🔢 mathematics

Variational formulae for entropy-like functionals for states in von Neumann algebras

This paper establishes variational formulae for entropy-like functionals, such as Segal and Rényi entropies, for normal states on semifinite von Neumann algebras, thereby generalizing existing results on relative entropy and elucidating the connection between quantum entropies and abelian subalgebras.

Original author: Andrzej Łuczak, Hanna Podsędkowska, Rafał Wieczorek

Published 2026-11-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original author: Andrzej Łuczak, Hanna Podsędkowska, Rafał Wieczorek

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the way we measure disorder or information depends heavily on the size and nature of the system we are observing. For simple, finite systems, like a collection of atoms in a box, physicists have long used a standard formula involving a "density matrix" to calculate entropy, a quantity that tells us how much information is hidden or how disordered a state is. This approach works perfectly when the system is small enough to be treated as a finite list of numbers. However, the universe is not always so tidy. In the extreme environments of black holes or the vast fields of quantum physics, the mathematical structures used to describe reality become infinite and unbounded. In these realms, the familiar density matrix breaks down because the objects representing the state of the system are no longer bounded or finite in the usual sense. Instead, they are affiliated with the algebra of observables in a way that allows them to be unbounded, requiring a completely different mathematical toolkit to describe their entropy.

This shift from finite to infinite systems is not just a technical detail; it is a fundamental change in how we must think about information in the universe. When physicists moved beyond the simple, finite models to these more complex, infinite algebras, they needed a new definition of entropy that could handle these unbounded operators. This is where the work of Andrzej Łuczak, Hanna Podsędkowska, and Rafał Wieczorek becomes essential. They have developed a set of new mathematical tools, called variational formulae, that allow researchers to calculate entropy-like quantities for these complex, infinite systems. Their work focuses on a specific type of entropy known as Segal entropy, as well as a family of related measures called Rényi entropies, which are crucial for understanding how information behaves in quantum field theory and black-hole physics.

The researchers approached this problem by asking a fundamental question: how can we express the entropy of a complex, infinite system using simpler, more manageable pieces? In the finite world, entropy can be calculated directly from the density matrix. In the infinite world, where the density matrix is replaced by an unbounded operator, direct calculation is often impossible. The team discovered that the entropy of such a system can be found by looking at all possible ways to break the system down into simpler, abelian subalgebras. Think of an abelian subalgebra as a collection of observables that behave like ordinary numbers, where the order of measurement does not matter, unlike the complex, non-commuting operators found in the full quantum system. The authors proved that the true entropy of the entire system is actually the highest value you can get by calculating the entropy of these simpler, abelian parts and then taking the best possible combination of them.

To reach this conclusion, the team utilized a powerful mathematical concept involving convex functions, which are shapes that curve outward like a bowl. They showed that for certain types of entropy functions, the value for the whole system is always less than or equal to the average of the values calculated for its parts, unless you find the specific arrangement of parts that maximizes this value. By proving that the maximum value is achieved when you look at all possible abelian subalgebras, they provided a way to define and calculate entropy without needing to solve the impossible problem of handling the unbounded operator directly. Instead, they transformed the problem into one of finding the optimal "slice" of the system that reveals the most information. This method works for both finite systems and the more difficult infinite systems, unifying the understanding of entropy across different scales of quantum reality.

The paper also extends this logic to other types of entropy, such as the Rényi entropies, which are used to measure different aspects of information distribution. For these measures, the researchers found that the calculation depends on whether the parameter used to define the entropy is greater than or less than one. If the parameter is greater than one, the entropy is found by taking the maximum value over all possible abelian subalgebras. If it is less than one, the entropy is found by taking the minimum value. This distinction is crucial because it reveals that different types of information measures behave differently when applied to infinite systems. The authors demonstrated that these new formulae are not just theoretical curiosities but are rigorous generalizations of known results, confirming that the relationship between entropy and the structure of abelian subalgebras holds true even in the most complex mathematical settings.

