Jensen's inequality for partial traces in von Neumann algebras

Motivated by recent finite-dimensional results, this paper establishes Jensen's inequality for partial traces within both semifinite and general non-tracial von Neumann algebras.

The Gist

Imagine you are a chef trying to perfect a complex recipe, but you have a twist: you have two kitchens (let's call them Kitchen A and Kitchen B) working together to create a single, massive dish.

In the world of mathematics and quantum physics, this "dish" is a complicated system of data or energy, and the "recipe" is a set of rules for how to process it. One of the most famous rules in math is called Jensen's Inequality.

The Basic Idea: The "Average" Rule

Think of Jensen's Inequality as a rule about averages.

• The Rule: If you take a bunch of numbers, average them, and then apply a curved function (like squaring them), you get a different result than if you apply the function to each number first and then average them.
• The Metaphor: Imagine you have a bag of apples.
  • Method 1 (Average then Cook): You mash all the apples into a single giant smoothie (the average), and then you bake it into a pie.
  • Method 2 (Cook then Average): You bake each individual apple into a tiny pie, and then you mash all those tiny pies together.
  • The Inequality: Because of the way baking works (the "curved" nature of heat), the giant pie made from the smoothie will always taste different (and mathematically, usually "smaller" or "larger" depending on the function) than the mashed-up tiny pies. Jensen's Inequality tells us exactly which one is bigger.

The Problem: The "Partial Trace"

In the real world (and in quantum physics), we often don't have the whole picture. We might only be able to taste the part of the dish coming from Kitchen A, while Kitchen B is hidden or too complex to measure directly.

In math, this is called a Partial Trace. It's like trying to figure out the flavor of the whole soup by only tasting the broth, ignoring the vegetables that are still floating in the pot.

For a long time, mathematicians knew how to apply the "Average Rule" (Jensen's Inequality) to simple, finite systems (like a small bowl of soup). But when they tried to apply it to infinite, complex systems (like the entire ocean of a quantum universe), the math broke down. The "kitchens" were too big, and the "ingredients" (operators) were too wild to handle with old tools.

The New Discovery: The "Universal Recipe"

The paper you provided, written by Mizanur Rahaman and Lyudmila Turowska, is like a master chef who has finally figured out how to apply the "Average Rule" to these massive, infinite kitchens.

Here is what they did, broken down simply:

1. The Old Way (Finite Dimensions): Before, if you wanted to compare the "Giant Pie" vs. the "Mashed Tiny Pies" in a small kitchen, you needed a very specific, heavy-handed tool (a "density matrix"). It was like needing a giant, industrial blender just to mix a cup of coffee. It worked, but it was clunky and didn't work for infinite oceans.
2. The New Way (Von Neumann Algebras): The authors proved that you can do this comparison in any size kitchen, even infinite ones, using a much more elegant tool.
   • They showed that even if you only look at a slice of the system (the "Partial Trace"), the rule still holds: Processing the whole system first and then looking at the slice is mathematically "safer" (or more predictable) than looking at the slice first and then processing it.

The Two Main Recipes in the Paper

The paper actually offers two versions of this new rule, depending on how "strict" your kitchen is:

1. The "Tracial" Kitchen (The Fair Kitchen)

• The Setting: Imagine a kitchen where every ingredient is weighed perfectly and fairly. There is a "trace" (a scale) that never lies.
• The Result: The authors proved that in this fair kitchen, you can swap the order of "averaging" and "cooking" (applying a convex function) without breaking the laws of physics. They showed that the inequality holds even if the ingredients are messy or infinite, as long as the kitchen has a fair scale.
• Why it's better: It's stronger than the old rule. The old rule required you to use a "square root" of your ingredients (a very specific, heavy tool). This new rule lets you use the ingredients directly.

2. The "State" Kitchen (The Subjective Kitchen)

• The Setting: Imagine a kitchen where there is no perfect scale, only a chef's opinion (a "state"). Maybe the chef thinks the soup is salty, even if it's not.
• The Result: Here, the math is trickier. To make the rule work, the "cooking" function (the convex function) has to be extra special. It can't just be a normal curve; it has to be an "Operator Convex" function.
• The Metaphor: Think of "Operator Convex" as a recipe that is so robust it can handle the chef's bias. If the function is strong enough, the rule still holds: The whole-system-first approach is still better than the slice-first approach.

