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Witnessing Quantum Coherence: from solid-state to biological systems
Che-Ming Li, Neill Lambert, Yueh-Nan Chen, Guang-Yin Chen, Franco Nori
TL;DR
Efficiently verifying quantum coherence in complex systems is difficult because access is limited and established tests can require noninvasive measurements or tomography. The paper introduces two time-domain quantum witnesses based on state preparation and correlation checks. Across five physical examples, the witnesses are reported to be robust, applicable to solid-state and biological systems, and finer in detection resolution than the LGI.
Problem
Limited system access and the experimental burden of noninvasive measurements or tomography make unambiguous verification of quantum coherence difficult, especially in complex and biological systems.
Method
The paper introduces two time-domain quantum witnesses that prepare a product system-environment state, allow it to evolve, and test two-time state correlations.
Results
The witnesses are applied to five examples spanning solid-state nanostructures, photonic systems, cavities, and biological energy transfer, with finer detection resolution than the LGI.
Takeaways & Limitations
The witnesses reduce the overhead and complexity of unambiguously detecting quantum phenomena and can be used to explore coherence in complex systems.
Takeaways & Limitations
Witness 1 does not distinguish system coherence from system-reservoir entanglement and can yield false positives from classical non-Markovian correlations without additional experimental overhead.
Abstract
from arXiv · showhide
Quantum coherence is one of the primary non-classical features of quantum systems. While protocols such as the Leggett-Garg inequality (LGI) and quantum tomography can be used to test for the existence of quantum coherence and dynamics in a given system, unambiguously detecting inherent "quantumness" still faces serious obstacles in terms of experimental feasibility and efficiency, particularly in complex systems. Here we introduce two "quantum witnesses" to efficiently verify quantum coherence and dynamics in the time domain, without the expense and burden of non-invasive measurements or full tomographic processes. Using several physical examples, including quantum transport in solid-state nanostructures and in biological organisms, we show that these quantum witnesses are robust and have a much finer resolution in their detection window than the LGI has. These robust quantum indicators may assist in reducing the experimental overhead in unambiguously verifying quantum coherence in complex systems.
Introduction
Quantum coherence is important for quantum technologies and proposed biological functions, but testing it in complex systems is difficult because access is limited and common methods impose substantial experimental burdens. The paper introduces two time-domain quantum witnesses based on correlation checks to reduce those burdens.
- Quantum coherence underlies phenomena including entanglement, quantum information processing, metrology, transport, and proposed biological energy transport.
- Biological tests are especially challenging because systems operate in hot, wet environments with limited full-system access and often indirect coherence signatures.
- The LGI classifies behavior through classical constraints but requires experimentally difficult noninvasive measurements, limiting reported tests.
- The paper introduces two time-domain quantum witnesses that avoid noninvasive measurements and quantum tomography, reducing experimental overhead and complexity.
- The witnesses prepare a known product state with the environment, allow joint evolution, and test two-time state correlations for non-classical properties.
- The correlation check compares a quantum correlator with pn(t0)Ωmn(t, t0), whose equality all classical dynamics satisfy; violations indicate quantum dynamics.
Witness 1
The first practical witness identifies quantum states from normalized population measurements, without noninvasive measurements or full tomography. It can reduce preparation overhead through partial summation, but does not by itself distinguish system coherence from system–reservoir quantum correlations.
- A positive WQ identifies the state at t0 as quantum, and WQ can be directly measured without ideal noninvasive measurements.It uses population-related expectation values and probabilities rather than requiring full correlation functions.
- The witness uses measured expectation values ⟨Qm(t)⟩ and initial populations {pn(t0)} to test the classical relation.
- Constructing the propagators may require preparing the system in each state or measuring every possible cross-correlation, which can be difficult in complex systems.The full state space may also be unknown.
- Because the summation terms are positive, the witness can be evaluated after a partial sum already exceeds ⟨Qm(t)⟩, substantially reducing experimental overhead.In one practical example, a single term was sufficient.
- The witness cannot distinguish isolated system coherence from entanglement between the system and its reservoir.Classical system–reservoir correlations can also produce false positives unless additional overhead is used to eliminate them.
