A generalization of Frenkel's formula

Imagine you are trying to measure the "distance" or "difference" between two complex objects, like two different recipes for a cake or two different maps of a city. In the world of quantum physics and advanced mathematics, these objects are called operators (think of them as complex machines that transform data), and the measure of their difference is called divergence.

For a long time, mathematicians had a specific, clever tool to measure this difference, discovered by a man named Frenkel. However, Frenkel's tool was a bit like a ruler that only worked perfectly if you were measuring the total "weight" (the trace) of the difference, rather than the difference itself. It was like knowing the total weight of the difference between two cakes, but not being able to see the actual shape of the difference.

Shmuel Friedland's paper is about upgrading that ruler. He has created a generalized formula that allows us to measure the "shape" of the difference directly, not just its total weight. This works even when the objects are very complex, infinite, or behave strangely.

Here is a breakdown of the paper's ideas using everyday analogies:

1. The Problem: Measuring the "Gap"

Imagine you have two smooth, glowing balls of light, Ball A and Ball B. You want to know how different they are.

In simple math, if the balls are just numbers, you can subtract them easily.
But in quantum physics, these balls are "operators." They are like complex machines that spin, stretch, and twist space. Subtracting them isn't as simple as $A - B$ . Sometimes, part of the machine might be "negative" (undoing work) and part "positive" (doing work).

Frenkel's old formula was a way to calculate the total energy of the difference between these two balls using a special integral (a fancy way of summing up infinite slices). It was brilliant, but it only gave you the "total score" (the trace).

2. The Solution: The "Divergence Operator"

Friedland asks: Can we write a formula that gives us the actual difference machine itself, not just its total score?

He says Yes. He introduces a new formula (Equation 7 in the paper) that acts like a 3D printer.

The Old Way: You could only calculate the total volume of the difference.
The New Way: You can now calculate the exact shape of the difference machine.

He calls this new thing the Divergence Operator ( $\Delta(A\|B)$ ). Think of it as a "Difference Engine." If you feed it two machines (A and B), it spits out a new machine that represents exactly how A differs from B, preserving all the complex details.

3. The "Layer Cake" Analogy

The paper uses a technique called the "Layer Cake Representation." Imagine you want to describe a complex cake (the difference between A and B).

Instead of describing the whole cake at once, you slice it horizontally into infinite thin layers.
Each layer represents a specific "threshold" of difference.
Frenkel's formula was a way to sum up the volume of all these layers.
Friedland's new formula says: "Let's keep the layers separate!" It shows that the entire difference machine is just the sum of these specific slices, weighted correctly.

This is powerful because it works even if the cake is infinite in size (infinite-dimensional space) or if the ingredients are weird (singular matrices).

4. The "Traffic Light" Rule (When things break)

The paper also discovers a crucial rule about when this measurement works and when it explodes.

Scenario A (The Smooth Road): If Machine A is "smaller" than Machine B (in a specific mathematical sense), the difference is well-behaved. The formula gives you a finite, usable result.
Scenario B (The Cliff): If Machine A has parts that Machine B completely lacks (like A has a feature B doesn't have at all), the "distance" between them becomes infinite.
- Analogy: Imagine trying to measure the difference between a Car and a Bicycle. If you try to measure the "engine difference," the car has a massive engine, and the bicycle has none. The difference is so huge it's effectively infinite.
- Friedland's paper proves that if this "infinite gap" exists, the formula correctly tells you: "The result is infinity." It doesn't break; it just warns you that the two objects are fundamentally incompatible in that specific way.

5. Why Should You Care?

This might sound like abstract math, but it's the backbone of Quantum Information Theory.

Quantum Computers: These machines rely on the precise manipulation of quantum states (the "balls of light"). To know how well a quantum computer is working, or how much information is lost, scientists need to measure the "distance" between the ideal state and the actual state.
Data Security: In cryptography, knowing the exact "divergence" between two data streams helps determine if a message has been hacked or if it's secure.

Summary

Shmuel Friedland took a famous mathematical shortcut (Frenkel's formula) that only measured the total weight of the difference between two quantum objects. He upgraded it to measure the entire shape of the difference.

He showed that:

You can calculate this complex difference directly using an integral (a sum of slices).
It works for both simple, finite objects and complex, infinite ones.
If the two objects are too different (one has parts the other lacks), the formula correctly screams "Infinity," telling us they are fundamentally incompatible.

It's like upgrading from a scale that only tells you how heavy a difference is, to a high-resolution scanner that shows you the exact 3D shape of the difference, complete with a warning light if the objects are too far apart to ever meet.

1. Problem Statement

The paper addresses the generalization of Frenkel's integral formula for the quantum relative entropy (divergence) from scalar traces to operator-valued expressions.

Context: In quantum information theory (QIT), the Umegaki relative entropy between two density matrices $\rho$ and $\sigma$ is defined as $D(\rho \| \sigma) = \text{Tr}(\rho(\log \rho - \log \sigma))$ .
Existing Knowledge: Frenkel established an integral representation for the trace of the divergence:
$D(A \| B) = \int_{-\infty}^{\infty} \frac{dt}{|t|(t-1)^2} \text{Tr}\left( ((1-t)A + tB)_- \right)$
where $(X)_-$ denotes the negative part of the operator $X$ .
The Gap: While the trace formula is well-known, a corresponding operator-level identity (an equality between operators rather than just their traces) has not been fully established for general non-commuting positive operators. The paper aims to derive an operator identity where the trace is removed, yielding an equality of operators.

