Abstract
The observation of genuine quantum effects in systems governed by nonHermitian Hamiltonians has been an outstanding challenge in the field. Here we simulate the evolution under such Hamiltonians in the quantum regime on a superconducting quantum processor by using a dilation procedure involving an ancillary qubit. We observe the parity–time (\({\mathcal{PT}}\))symmetry breaking phase transition at the exceptional points, obtain the critical exponent, and show that this transition is associated with a loss of state distinguishability. In a twoqubit setting, we show that the entanglement can be modified by local operations.
Introduction
The Hermiticity of physical observables is a fundamental tenant of standard quantum physics, guaranteeing real eigenspectra and leading to the generation of unitary dynamics in closed quantum systems. However, this is needlessly restrictive: it has been shown^{1} that nonHermitian Hamiltonians endowed with parity and time (\({\mathcal{PT}}\)) symmetry possess real positive eigenvalues and eigenvectors with positive norm. Experimental platforms where nonHermitian Hamiltonians can be implemented to comprise optical waveguides^{2,3,4}, polarized photons^{5}, nuclear spins^{6,7}, superconducting circuits^{8,9}, mechanical oscillators^{10}, nitrogenvacancy centers in diamond^{11,12}, fiberoptic networks^{13}, and ultracold Fermi gases^{14}. Open systems are a natural candidate for realizing these Hamiltonians, since nonHermitian terms appear naturally as a consequence of energy being injected or lost^{15}.
However, a major drawback of opensystem approaches is the need to control precisely the gain and the dissipation: these experiments require complicated setups with gain and alternating losses^{5}, and yet only wavelike effects can be observed. Moreover, employing gainloss systems for the study of quantum properties such as entropy, entanglement, and correlations is fundamentally impossible because gain inevitably adds noise^{16}. Thus, in order to make progress one would need genuine realizations of nonHermitian dynamics in the quantum regime^{4,9,12}, which maintain and allow the measurement of delicate quantum effects.
Here we show that the nonHermitian dynamics can be simulated digitally^{17} in a superconducting quantum processor by extending the Hilbert space with the use of an ancilla qubit and under the action of appropriately defined gates. To achieve this, we combine two techniques: a dilation method that is universal (applicable to any Hamiltonian)^{12} and an optimal method for generating any twoqubit gate with combinations of singlequbit gates and at most three CNOT (controlledNOT) gates^{18,19}. This combination enables us to observe and fully characterize the broken \({\mathcal{PT}}\)symmetry transition and to settle decisively the relationship between nonHermitian quantum mechanics and nogo theorems on state distinguishability and monotony of entanglement^{20,21,22,23}. We achieve this by making use of the emergent technology of superconducting processors, on which significant technical progress has been shown in recent times by IBM^{24}. Although still imperfect, these devices have already enabled important results such as quantum error correction^{25}, faulttolerant gates^{26}, proofs of violation of Mermin^{27} and Leggett–Garg^{28} inequalities, demonstrations of nonlocal parity measurements^{29,30}, simulations of paradigmatic models in open quantum systems^{31}, the creation of highly entangled graph states^{32}, the determination of molecular groundstate energies^{33}, the implementation of quantum witnesses^{34}, and quantumenhanced solutions to large systems of linear equations^{35}. The use of a superconducting quantum processor offers the possibility of extracting all relevant quantum correlations and of designing and programming efficiently the required gates, adapted to the particular topology of the machine.
We consider the generic system qubit nonHermitian Hamiltonian^{36} with natural units ℏ = 1
where r is a real parameter and σ_{x} and σ_{z} are the Pauli matrices. The eigenvalues are \(\pm\! \sqrt{1{r}^{2}}\), and the condition for nonHermiticity is simply r ≠ 0. The parity operator is \({\mathcal{P}}={\sigma }_{x}\) and the timeinversion operator is the complex conjugation operator \({\mathcal{T}}=\star\). This Hamiltonian has an exceptional point at ∣r∣ = 1, where the two eigenvectors coalesce and the eigenvalues become parallel. For ∣r∣ < 1 the eigenvalues are real, corresponding to distinct eigenvectors, and the Hamiltonian satisfies \({\mathcal{PT}}\)symmetry \([{\mathcal{PT}},{H}_{{\rm{q}}}]=0\) (see Supplementary Note 1); for ∣r∣ > 1, the eigenvalues become imaginary and the \({\mathcal{PT}}\) symmetry is broken. The Hamiltonian Eq. (1) can be understood as the standard nonHermitian form providing equal coupling (offdiagonal terms) between the basis states as well as equal gain and loss via the complex diagonal terms required for \({\mathcal{PT}}\) symmetry^{37}.
We realize the Hamiltonian Eq. (1) in a dilated space with the help of an ancilla, observing singlequbit dynamics under PTsymmetric Hamiltonians and witnessing the coalesce of eigenvectors at the exceptional points. Further, by allowing different quantum states to evolve under the same set of operations generated by nonHermitian generators, we show that the trace distance between arbitrary states is modified—a task that is forbidden in Hermitian quantum mechanics. We extract the critical exponent of the transition, obtaining a value in agreement with theoretical predictions. We also observe an apparent violation of entanglement monotonicity in a twoqubit system, where one of the qubits is driven by a nonHermitian Hamiltonian. Finally, we conclude by providing the complete dynamics of correlations developed between system qubits and the ancilla.
