🔢 mathematics

Finite relative entropy for locally squeezed states

This paper extends closed-form relative entropy calculations to multi-mode squeezed states in a free scalar quantum field, demonstrating that while spatially local squeezing yields unphysical divergences, spacetime-local squeezing produces a physically reasonable class of states with finite relative entropy.

Original authors: Daniela Cadamuro, Markus B. Fröb, Dimitrios Katsinis, Jan Mandrysch

Published 2026-09-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniela Cadamuro, Markus B. Fröb, Dimitrios Katsinis, Jan Mandrysch

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible landscape of quantum physics, particles are not just tiny billiard balls but also waves of probability that can be stretched, squeezed, and reshaped. One of the most powerful tools physicists use to understand these waves is a concept called relative entropy. Think of it as a precise ruler for measuring how different two quantum states are from one another. While this ruler is fundamental to understanding information and the structure of the universe, it has been notoriously difficult to use. For decades, scientists could only calculate this difference for very simple, idealized situations, such as the calm, empty vacuum of space or states where the waves are perfectly aligned. When the waves become more complex, the math usually breaks down, leading to infinite, meaningless answers that tell us nothing about reality.

A team of researchers has now extended the reach of this ruler into a new, more complicated territory. They investigated a specific type of quantum state known as a "squeezed state." In the world of quantum optics, squeezing is like taking a balloon and pressing it from the sides; it makes the balloon thinner in one direction but fatter in another. This process reduces uncertainty in one property of the system while increasing it in another, a trade-off that is incredibly useful for precision measurements and quantum communication. While scientists have long studied squeezing in simple, single-mode systems, the researchers wanted to see what happens when this squeezing is applied to a field that exists everywhere in space and time, involving an infinite number of modes simultaneously.

The team focused on a real, massive scalar field, a theoretical model that behaves like a simple version of the fields that make up our universe. They asked a critical question: what happens if we try to squeeze this field locally? By "locally," they meant applying the squeezing force only within a specific region, rather than everywhere at once. They explored two different ways to do this. The first method involved squeezing the field only at a single moment in time, across a specific slice of space. The second method involved squeezing the field within a specific region of space-time, meaning the force was applied over a duration and a volume.

The results of this investigation revealed a sharp and surprising divide between the two methods. When the researchers applied the squeezing only in space, at a single instant, the universe reacted with chaos. The energy density of the field, which measures how much energy is packed into a given spot, exploded to infinity. Consequently, the relative entropy, the measure of difference between this new state and the empty vacuum, also became infinite. This finding effectively rules out the idea that one can create a physically reasonable, squeezed state by acting on a field at a single moment in time. Such a state would require infinite energy to create and would be impossible to exist in the real world.

However, the story changed completely when the researchers applied the squeezing across both space and time. By spreading the squeezing operation out over a region of space-time, they found that the resulting state was perfectly well-behaved. The energy density remained finite, and the changes to the vacuum were smooth and manageable. Most importantly, the relative entropy between this new state and the empty vacuum was finite and could be calculated explicitly. This proves that it is possible to create a class of "locally squeezed" states that are physically realizable, provided the squeezing is applied in a way that respects the flow of time as well as the spread of space.

The researchers achieved this by using a mathematical tool called the Wick square, which allows them to handle the infinite values that naturally arise in quantum fields without breaking the math. They demonstrated that these states are "quasi-free," meaning they retain a simple, predictable structure that makes them easier to study, and they belong to the same family of states as the standard vacuum. This work is significant because it provides a rigorous foundation for understanding how quantum fields can be manipulated locally without destroying the fabric of the theory. It shows that while nature forbids certain instantaneous, space-only manipulations, it allows for a rich class of states created by local operations in space-time, opening the door for future calculations of entropy and information in complex quantum systems.

Problem Statement
Relative entropy serves as a fundamental measure of state distinguishability in both quantum information theory and relativistic quantum field theory (QFT). Despite its conceptual importance, explicit computations of relative entropy remain notoriously difficult. To date, closed-form results have been restricted to ground states, coherent states, and, more recently, single-mode squeezed states. These existing results typically rely on restricting state comparisons to simple spacetime regions where relative entropy relates to geometric quantities like energy density.

