A new hydrodynamic description accurately predicts entanglement entropy for both a quantum system and its surrounding environment during free expansion; previously calculations failed to capture how much information resides in that environment when dealing with finite systems. Atharva Naik and colleagues refined how phase space density, a measure of particle distribution, is used within an existing formula, known as Yang-Yang entropy. The team refined a method to calculate entanglement, how connected quantum systems “know” about each other, when one part expands into empty space; previous calculations struggled with finite-sized systems.

The work improves upon existing formulas by accurately describing phase space density, which measures particle distribution within a system. Consequently, scientists can now more precisely predict behaviour even at non-zero temperatures, aiding understanding of complex interactions in larger quantum scenarios; accurate modelling of these connections advances theoretical knowledge of such phenomena. The team has refined calculations predicting how entanglement, a quantum connection between systems, evolves during free expansion; previous methods struggled to accurately account for finite-sized setups.

They achieved this by improving upon existing formulas using what’s known as hydrodynamic description, which treats particles not as individual points but as a fluid flowing through space simplifying complex interactions. This approach allows scientists to more precisely predict behaviour even at non-zero temperatures, key for understanding larger quantum scenarios where information shared between a system and its environment changes over time, visualised as a ‘Page curve’.

The Yang-Yang formula, a mathematical tool estimating entanglement entropy based on particle distribution, is central to their work. However, applying it to the surrounding “reservoir” required modification to properly capture how much information resides there; does this new refinement unlock deeper insights into these fundamental connections.

Hydrodynamic Modelling Reveals Complete Entanglement Quantification with Environmental Reservoirs

Entanglement measures now reflect reservoir information content; previously, calculations could only predict entanglement within a quantum system, missing the quantification shared with its environment by a factor of two. A refined hydrodynamic description, treating particles as fluids rather than individual points, enabled researchers from Tata Institute of Fundamental Research and Princeton University to achieve this breakthrough. The new approach resolves an outstanding problem in finite-sized systems by accurately relating phase space density to correlation matrix eigenvalues.

It successfully predicts von Neumann entropy for both the expanding system itself *and* the surrounding “reservoir”, even at non-zero temperatures where previous methods faltered due to inaccuracies capturing coherences. Unlike prior calculations, their work quantifies entropy for both the expanding system *and* the reservoir it expands into; earlier methods only predicted entanglement within the initial system.

Agreement with microscopic computations of entanglement using Yang-Yang entropy validated this approach, confirming the dynamic Page curve, a key indicator of information preservation in black hole physics. This provides insight into how accurately the model captures quantum coherences previously difficult to account for.

Successful Entanglement Prediction Validates Approximations Despite Mathematical Limitations

While predicting entanglement between expanding quantum systems and their environments, something previous calculations struggled with in finite setups, this research currently relies on what researchers term a “heuristic derivation”. Establishing solid proof is vital before applying these findings more broadly, as its mathematical basis isn’t yet fully rigorous and could depend on unproven assumptions.

Still, despite relying on an unproven approach, this work offers valuable insight into entanglement between quantum systems and their surroundings. The team successfully predicts connection evolution in expanding scenarios where prior methods failed; detailed microscopic calculations align with simpler approximations using semiclassical hydrodynamics.

This advancement builds upon refining how phase space density, the particle distribution used in calculations relating to correlation matrix eigenvalues, is utilised. Accurately modelling this connection even at non-zero temperatures provides a more complete picture of active interactions during expansion. Consequently, better understanding is gained regarding how system properties and environmental influence affect the evolution of entanglement.

This research accurately predicted entanglement entropy for an expanding one-dimensional Fermi gas, a quantum system evolving into a vacuum, by connecting hydrodynamic descriptions to microscopically computed values like Yang-Yang entropy. This matters because it extends our ability to quantify information flow between systems and their environments, something previously limited to finite setups; researchers were able to calculate entropy for both the expanding system *and* its surrounding reservoir.

The study also produced a Levitov-Lesovik formula that describes particle fluctuations during this expansion, offering insight into capturing quantum coherences within the model. These findings validate approximations despite current mathematical limitations in fully proving the approach.

👉 More information
🗞 Hydrodynamic description of correlation matrix eigenvalues in a finite temperature expanding free fermionic gas
✍️ Atharva Naik, Madhumita Saha, Manas Kulkarni and Abhishek Dhar (Tata Institute of Fundamental Research); David A. Huse (Princeton University)
🧠 ArXiv: https://arxiv.org/abs/2610.30062

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