Quantifying non-Hermiticity using single- and many-particle quantum properties

TL;DR

"Quantifying non-Hermiticity using single- and many-particle quantum properties" develops two complementary measures of non-Hermiticity: the Hamiltonian-based distance $\mathcal{D}=\frac{||\hat{H}_{\mathrm{nh}}-\hat{H}_{\mathrm{nh}}^\dagger||}{||\hat{H}_{\mathrm{nh}}||}$ and the observable-based non-Hermiticity score $SC^{\mathcal{F}}_{\mathrm{nh}}=|\mathcal{F}_{RR}[\rho_{RR}]-\mathcal{F}_{LL}[\rho_{LL}]|$, which compare the right- and left-evolved ensembles. These quantifiers are applied to two systems—the imperfect Bell-state construction and the interacting Hatano--Nelson model—to show that Hamiltonian non-Hermiticity and observable non-Hermiticity can diverge across parameter regimes and can qualitatively mark PT-symmetry transitions. The findings reveal that single- and multi-site observables, as well as dynamical quantities like purity and entanglement, may reflect non-Hermiticity differently than the Hamiltonian itself, highlighting the need to analyze both notions to fully characterize non-Hermitian physics. The framework provides a tool to identify exotic non-Hermitian phases and informs strategies for preparing resourceful states for quantum technologies.

Abstract

The non-Hermitian paradigm of quantum systems displays salient features drastically different from Hermitian counterparts. In this work, we focus on one such aspect, the difference of evolving quantum ensembles under $H_{\mathrm{nh}}$ (right ensemble) versus its Hermitian conjugate, $H_{\mathrm{nh}}^{\dagger}$ (left ensemble). We propose a formalism that quantifies the (dis-)similarity of these right and left ensembles, for single- as well as many-particle quantum properties. Such a comparison gives us a scope to measure the extent to which non-Hermiticity gets translated from the Hamiltonian into physically observable properties. We test the formalism in two cases: First, we construct a non-Hermitian Hamiltonian using a set of imperfect Bell states, showing that the non-Hermiticity of the Hamiltonian does not automatically comply with the non-Hermiticity at the level of observables. Second, we study the interacting Hatano--Nelson model with asymmetric hopping as a paradigmatic quantum many-body Hamiltonian. Interestingly, we identify situations where the measures of non-Hermiticity computed for the Hamiltonian, for single-, and for many-particle quantum properties behave distinctly from each other. Thus, different notions of non-Hermiticity can become useful in different physical scenarios. Furthermore, we demonstrate that the measures can qualitatively mark the model's Parity--Time (PT) symmetry-breaking transition. Our findings can be instrumental in unveiling new exotic quantum phases of non-Hermitian quantum many-body systems as well as in preparing resourceful states for quantum technologies.

Figures (8)