Results
To realize the Hamiltonian Eq. (1) we use a Naimark dilation procedure employing an additional ancilla qubit and a Hermitian operator \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) acting on the total qubit–ancilla Hilbert space. The dynamics under \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) is determined by the Schrödinger equation
whose solution is given by
where \({\left\psi (t)\right\rangle }_{{\rm{q}}}\) is the solution of \(i\frac{\mathrm{d}}{{\mathrm{d}}t}{\left\psi (t)\right\rangle }_{{\rm{q}}}={H}_{{\rm{q}}}{\left\psi (t)\right\rangle }_{{\rm{q}}}\). Here \({\left\tilde{\psi }(t)\right\rangle }_{{\rm{q}}}=\eta (t){\left\psi (t)\right\rangle }_{{\rm{q}}}\), where η(t) is a positive linear operator given by \(\eta (t)={[(1+{\eta }_{0}^{2})\exp (i{H}_{{\rm{q}}}^{\dagger }t)\exp (i{H}_{{\rm{q}}}t){\Bbb{I}}]}^{1/2}\)^{12}, with \(\eta (0)={\eta }_{0}{\Bbb{I}}\) at the initial time t = 0 (see “Methods”).
In order to obtain a solution of the form Eq. (3), the ancilla and the qubit are initialized in the state \({\left{{\Psi }}(0)\right\rangle }_{{\rm{a}},{\rm{q}}}=({\left0\right\rangle }_{{\rm{a}}}+{\eta }_{0}{\left1\right\rangle }_{{\rm{a}}})\otimes {\left\psi (0)\right\rangle }_{{\rm{q}}}\), as shown in Fig. 1a, by using a rotation R_{y}(θ) on the ancilla qubit, where \(\theta =2\,{\tan }^{1}\,{\eta }_{0}\) and \({\left\psi (0)\right\rangle }_{{\rm{q}}}={\left0\right\rangle }_{{\rm{q}}}\). Thus, the dynamics of the system qubit in the subspace with the ancilla in state \({\left0\right\rangle }_{{\rm{a}}}\) satisfies the desired evolution by the nonHermitian Hamiltonian Eq. (1). For an arbitrary time t and for any given r the corresponding unitary operator \({U}_{{\rm{a}},{\rm{q}}}(t)={\rm{T}}\exp [i\mathop{\int}\nolimits_{0}^{t}{{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(\tau )\mathrm{d}\tau ]\) is obtained numerically as described below.
Parity–time symmetry breaking in a single qubit
We implement the unitary operator U_{a,q}(t) using the q[0] and q[1] qubits of a fivequbit IBM quantum processor for different values of the Hamiltonian parameter r. We start by presenting in Fig. 1b the expected theoretical results for the groundstate population obtained under H_{q}, that is, generated by the nonunitary evolution operator \(\exp (i{H}_{{\rm{q}}}t)\) (see “Methods”). The breaking of \({\mathcal{PT}}\) symmetry, as one crosses the exceptional point r = 1, is clearly visible. Indeed, for r < 1 the eigenvalues of H_{q} are real and one observes Rabi oscillations. When ∣r∣ exceeds 1, the eigenvalues of H_{q} become imaginary, the \({\mathcal{PT}}\) symmetry is broken, and what one observes is the decay of the population. Figure 1c–e presents the results from the experimental realization of H_{q} on IBM quantum experience for three different values of r. We note that the agreement with the theoretical values is excellent. Each experiment is repeated 8192 times. Thus, the statistical errors here are of the order of \(1/\sqrt{8192}=0.01\), which lead to the error bars too small to be shown distinctly in the experimental plots. In addition, in various experiments presented here, we track the systematic errors in terms of measurement corrections and incorporate these corrections in respective experimental datasets as described in “Methods”, with further details given in the Supplementary Note 5.
The details of the implementation are shown in Fig. 1a. We start with the qubit and the ancilla both initialized in the state \(\left0\right\rangle\), after which the ancilla alone is subjected to a rotation along the yaxis by an angle θ, which initializes the ancilla subspace θ^{12}. The explicit form of the operator U_{a,q}(t) at any arbitrary time t is found by a numerical decomposition into single and twoqubit gates^{18}. This decomposition U_{num}(t) = U_{n}…U_{1} matches with the desired unitary operator U_{a,q}(t) with a fidelity F_{U} > 0.99, where the fidelity is defined as F_{U} = 1 − ∣∣U_{num}(t) − U_{a,q}(t)∣∣/∣∣U_{a,q}(t)∣∣ (also see Supplementary Note 2). The quantum circuit that implements the decomposition of U_{num}(t) (see inset of Fig. 1a) comprises a sequence of singlequbit rotations \({{\mathcal{U}}}_{{\rm{q}}({\rm{a}})}^{j}\), each of them having up to three degrees of freedom, and three twoqubit CNOT gates^{18,19}. The width of this circuit is 2 and the depth is 8. Specifically, the singlequbit gates are general rotations given by \({{\mathcal{U}}}_{{\rm{q}}({\rm{a}})}^{j}(\alpha ,\beta ,\gamma )={R}_{z}{(\alpha )}_{{\rm{q}}({\rm{a}})}^{j}{R}_{y}{(\beta )}_{{\rm{q}}({\rm{a}})}^{j}{R}_{z}{(\gamma )}_{{\rm{q}}({\rm{a}})}^{j}\), where α, β, and γ are the angles of rotations and the operators R_{y}, R_{z} correspond to the rotations generated by the Pauli operators σ_{y} and σ_{z}, respectively. The operator \({{\mathcal{U}}}_{{\rm{q}}({\rm{a}})}^{j}(\alpha ,\beta ,\gamma )\) has a direct correspondence with the singlequbit operator U3, as defined by IBM. For instance, given r = 0.6 and t = 0.5 (see Fig. 1a), we have the following set of operations: \({{\mathcal{U}}}_{{\rm{a}}}^{1}(2.83,0.55,3.72),\ {{\mathcal{U}}}_{{\rm{q}}}^{1}(0.51,2.98,\,1.63)\), \({{\mathcal{U}}}_{{\rm{a}}}^{2}(1.75,\,3.34,\,4.60),\ {{\mathcal{U}}}_{{\rm{q}}}^{2}(0.00,\,0.00,\,4.02)\), \({{\mathcal{U}}}_{{\rm{a}}}^{3}(4.81,3.08,1.02),\ {{\mathcal{U}}}_{{\rm{q}}}^{3}(0.01,0.29,0.04)\), and \({{\mathcal{U}}}_{{\rm{a}}}^{4}(0.00,5.19,0.50),\ {{\mathcal{U}}}_{{\rm{q}}}^{4}(0.46,1.51,0.37)\). After the U_{num}(t) implementation, the postselected subspace of our interest corresponds to the ancilla in state \({\left0\right\rangle }_{a}\). At the end of the algorithm, we measure the probabilities P_{kl} of the qubit–ancilla state in the computational basis \(\{{\leftkl\right\rangle }_{{\rm{a}},{\rm{q}}}\equiv {\leftk\right\rangle }_{{\rm{a}}}{\leftl\right\rangle }_{{\rm{q}}}\}\) with k, l ∈ {0,1}. Finally, the groundstate population in the desired postselected subspace of the system qubit is given by, p_{0}(t) = P_{00}/(P_{00} + P_{01}), which can be obtained directly from the experiments. These are shown with red dots in Fig. 1c–e and follow very closely the results for the population in the \({\left0\right\rangle }_{{\rm{q}}}\) state of the qubit under the nonHermitian Hamiltonian Eq. (1). The results demonstrate a highfidelity simulation of the \({\mathcal{PT}}\)symmetry breaking in a single qubit.
Quantum state distinguishability
Next, we demonstrate an unexpected consequence of nonHermitian dynamics concerning state distinguishability. Designing a general protocol to distinguish two (or more) arbitrary quantum states is a challenge in standard Hermitian quantum mechanics. On the other hand, the evolution of an arbitrary pair of states under a nonHermitian operator can alter the distance between them, and may even make the arbitrary pair of quantum states orthogonal^{20,22,23}. To observe this unusual feature of nonHermitian dynamics, we use the quantum circuit in Fig. 1a to evolve the system qubit, initialized, respectively, in the orthogonal states \({\left0\right\rangle }_{{\rm{q}}}\) and \({\left1\right\rangle }_{{\rm{q}}}\). At various different instances of time, the state of the system qubit in the postselected subspace with ancilla in state \({\left0\right\rangle }_{{\rm{a}}}\) is obtained and the trace distance
between the respective states is determined, where ρ_{diff}(t) = ρ_{1q}(t) − ρ_{2q}(t) and \({\rho }_{i{\rm{q}}}(t)={\left{\psi }_{i}(t)\right\rangle }_{{\rm{q}}}{\left\langle {\psi }_{i}(t)\right}_{{\rm{q}}}\). For the given pair of initial states, the expected pattern for the variation of \({\mathcal{D}}\) with r and t is shown in Fig. 2a. The characteristic recurrence time T_{R} in the \({\mathcal{PT}}\)symmetric phase and the decay time \({\tau}_{\rm{D}}\) in the brokensymmetry phase are plotted in Fig. 2b and compared to their analytical expressions (\({T}_{\mathrm{R}}=\pi /\sqrt{1{r}^{2}}\) and \(\tau_{\rm{D}} =1/2\sqrt{{r}^{2}1}\) from Supplementary Eqs. (S11) and (S12)). Note that these times reflect the delicate balance between gain and loss, which is encoded in the structure of the Hamiltonian (see Supplementary Note 4). In Fig. 2c–e we show the experimentally obtained variation of the trace distance for three different values of r. An oscillating pattern in the trace distance is obtained for r < 1, which is a signature of information exchange between the system and the environment, while for r ≥ 1 we measure a decay pattern, which corresponds to loss of information to the environment. Interestingly, the oscillations in distinguishability correspond to oscillations in entanglement of qubit–ancilla state^{22} (see Supplementary Fig. 2). For r = 1 (exceptional point) these timescales diverge, and one cannot define anymore a characteristic time of the system. Instead, in close analogy with phase transitions in manybody systems, the distinguishability follows asymptotically a powerlaw \({\mathcal{D}} \sim {t}^{\delta }\)^{22}, where the critical exponent δ = 2 corresponds to two coalescing eigenstates. We have first checked numerically that for t ≫ 1 the distinguishability indeed displays this powerlaw behavior, with the critical exponent very close to 2. We can verify this scaling also experimentally, with the caveat that for t ≳ 3 the distingusihability becomes already smaller than the precision that we can reach on the IBM machine. Still, we can identify an interval t ∈ [1, 3] where the theoretical plot \(\mathrm{ln}\,{\mathcal{D}}\) versus \(\mathrm{ln}\,t\) starts to be approximately linear, with slope δ = 1.93 ± 0.08, see inset of Fig. 2d. In this region, we obtain by fitting the experimental data δ = 1.75 ± 0.15 (dashed red line in the inset), a reasonably close value.
Evaluating the distinguishability requires a complete characterization of the singlequbit density matrices ρ_{1q}(t) and ρ_{2q}(t), which is done by a set of three operations that independently fetch the elements of the density matrix. Each of these experiments is repeated 8192 times, such that even after evaluating Eq. (4), the statistical error in the measure of distinguishability remains small.
