The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.
The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.
Non-unitary dynamics in quantum mechanics arise in open quantum systems, where the interaction between the system and its environment is not negligible. One way this coupling can be realised is through measurement, where the backaction from the measurement apparatus induces non-unitary dynamics in the system. An effective non-Hermitian description arises within the no-jump approximation of a measured system, in which the time-evolved states are explicitly normalised at all times. Non-Hermiticity in quantum Hamiltonians gives rise to non-unitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this thesis, we explore these two interconnected concepts related to non-unitary dynamics: non-Hermiticity and measurements. However, instead of effective non-Hermiticity, we consider the cases where non-Hermiticity is realised exactly. This setup necessitates a treatment under the self-consistent frameworks of non-Hermitian quantum mechanics in order to restore its probabilistic interpretation. Concretely, the self-consistent metric formalism restores unitarity of the evolution through a modified Hilbert space endowed with a dynamical metric. Through comparative studies using exactly solvable models, we find that the physics of the two approaches of non-Hermitian quantum mechanics is fundamentally different. This has a profound physical impact in terms of defect production in quenches across an exceptional point (EP). We demonstrate this through a non-Hermitian system hosting both PT-symmetric and PT-broken modes subject to a linear quench. Our results highlight the role played by the EP and the spontaneous breaking of the PT-symmetry in contributing to a non-zero defect density, even in the limit of an infinitely slow drive. Further, we uncover generalised transformations underpinning the results, which are incorporated in the dynamical metric. Going beyond purely theoretical considerations, we implement two novel dilation schemes to emulate the metric dynamics in a digital quantum simulator. Our protocols enable controlled access to all PT-symmetry regimes and to a direct tomography of the dynamical metric, opening the possibility of studying a new kind of non-Hermitian physics in engineered matter. To explore the second theme of the thesis, we investigate the dynamics of entanglement entropy in the continuously monitored Bose-Hubbard model. Our numerical data reveals a counterintuitive enhancement of superfluid correlations in a parameter regime, in which the entanglement entropy increases with the onsite repulsion strength in the presence of measurement. This phenomenon arising from the interplay between hopping, interaction, and measurement, is also present in the effective central charge extracted from a fit of the logarithmic scaling form of the entropy. In summary, this thesis reveals the richness of non-unitary quantum dynamics. Through an in-depth study of the different aspects of non-Hermitian systems, we illustrate the non-trivial physics uncovered only with the consideration of the dynamical metric. By bridging theoretical insights and physical realisation through novel dilation protocols realisable in engineered matter, our work lays the groundwork for further exploration of dynamical aspects in non-Hermitian and monitored quantum systems.
The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constraint of parity-time (PT) symmetry, we show that the fidelity F is always real for the PT-unbroken states. For the PT-broken states, the real part of the fidelity susceptibility Re[XF] is corresponding to considering both the PT partner states, and the negative infinity is explored by the perturbation theory when the parameter approaches the exceptional point (EP). Moreover, at the second-order EP, we prove that the real part of the fidelity between PT-unbroken and PT-broken states is ReF=12 . Based on these general properties, we study the two-legged non-Hermitian Su-Schrieffer-Heeger (SSH) model and the non-Hermitian XXZ spin chain. We find that for both interacting and non-interacting systems, the real part of fidelity susceptibility density goes to negative infinity when the parameter approaches the EP, and verifies it is a second-order EP by ReF=12 .
Since the realization of quantum systems described by non-Hermitian Hamiltonians with parity-time (PT) symmetry, interest in non-Hermitian, quantum many-body models has steadily grown. Most studies to date map to traditional quantum spin models with a non-Hermiticity that arises from making the model parameters complex or purely imaginary. Here we present a set of models with non-Hermiticity generated by splitting a Hermitian term into two Jordan-normal form parts, and the perturbations are confined to one or two sites. We present exact diagonalization results for a finite PT threshold in such models and provide an analytical approach for understanding the numerical results. Surprisingly, with non-Hermitian potentials confined to two or even a single site, the PT threshold seems insensitive to the size of the quantum spin chain. Our results provide a pathway to experimentally feasible non-Hermitian quantum spin chains where the confluence of many-body effects and non-Hermiticity effects can be observed.
Combining non-Hermiticity and interactions yields unique effects in open quantum many-body systems. Here, we develop the generalized Schrieffer-Wolff transformation, discuss its symmetry properties, and derive the effective Hamiltonian suitable for various quasidegenerate non-Hermitian systems. We apply our results to an exemplary PT-symmetric circuit QED composed of two non-Hermitian qubits embedded in a lossless resonator. We consider a resonant quantum circuit as $|{\ensuremath{\omega}}_{r}\ensuremath{-}\mathrm{\ensuremath{\Omega}}|\ensuremath{\ll}{\ensuremath{\omega}}_{r}$, where $\mathrm{\ensuremath{\Omega}}$ and ${\ensuremath{\omega}}_{r}$ are qubits and resonator frequencies, respectively, providing well-defined groups of quasidegenerate resonant states. For such a system, using direct numerical diagonalization we obtain the dependence of the low-lying eigenspectrum on the interaction strength between a single qubit and the resonator g and the gain (loss) parameter $\ensuremath{\gamma}$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify PT-symmetry broken and unbroken phases, trace the formation of exceptional points of the second and the third order, and provide a complete phase diagram $g\ensuremath{-}\ensuremath{\gamma}$ of low-lying resonant states. We relate the formation of exceptional points to the additional P-pseudo-Hermitian symmetry of the system and show that non-Hermiticity mixes the ``dark'' and the ``bright'' states, which has a direct experimental consequence.
As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.
For decades, Hermiticity was considered an immutable axiom of quantum mechanics, essential for ensuring real energies and unitary evolution. This perspective has shifted radically, driven by the realization that non-Hermitian Hamiltonians provide a powerful effective description of open quantum systems, granting access to unique phenomena such as Exceptional Points and the Non-Hermitian Skin Effect. In this Perspective, we chart the trajectory of this field, moving from its established foundations in single-particle, linear models to the emerging frontier of interacting many-body systems. We first clarify the physical origins of non-Hermitian dynamics, distinguishing between mean-field approximations, conditional "no-click" evolution, and exact Liouvillian dynamics. We then focus on the rich phenomenology arising from the interplay of non-Hermiticity and interactions. We discuss interaction-induced topological phases, the generalization of skin effects to the many-body Hilbert space, and the distinct signatures of dissipative quantum chaos and complexity. Finally, we highlight collective phenomena in nonlinear regimes, including skin solitons and dissipative phase transitions. We also comment on measurement-induced entanglement transitions and their relation to non-Hermitian spectra and topology. By synthesizing these diverse developments, we provide a roadmap for the future of non-Hermitian physics.