Researchers at the Indian Institute of Technology Bombay and the S. N. Bose National Centre for Basic Sciences have developed a new framework for quantifying multipartite entanglement in complex quantum systems.

The work circumvents limitations of existing methods when applied to systems with strong interactions between particles, using controlled quenching of interactions or local observable measurements to evaluate ergotropy, the maximum extractable work from an isolated system. This approach correctly estimates entanglement in both stable and evolving states, demonstrated through simulations of the Tavis-Cummings, Dicke, and transverse-field Ising models, and offers a versatile tool for characterizing entanglement in emerging quantum simulators.

Ergotropy and Entanglement: Linking Thermodynamic Work to Quantum Correlations

Evaluating the maximum work obtainable from a quantum system offers a new pathway for quantifying multipartite entanglement, circumventing limitations inherent in existing methods. The approach relies on either rapidly altering interactions within the system, a technique called controlled quenching, or precisely measuring local observable properties. This advance addresses a longstanding challenge in characterizing entanglement, particularly in complex quantum systems relevant to emerging quantum technologies. Traditional methods often require complete knowledge of a system’s quantum state, a process known as full state tomography, which becomes increasingly difficult as the number of interacting particles grows.

The new framework, however, sidesteps this need by focusing on measurable thermodynamic quantities, offering a more experimentally viable route to assessing entanglement. Researchers found that ergotropy, when carefully evaluated, reveals the degree of entanglement even when the system’s fundamental state isn’t easily separable.

The ability to accurately gauge entanglement is important for validating the performance of quantum simulators and near-term quantum computers. These devices, while not yet capable of fully fault-tolerant computation, rely on entangled states to perform calculations. The framework’s versatility is demonstrated through its successful application to several established physical models, including the Tavis-Cummings model, the three-level Dicke model and the transverse-field Ising model. These simulations highlight the method’s potential for characterizing entanglement in realistic, albeit simplified, physical systems.

The concept of ergotropy itself builds on the established connection between quantum thermodynamics and entanglement. Extending this to multiple particles, however, requires overcoming the challenges posed by strong interactions, where local operations often fail to capture the full extent of quantum correlations.

The researchers demonstrate that by either “quenching the system or making local measurements,” they can accurately capture both global and local ergotropy, even in the presence of strong interparticle interactions. Controlled quenching involves rapidly changing the strength of interactions, allowing researchers to observe how the system responds and infer its entanglement properties. Alternatively, measuring carefully chosen local observables provides a similar insight without requiring full knowledge of the system’s global state.

This flexibility is particularly valuable for quantum simulators, where operators and measurements can be precisely controlled. The framework’s practicality is further enhanced by its compatibility with quantum circuits, the building blocks of many quantum computers. The team implemented their approach using variational quantum algorithms, allowing them to estimate ergotropy and entanglement in simulated quantum circuits with varying levels of complexity and noise.

This is significant because real-world quantum devices are inherently noisy, and any entanglement measure must be robust enough to function in such environments. “On introducing noise into the system, we see deviation in the exact and simulated results,” the researchers note, acknowledging the challenges of simulating noisy quantum systems. The study highlights the potential of ergotropy as a resource for quantum information processing. The ability to quantify entanglement using a thermodynamic measure opens doors for developing new protocols for quantum communication and computation.

Quenching Interactions and Local Observables for Entanglement Evaluation

The framework relies on assessing either the global or local ergotropy, the maximum work obtainable from an isolated system, through precisely controlled alterations to interactions or by measuring only local properties. This approach allows for accurate estimation of genuine multipartite entanglement in both stable and time-dependent states, even when interactions are substantial, a challenge for earlier methods.

Researchers addressed a limitation of traditional entanglement quantification methods, which diminishes when applied to systems where the fundamental, or ground, state is already entangled, leading to inaccurate readings even for seemingly separable states. They recognized that ergotropy, as an intrinsic property of a quantum state, is independent of the specific interaction structure governing the system.

This independence allows for the development of a formalism where the ergotropic gap, the difference between global and local ergotropies, can be reliably calculated even with strong interactions present. The team demonstrated that the ergotropic gap can be simplified under specific conditions, reducing to a calculation based on passive states and local Hamiltonian terms.

This simplification is particularly relevant to the growing field of quantum simulation, where complex quantum systems are emulated using controllable platforms like trapped ions or superconducting circuits. Within these simulators, the energy of a quantum state can be experimentally determined by either rapidly altering the interactions between particles, a process called quenching, or by measuring only local observables.

The evaluation of the ergotropic gap remains independent of the specific interaction structure or measurement scheme employed. The framework’s utility extends to quantifying multipartite entanglement, a measure of correlation between multiple quantum particles. By focusing on the ergotropic volume, a metric capturing genuine multiparty entanglement, the team derived analytical expressions applicable to stationary states found in hybrid cavity QED systems and many-body spin models. The approach allows for the assessment of entanglement without needing to fully characterize the quantum state, a significant advantage for complex systems.