Bipartite systems under nonHermitian evolution
Next, we observe the dynamics of entanglement in a bipartite system when one of the parties undergo a local operation generated by H_{q}, for different values of r. Such scenarios have been studied theoretically^{21,22}, and it was shown that entanglement restoration and information recovery can happen in the \({\mathcal{PT}}\)symmetric phase. This breaks entanglement monotonicity, allowing the creation of entanglement by a local operation. This unusual effect is due to the modified evolution in the postselected subspace due to mere existence of a component of the total wavefunction outside this subspace^{4,38,39}.
To study this phenomenon, we consider a system consisting of two qubits q and \({\rm{q}}^{\prime}\) initialized in a maximally entangled Bell state, \({\left{{{\Phi }}}^{+}\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }=({\left00\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }+{\left11\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} })/\sqrt{2}\). One system qubit (say q) undergoes a nonHermitian evolution by \({{\Bbb{H}}}_{{\rm{q}},{\rm{q}}^{\prime} }={H}_{{\rm{q}}}\otimes {{\Bbb{I}}}_{{\rm{q}}^{\prime} }\) with the help of an ancillary qubit a, such that the total Hamiltonian including the dilation is \({{\Bbb{H}}}_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }={{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}\otimes {{\Bbb{I}}}_{{\rm{q}}^{\prime} }\) leading to a unitary evolution, \({{\Bbb{U}}}_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }={U}_{{\rm{a}},{\rm{q}}}\otimes {{\Bbb{I}}}_{{\rm{q}}^{\prime} }\). Finally, the threepartite state of the system is measured and postselected subject to the state of the ancilla being \({\left0\right\rangle }_{{\rm{a}}}\).
The experimental implementation on the IBM quantum processor is carried out using three qubits, as shown in Fig. 3a. As before, we average over 8192 realizations. We perform the complete quantum state tomography of the twoqubit system in the postselected subspace at various different values of time t. This is done using a set of seven experiments on the system qubits q and \({\rm{q}}^{\prime}\), followed by σ_{z} measurements of all three qubits as shown in Fig. 3a—see Supplementary Note 5 for further details. At the end of each of these tomography measurements, the populations are obtained as \({p}_{kl}={P}_{0kl}/\mathop{\sum }\nolimits_{m,n = 0}^{1}{P}_{0mn}\), where k, l ∈ {0, 1}, from which the desired density operator of the system qubits \({\rho }_{{\rm{q}},{\rm{q}}^{\prime} }^{(0)}\) in the postselected subspace is obtained. To study the entanglement dynamics, we use the concurrence^{40,41} as a measure of entanglement, given by \({{\mathcal{C}}}_{{\rm{q}},{\rm{q}}^{\prime} }^{(0)}=\max \{0,\sqrt{{\lambda }_{1}}\sqrt{{\lambda }_{2}}\sqrt{{\lambda }_{3}}\sqrt{{\lambda }_{4}}\}\), where λ_{i}’s are the eigenvalues of the operator \({\rho }_{{\rm{q}},{\rm{q}}^{\prime} }^{(0)}({\sigma }_{y}\otimes {\sigma }_{y}){({\rho }_{{\rm{q}},{\rm{q}}^{\prime} }^{(0)})}^{\star }({\sigma }_{y}\otimes {\sigma }_{y})\) written in decreasing order.
The change of concurrence with time is then observed for different values of r, as shown in Fig. 3b–d. For r = 0 we have checked that the dynamics is unitary and there is no variation in the entanglement values. In this case, the standard result that the entanglement is not changed by local operations is confirmed. However, for 0 < r < 1, the concurrence is found to be oscillating, which is clearly seen in Fig. 3b, while for r > 1 the Hamiltonian H_{q} governing local evolution ceases to obey the symmetry, and the entanglement gradually decays with time, as shown in Fig. 3d. For r = 1 we find the same theoretical asymptotic scaling as for distinguishability \({{\mathcal{C}}}_{{\rm{q}},{\rm{q}}^{\prime} }^{(0)} \sim {t}^{\delta }\), where δ = 2. To compare with the experiment, again we restrict the time to t ∈ [1, 3], and find δ = −1.71 ± 0.01 from the theoretical curve and δ = −1.93 ± 0.27 from the measured data (see inset in Fig. 3c). In Fig. 3f we present the corresponding theoretical curves for the time variation of concurrence for a wider range of r parameters.
The obtained variation in concurrence under a local operation contradicts at first sight the wellknown property of monotonicity of entanglement. To make this effect even more striking, we have performed another experiment where we observe the increase in entanglement between the qubits q and \({\rm{q}}^{\prime}\) under the action of the \({\mathcal{PT}}\)symmetric nonHermitian Hamiltonian. Specifically, at t = 0 we prepare the state \(\cos (\vartheta ){\left{{{\Phi }}}^{}\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }i\sin (\vartheta ){\left{{{\Psi }}}^{+}\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }\), where \({\left{{{\Psi }}}^{\pm }\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }=({\left01\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }\pm {\left10\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} })/\sqrt{2}\) and \({\left{{{\Phi }}}^{\pm }\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }=({\left00\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} }\pm {\left11\right\rangle }_{{\rm{q}},{\rm{q}}^{\prime} })/\sqrt{2}\) are the standard maximally entangled Bell states. The angle ϑ defines the concurrence \( \cos (2\vartheta )\) of this state. For the experiment—shown in Fig. 3e—we took ϑ = 59.185°, yielding a concurrence of 0.475 at t = 0. The preparation of this state is done by single and twoqubit gates acting on the two qubits; the ancilla is not involved and remains separate in the state \({\left0\right\rangle }_{{\rm{a}}}\). Next, we simulate the action of the nonHermitian Hamiltonian for 0 ≥ t < 2 and r = 0.3, r = 1, r = 1.3, see Fig. 3e. In the first case, the entanglement increases up to ~0.8 (and would continue to oscillate at longer times), while for r = 1, r = 1.3 it decreases monotonously.
