Hermiticity has long been a cornerstone of quantum theory, ensuring the conservation of probability and real energy spectra. Recent advances in non-Hermitian physics have revealed that this requirement can be relaxed, giving rise to richer dynamics where the Hilbert space is endowed with a time-dependent inner product defined by a metric operator ρ ( t ) . This dynamical metric fundamentally reshapes quantum predictions, yet direct access to it on physical devices has remained elusive. Here, we bridge this gap by presenting general, platform-agnostic operator-dilation schemes that embed the metric formalism into closed Hermitian systems. Our proposed framework converts any time-dependent non-Hermitian Hamiltonian into a larger unitary evolution, bringing the theory from a purely mathematical curiosity into the experimentally testable realm. This permits the first controlled study of PT -symmetric and symmetry-broken regimes at arbitrarily long timescales. We demonstrate the power of this approach on a digital quantum simulator, showing that cutting-edge questions about non-Hermiticity are already within reach of today's noisy intermediate-scale quantum processors. Our results provide the first direct evidence of a dynamically evolving metric in a single qubit, establishing a new paradigm for simulating non-Hermitian quantum mechanics and opening the door to its realization in the laboratory.
Hermiticity has long been a cornerstone of quantum theory, ensuring the conservation of probability and real energy spectra. Recent advances in non-Hermitian physics have revealed that this requirement can be relaxed, giving rise to richer dynamics where the Hilbert space is endowed with a time-dependent inner product defined by a metric operator ρ ( t ) . This dynamical metric fundamentally reshapes quantum predictions, yet direct access to it on physical devices has remained elusive. Here, we bridge this gap by presenting general, platform-agnostic operator-dilation schemes that embed the metric formalism into closed Hermitian systems. Our proposed framework converts any time-dependent non-Hermitian Hamiltonian into a larger unitary evolution, bringing the theory from a purely mathematical curiosity into the experimentally testable realm. This permits the first controlled study of PT -symmetric and symmetry-broken regimes at arbitrarily long timescales. We demonstrate the power of this approach on a digital quantum simulator, showing that cutting-edge questions about non-Hermiticity are already within reach of today's noisy intermediate-scale quantum processors. Our results provide the first direct evidence of a dynamically evolving metric in a single qubit, establishing a new paradigm for simulating non-Hermitian quantum mechanics and opening the door to its realization in the laboratory.
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.
Non-Hermiticity in quantum Hamiltonians leads to nonunitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT-symmetric and PT-broken modes subject to a linear quench. Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that PT-broken time evolution leads to defect freezing and hence the violation of adiabaticity. This physics necessitates the so-called metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide class of experimental systems.
Non-Hermiticity in quantum Hamiltonians leads to nonunitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this work, we study the dynamics of an exactly solvable non-Hermitian system, hosting both PT-symmetric and PT-broken modes subject to a linear quench. Employing a fully consistent framework, in which the Hilbert space is endowed with a nontrivial dynamical metric, we analyze the dynamics of the generated defects. In contrast to Hermitian systems, our study reveals that PT -broken time evolution leads to defect freezing and hence the violation of adiabaticity. This physics necessitates the so-called metric framework, as it is missed by the oft used approach of normalizing quantities by the time-dependent norm of the state. Our results are relevant for a wide class of experimental systems.
Despite acute interest in the dynamics of non-Hermitian systems, there is a lack of consensus in the mathematical formulation of non-Hermitian quantum mechanics in the community. Different methodologies are used in the literature to study non-Hermitian dynamics. This ranges from consistent frameworks like biorthogonal quantum mechanics and metric approach characterized by modified inner products, to normalization by time-dependent norms inspired by open quantum systems. In this work, we systematically explore the similarities and differences among these various methods. Utilizing illustrative models with exact solutions, we demonstrate that these methods produce not only quantitatively different results but also distinct physical interpretations. For dissipative systems where non-Hermiticity arises as an approximation, we find that simply dividing by the norm in the PT -broken regime closely aligns with the full master equation solutions. In contrast, for quantum systems where non-Hermiticity can be engineered exactly, incorporating metric dynamics is crucial for the probabilistic interpretation of quantum mechanics, necessitating the generalizations of similarity transformations and unitarity to non-Hermitian systems. This study lays the groundwork for further exploration of non-Hermitian Hamiltonians, potentially leveraging generalized transformations for novel physical phenomena.
A conceptual framework extending (time-independent) PT-symmetric quantum mechanics into the time-dependent domain is presented. It is built upon a nontrivial time-dependent metric operator identified here and works for generic finite-dimensional non-Hermitian systems. All the ingredients of our framework, such as the time-dependent Hilbert space, the observable, and the measurement postulate, can be ``realized'' by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent PT-symmetric systems. Possible applications of our framework are illustrated with well-known examples in quantum thermodynamics.
A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.