Quantum state tomography reveals a qubit’s broken linearity

Quantum state tomography revealed a breakdown of the superposition principle within a superconducting qubit, challenging a core tenet of quantum theory, according to research from Washington University and the University of California, Berkeley. Orion Lee, Qian Cao and colleagues theoretically characterized and experimentally investigated this deviation from linearity in a dissipative superconducting transmon circuit, finding that “linearity is a foundational property of quantum dynamics.” Their work demonstrates that no-jump postselection generates an anti-Hermitian nonlinearity, causing a measurable non-linear response in the qubit’s evolution and a violation of linearity quantified across the exceptional-point degeneracy of the non-Hermitian Hamiltonian.

Postselection Isolates Nonlinear Transmon Dynamics

This postselection, which focuses on trajectories where the qubit does not “jump” to a lower energy state, effectively reshapes the system’s dynamics, leading to a deviation from the standard linear behavior expected in quantum mechanics. The experimental setup involved engineering the decay rates of the transmon’s energy levels, specifically making the decay from the |e⟩ to |g⟩ state significantly faster than the decay from |f⟩ to |e⟩.

This asymmetry, combined with coherent coupling between the |e⟩ and |f⟩ states, sets the stage for the non-Hermitian dynamics. By preparing different initial states, basis states and superposition states, the researchers could then observe how these states evolved under the influence of the effective, nonlinear Hamiltonian created by the postselection process.

By comparing the evolution of a superposition state to a superposition of individually-evolved basis states, they tested linearity and observed clear violations which they quantified across the exceptional-point (EP) degeneracy of the non-Hermitian Hamiltonian. “We demonstrate a practical tool to realize anti-Hermitian nonlinearity at the quantum level, and thereby access and control nonlinear quantum dynamics,” the authors wrote in their published work. Further analysis extended to density matrices, revealing that linearity breaks down specifically within the two-level subspace defined by the |e⟩ and |f⟩ states, while remaining preserved in the full three-level system.

Dissipative Superconducting Qubit & Effective Hamiltonian

Within the three-level manifold of a superconducting transmon circuit, a specific form of postselection generates an effective non-Hermitian Hamiltonian, influencing the qubit’s evolution in a way that deviates from standard quantum linearity. This carefully controlled dissipation, combined with a driving Hamiltonian H detuned by Δ with an amplitude J, allows for the realization of effective non-Hermitian dynamics.

The resulting system exhibits a transition analogous to PT-symmetry breaking, where J > Γ/4 indicates an unbroken symmetry and J < Γ/4 signifies a broken regime, with J = Γ/4 marking an exceptional point (EP) degeneracy. Quantum state tomography was then employed to track the evolution of different initial states under this effective, non-linear Hamiltonian, revealing a clear departure from expected linear behavior.

Three-Level Manifold Defines Non-Hermitian Evolution

This postselection, as detailed in the work, generates an effective non-Hermitian Hamiltonian governing the excited two-level subspace alongside an anti-Hermitian nonlinearity, a previously unobserved effect in quantum systems. By focusing on trajectories that remain within the manifold and excluding quantum jumps to the |g⟩ state, the researchers effectively renormalized the ensemble, creating conditions for observing the non-Hermitian dynamics.

Initial observations revealed asymmetric oscillations favoring population in |f⟩ over |e⟩, a characteristic signature of the non-Hermitian evolution. Further analysis focused on three initial states: ${\rho }_e$ =, ${\rho }_f$ and a classical mixture ${\rho }_m$ with equal parts preparations in | + x⟩ and |-x⟩.

The observed dynamics of populations within the manifold confirmed the non-Hermitian evolution, with the mixture’s evolution diverging from a simple superposition of the individual |e⟩ and |f⟩ curves.

Quantum State Tomography Reveals Linearity Violations

The investigation focused on dynamics generated by a process that creates an anti-Hermitian nonlinearity not typically observed in quantum systems. This nonlinearity manifests as a deviation from the linear Schrödinger equation, a cornerstone of quantum mechanics. “As the post-selected state |ψ(t)⟩ is always normalized, it satisfies a nonlinear Schrödinger equation,” the researchers write, highlighting the theoretical basis for this unexpected behavior.

The experimental setup involved a three-transmon subsection of a multi-transmon chip, where two transmons were dispersively coupled to microwave readout resonators. These resonators enabled precise state measurement of the transmons, which was essential for reconstructing the system’s density matrix using quantum state tomography.

The process involved iterative Bayesian updates to correct raw probabilities obtained from the measurements, ultimately allowing the team to map the evolution of the qubit’s state with high fidelity. This detailed characterization of the system’s dynamics was essential for detecting the subtle deviations from linear behavior. These results provide direct evidence of nonlinearity in non-Hermitian quantum evolution, highlighting unique features that are absent in classical non-Hermitian systems.

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