Extension of Noether’s theorem in $\mathcal{PT}$-symmetric systems and its experimental demonstration in an optical setup

The subject of finding the symmetries of dynamics is of fundamental interest and has broad applications in physics, e.g., high-energy scattering experiments, control issues in mesoscopic physics and quantum cosmology Symmetry1 ; chiralsymmetry1 ; chiralsymmetry2 ; chiralsymmetry3 ; chiralsymmetry4 ; topological2 . On the other hand, by means of symmetries, one can generally make non-trivial inferences from complex systems, such as many-body systems, dissipative systems and non-Hermitian systems. As an important theorem which is related to symmetries, Noether’s theorem Noether1 has important applications in quantum physics and quantum information science Noether2 ; Ehrenfest1 ; Ehrenfest2 ; Ehrenfest3 ; Noetheranalysis ; Noethercurrents . Noether’s theorem states that every symmetry of dynamics implies a conservation law, and it was originally applied in Lagrangian approach in classical mechanics to uncover conserved quantities from symmetries of the Lagrangian. In many cases, the existence of these conserved quantities is very important for understanding the physical states and the properties of the systems Noether2 ; Ehrenfest2 ; Noetheranalysis ; Noethercurrents . The theorem applies also in quantum mechanics, and the most prominent example of Noether’s theorem is Ehrenfest’s theorem in closed systems Ehrenfest1 ; Ehrenfest

For an operator $\hat{F}$ without explicit time dependence, it then follows that its expectation value $\langle{\hat{F}}\rangle$ is a constant of motion if it commutes with the Hermitian Hamiltonian $\hat{H}$. However, Ehrenfest’s theorem is not applicable for open systems Ehrenfest3 ; Ehrenfest ; open1 ; r2 ; r5 . Furthermore, even in closed systems, Ehrenfest’s conservation law cannot capture all features of symmetry when mixed states are considered Ehrenfest2 .

A natural extension of Noether’s theorem in non-Hermitian systems is to replace the Hermitian Hamiltonian $\hat{H}$ with a non-Hermitian Hamiltonian $\hat{H}^{\dagger}$, which turns eq. (1) into ${d\langle{\hat{F}}\rangle}/{dt}=\frac{1}{i\hbar}\langle{[\hat{F}\hat{H}-\hat{H}^{\dagger}{\hat{F}}]}\rangle+\langle{{d\hat{F}}/{dt}}\rangle$ conservationlaws2 ; interwining1 ; interwining2 ; SRM . Up to now, based on the important intertwining relation $\hat{F}\hat{H}=\hat{H}^{\dagger}{\hat{F}}$ interwining1 ; interwining2 ; SRM , several methods have been proposed to obtain conserved quantities, including spectral decomposition methods SDM1 ; SDM2 , recursive construction of intertwining operators RCM , sum-rules method SRM , Stokes parametrization approach SPA , and so on. Recently, the authors in ref. Ehrenfest4 investigated a manifestation of Noether’s theorem in non-Hermitian systems, where an inner product was defined as $(\varphi,\psi)\equiv\varphi_{u}^{T}\psi_{v}$ without its complex conjugation. In their framework, a generalized symmetry, which they termed pseudochirality, emerges naturally as the counterpart of the symmetry defined by the commutation relation in quantum mechanics. Some existing works Ehrenfest1 ; Ehrenfest2 ; Ehrenfest3 ; Ehrenfest ; Ehrenfest4 ; Noetheranalysis ; Noethercurrents ; conservationlaws2 ; open1 ; interwining1 ; interwining2 ; SRM ; SDM1 ; SDM2 ; RCM ; SPA ; r2 ; r5 enrich the understanding of obtaining conserved quantity beyond the Hermitian framework, whereas a full understanding of the relation between symmetry and conservation law, and practical methods for extracting expectation values in non-Hermitian systems, remain elusive. Therefore, in order to properly deal with conservation problems using Noether’s theorem and explore its potential applications in non-Hermitian systems, there is an urgent need to extend Noether’s theorem to non-Hermitian systems.

