Wasserstein distances and divergences of order $p$ by quantum channels

This paper introduces a non-quadratic generalization of the quantum optimal transport problem using quantum channels to define $p$-Wasserstein distances and divergences, while proving a generalized triangle inequality for quadratic Wasserstein divergences under the condition that at least one involved state is pure.

The Gist

Imagine you are trying to move a pile of sand from one location to another. In the classical world, this is easy: you just calculate the shortest path and the least amount of effort required. This is what mathematicians call "Optimal Transport."

But what if the "sand" isn't made of grains, but of quantum information? In the quantum world, things don't have fixed positions; they exist in a fuzzy state of possibilities. You can't just pick up a grain of sand; you have to move a "probability cloud."

This paper, written by Bunth and his colleagues, explores how to do "Optimal Transport" in this strange, fuzzy quantum realm.

1. The Problem: Moving "Fuzzy" Clouds

In classical math, if you want to move a pile of sand from point A to point B, you use a "cost function" (like how much fuel your truck uses). Usually, we use a "squared distance" cost—the further you go, the much more expensive it gets.

In the quantum world, the authors are looking at Quantum Channels. Think of a quantum channel as a specialized "conveyor belt" that doesn't just move matter, but reshapes the very nature of the information being moved. The researchers wanted to see what happens if we change the "cost" of the conveyor belt. Instead of just using the standard "squared distance," they introduced a new way to measure cost called $p$-Wasserstein distances.

2. The Analogy: The Shape of the Cost

Imagine you are a delivery driver.

• The Quadratic Case ($p=2$): This is like a standard taxi meter. The cost grows predictably as you travel. This is the "old" way scientists studied quantum transport.
• The $p$-Wasserstein Case ($p \neq 2$): This is like a delivery service that has weird rules. Maybe for short trips, it’s incredibly cheap, but for long trips, the price skyrockets exponentially. Or maybe it’s a flat fee regardless of distance.

The authors are essentially building a "menu" of different ways to price quantum transport, allowing scientists to study quantum systems under many different "economic" conditions.

3. The "Glitch" in the System (The Divergence Problem)

In normal math, the distance from Point A to Point A is always zero. But in the quantum version, because of the "fuzziness," the math sometimes says the distance from a state to itself is a positive number! This is like saying if you stand perfectly still, you’ve actually traveled five miles.

To fix this "glitch," the authors use something called Divergences. Think of a divergence as a "corrected" distance—it’s a mathematical adjustment that subtracts the "self-travel" so that the distance from a state to itself is a clean, logical zero.

4. The Big Discovery: The Triangle Inequality

One of the most important rules in any kind of distance is the Triangle Inequality. It simply says: "The direct path from A to C cannot be longer than the path from A to B plus the path from B to C." If this rule breaks, your "distance" isn't really a distance anymore; it's just a random number.

The authors found something fascinating:

• For most "weird" quantum costs ($p > 2$), the triangle inequality can break. It’s like finding a shortcut that is actually longer than taking the long way around!
• However, they proved that if at least one of the quantum states involved is "pure" (meaning it is as clear and non-fuzzy as a quantum state can possibly be), the rule holds true.

Summary: Why does this matter?

By creating this new mathematical toolkit, the authors have given physicists a more flexible way to measure how much "effort" it takes to transform one quantum state into another. Whether they are studying how quantum computers process information or how particles interact, they now have a more precise "ruler" to measure the cost of change in the quantum universe.

Technical Summary

This paper, titled "Wasserstein Distances and Divergences of Order $p$ by Quantum Channels," presents a significant theoretical advancement in the field of non-commutative optimal transport. It generalizes the existing framework of quadratic quantum Wasserstein distances to a broader class of $p$-order distances and divergences.

Below is a detailed technical summary of the work.

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1. The Problem: Non-Commutative Optimal Transport

In classical optimal transport, one seeks a coupling (transport plan) between two probability measures that minimizes the expected cost of moving mass. In the quantum setting, the "mass" consists of density operators (states), and the "transport" is realized by quantum channels (completely positive trace-preserving maps).