2. Methodology

Friedland employs a combination of spectral theory, operator calculus, and approximation techniques in both finite and infinite-dimensional Hilbert spaces.

Definitions and Notation:
- Defines the divergence operator $\Delta(A \| B)$ as:
  $\Delta(A \| B) := A(\log A - \log B) - B D_{\log}[B](A) + B$
  where $D_{\log}[B](A)$ is the directional derivative of the logarithm.
- Uses the spectral decomposition and the positive/negative parts of operators ( $T_+, T_-$ ).
- Introduces the operator $O_\gamma(A \| B) = (A - \gamma B)_+$ .
Finite-Dimensional Approach ( $n \times n$ Matrices):
- Analyticity: Utilizes the fact that the positive part of a Hermitian pencil $(A + tB)_+$ is analytic almost everywhere on $\mathbb{R}$ , with singularities only at a finite set of points.
- Change of Variables: Transforms the integral over $t \in (-\infty, \infty)$ into an integral over $\gamma \in (1, \infty)$ using the substitution $\gamma = \frac{t}{t-1}$ .
- Decomposition: Splits the integral into regions $t \in [0, 1]$ , $t \in (1, \infty)$ , and $t \in (-\infty, 0)$ to relate the integral of the negative part $((1-t)A + tB)_-$ to integrals involving $O_\gamma(A \| B)$ and $O_\gamma(B \| A)$ .
- Limit Arguments: Proves the identity for strictly positive definite matrices ( $A, B > 0$ ) and extends it to positive semidefinite matrices ( $A \geq 0$ ) via a limiting process ( $A_k \to A$ ).
Infinite-Dimensional Approach (Hilbert Space):
- Extends the results to bounded self-adjoint operators and $p$ -Schatten class operators ( $S_p$ ).
- Uses finite-rank approximations ( $P_n A P_n$ ) converging in the strong or norm topology.
- Relies on Kato's inequalities regarding the continuity of the map $T \mapsto |T|$ and the spectral projections.
- Applies Rellich's theorem to ensure the analyticity of eigenvalues and eigenvectors of Hermitian pencils, which is crucial for defining the integrals.

3. Key Contributions and Results

A. The Main Operator Identity

The primary result is the generalization of Frenkel's formula to an operator equality. For $A \geq 0$ and $B > 0$ :
$A(\log A - \log B) - B D_{\log}[B](A) + B = \int_{-\infty}^{\infty} \frac{dt}{|t|(t-1)^2} ((1-t)A + tB)_-$
This is equivalent to the "layer cake" representation:
$\Delta(A \| B) = \int_{1}^{\infty} \left( \gamma^{-1} O_\gamma(A \| B) + \gamma^{-2} O_\gamma(B \| A) \right) d\gamma$
where $O_\gamma(A \| B) = (A - \gamma B)_+$ .

B. Dichotomy of Convergence (Theorem 1 & 2)

The paper establishes a critical dichotomy regarding the convergence of these integrals:

Bounded Case: If there exists $\tau \geq 1$ such that $A \leq \tau B$ (i.e., the support of $A$ is contained in the support of $B$ and $A$ is bounded relative to $B$ ), then the integral converges to a bounded, non-negative operator $\Delta(A \| B)$ .
Unbounded Case: If no such $\tau$ exists (specifically if $\text{supp}(A) \not\subseteq \text{supp}(B)$ ), the integral diverges to infinity (in the sense that the norm of the partial integrals tends to infinity). In this case, both sides of the divergence formula are effectively infinite.

C. Extension to $p$ -Schatten Classes (Theorem 3)

The results are extended to infinite-dimensional separable Hilbert spaces for operators in the $p$ -Schatten class ( $S_p$ ).

The paper defines the operator divergence via limits of finite-dimensional projections.
It proves that if the integral of the norms $\int \gamma^{-1} \|O_\gamma(A \| B)\|_p d\gamma$ is finite, then the operator identity holds in the $p$ -norm topology.
It provides conditions under which the sequence of finite-dimensional approximations converges to the infinite-dimensional operator divergence.

D. Auxiliary Integral Representations

The paper provides new integral representations for the terms $B D_{\log}[B](A)$ and $A(\log A - \log B)$ under specific boundedness conditions (e.g., $A^2 \leq \alpha^2 B^2$ ), ensuring the convergence of these operator integrals without relying on spectral decompositions.

4. Significance

Theoretical Advancement: This work moves beyond scalar trace inequalities to operator inequalities. It confirms that the structural relationship between relative entropy and the "layer cake" integral representation holds at the operator level, not just after taking the trace.
Quantum Information Theory (QIT): The formula provides a rigorous tool for analyzing quantum divergences without assuming commutativity ( $AB \neq BA$ ). This is essential for non-commutative probability and quantum channel analysis.
Computational and Analytical Utility: By expressing the divergence operator as an integral of positive parts of linear combinations of operators, the paper offers a new perspective for numerical approximations and theoretical proofs involving operator monotonicity and convexity.
Generalization of Known Results: It unifies and generalizes previous results by Frenkel, Cheng, Liu, and others, clarifying the exact conditions (support containment) under which these operator divergences are well-defined and finite.

In summary, Friedland successfully lifts Frenkel's integral formula from the realm of traces to the realm of operators, establishing a robust framework for defining and analyzing the "divergence operator" in both finite and infinite-dimensional quantum systems.