A significant gap exists in the understanding of "infinite-mode" squeezed states, particularly those generated by local operations. While infinite-mode squeezing has been studied in contexts such as conformal field theory, "locally squeezed states"—defined by the action of unitary operators generated by local, quadratic Hamiltonians—have received little attention. The central problem addressed in this work is to establish a well-behaved class of such states with finite and computable relative entropy, distinguishing between two distinct notions of locality: squeezing that is local in space versus squeezing that is local in spacetime.

Methodology
The authors analyze a real linear massive scalar field $\Phi$ on Minkowski spacetime in arbitrary dimensions, focusing on states of the form $W_2(h)\Omega$ , where $\Omega$ is the Fock vacuum and $W_2(h) = \exp[i/2 : \Phi^2:(h)]$ . Here, $:\Phi^2:(h)$ denotes the Wick square smeared with a Schwartz function $h$ . The analysis proceeds by distinguishing two formalisms:

Local Squeezing in Space: The authors first examine squeezing defined on a constant-time Cauchy surface using canonical variables $(\phi, \pi)$ . They construct squeezing operators $S(k)$ generated by Hamiltonian densities quadratic in these time-zero fields. The analysis utilizes the initial value formalism to compute the transformation of fields and their variances.
Local Squeezing in Spacetime: The primary analysis employs a covariant formalism. The authors treat the squeezing generator as the imaginary exponential of a spacetime-smeared Wick square. They rigorously establish the unitary implementability of the resulting Bogoliubov transformations by proving the essential self-adjointness of the relevant operators on the finite-particle subspace of the Fock space. This involves adapting and generalizing previous results on the self-adjointness of the Wick square.

The authors utilize tools from functional analysis, including the Riesz–Thorin interpolation theorem for norm estimates, spectral calculus for defining unitary operators, and detailed estimates on integral kernels to demonstrate the smoothness of correlation functions. They specifically investigate whether these states preserve the Hadamard condition and whether the relative entropy with respect to the Minkowski vacuum (restricted to a wedge algebra) remains finite.

Key Contributions and Results

Divergence of Space-Local Squeezing: The authors demonstrate that squeezing defined locally in space (via time-zero field operators) leads to physically unreasonable states. Specifically, the vacuum expectation value of the energy density for such states diverges ( $\langle \Omega | T^{00}_S | \Omega \rangle = \infty$ ) unless the squeezing is trivial. Consequently, the total energy and the relative entropy for these states are infinite. This result suggests that space-local squeezing, as defined in the canonical formalism, does not yield a physically viable class of states in the continuum limit.
Well-Defined Spacetime-Local Squeezing: In contrast, the authors prove that squeezing defined locally in spacetime (via the smeared Wick square $:\Phi^2:(h)$ $: Φ^{2} : (h)$ ) yields a well-behaved class of quasi-free states.
- Unitary Implementability: They establish that the Bogoliubov transformation induced by spacetime-local squeezing is unitarily implementable on the Fock space. The resulting states lie in the same folium as the Minkowski vacuum.
- Hadamard Condition: The authors show that these states preserve the Hadamard condition, ensuring that the change in vacuum correlations and energy density is smooth.
- Finite Relative Entropy: A central result is the proof that for a Schwartz function $h$ supported in a wedge region $W$ , the relative entropy between the locally squeezed state and the Minkowski vacuum, restricted to the wedge algebra $A_W$ , is finite. This is achieved by showing that the total energy and the integral of the energy density weighted by the wedge coordinate are finite.

Significance
The paper establishes a rigorous foundation for "locally squeezed states" in QFT, clarifying that the definition of locality is critical for physical viability. The work demonstrates that while space-local squeezing leads to divergent energies and entropies, a covariant definition of local squeezing via the Wick square produces physically reasonable states with finite relative entropy.

This extends the known class of states for which relative entropy can be computed in closed form beyond ground, thermal, and single-mode squeezed states. By proving that local-in-spacetime squeezing yields finite relative entropy, the authors provide a new tool for investigating state distinguishability in QFT. The results connect to broader programs in computing relative entropies, which are essential for understanding entanglement, energy conditions, and the thermodynamic properties of quantum fields, without requiring artificial UV cutoffs that render the continuum limit divergent. The authors note that while their construction focuses on the massive scalar field, the framework suggests potential generalizations to other fields and curved spacetimes, provided suitable self-adjointness results for the Wick square hold.

More like this