What makes this work particularly significant is that it bridges the gap between the abstract mathematics of operator algebras and the physical reality of quantum information. By showing that the entropy of a state in a semifinite von Neumann algebra can be determined by looking at its restrictions to abelian subalgebras, the authors provide a concrete method for physicists to calculate these values. This is especially important for fields like black-hole physics, where the algebras of observables are of a type that requires these infinite-dimensional techniques. The results confirm that the connection between quantum entropies and the structure of simpler, commutative subalgebras is a fundamental property of these systems, offering new interpretations for how information is stored and processed in the quantum universe.

The researchers did not stop at just proving these formulae exist; they also showed how to construct the specific operators needed to achieve the maximum or minimum values. They demonstrated that by carefully choosing a sequence of projections, which are like mathematical filters that select specific parts of the system, one can approximate the true entropy of the system as closely as desired. This constructive approach means that the formulae are not just abstract statements but can be applied in practice. The team's work covers both finite and infinite algebras, ensuring that the results are robust and applicable to a wide range of physical scenarios. By generalizing the known results for relative entropy, they have provided a unified framework that simplifies the calculation of entropy-like functionals in complex quantum systems.

In the end, this paper offers a clear path through a mathematical landscape that was previously difficult to navigate. It establishes that the entropy of a quantum state in an infinite system is not an isolated, unapproachable quantity, but rather something that can be understood by examining the system through the lens of its simpler, commutative components. The findings confirm that the structure of abelian subalgebras plays a central role in defining entropy, providing a new perspective on how information is organized in the quantum world. For physicists working on the frontiers of quantum field theory and gravity, these variational formulae offer a reliable tool to calculate and understand the entropy of states that were previously too complex to analyze directly. The work stands as a testament to the power of mathematical rigor in uncovering the deep connections between the structure of quantum systems and the flow of information.

Technical Summary: Variational Formulae for Entropy-like Functionals in Semifinite von Neumann Algebras

Problem Statement
The paper addresses the formulation of entropy-like functionals for normal states on semifinite von Neumann algebras (types $II_1$ and $II_\infty$ ). In the standard type I setting (finite-dimensional or $B(H)$ ), entropy is defined via density matrices $\rho$ and the trace formula $S(\rho) = -\text{tr}(\rho \log \rho)$ . However, in semifinite algebras equipped with a normal faithful semifinite trace $\tau$ , the analogue of a density matrix is an unbounded positive self-adjoint operator $h_\omega$ affiliated with the algebra, belonging to the non-commutative $L^1$ space $L^1(M, \tau)$ .

The primary challenge is to establish variational formulae for functionals of the form $\tau(f(h))$ , where $f$ is a convex or concave function, specifically for Segal entropy ( $f(t) = t \log t$ ) and Rényi entropies ( $f(t) = t^\alpha$ ). While variational characterizations exist for relative entropy in finite algebras, extending these to the semifinite case—where operators may be unbounded and the trace is not finite—requires new analytical tools and careful handling of the domain of definition.

Methodology
The authors employ a combination of non-commutative integration theory, spectral analysis, and convex analysis. The core methodology involves:

Operator Affiliation and $L^1$ Spaces: The framework utilizes the correspondence between normal states $\omega$ and density operators $D_\omega \in L^1(M, \tau)_+$ . The functionals are defined via the trace $\tau(f(D_\omega))$ .
Variational Principles via Jensen's Inequality: The authors leverage Jensen's inequality for operator convex/concave functions. Specifically, they utilize the weaker form $\tau(f(\Phi(h))) \leq \tau(\Phi(f(h)))$ for positive linear unital maps $\Phi$ preserving the trace.
Abelian Subalgebras and Conditional Expectations: A central technique is the restriction of the problem to abelian von Neumann subalgebras $A \subset M$ . By considering the conditional expectation $E_A: M \to A$ (extended to $L^1$ ), the authors reduce the non-commutative problem to a commutative one involving spectral measures.
Spectral Decomposition and Measure Theory: The proofs rely heavily on the spectral decomposition of the density operator $D_\omega = \int_0^\infty t \, e(dt)$ . The authors construct specific abelian subalgebras generated by spectral projections or countable resolutions of identity to approximate the operator $h$ .
Approximation Arguments: For the Segal entropy, the authors construct specific operators $h$ (combining spectral parts and projections on the kernel of the density) to demonstrate that the supremum in the variational formula can be approached arbitrarily closely, even when the density has a non-trivial kernel or infinite trace support.