Entanglement correlations between system and ancilla
The simulation of nonHermiticity by the dilation method allows us to get an indepth understanding of this phenomenon. Let us look at the complete picture in the eightdimensional Hilbert space of this tripartite system (initialized in the state \(\left0\right\rangle \otimes \left{{{\Phi }}}^{+}\right\rangle\)), where, as we have seen in Fig. 3a, one of the system qubits q along with the ancillary qubit “a” undergo the unitary evolution U_{a,q}. The relevant correlations for the ensuing analysis are plotted in Fig. 3g for r = 0.6. We define the singlequbit reduced states by tracing out the other qubits \({\rho }_{i}={{\rm{Tr}}}_{j,h}[{\rho }_{i,j,h}]\), while the twoqubit reduced density operators are obtained by a single partial trace operation \({\rho }_{i,j}={{\rm{Tr}}}_{h}[{\rho }_{i,j,h}]\), where the three qubits are labeled by \(i,j,h\in \{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} \}\). The concurrence associated with the state ρ_{i,j} is denoted by \({{\mathcal{C}}}_{i,j}\) and it is calculated using the formula for mixed twoqubit states mentioned earlier. It is interesting to note that q and \({\rm{q}}^{\prime}\) are always in the permutation symmetric subspace of the twoqubit Hilbert space as one of the qubits evolves under H_{q} (see Supplementary Note 3). Therefore, it is enough to observe any one of the system qubits. Analyzing first the singlequbit states, we find that the singlequbit reduced density operators ρ_{q} and \({\rho }_{{\rm{q}}^{\prime} }\) remain maximally mixed all through the evolution, with a constant value of linear entropy \({s}_{{\rm{q}}}=1{\rm{Tr}}({\rho }_{q}^{2})=0.5\).
Next, we observe that the concurrences \({{\mathcal{C}}}_{{\rm{a}},{\rm{q}}}\) and \({{\mathcal{C}}}_{{\rm{a}},{\rm{q}}^{\prime} }\) between the ancilla and the respective system qubits, that is, a and q (or a and \({\rm{q}}^{\prime}\)) always remain zero. This shows that the dynamics under \({{\mathbb{U}}}_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }\) does not develop bipartite correlations between the respective system qubits and the ancilla qubit. Therefore, the creation of a tripartite correlation between the system and the ancilla can happen only through entangling correlations between the twoqubit reduced state of \({\rm{q}},{\rm{q}}^{\prime}\) and the ancilla. To quantify this tripartite correlation, we use the threetangle for pure states^{41}
where in the last equation we used the invariance of the tangle under permutations. As shown by Eq. (5), the maximum value of the three tangle is obtained in the absence of concurrence between the individual components. Here \({{\mathcal{C}}}_{{\rm{q}}:{\rm{a}}}\equiv {{\mathcal{C}}}_{{\rm{q}},{\rm{a}}}\) and \({{\mathcal{C}}}_{{\rm{q}}:{\rm{q}}^{\prime} }\equiv {{\mathcal{C}}}_{{\rm{q}},{\rm{q}}^{\prime} }\) are the concurrences of the twoparty reduced states ρ_{a,q} and \({\rho }_{{\rm{q}},{\rm{q}}^{\prime} }\). The first term on the righthand side of Eq. (5) is the square of concurrence between the bipartitions \({\rho }_{{\rm{q}}}:{\rho }_{{\rm{a}},{\rm{q}}^{\prime} }\), where one partition consists of the qubit q, while the other partition is formed by the ancilla a and the system qubit \({\rm{q}}^{\prime}\). For a pure threequbit state \({\rho }_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }\), the quantity \({{\mathcal{C}}}_{{\rm{q}}:{\rm{a}},{\rm{q}}^{\prime} }\) is effectively related to the mixedness of its bipartitions. More specifically, the square of concurrence between the partitions ρ_{q} and \({\rho }_{{\rm{a}},{\rm{q}}^{\prime} }\) is twice the linear entropy of the reduced density operator of either partition, given by \(2(1{\rm{Tr}}{\rho }_{q}^{2})\) or \(2(1{\rm{Tr}}{\rho }_{{\rm{a}},{\rm{q}}^{\prime} }^{2})\). We now know from the simulated dynamics that the linear entropy \({s}_{{\rm{q}}({\rm{q}}^{\prime} )}=1{\rm{Tr}}{\rho }_{{\rm{q}}({\rm{q}}^{\prime} )}^{2}=0.5\); therefore, at all times \({{\mathcal{C}}}_{{\rm{q}}:{\rm{a}},{\rm{q}}^{\prime} }^{2}=1\). Further, as shown in Fig. 3g, there is no bipartite entanglement between the ancilla and the respective system qubits q (or q′), which implies that \({{\mathcal{C}}}_{{\rm{q}},{\rm{a}}({\rm{a}},{\rm{q}}^{\prime} )}=0\). From Eq. (5), we obtain,
Thus, the three tangle among system qubits and ancilla and the concurrence between the system qubits are complementary to each other (see Supplementary Note 3). By permuting the partitions in Eq. (5), it is easy to obtain \({\tau }_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }=2{s}_{{\rm{a}}}\), where \({s}_{{\rm{a}}}=1{\rm{Tr}}({\rho }_{{\rm{a}}}^{2})\). The unitary \({{\Bbb{U}}}_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }\), which induces a local nonHermitian drive of qubit q in the postselected subspace of the ancilla, is in fact a nonlocal operation on the system qubit q and the ancilla a. Under \({{\Bbb{U}}}_{{\rm{a}},{\rm{q}},{\rm{q}}^{\prime} }\), as the ancilla entangles and disentangles with the joint state of the system qubits, we see the resulting oscillations of various correlations in time. These oscillating correlations with rdependent characteristic times, when postselected in the ancilla subspace, produce an apparent violation of entanglement monotonicity. While we observe experimentally the variation in entanglement under local operations in only one postselected subspace, other subspaces of the same system also witnesses similar patterns for the variation of entanglement under local operations as shown in Supplementary Fig. 3.