The authors address two primary limitations in the current literature:

1. Quadratic Restriction: Most existing quantum Wasserstein models are restricted to a quadratic cost ($p=2$), which limits their applicability to non-linear phenomena.
2. Metric Properties: Many quantum Wasserstein definitions fail to be genuine metrics (e.g., the distance from a state to itself might be non-zero). While "divergences" were introduced to fix this, their geometric properties (like the triangle inequality) are not well-understood for $p \neq 2$.

2. Methodology

The authors propose two distinct mathematical frameworks to generalize the transport problem:

A. The Independent Copy Framework (Discrete Observables)

The authors consider a finite collection of observables $\mathcal{A} = \{A_1, \dots, A_K\}$. To define the cost, they consider $K$ independent copies of a quantum coupling $\Pi$. By measuring $A_k$ on the first subsystem and its transpose $A_k^T$ on the second, they generate a joint probability distribution of random variables. The cost is the expectation of a classical cost function $c(x, y)$ (specifically the $l_q$ norm raised to power $p$) over these measurement outcomes.

B. The Position Operator Framework (Continuous Observables)

To bridge the gap with classical Euclidean transport, the authors use the position operator $Q$ on the Hilbert space $L^2(\mathbb{R}^K)$. Unlike the first method, this approach allows for correlations between the $K$ coordinates, providing a more direct analogue to classical optimal transport on $\mathbb{R}^K$.

3. Key Contributions and Results

I. Definition of $p$-Wasserstein Distances and Divergences

The paper introduces the $(p, q)$-Wasserstein distance $D_{\mathcal{A}, p, q}(\rho, \omega)$ and a corresponding $(A, p)$-Wasserstein divergence $d_{A, p}(\rho, \omega)$. The divergence is constructed to "center" the distance, effectively subtracting the self-distance of the states to improve metric-like behavior.

II. Geometric Properties and the "Threshold" of $p=2$

A major portion of the paper is dedicated to the behavior of these distances as the parameter $p$ changes:

• The $p=2$ Threshold: The authors prove that the triangle inequality for divergences is highly sensitive to $p$. They demonstrate that for $p > 2$, the divergence can actually become negative (Proposition 3), meaning it fails to behave like a distance.
• Counterexamples for $p < 2$: They show that for any $p < 2$, there exist states and observables where the triangle inequality fails (Proposition 19).
• The Qubit Case: For a two-level system (qubit), they prove that the triangle inequality holds for all $p \geq 2$ (Theorem 18).

III. Characterization of Cost Operators

The authors investigate the sets of all possible cost operators $\mathcal{C}_p(H)$ generated by different $p$ values. They prove:

• Monotonicity: The sets of cost operators are nested: $\mathcal{C}_1(H) \subseteq \mathcal{C}_p(H) \subseteq \mathcal{C}_\infty(H)$.
• Strict Monotonicity: For Hilbert spaces with dimension $\geq 3$, the sets are strictly increasing for $p > 1$.
• Invariance for Qubits: For $H = \mathbb{C}^2$, the set of cost operators is the same for all $p > 0$.

IV. Generalization of the Triangle Inequality

The paper provides a new proof for the triangle inequality of quadratic ($p=2$) divergences under the condition that at least one of the three states involved is pure (Theorem 20). This generalizes previous results that required the intermediate state to be pure.

4. Significance

This work is significant for several reasons:

1. Mathematical Completeness: It provides a rigorous foundation for $p$-order quantum optimal transport, filling a gap left by the quadratic-only models.
2. Classification of Metrics: By identifying $p=2$ as a critical threshold, the paper provides a roadmap for which quantum transport models are geometrically "well-behaved" and which are not.
3. Connection to Classical Theory: The use of the position operator ensures that this quantum theory is a true generalization of classical optimal transport, making it relevant for researchers in both quantum information and classical probability.
4. Foundational Tool: The results regarding cost operators and the structure of divergences provide essential tools for future researchers working on quantum machine learning, data classification, and quantum many-body problems.