Over the past decades, there is considerable interest in the study of the dynamic properties of parity-time ($\mathcal{PT}$)-symmetric non-Hermitian systems Non-Hermitian1 ; Non-Hermitian2 ; Non-Hermitian3 ; Non-Hermitian4 ; Non-Hermitian5 ; Non-Hermitianadd ; r1 ; r6 . The unique properties of $\mathcal{PT}$-symmetric systems and their applications have been investigated in various physical systems topological1 ; Non-Hermitian6 ; optomechanics1 ; optomechanics2 ; photonics1 ; photonics2 ; microwave1 ; microwave2 ; r3 ; r4 . Moreover, many remarkable and unexpected quantum phenomena have been observed in $\mathcal{PT}$-symmetric systems, such as critical phenomena CriticalPhenomena1 ; CriticalPhenomena2 , chiral population transfer energytransfer1 ; energytransfer2 , information retrieval InformationRetrieva1 ; InformationRetrieva2 , coherence flow InformationRetrieva3 and topological invariants invariant1 ; invariant2 . A complete characterization of conservation laws in $\mathcal{PT}$-symmetric systems has been intensely explored RCM ; SDM2 . For example, based on the intertwining relation interwining1 ; interwining2 ; SRM , reference RCM has presented a complete set of conserved observables for a class of $\mathcal{PT}$-symmetric Hamiltonians in a single-photon linear optical circuit. Moreover, in the pseudo-Hermitian representation of quantum mechanics SDM1 , reference APT has further implemented a model circuit of a generic anti-$\mathcal{PT}$-symmetric system. A counterintuitive energy-difference conserving dynamics has been observed APT , which is in stark contrast to the standard Hermitian dynamics keeping the system’s total energy constant. However, based on biorthogonal quantum mechanics, the manifestation of Noether’s theorem and a complete observation of conserved quantities in $\mathcal{PT}$-symmetric systems and their consequences are still lacking both theoretically and experimentally.

In this work, we extend Noether’s theorem to a class of significant $\mathcal{PT}$-symmetric non-Hermitian systems and introduce a generalized expectation value of a time-independent operator based on biorthogonal quantum mechanics Biorthogonal1 ; Biorthogonal2 ; Biorthogonal3 ; Biorthogonal4 . For the $\mathcal{PT}$-symmetric systems considered here, the eigenvalues of the $\mathcal{PT}$-symmetric Hamiltonian $\hat{H}_{\mathcal{PT}}$ change from purely real numbers to purely imaginary numbers. Such $\mathcal{PT}$-symmetric systems have been widely used to investigate the dynamics of non-Hermitian systems in the presence of balanced gain and loss RCM ; Ehrenfest4 ; photonics1 ; CriticalPhenomena1 ; InformationRetrieva1 ; InformationRetrieva2 ; InformationRetrieva3 . Our work shows that the extended Noether’s theorem can be used to deal with conservation law problems about pure states and mixed states. Remarkably, we find that for an operator $\hat{F}$ without explicit time dependence, its generalized expectation value is a constant of motion if $\hat{F}$ presents a standard symmetry in the $\mathcal{PT}$-symmetry unbroken regime, or a chiral symmetry in the $\mathcal{PT}$-symmetry broken regime. In addition, we experimentally investigate the extended Noether’s theorem in $\mathcal{PT}$-symmetric single-qubit and two-qubit systems using an optical setup. Several novel results are found. First, our experiment demonstrates the existence of the constant of motion. Second, our experiment reveals that the constant of motion can be used to judge whether the $\mathcal{PT}$ symmetry of a system is broken. Last, our experiment reveals the phenomenon of masking quantum information masking1 ; masking2 in a $\mathcal{PT}$-symmetric two-qubit system.

To extend Noether’s theorem to $\mathcal{PT}$-symmetric systems, the biorthogonal quantum mechanics Biorthogonal1 ; Biorthogonal2 ; Biorthogonal3 ; Biorthogonal4 is applied. In biorthogonal quantum mechanics, the inner product is defined as

where $|\psi\rangle=\Sigma_{l}c_{l}|{\phi_{l}}\rangle$ ($|\varphi\rangle=\Sigma_{k}d_{k}|{\phi_{k}}\rangle$) is an arbitrary pure state with its associated state $\langle\widehat{\psi}|\equiv\Sigma_{l}c^{*}_{l}\langle\widehat{{\phi_{l}}}|$ ($\langle\widehat{\varphi}|\equiv\Sigma_{k}d^{*}_{k}\langle\widehat{{\phi_{k}}}|$), and $\{\langle\widehat{{\phi_{l(k)}}}|\}$ and $\{{|\phi_{l(k)}}\rangle\}$ are left and right eigenstates of a non-Hermitian Hamiltonian (Appendixes A.1 and A.2).