The team’s work builds on previous studies that linked work extraction to underlying quantum correlations, offering a more operationally accessible method for entanglement quantification. The researchers specifically highlight the applicability of their method to systems governed by interacting Hamiltonians, expressed as the sum of local Hamiltonians and an interaction term.

Genuine Multipartite Entanglement Quantification in Strongly Interacting Systems

The ability to accurately assess entanglement extends to systems where interactions between particles are strong, according to a new framework detailed in recent work. This advancement addresses a key limitation of existing entanglement quantification techniques, which often struggle with the complexities introduced by strong collective or interparticle interactions within quantum systems.

This methodology bypasses computational demands that have previously hindered the analysis of multipartite entanglement, particularly genuine multipartite entanglement (GME), which characterizes states inseparable across all possible partitions. Analytical expressions for the ergotropic volume, a quantity capturing genuine multiparty entanglement, can be derived using this method, and these align with numerical values obtained from established geometric measures of entanglement.

The team’s analysis confirms that even a local Hamiltonian can accurately reflect the ergotropic gap and quantify entanglement, a finding validated through comparisons with the original measure in regimes where it remains applicable. The new approach successfully quantifies entanglement in scenarios where the original measure fails, demonstrating its robustness and expanded range of application. These models serve as benchmarks for assessing the framework’s performance and highlighting its versatility in characterizing entanglement in near-term quantum simulators.

The consistency between the ergotropic volume and other geometric measures of entanglement reinforces the reliability of this thermodynamic approach. The work also demonstrates the ability to capture physical properties of the quantum state, such as quantum phase transitions and critical behavior, solidifying its value as a figure of merit for studying entanglement in strongly interacting systems.

A protocol is presented detailing how the measurement of both the ergotropic gap and the ergotropic volume can be integrated into a quantum circuit and estimated using variational quantum algorithms. This integration allows for the assessment of multipartite entanglement at various stages of a quantum simulation, particularly relevant when conducting such simulations on NISQ devices. The circuit-level formulation highlights the practicality and experimental relevance of the method, making it a valuable tool for researchers working with current quantum technologies.

The team’s findings demonstrate that the ergotropic gap can be used to certify genuine entanglement in multipartite states, offering a new operational thermodynamic measure consistent with established computational measures like generalized geometric measures and multipartite concurrence. The definition of the ergotropic volume, calculated as the volume of a hyper-cuboid with sides representing the ergotropic gap across different partitions, provides a quantifiable metric for assessing the degree of multipartite entanglement.

For an N-partite quantum system, this volume is calculated across D = $2^{N - 1} - 1$ different partitions, where each partition divides the system into two disjoint groups of subsystems. The ability to limit ergotropic measurements to quenched Hamiltonians or local observables is a key strength of this approach, simplifying experimental implementation and broadening its applicability to a wider range of quantum systems.

Extending Entanglement Measurement to Time-Evolved Quantum Circuits

Evaluating entanglement in quantum systems evolving over time presents a significant challenge, now addressed by a framework enabling both analytical and numerical characterization of genuine multipartite entanglement (GME) in stationary and time-evolved states. This approach extends to systems exhibiting strong interactions, including those simulated on quantum circuits with varying complexity and the introduction of noise, a common feature of near-term quantum devices. These models are chosen because they encompass diverse interaction types and collective excitations relevant in both cavity QED and spin-lattice systems, broadening the scope of entanglement assessment.

The ergotropic gap, defined as the difference between ergotropies obtained under global and local unitary optimizations, is a faithful quantifier of entanglement, satisfying the requirements of an entanglement monotone. This allows for the certification of GME in mixed states, though the study focuses on pure states experiencing strong interactions.

A key innovation lies in the implementation of a variational quantum algorithm to estimate entanglement in time-evolved states of quantum spin models simulated on a quantum circuit. The algorithm measures ergotropy using a quantum circuit designed to evolve an initial state, employing Suzuki-Trotter decomposition to minimize errors. A corresponding ansatz circuit then seeks the optimal unitary transformation that minimizes the energy of the time-evolved state, a process facilitated by a classical optimizer that iteratively adjusts parameters and runs the quantum circuit.

The key strength of ergotropic volume as an entanglement measure is that it can be readily measured in experiments where energy observables are more accessible, including quantum circuits implemented in noisy intermediate scale quantum (NISQ) devices. Harsh Sharma, Sampriti Saha, A. S. Majumdar, Manik Banik, Himadri Shekhar Dhar highlighted its potential as a versatile tool for characterizing entanglement in near-term quantum simulators.

The framework’s versatility extends to both analytical and numerical characterization, revealing correlations that are otherwise difficult to capture. This capability is particularly valuable for understanding the behavior of complex quantum systems where traditional methods struggle. The researchers note that the method’s extension to time-evolved states implemented on quantum circuits represents an advancement.

👉 More information
🗞 Thermodynamic probes of multipartite entanglement in strongly interacting quantum systems
✍️ Harsh Sharma, Sampriti Saha, A. S. Majumdar, Manik Banik and Himadri Shekhar Dhar
🧠 DOI: http://link.aps.org/doi/10.1103/psph-rgkg

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