• Figure 1: Behavior of degree of non-Hermiticity computed using $\mathcal{D}=\frac{||\hat{H}_{\mathrm{nh}}-\hat{H}_{\mathrm{nh}}^\dagger||}{||\hat{H} _{\mathrm{nh}}||}$ for two different non-Hermitian Hamiltonians. (a) Behavior of $\mathcal{D}$ for $\hat{H}^{\mathrm{Bell}}_{\mathrm{nh}}$ constructed using a set of imperfect Bell states as defined in Eqs. (\ref{['eqn:right_vec']}) and (\ref{['eqn:left_vec']}). For each value of the non-Hermiticity parameter $\bar{\alpha}=1-\alpha$, we plot the maximum ($\mathcal{D}_{\max}$, blue) and the minimum ($\mathcal{D}_{\min}$, red) within 1000 random realizations of the Hamiltonian obtained by choosing its energy eigenvalues $\lbrace\lambda_m \rbrace$ from a Gaussian distribution with zero mean and unit variance. While $\mathcal{D}_{\min}$ increases only slowly and under strong fluctuations, $\mathcal{D}_{\max}$ exhibits a clear monotonic growth with $\bar{\alpha}$. (b) Behavior of $\mathcal{D}$ for the interacting non-Hermitian Hatano--Nelson model ($\hat{H}^{\mathrm{HN}}_{\mathrm{nh}}$). At low values of interaction $V$, $\mathcal{D}$ increases monotonically with asymmetric coupling $\chi$ and saturates to $\mathcal{D}\approx 2$. However, even for moderately large values of $V$, $\mathcal{D}$ takes a significant non-zero value only beyond a threshold coupling, $\chi_c$. Further, beyond $V\gtrsim 10^4$, $\mathcal{D}$ remains zero for all values of $\chi$ considered in our analysis. In other words, under strong interactions the distance between $\hat{H}_{\mathrm{nh}}^{\mathrm{HN}}$ and $(\hat{H}_{\mathrm{nh}}^{\mathrm{HN}})^\dagger$ becomes insignificant in comparison to the norm of $\hat{H}_{\mathrm{nh}}^{\mathrm{HN}}$ (see also Fig. \ref{['fig:Ham_unnorm']} in Appendix \ref{['appendixC']}). The white circles correspond to the PT symmetry-breaking transition of the model marked by non-analytic behavior of the finite-size level-spacing $\Delta_{01}:=\mathrm{Re}(\lambda_1 - \lambda_0)$ (inset: $\Delta_{01}$ versus $V$ for fixed $\chi=2.5$). There is a qualitative agreement between the regions where $\mathcal{D}$ drops and non-analyticities in $\Delta_{01}$. The data is reported for $N = 12$ sites, half-filling, and with anti-periodic boundary conditions. Note that in (b), and in all subsequent plots where the parameter $V$ is considered, it is presented on a logarithmic scale.
• Figure 2: Non-Hermiticity at the level of single- and two-site quantum properties of the model constructed using imperfect Bell states. (a) Non-Hermiticity score $SC^{m_z}$ of a single-site quantum property, the $z$-magnetization, computed for both right as well as left eigenvectors. The maximum and the minimum ($SC^{m_z}_k$) values of the score are obtained for $k \in 1,2$ and $k\in 3,4$, respectively. In both cases, the behavior remains qualitatively the same as the Hamiltonian non-Hermiticity shown in Fig. \ref{['fig:Ham']}(a). (b) Entanglement score, $SC^{\mathcal{S}}$, obtained for the von Neumann entropy. Interestingly, the behavior obtained for quantum entanglement differs from the single-site score. In particular, unlike $SC^{m_z}$, $SC^{\mathcal{S}}_k$ attains its minimum value for $k=1,2$ and becomes maximum for $k=3, 4$. Moreover, though $SC^{\mathcal{S}}_{k=3,4}$ shows monotonic growth similar to $SC^{m_z}$ and $\mathcal{D}_{\min,\max}$, $SC^{S}_{k=1,2}$ vanishes for low as well as high values of $\bar{\alpha}$ and attains a maximum around $\bar{\alpha} = 0.449$.
• Figure 3: Behavior of non-Hermiticity score with time, for global purity $\mathcal{P}$ and von Neumann entropy $\mathcal{S}$, computed for an initial state $\hat{\rho}_W=\delta |\Psi^-\rangle \langle \Psi^-|+ (1-\delta)\frac{\mathbb{I}}{4}$ subjected to evolution under $\hat{H}^{\mathrm{Bell}}_{\mathrm{nh}}$ and $(\hat{H}^{\mathrm{Bell}}_{\mathrm{nh}})^\dagger$. We probe the cases for $\alpha$=0.1, 0.3, 0.6, 0.9 as shown using curves with darker to lighter shades of blue, and in all cases fix the Hamiltonian eigenvalues to $\lambda_1=0.1,\ \lambda_2=0.2, \ \lambda_3=0.3, \ \lambda_4=0.4$. (a)-(c) Purity score, $SC^{\mathcal{P}}(t)$ for (a) $\delta=0.1$, (b) $\delta=0.5$, and (c) $\delta=0.9$. For large $\delta$, the amplitude of $SC^{\mathcal{P}}(t)$ decreases with increasing $\alpha$, while for small $\delta$ it attains a maximum at intermediate values of $\alpha$. (d)-(f) Non-Hermiticity score for the VNE, $SC^{\mathcal{S}}(t)$, for same choices of $\delta$.
• Figure 4: Comparison of non-Hermiticity at the level of the Hamiltonian and single- and multi-party physical properties of the non-Hermitian Hatano--Nelson model ($\hat{H}_{\mathrm{nh}}^{\mathrm{HN}}$). We compute the non-Hermiticty score for both the number operator $\hat{n}$ and half-chain VNE $\mathcal{S}$ for all the right and left eigenvectors of the model and finally compute the maximum among all of them. (a) Depicts the behavior of $SC^{\hat{n}}_{\max}$ in the $V-\chi$ plane. We note that for a large region in the parameter space, the behavior of $SC^{\hat{n}}_{\max}$ remains almost complementary to $\mathcal{D}$, as shown in Fig. \ref{['fig:Ham']}(b). A similar plot for the half-chain VNE is presented in (b), where for the same set of parameters, we plot the behavior of $SC^{\mathcal{S}}_{\max}$. We note that $SC^{\mathcal{S}}_{\max}$ behaves differently from both $SC^{\hat{n}}_{\max}$ as well as the degree of non-Hermiticity ($\mathcal{D}$). In particular, $SC^{\mathcal{S}}_{\max}$ attains significant value only when both $V$ and $\chi$ take large values. However, closer inspection shows the qualitative agreement between the trend of $SC^{\hat{n}}_{\max}$ and $SC^{\mathcal{S}}_{\max}$. The white circles correspond to the PT symmetry-breaking transition marked by the non-analytic behavior of $\Delta_{01}$ defined in Sec. \ref{['subsec:hn']} and shown in the inset of Fig. \ref{['fig:Ham']}. We notice the qualitative agreement between the PT symmetry-breaking transition and domain for vanishing scores. The data is reported for $N = 12$ sites, half-filling, and with anti-periodic boundary conditions.
• Figure 5: Global characteristics of $\overline{{SC}^\mathcal{F}_{\mathrm{nh}}}$ for the non-Hermitian Hatano--Nelson model ($H_{\mathrm{nh}}^{\mathrm{HN}}$). We compute the total number of elements of $\overline{{SC}^{\hat{n}}_{\mathrm{nh}}}$ and $\overline{{SC}^{\mathcal{S}}_{\mathrm{nh}}}$ with the conditions ${SC}^{\hat{n}}_{\mathrm{nh}}[k]\geq \mathcal{E}_{\mathrm{Th}}^{\hat{n}}$ and ${SC}^{\mathcal{S}}_{\mathrm{nh}}[k]\geq \mathcal{E}_{\mathrm{Th}}^{\mathcal{S}}$, respectively, where $k = 1,2,...,D$ with $D$ being the Hilbert space dimension. These quantities, scaled by $D$, are shown in (a) and (b). We obtain a similar trend as in Fig. \ref{['fig:Hatano_nelson']}. Data for threshold values $\mathcal{E}_{\mathrm Th}^{\hat{n}} = \mathcal{E}_{\mathrm Th}^{\mathcal{S}} = 0.1$. Similar to Figs. \ref{['fig:Ham']} and \ref{['fig:Hatano_nelson']}, the white circles correspond to the PT symmetry-breaking transition. The data is reported for $N = 12$ sites, half-filling, and with anti-periodic boundary conditions.