Here, we use ${\hat{\rho}}$ (${\hat{\rho}}_{b}$) to denote a density operator in standard (biorthogonal) quantum mechanics. Without loss of generality, let us consider the $\mathcal{PT}$-symmetric system to be in a mixed state ${\hat{\rho}}_{b}(t)=\sum_{n=1}^{N}p_{n}|\psi_{n}(t)\rangle\langle\widehat{\psi_{n}(t)}|$, $p_{n}$ is the probability of the system being in a pure state $|\psi_{n}(t)\rangle$, with $\langle\widehat{\psi_{n}}(t)|\psi_{n}(t)\rangle=1$. With the inner product introduced in eq. (2), a generalized expectation value $(\hat{F})$ of an operator $\hat{F}$ can be defined (see Appendix A.3)

where $\langle\widehat{\psi_{n}(t)}|{\hat{F}}|\psi_{n}(t)\rangle$ is the generalized expectation value $(\hat{F})$ of the operator $\hat{F}$ for an arbitrary pure state $|\psi_{n}(t)\rangle$. Equation (3) provides a natural generalization of expectation value of an operator $\hat{F}$ for an arbitrary quantum state, either a mixed state or a pure state.

As one of the main contributions of this work, we find that the temporal evolution of the expectation value $(\hat{F})$ of the operator $\hat{F}$ follows two different forms (see Appendix A.3 for the detailed derivation)

where $(\cdot)_{n}=\langle\widehat{\psi_{n}(t)}|\cdot|\psi_{n}(t)\rangle$. Equation (5) corresponds to the case when the system works in the $\mathcal{PT}$-symmetry unbroken regime, while eq. (6) corresponds to the case when the system works in the $\mathcal{PT}$-symmetry broken regime. From eq. (5), one can see that the expectation value $(\hat{F})$ is a constant of motion if the Hamiltonian $\hat{H}_{\mathcal{PT}}$ and the time-independent operator $\hat{F}$ satisfy the commutation relation $[\hat{F},\hat{H}_{\mathcal{PT}}]=0$, i.e., the operator $\hat{F}$ presents a standard symmetry in the $\mathcal{PT}$-symmetry unbroken regime Symmetry1 . On the other hand, eq. (6) implies that the expectation value $(\hat{F})$ is also a constant of motion if $\hat{H}_{\mathcal{PT}}$ and $\hat{F}$ satisfy the anti-commutation relation $\{\hat{F},\hat{H}_{\mathcal{PT}}\}=0$, i.e., $\hat{F}$ presents a chiral symmetry in the $\mathcal{PT}$-symmetry broken regime chiralsymmetry1 .

To understand the above results intuitively, let us consider a $\mathcal{PT}$-symmetric single-qubit system where the eigenvalues of the Hamiltonian change from real (in the $\mathcal{PT}$-symmetry unbroken regime), to purely imaginary (in the $\mathcal{PT}$-symmetry broken regime). The Hamiltonian for this system is given by (hereafter, we assume $\hbar=1$)

where $i\gamma\hat{\sigma}_{z}$ is the non-Hermitian part of the Hamiltonian governing gain and loss Non-Hermitian1 ; non-Hermitianadd . The parameter $s>0$ is an energy scale, $a=\gamma/s>0$ is a coefficient representing the degree of non-Hermiticity, $\hat{\sigma}_{x}$ and $\hat{\sigma}_{z}$ are the standard Pauli operators. The eigenvalues of $\hat{H}_{\mathcal{PT}}$ are given by $E_{1}=s\sqrt{1-a^{2}}$ and $E_{2}=-s\sqrt{1-a^{2}}$, which are real numbers for $0<a<1$ (the $\mathcal{PT}$-symmetry unbroken regime), while purely imaginary numbers for $a>1$ (the $\mathcal{PT}$-symmetry broken regime). The right eigenvectors of $\hat{H}_{\mathcal{PT}}$ are $|\phi_{1}\rangle=f_{1}\times(A_{1}|0\rangle+|1\rangle)$ and $|\phi_{2}\rangle=f_{2}\times(A_{2}|0\rangle+|1\rangle)$, while the left eigenvectors of $\hat{H}_{\mathcal{PT}}$ are $\langle\widehat{\phi}_{1}|=f_{3}^{*}\times(-A_{2}^{*}\langle 0|+\langle 1|)$ and $\langle\widehat{\phi}_{2}|=f_{4}^{*}\times(-A_{1}^{*}\langle 0|+\langle 1|)$ (Appendix A.2). Here, $A_{1}=ia+\sqrt{1-a^{2}}$, $A_{2}=ia-\sqrt{1-a^{2}}$, and $f_{1},f_{2},f_{3},f_{4}$ satisfy $f_{1}\cdot{f}_{3}^{*}\times(1-A^{*}_{2}A_{1})=f_{2}\cdot{f}_{4}^{*}\times(1-A^{*}_{1}A_{2})=1$ to satisfy the biorthogonality and closure relations.