Why Should You Care?

You might ask, "I don't cook infinite quantum soups. Why does this matter?"

• Quantum Computers: These machines rely on manipulating massive, complex states of information. Understanding how to "average" or "measure" parts of these systems without breaking the math is crucial for building stable quantum computers.
• Energy and Physics: The paper mentions "eigenvalue asymptotics," which is a fancy way of saying "predicting how energy levels behave in huge systems." This helps physicists understand how stars burn or how materials conduct electricity at the atomic level.
• The Big Picture: This paper is a bridge. It takes a rule that worked for small, simple things and proves it works for the infinite, chaotic universe. It tells us that even in the most complex, infinite systems, there is an underlying order to how averages and functions interact.

The Takeaway

Rahaman and Turowska have handed us a universal key. They proved that no matter how big or complex your "kitchen" (quantum system) is, you can trust a specific mathematical rule to tell you how the "whole" relates to the "part." They removed the need for clunky, heavy tools and showed that the rule works naturally, even in the infinite void of the quantum world.

Technical Summary

Here is a detailed technical summary of the paper "Jensen's inequality for partial traces in von Neumann algebras" by Mizanur Rahaman and Lyudmila Turowska.

1. Problem Statement and Motivation

The paper addresses the extension of Jensen's inequality from finite-dimensional Hilbert spaces to the infinite-dimensional setting of von Neumann algebras.

• Background: Jensen's inequality is a cornerstone of convex analysis and probability. In operator theory, it has been generalized to positive maps and operator convex functions (Davis, Choi, Hansen–Pedersen).
• Recent Context: A recent result by Carlen, Frank, and Larson (2025) established a Jensen's inequality for partial traces on finite-dimensional Hilbert spaces. Specifically, for a density matrix $\rho$ and a self-adjoint operator $H$, they proved:
  $$\text{Tr}_2 f (\text{Tr}_1(\rho^{1/2} H \rho^{1/2})) \leq \text{Tr}_1(\rho^{1/2} \text{Tr}_2 f(H) \rho^{1/2})$$
• The Gap: While trace Jensen's inequalities exist for general positive maps in operator algebras (e.g., Brown–Kosaki, Petz), a specific inequality for partial traces involving the conjugation by an operator $a$ (analogous to the density matrix $\rho$) had not been established in the general von Neumann algebra framework.
• Objective: The authors aim to generalize the Carlen-Frank-Larson result to:
  1. Semifinite von Neumann algebras (tracial case).
  2. General (non-tracial) von Neumann algebras (state case).

2. Methodology

The authors employ a combination of non-commutative integration theory, spectral analysis, and operator convexity.

A. Semifinite Case (Tracial)

• Framework: They work with semifinite von Neumann algebras $(M, \tau)$ equipped with a normal faithful semifinite trace. They utilize non-commutative $L_p$ spaces ($L_1, L_2$) and the space of measurable operators $L_0(M, \tau)$.
• Key Technical Step (Theorem 4): Before proving the main partial trace result, they generalize Petz's Theorem (1987). Petz's original result applied to positive elements. The authors prove that for a unital positive map $\Phi$ and a self-adjoint element $x$ (not necessarily positive), the inequality $\tau(f(\Phi(x))) \leq \tau(\Phi(f(x)))$ holds, provided the traces are defined.
  • Proof Strategy: They decompose the self-adjoint operator $x$ based on the monotonicity of the convex function $f$ over specific intervals. They utilize the spectral pre-order ($\lesssim$) and Jordan decomposition to compare the positive and negative parts of $f(\Phi(x))$ and $\Phi(f(x))$.
• Partial Trace Construction: They define the partial trace operation using slice maps. For $a \in L_2(M_1, \tau_1)$, they construct a map $\Phi(X) = (\tau_1 \otimes \text{id})((a^* \otimes 1)X(a \otimes 1))$. They show this map is unital and completely positive.