Key Contributions and Results

Variational Formula for Segal Entropy (Theorem 1 & 2):
The paper establishes that for a normalised state $\omega$ with finite Segal entropy on a semifinite algebra $M$ :
$H(\omega) = \sup_{h^*=h \eta M} \left\{ \tau(D_\omega h) - \log \tau(e^h) : D_\omega h, e^h \in L^1(M, \tau) \right\}$
This generalizes the known finite-algebra result $H(\omega) = \sup_{h \in M_h} (\omega(h) - \log \tau(e^h))$ to the semifinite case, removing the requirement for $\omega$ to be faithful and expanding the supremum to unbounded affiliated operators.
Variational Formulae for General Convex/Concave Functions (Propositions 3 & 4, Theorems 5 & 6):
The authors prove that for a continuous convex function $f \in \mathcal{J}$ (a class including $t \log t$ and $t^\alpha$ ) and $h \in L^1(M, \tau)_+$ :
$\tau(f(h)) = \sup \{ \tau(f(E_A h)) : A \text{ is an abelian subalgebra with } \tau|_A \text{ semifinite} \}$
Conversely, for concave functions, the equality holds with an infimum. This result connects the non-commutative functional $\tau(f(h))$ directly to its commutative restrictions.
Discrete Variational Characterization (Theorems 5 & 6):
The supremum/infimum over abelian subalgebras is further refined to a discrete optimization over countable resolutions of identity $\{p_i\}$ :
$\tau(f(h)) = \sup_{\{p_i\}} \sum_{i=1}^\infty f\left( \frac{\tau(p_i h)}{\tau(p_i)} \right) \tau(p_i)$
where the sum runs over projections $p_i$ such that $\sum p_i = 1$ and $\tau(p_i) < \infty$ .
Specific Applications to Entropies (Corollaries 7 & 8):
Applying the general theorems to specific functions yields variational formulae for:
- Segal Entropy: Expressed as a supremum of classical entropies over abelian subalgebras.
- Rényi Entropies ( $R_\alpha$ ): Expressed as infima (for $\alpha < 1$ ) or suprema (for $\alpha > 1$ ) of sums involving powers of state probabilities and trace weights.
Structural Interpretation (Theorem 9):
The paper demonstrates that the Segal entropy of a state $\omega$ is the supremum of the entropies of its restrictions to all abelian subalgebras $A$ (where $\tau|_A$ is semifinite). This highlights that the "quantum" entropy is fully determined by the "classical" entropies of its abelian projections.

Significance and Claims
The authors claim that these results generalize known variational formulae for relative entropy and Segal entropy from finite to semifinite von Neumann algebras. The work provides a rigorous mathematical foundation for entropy in settings relevant to quantum field theory and black-hole physics, where type II algebras are fundamental.

A key conceptual contribution highlighted is the connection between quantum entropies and the structure of abelian subalgebras. The paper posits that the entropy of a state in a semifinite algebra can be interpreted entirely through the lens of its restrictions to abelian subalgebras, offering a new perspective on the "classical" content of quantum information in infinite-dimensional systems. The authors do not propose new experimental applications but emphasize the necessity of these generalizations for the algebraic formulation of quantum statistical mechanics in non-type I settings.

Technical Summary: Variational Formulae for Entropy-like Functionals in Semifinite von Neumann Algebras

More like this