Witness 2
The second witness assumes time-independent propagators within a Markovian, weak-coupling subset of processes. It avoids explicit propagator measurement and exact state initialization, using invasive population measurements, but its interpretation depends on those assumptions.
- Witness 2: Under the Markovian propagator assumption, quantum properties can be identified without explicitly measuring propagators or exactly initializing every state.The assumption is Ωmn(t,t0) = Ωmn(t′,t′0) for equal time intervals.
- Witness 2: The relevant process subset has weak system–reservoir coupling, product system–reservoir states, and a time-invariant reservoir state.
- Witness 2: The extended procedure replaces state preparation with repeated expectation-value measurements whose number scales linearly with system size.
- Witness 2: The matrices Pj encode populations at paired times, while Ωmj and Qmj represent propagator and expectation-value vectors used to solve for Ωmj.For nonzero det(Pj), the propagator component is obtained algebraically by replacing a column of Pj with Qmj.
- Witness 2: Comparing propagators from different time-domain sets provides a witness when classical systems in the assumed subset should have identical propagators.Classical system–reservoir correlations violate that identical-propagator assumption.
- Witness 2: A positive WΩmn indicates that some initial states can be considered quantum, while requiring only state-population information and allowing invasive observations.
Examples
The paper applies its quantum witnesses across five systems, selecting the more appropriate witness for each example. These include solid-state, biological, optical, and photonic quantum dynamics.
- Examples: Five examples span Cooper-pair tunneling, transmon evolution, double-quantum-dot transport, photosynthetic energy transfer, lossy-cavity Rabi oscillations, and photonic-qubit rotations.For each system, the authors choose the witness they consider more appropriate given its properties.
Rabi oscillations in superconducting qubits
Population oscillations alone do not definitively establish quantum dynamics because classical rate equations can mimic them. The witnesses detect coherent evolution in superconducting systems using experimentally accessible measurements, including a witness value of approximately 0.45 for a single-qubit gate.
- Population oscillations alone are not definitive evidence of quantum coherent dynamics because classical autonomous rate equations can reproduce them.
- The witnesses are applied to two-level Cooper-pair-box dynamics, where WΩ21 detects quantumness in Cooper-pair tunnelling using only state-population information.The approach therefore uses simple invasive measurements rather than non-invasive measurements.
- Single- and multiple-transmon qubits coupled to transmission lines provide another superconducting-qubit application of the witnesses.Qubit-state measurements are performed by monitoring microwave-cavity transmission.
- A microwave pulse ε(t) drives transitions between qubit states, and optimal-control design produces a Hadamard process with approximately 94% process fidelity.The pulse is designed using quantum-process-tomography-based optimal control theory.
- WQ ≈0.45 certifies the quantumness of the Hadamard process EH when the input state is |0⟩.
Quantum transport in quantum dots
The paper models single-electron transport through double quantum dots to test whether time-domain witnesses can distinguish quantum from classical transport under invasive measurements.
- Single-electron double-quantum-dot transport illustrates that the witnesses remain applicable with invasive measurements.The model addresses the experimental challenge of distinguishing quantum from classical nanostructure transport.
Energy transfer in a light-harvesting complex
The FMO pigment-protein complex models energy transport with strong system-bath interactions and is detected as quantum by the first witness, including at room temperature. The analysis also addresses classical system-bath correlations and experimental overhead.
- System-bath correlations: Classical system-bath correlations can create a false positive for the first witness unless the propagator construction accounts for them.The second witness is not valid in this strong-interaction regime.
- System-bath correlations: The proposed procedure prepares a separable system-bath state, projects the system onto a state without preserving coherence, and measures later occupations to deduce propagator terms.This construction retains the post-measurement system-bath state to account for classical correlations.
- System and model: The FMO complex is a seven-site structure that transfers excitations from a light-harvesting antenna to a reaction center.The model targets non-Markovian and non-perturbative system-bath interactions using hierarchical equations of motion.
- Detection: The first witness detects the FMO model as quantum even at room temperature.The analysis discards physical density-matrix coherence terms while retaining the bath state when constructing the propagator terms.
- Detection: At 77 K, the full witness detects coherence beyond t0 = 0.3 ps, compared with an LGI detection window of 0.035 ps for the same parameters.Only partial propagator information is needed for detection at small times, reducing experimental overhead.