The Hamiltonian (7) can be considered as a deformed Pauli operator, $\hat{H}_{\mathcal{PT}}=E_{1}|{\phi_{1}}\rangle\langle\widehat{{\phi_{1}}}|-E_{1}|{\phi_{2}}\rangle\langle\widehat{{\phi_{2}}}|$, in view of the biorthogonal partners $\{|{\phi_{1}}\rangle,|{\phi_{2}}\rangle\}$ and $\{\langle\widehat{{\phi_{1}}}|,\langle\widehat{{\phi_{2}}}|\}$ (Appendixes A.1 and A.2). If a time-independent operator $\hat{F}$ can be expressed in the form

where $c_{1}$ and $c_{2}$ are arbitrary nonzero coefficients, one can easily verify $[\hat{F},\hat{H}_{\mathcal{PT}}]=0$. Thus, according to eq. (5), the expectation value $(\hat{F})$ is a constant of motion in the $\mathcal{PT}$-symmetry unbroken regime. On the other hand, if a time-independent operator $\hat{F}$ can be expressed in the form

where $\tilde{c}_{1}$ is an arbitrary nonzero coefficient, one can obtain $\{\hat{F},\hat{H}_{\mathcal{PT}}\}=0$. In this case, according to eq. (6), the expectation value $(\hat{F})$ is a constant of motion in the $\mathcal{PT}$-symmetry broken regime.

From an experimental point of view, in order to keep the expectation value $(\hat{F})$ as a real number, the chosen operator $\hat{F}$ should be Hermitian in biorthogonal quantum mechanics (see Appendix A.4). Therefore, in the subsequent discussion, the coefficients $c_{1}$ and $c_{2}$ in eq. (8) are chosen as real numbers, and the coefficient $\tilde{c}_{1}$ in eq. (9) is chosen as a purely imaginary number.

We have extended Noether’s theorem to a class of significant $\mathcal{PT}$-symmetric non-Hermitian systems and introduced a generalized expectation value of a time-independent operator based on biorthogonal quantum mechanics. We have demonstrated that in the $\mathcal{PT}$-symmetry unbroken regime, the generalized expectation value of a time-independent operator is a constant of motion, if the time-independent operator and the non-Hermitian Hamiltonian satisfy the commutation relation, i.e., the operator presents a standard symmetry. Moreover, even in the $\mathcal{PT}$-symmetry broken regime, the expectation value of a time-independent operator is still a constant of motion provided the operator and the non-Hermitian Hamiltonian satisfy the anti-commutation relation, i.e., the operator presents a chiral symmetry. Furthermore, we have experimentally confirmed our predictions in $\mathcal{PT}$-symmetric single-qubit and two-qubit systems by using an optical setup. Our experiment has demonstrated the existence of the predicted constant of motion. Meanwhile, a novel phenomenon of masking quantum information is first observed in a $\mathcal{PT}$-symmetric two-qubit system. The extended Noether’s theorem not only contributes to a full understanding of the relation between symmetry and conservation law in $\mathcal{PT}$-symmetric physics, but also has potential applications in quantum information theory and quantum communication protocols.

