configuration consisting of two D × D subsystems (with D being the Hilbert space dimension). As required by the global thermalization (chaos), one of the two subsystems is fully ranked, while the other is rank deficient. For the latter (rank-deficient) subsystem, there is a block with non-defective degeneracies that contains non-local symmetries, as well as the accumulation effect of the linear map in adjacent eigenvectors. The maximally mixed state formed by the eigenvectors of this special region does not exhibit thermal ensemble behavior (neither canonical or Gibbs), and displays similar characteristics to the corresponding reduced density, which can be verified through the Loschmidt echo and variance of the imaginary spectrum. This non-defective degeneracy region partially meets the Lemma in 10.1103/PhysRevLett.122.220603 and theorem in 10.1103/PhysRevLett.120.150603. The coexistence of strong entanglement and initial state fidelity in this region make it possible to achieve a maximally mixed density, which, however, does not correspond to a thermal canonical ensemble (with complete insensitivity to the environmental energy or temperature). Outside this region, the collection of eigenstates (reduced density) always exhibit restriction on the corresponding Hilbert space dimension (with, e.g., infinite number of bound states), and thus suppress the global thermalization. There are abundant physics for those densities in Hermitian and non-Hermitian bases, which we investigate separately in this work. For example, the disentangling effect originates from non-Hermitian skin effect where the coherence exists along the direction orthogonal to the entangled boundaries of the Loschmidt echo spectrum in the Hermitian basis, while it originates from the many-body localization with the coherence among echo boundaries in the non-Hermitian basis which is disorder-free.
Theoretical analysis of a prototypical two-qubit effective non-Hermitian system characterized by asymmetric Heisenberg XY interactions in the absence of external magnetic fields demonstrates that maximal bipartite entanglement and quantum phase transitions can be induced exclusively through non-Hermiticity. At thermal equilibrium as T→0, the system attains maximal entanglement C=1 for values of the non-Hermiticity parameter greater than a critical value γ>γc=J(1−δ2), where J denotes the exchange interaction and δ represents the anisotropy of the system; conversely, for γ<γc, entanglement is nonmaximal and given by C=(1−(γ/J)2). The entanglement undergoes a discontinuous transition to zero precisely at γ=γc. This phase transition originates from the closing of the energy gap at a non-Hermiticity-driven ground state degeneracy, which is fundamentally different from an exceptional point. This work suggests the use of singular-value-decomposition generalized density matrix for the computation of entanglement in bi-orthogonal systems.
Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian, while much less has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, noninteracting model of fermions which is invariant under combined PT transformation. Our models show a phase transition from a PT unbroken phase to broken phase as we tune the Hermiticity-breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently, only left) eigenstates or a combination of both left and right eigenstates which form a complete set of biorthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states shows some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the PT-unbroken phase, the entanglement entropy obtained from only the right (or equivalently, left) eigenstates shows identical behavior to an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.
Non-Hermitian quantum systems exhibit qualitatively distinct physical behavior compared to Hermitian systems, a prime example being spectral singularities known as exceptional points. Their relevance in, e.g., quantum sensing, unidirectional transport, and robust lasing makes it important to be able to identify exceptional points through observable features of a many-body system. Here, using as an example a one-dimensional complex XY spin chain realizing both rotation-time RT- and parity-time PT-symmetric regimes, we develop a framework for detecting exceptional points based on the distinction between left and right eigenvectors of the Hamiltonian, which in a non-Hermitian system are no longer the adjoint of each other. We first show that a global measure constructed from the difference between the Hamiltonian and its adjoint locates exceptional points via distinct non-analytic behavior. At the level of observables, differences in local spin correlations evaluated on the right and left eigenstates provide a reliable static detection scheme. In contrast, static bipartite entanglement measures fail to capture this distinction, urging us to study the quantum dynamics of the model. Following a sudden quench, we demonstrate that the time-averaged right-left entanglement entropy difference directly encodes signatures of the exceptional point. In the RT-symmetric regime, it exhibits a pronounced peak at the exceptional point, whereas in the PT-symmetric regime it behaves as an order-parameter-like quantity, remaining finite in one phase and vanishing at the transition. Our results establish a direct link between the structure of non-Hermitian eigenstates and observable signatures of exceptional points, providing a practical route to identify them in existing quantum simulators.
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=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 study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.