Conclusion
We have realized a quantum simulation of a singlequbit under a nonHermitian Hamiltonian, observing the \({\mathcal{PT}}\)symmetry breaking as the exceptional point is crossed and the associated change in distinguishability. The use of a quantum processor for the simulation has the advantage that more complex scenarios can be studied, such as a bipartite system with one of the qubits driven by a nonHermitian Hamiltonian. In this case, we observe the violation of the entanglement monotonicity nogo result from standard quantum mechanics. We also note that, while our method relies on dilation by the use of an ancilla, another approach to nonHermitian evolution exists, where the dimension of the Hilbert space remains the same but the standard inner product is modified (see “Methods”). These two methods can be put in an exact correspondence—the metric used in the latter approach can be identified as the operator \({\eta }^{2}+{\Bbb{I}}\) from the dilation approach.
The simulation of phenomena governed by \({\mathcal{PT}}\)symmetry at the singlequantum level open up several novel perspectives. Our scheme provides a systematic way of studying more complex nonHermitian manyqubit systems. It is important to realize that for a system of N qubits the overhead in the width of the circuit is just one ancilla qubit. For example, it would be straightforward to generalize to the study of entanglement that we have performed to one nonHermitian qubit and N − 1 Hermitian ones, in which case the depth of the circuit remains equal to 8. Furthermore, because we have access to the quantum regime, our scheme enables the study of quantum fluctuations. Since these systems are open—connected to a source of energy providing gain and reservoir for dumping this energy—they naturally will lead to new insights into quantum thermodynamics.
Methods
Simulating the nonHermitian Hamiltonian in the dilated space
The singlequbit evolution under a general timedependent nonHermitian Hamiltonian H_{q}(t) is obtained in a certain subspace of an ancilla–qubit system undergoing a unitary evolution generated by \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\). The Hamiltonian \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) in a fourdimensional Hilbert space can be obtained from H_{q} by Naimark dilation^{12}. Using this method, we can write the Hamiltonian \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) in the form:
with
where η(t) and M(t) are Hermitian operators; T and \(\widetilde{{{T}}}\) are timeordering and antitimeordering operators, respectively, and \({\Bbb{I}}\) is the 2 × 2 identity operator. The Hamiltonian \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) can be obtained as follows. First, the Eqs. (10) and (11)) for η(t) and M(t) reflect the invariance of the norm of \({\left{{\Psi }}\right\rangle }_{{\rm{a}},{\rm{q}}}(t)\) in the form Eq. (3) on the dilated space under evolution (see also the discussion about metric below). Then, the Schrödinger equations \(i({\mathrm{d}}/{\mathrm{d}}t){\left{{\Psi }}\right\rangle }_{{\rm{a}},{\rm{q}}}={{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t){\left{{\Psi }}\right\rangle }_{{\rm{a}},{\rm{q}}}\), and \(i({\mathrm{d}}/{\mathrm{d}}t){\left\psi (t)\right\rangle }_{{\rm{q}}}={H}_{{\rm{q}}}(t){\left\psi (t)\right\rangle }_{{\rm{q}}}\) together with Eqs. (7) and (3) produce a linear system of equations in the unknown operators Λ(t) and Γ(t),
By multiplying the second equation to the right with η(t) and with −η^{−1}(t) and adding the results to the first equation, we obtain immediately the solution as given by Eqs. (8) and (9). Note also that for Hermitian Hamiltonians H_{q} the second term in Eq. (7), which is the qubit–ancilla interaction, becomes zero.