B. General Case (Non-Tracial)

• Framework: For general von Neumann algebras without a trace, they replace the trace with normal states $\rho_1, \rho_2$.
• Constraint: In the absence of a trace, the standard convexity of $f$ is insufficient. The authors require $f$ to be operator convex.
• Technique: They utilize the Hansen–Pedersen inequality (operator Jensen's inequality for contractive maps) and the properties of slice maps defined by states.

3. Key Contributions and Results

Main Result 1: Tracial Jensen's Inequality for Partial Traces (Theorem 2)

Let $(M_1, \tau_1)$ and $(M_2, \tau_2)$ be semifinite von Neumann algebras. Let $H \in M_1 \bar{\otimes} M_2$ be self-adjoint, $f: \mathbb{R} \to \mathbb{R}$ be continuous and convex, and $a \in L_2(M_1, \tau_1)$ with $\tau_1(a^*a) = 1$. Then:
$$\tau_2 \left( f \left( (\tau_1 \otimes \text{id})[(a^* \otimes 1)H(a \otimes 1)] \right) \right) \leq \tau_1 \left[ a^* (\text{id} \otimes \tau_2) f(H) a \right]$$

• Significance: This generalizes the Carlen-Frank-Larson theorem.
  • It removes the requirement for $a$ to be a density matrix square root ($\rho^{1/2}$) in the finite-dimensional sense; $a$ is simply an $L_2$ element with unit norm.
  • It handles the infinite-dimensional setting where operators may not be trace-class.
  • It extends the result to general semifinite traces, not just finite traces.

Main Result 2: Non-Tracial Jensen's Inequality (Theorem 8)

Let $M_1, M_2$ be von Neumann algebras with normal states $\rho_1, \rho_2$. Let $H$ be self-adjoint in $M_1 \bar{\otimes} M_2$, $a \in M_1$ be a contraction, and $f$ be operator convex. Then:
$$\rho_2 \left( f \left[ (\rho_1 \otimes \text{id})((a^* \otimes 1)H(a \otimes 1)) \right] \right) \leq \rho_1 \left( a^* (\text{id} \otimes \rho_2) f(H) a \right)$$

• Significance: This establishes the inequality for general states where no trace exists, at the cost of requiring operator convexity.

Technical Lemma (Theorem 4)

The authors prove that Petz's trace Jensen inequality holds for self-adjoint (not just positive) elements. This is a non-trivial extension because the standard proof relies on the positivity of the argument, which is lost when $x$ is merely self-adjoint. They achieve this by carefully analyzing the spectral projections and the Jordan decomposition of the resulting operators.

4. Significance and Applications

1. Mathematical Rigor in Operator Algebras: The paper fills a specific gap in the literature regarding partial traces in infinite-dimensional operator algebras. It bridges the gap between finite-dimensional quantum information results and the general theory of von Neumann algebras.
2. Eigenvalue Asymptotics: The authors note that the finite-dimensional version of this inequality was used to study eigenvalue asymptotics of operators with homogeneous potentials. The infinite-dimensional generalization suggests potential applications in spectral theory for unbounded operators in quantum mechanics.
3. Quantum Information Theory: Jensen's inequalities are fundamental to the study of quantum entropy and relative entropy. By extending these inequalities to partial traces in general algebras, the results provide new tools for analyzing quantum systems with infinite degrees of freedom, potentially impacting the study of quantum entropy in non-tracial settings.
4. Refinement of Existing Bounds: In the finite-dimensional case, the authors' result is slightly stronger than the Carlen-Frank-Larson theorem because it does not require the conjugating element to be a square root of a density matrix, but rather any $L_2$ element of unit norm.

5. Conclusion

Rahaman and Turowska successfully extend a recent breakthrough in finite-dimensional quantum information to the broad framework of von Neumann algebras. Their work relies on a sophisticated generalization of Petz's theorem to self-adjoint operators and the careful application of slice maps. The results provide a robust mathematical foundation for analyzing convex functions of partial traces in both tracial and non-tracial quantum systems.