Vacuum Rabi oscillation in a lossy cavity
A Rydberg atom resonantly coupled to a single-mode cavity evolves through atom-field states under photon loss. The second witness detects damped coherent oscillations in a high-Q cavity but vanishes when irreversible emission dominates.
- System dynamics: The atom-cavity system uses adjacent circular Rydberg states |e⟩ and |g⟩, with photon loss driving evolution toward |g⟩|0⟩p.The evolution is described by a master equation.
- High-Q cavity: When 2ωR ≫ ω0/Q, the second witness detects damped coherent oscillations of the atom-cavity state.This is the high-Q regime illustrated using experimental parameters.
- Measurement: Field-ionization detectors can provide the required measurements on atom states |e⟩ and |g⟩.
Coherent rotations of photonic quantum bits
The first witness is adapted from time to wave-plate-angle settings to detect coherent rotations between horizontal and vertical photonic polarization states. It detects coherence across almost the full prepared-state range while requiring one local measurement setting instead of three for tomography.
- Photonic qubits: Polarization states |H⟩ and |V⟩ form a photonic qubit that can be coherently manipulated with half-wave and quarter-wave plates.These elements implement arbitrary qubit rotations.
- Transformations: A half-wave plate set to φ = π/8 creates a photonic Hadamard gate.
- Witness construction: The rotation R(φ, θ) = Qwp(θ)Hwp(φ) is probed by using the first witness to test coherence between |H⟩ and |V⟩.The time-domain witness is rephrased in terms of wave-plate settings.
- Witness construction: The initial state is set as ρ0 = R†(φ, θ) |m⟩⟨m| R(φ, θ), with m = H and n = V as measurement-basis states.
- Measurement overhead: One local measurement setting suffices for the first witness, compared with three local settings for single-qubit state tomography.
Discussion
The paper concludes that its quantum witnesses efficiently detect coherence without non-invasive measurements and offer broader detection windows than several existing methods. Five examples support their robustness across complex systems.
- Contribution: The witnesses detect quantum coherence without the restriction of non-invasive measurements.
- Comparison: Compared with the Leggett-Garg inequality and general quantum tomography, the approach reduces experimental overhead and complexity while providing a larger detection window.
- Scope: Five physical examples illustrate robust use across nanostructure transport, biological systems, cavities, and photonic qubits.The examples include Cooper-pair boxes, transmons, double quantum dots, FMO energy transfer, vacuum Rabi oscillations, and coherent photonic rotations.
Methods
The methods model FMO dynamics with hierarchical equations of motion and derive quantum two-time correlations alongside their classical-mixture counterpart. The framework assumes separable system and bath states, independent Drude baths, and hierarchy truncation checked through convergence.
- Correlation construction: Quantum two-time state-state correlations are compared with correlations generated from an initially classical system–reservoir mixture.For the classical mixture, the correlation is expressed through conditional propagators, whose value is the probability of finding state m at time t after initializing state n at t0.
- Correlation construction: The system evolution for the classical-mixture correlation is represented with an operator-sum construction after tracing over the reservoir.The reservoir state is decomposed into eigenstates with probabilities prk, which define the corresponding evolution operators Ej(τ).
- Hierarchy model for FMO: The FMO model uses hierarchical equations of motion to describe system–bath dynamics, with the Liouvillian governing Hamiltonian evolution.The system starts separable from a thermal bath, and the hierarchy includes physical and auxiliary density matrices.
- Bath model: Each of FMO’s seven sites has an independent Drude bath characterized by decay constant γj and reorganisation energy λj.The reorganisation energy is proportional to system–bath coupling strength, while the bath correlation frequencies include µj,0 = γj and µj,m = 2πm/ℏβ for m ≥ 1.
- Hierarchy structure: The hierarchy label n contains non-negative integers that identify equations, with n = 0 representing the physical system density matrix and other labels representing auxiliary density matrices.Increasing or decreasing the integer at position j,m selects the corresponding neighboring density matrix in the hierarchy.
- Numerical truncation: The hierarchy is truncated at a maximum total label order Nc, called the tier, and Nc is selected by checking convergence of the system dynamics.The Ishizaki–Tanimura boundary condition is also used at the hierarchy boundary.