Initial conditions
To obtain an explicit form of \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\), one should choose the operator M(t) at time t = 0 such that \({{M}}(t){\Bbb{I}}\) is positive for all t in the desired time interval. As a preliminary choice, we can take
where m_{0} > 1, may be chosen arbitrarily. Further, we obtain the eigenvalues of M(t) in the desired time interval. Fixing the value of r, at any arbitrary time t, the eigenvalues of M(t) are labeled as μ_{1}(t) and μ_{2}(t), from where we numerically obtain \({\mu }_{\min }(t)=\min \{{\mu }_{1}(t),{\mu }_{2}(t)\}\). Interestingly, for r = 0, H_{q} is Hermitian and
with eigenvalues m_{0}, which is the maximum value that μ_{min} can assume. Therefore, for any arbitrary r and t, \({m}_{0}/{\mu }_{\min }\ge 1\). Thus, at t = 0, M(t) is chosen to be,
where f > 1, which ensures that \({{M}}(t){\Bbb{I}}\) remains positive for all t. From Eq. (10) we have
The dynamics of the total ancilla–qubit system under \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) is obtained from the Schrödinger equation
whose solution is given by
where \({\left\psi (t)\right\rangle }_{{\rm{q}}}\) is the solution of \(i\frac{\mathrm{d}}{{\mathrm{d}}t}{\left\psi (t)\right\rangle }_{{\rm{q}}}={H}_{{\rm{q}}}{\left\psi (t)\right\rangle }_{{\rm{q}}}\), and \({\left\tilde{\psi }(t)\right\rangle }_{{\rm{q}}}=\eta (t){\left\psi (t)\right\rangle }_{{\rm{q}}}\). At t = 0 the state of the total system is
which is a separable state of the ancilla \({\left\psi (0)\right\rangle }_{{\rm{a}}}\) and the system qubit \({\left\psi (0)\right\rangle }_{{\rm{q}}}\). For preparing the initial state Eq. (19) the ancilla is taken in one of the eigenvectors of σ_{z}, say \({\left0\right\rangle }_{{\rm{a}}}\). This is then subjected to a rotation by an angle θ around the yaxis, \({R}_{y}(\theta )=\exp (i\theta {\sigma }_{y}/2)\), where, \(\theta =2{\tan }^{1}{\eta }_{0}\). This leads to
which is the initial state as defined by the protocol. On the other hand, the qubit q may be initialized in any arbitrary state \({\left\psi (0)\right\rangle }_{{\rm{q}}}\). For the case of a single qubit, as discussed in the first part of the paper, we considered two different values of the state of the qubit: (i) \({\left\psi (0)\right\rangle }_{{\rm{q}}}=\left0\right\rangle\) and (ii) \({\left\psi (0)\right\rangle }_{{\rm{q}}}=\left1\right\rangle\). The same formalism applies also in the case of the twoqubit system discussed in the second part of the paper, in which case the qubit–qubit system is initialized in a maximally entangled Bell state \({\left\psi (0)\right\rangle }_{{\rm{q}}}\to \left{{{\Phi }}}^{+}\right\rangle =\frac{1}{\sqrt{2}}(\left00\right\rangle +\left11\right\rangle )\).
For instance, in case (i) we choose m_{0} = 2 and f = 1.01. At r = 0.6 in the time range t ∈ [0, 8], we obtain η_{0} = 1.7436 and θ = 2.1001 (radians). At the exceptional point, that is, r = 1, in the same time range t ∈ [0, 8], we get η_{0} = 16.1112 and θ = 3.0176 radians. Further, for r = 1.3, \({\mu }_{\min }\) is obtained separately for various time intervals to increase the probability of success. This then led to different values of η_{0} and hence θ for each time point.
The metric
NonHermitian quantum dynamics can be alternatively formulated by using Hilbert spaces with a modified bra vector, resulting in a redefinition of the inner product^{23,36,42}. Here we make an explicit connection with this approach, showing that the Hermitian operator \(M(t)=\eta {(t)}^{2}+{\Bbb{I}}\) can be identified as the metric that plays a key role in this formalism.
Indeed, from Eq. (18) we can calculate the norm of the dilated vector \({\left{{\Psi }}(t)\right\rangle }_{{\rm{a}},{\rm{q}}}\),
This norm has to be conserved during the time evolution. By taking the time derivative of Eq. (21) and using Eq. (2) we get
This is the defining relation for the metric^{23}. Note that this can also be obtained in a straigthforward way from Eq. (11). Thus, in this approach to nonHermitian quantum mechanics for every vector \({\left\psi (t)\right\rangle }_{{\rm{q}}}\) in the Hilbert space we define the covector as \({\,}_{{\rm{q}}}\left\langle \psi (t)\rightM(t)\), which ensures that the inner product _{q}〈ψ(t)∣M(t)∣ψ(t)〉_{q} from Eq. (21) has the meaning of a conserved probability.
For the particular case of the Hamiltonian H_{q} studied in this work, the metric M(t) can be obtained analytically by employing the properties of 2 × 2 matrices (see also Supplementary Eq. (3)). In the \({\mathcal{PT}}\)symmetric phase we obtain an exact formula for the metric,
One can check also that M(t) is positively defined for r < 1, while for r = 0 we recover the standard Hermitian quantum mechanics with M(t) = M_{0}. The result above Eq. (23) can be also obtained from the generic formula for the metric, as per ref. ^{23}, for parameters A = 0, B = − r/(1 − r^{2}), C = 1/(1 − r^{2}), and D = 0.
It is interesting to remark how the main problem of nonHermitian quantum mechanics, that of nonconservation of probability, has been dealt with in completely different ways: either by the introduction of a metric and modifying the inner product, or, in the dilation method, by adding an ancilla that absorbs the excess population.
Evolution and measurement in the dilated space
Let us consider the evolution of an arbitrary state of a twoqubit system under the Hamiltonian \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) in Eq. (7),
where ψ(0) is the initial state at t = 0. This may also be written as
where T is the timeordering operator. For a given set of values of r and t, it is useful to obtain an explicit form of the unitary operator U_{a,q}(t). This is done by observing the respective \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) evolutions of the complete set of twoqubit basis states. To find U_{a,q}(t) we solve the Schrödinger equation numerically for different initial states,
where \({\left00\right\rangle }_{{\rm{a}},{\rm{q}}}\), \({\left01\right\rangle }_{{\rm{a}},{\rm{q}}}\), \({\left10\right\rangle }_{{\rm{a}},{\rm{q}}}\), and \({\left11\right\rangle }_{{\rm{a}},{\rm{q}}}\) correspond to the complete set of basis vectors in the fourdimensional Hilbert space. The system qubit and the ancilla are initialized in all four bases states, respectively, and then evolved numerically under \({{\mathcal{H}}}_{{\rm{a}},{\rm{q}}}(t)\) for a given time. Then, after solving this equation for \(\left{\psi }_{00}(t)\right\rangle\), \(\left{\psi }_{01}(t)\right\rangle\), \(\left{\psi }_{10}(t)\right\rangle\), and \(\left{\psi }_{11}(t)\right\rangle\), we obtain the closed form of the unitary operator at an arbitrary time t, given by
For different values of time, U_{a,q}(t) is obtained, which is a general unitary operator in the fourdimensional Hilbert space. Each U_{a,q}(t) at a given time is then decomposed numerically in the form of singlequbit rotations and twoqubit CNOT gates, as shown in Fig. 1e.