The present work has some elements in common with previous works on obtaining conserved quantity in non-Hermitian systems, especially the idea of using pseudo-Hermiticity (equivalently, the intertwining relation) interwining1 ; interwining2 ; SRM ; SDM1 ; SDM2 . Therefore, we here address the difference between our work and previous works. As shown in Refs. interwining1 ; interwining2 , every Hamiltonian with a real spectrum is pseudo-Hermitian, and all the $\mathcal{PT}$-symmetric non-Hermitian Hamiltonians belong to the so-called pseudo-Hermitian Hamiltonians. In the pseudo-Hermitian representation of quantum mechanics, the expectation value $\langle{\hat{F}}\rangle$ of a time-independent operator ${\hat{F}}$ is a conserved quantity provided the intertwining relation, $\hat{F}\hat{H}=\hat{H}^{\dagger}{\hat{F}}$, is satisfied. In principle, a complete set of conserved observables can be obtained by numerically solving a set of $N^{2}$-dimensional linear intertwining relation SRM ; SDM1 ; SDM2 ; RCM ; SPA . However, a common problem, which one may encounter via pseudo-Hermiticity (intertwining relation), is how to connect the conserved quantities with the symmetries of dynamics. Compared with previous works interwining1 ; interwining2 ; SRM ; SDM1 ; SDM2 ; RCM ; SPA , the main difference of our work is that by introducing a generalized expectation value of an operator based on biorthogonal quantum mechanics, we connect two important symmetries with conserved operators in the $\mathcal{PT}$-symmetry unbroken and broken regimes, respectively. We remark that the proposed standard symmetry $\hat{F}\hat{H}=\hat{H}{\hat{F}}$ and the chiral symmetry $\hat{F}\hat{H}=-\hat{H}{\hat{F}}$ are essentially different from the intertwining relation $\hat{F}\hat{H}=\hat{H}^{\dagger}{\hat{F}}$, because of $\hat{H}\neq\hat{H}^{\dagger}$ and $\hat{H}\neq-\hat{H}^{\dagger}$ in $\mathcal{PT}$-symmetric systems.

We note that the extended Noether’s theorem is always valid for such $\mathcal{PT}$-symmetric systems provided the eigenvalues of $\hat{H}_{\mathcal{PT}}$ change from purely real numbers to purely imaginary numbers; or equivalently, $\hat{H}_{\mathcal{PT}}$ exhibits an exceptional point of the order of the system’s dimension. As an example, consider a 3-dimensional $\mathcal{PT}$-symmetric system SDM2 , for which the Hamiltonian reads $H_{\mathcal{PT}}=sJ_{x}+i\gamma J_{z}$, where $J_{x}$ and $J_{z}$ are the 3-dimensional angular momentum operators. Such a $\mathcal{PT}$-symmetric Hamiltonian has a third-order exceptional point at $\gamma=s$ and its spectrum also changes from real to purely imaginary SDM2 . Then, based on the extended Noether’s theorem, one can quickly find its conserved quantities in the $\mathcal{PT}$-symmetry unbroken and broken regimes, respectively.

For any quantum system, whose Hamiltonian can be simplified to the form in eq. (7), the extended Noether’s theorem presented in this work can be implemented straightforwardly. Note that for the simplified Hamiltonian, arbitrary dressed states can be chosen as basis states as long as the dressed states satisfy the biorthogonality and closure relations. This might lead to a useful step toward realizing fast symmetry discrimination and conserved quantity acquisition for multi-qubit $\mathcal{PT}$-symmetric systems. Moreover, in above discussion, we focus on the case of an operator $\hat{F}$ without explicit time dependence. However, the derived equations (5,6) also work well in a general case i.e., the operator $\hat{F}(t)$ is time-dependent. Then, one may obtain constant of motion for a time-dependent operator in a time-dependent $\mathcal{PT}$-symmetric system, which may be interesting and attractive. Furthermore, in some sense, the $\mathcal{PT}$-symmetric Hamiltonian in eq. (7) has parallels with non-Hermitian topological phases topological1 ; topological2 and the extended classification of topological classes chiralsymmetry3 ; chiralsymmetry4 . The discovery of the relation between conserved quantities and non-Hermitian topological invariants invariant1 ; invariant2 is also interesting and attractive, which is a fascinating field where further extension of this work may be explored.

This work was supported by the National Natural Science Foundation of China (NSFC) (Grants Nos. 12264040, 12204311, 11804228, 11865013 and U21A20436), Jiangxi Natural Science Foundation (20212BAB211018, 20192ACBL20051), the project of Jiangxi Province Higher educational Science and Technology Program (Grant Nos. GJJ190891, GJJ211735), and Key-Area Research and Development Program of Guang Dong province (2018B03-0326001). F.N. is supported in part by Nippon Telegraph and Telephone Corporation (NTT) Research, the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R$\&$D Grant Number JPMJMS2061], the Japan Society for the Promotion of Science (JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the Asian Office of Aerospace Research and Development (AOARD) (via Grant No. FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06.