This quantum circuit decomposition gives rise to U_{num}(t), whose operation is very close to the theoretical U_{a,q}(t). To characterize this, we calculate the error function err_{U}(t) = ∣∣U_{a,q}(t) − U^{num}(t)∣∣_{2}/∣∣U_{a,q}(t)∣∣_{2}, with the 2norm defined by \(  A { }_{2}=\sqrt{{\lambda }_{\max }}\), where \({\lambda }_{\max }\) is the largest eigenvalue of the matrix A^{*}A. Here U^{num}(t) is an unitary operator generated by the circuit in the inset of Fig. 1a, where the parameters α, β, γ of \({{\mathcal{U}}}_{{\rm{q}}({\rm{a}})}^{j}(\alpha ,\beta ,\gamma )\) are chosen to minimize the expression ∣∣U_{a,q}(t) − U^{num}(t)∣∣_{2}. Typically, we find err_{U}(t) = ∣∣U_{a,q}(t) − U^{num}(t)∣∣_{2}/∣∣U_{a,q}(t)∣∣_{2} to be of the order of 10^{−4}, which demonstrates the high accuracy of our U_{a,q} implementation. The accuracy with which our gate decomposition and the U_{a,q}(t) operator match with each other is presented by an example data set in the Supplementary Table 1. U_{a,q}(t) for arbitrary values of (r, t) and the corresponding U^{num}(t) can be obtained from a GitHub code repository^{43}.
Quantum state reconstruction
For singlequbit tomography we take 4 − 1 = 3 measurements, corresponding to the set of Pauli operators σ_{x}, σ_{y}, σ_{z}. For higherdimensional quantum systems of n qubits, we need 2^{2n} − 1 measurements corresponding to combinations of σ_{x}, σ_{y}, σ_{z} and the identity matrix \({\Bbb{I}}\) of the two qubits. Therefore, a complete quantum state tomography of a twoqubit system requires a set of (16 − 1) experiments, which correspond to determining the expectation values of all the twoqubit operators formed by products of Pauli operators and the identity. In the present work, we need only to examine the postselected subspace of the total system with the ancilla in state \(\left0\right\rangle\). Therefore, we circumvent the complexities of threequbit tomography by restricting our measurement to a 4 × 4 block of the complete 8 × 8 threequbit density operator.
We perform a complete quantum state tomography of the system qubits by applying the following seven operators, namely \({{\Bbb{T}}}_{1} = {\Bbb{I}}\otimes {\Bbb{I}}, {{\Bbb{T}}}_{2} = H\otimes {\Bbb{I}}, {{\Bbb{T}}}_{3} = {R}_{x}(\pi /2)\otimes {\Bbb{I}},{{\Bbb{T}}}_{4} = {\Bbb{I}}\otimes H,{{\Bbb{T}}}_{5}={\Bbb{I}}\otimes {R}_{x}(\pi /2),{{\Bbb{T}}}_{6}=({\Bbb{I}}\otimes H){\rm{CNOT}}, {{\Bbb{T}}}_{7}=({\Bbb{I}}\otimes {R}_{x}(\pi /2)){\rm{CNOT}}\). The application of each of these operators is followed by the measurement in the σ_{z} bases and postselection of the desired subspace. Thus, in each of these experiments, we measure all three qubits, and obtain eight diagonal elements p_{i,j,k} = ∣c_{i,j,k}∣^{2}. Finally, the corresponding populations of the twoqubit reduced density operator in the postselected subspace with ancilla in state \({\left0\right\rangle }_{{\rm{a}}}\) are given by
Next, these populations are corrected for measurement errors, and the postselected twoqubit density operators obtained further undergo convex optimization^{44,45} (see Supplementary Note 5 for further details).
Data availability
The data that support the findings of this study are available from authors upon reasonable request.
Code availability
The codes used for the simulations can be found in the GitHub repository^{43}.
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Acknowledgements
We acknowledge support from the Foundational Questions Institute Fund (FQXi) via the Grant No. FQXiIAF1906, from the EU Horizon 2020 research and innovation program (grant agreement no. 862644, FET Open QUARTET), and from the Academy of Finland through project no. 328193 and through the “Finnish Center of Excellence in Quantum Technology QTF” project no. 312296. In addition, we would like to thank the Scientific Advisory Board for Defence (Finland) and Saab. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.
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S.D. and G.S.P. initiated the project and obtained the key analytical results. A.A.M. designed the quantum circuits and performed the simulations on the IBM Q machine with input from S.D. All authors discussed the results. S.D. and G.S.P. wrote the manuscript, with contributions also from A.A.M.
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Dogra, S., Melnikov, A.A. & Paraoanu, G.S. Quantum simulation of parity–time symmetry breaking with a superconducting quantum processor. Commun Phys 4, 26 (2021). https://doi.org/10.1038/s42005021005342
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DOI: https://doi.org/10.1038/s42005021005342
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