Relations between different definitions of the quantum Wasserstein distance for qubits

This paper demonstrates that for qubits, the quantum Wasserstein distances defined by Golse et al. and by De Palma and Trevisan coincide when the cost function involves a single operator, implying that the self-distance in this scenario equals the Wigner-Yanase skew information.

The Gist

Imagine you are trying to measure the "distance" between two different quantum states. In the classical world, if you have a pile of sand and you want to move it to a new shape, the "Wasserstein distance" (often called the Earth Mover's Distance) is simply the minimum amount of work required to move the grains from the first shape to the second. If the two shapes are identical, the work required is zero.

But in the quantum world, things get weird. Quantum states are fuzzy, probabilistic, and can be "entangled." Because of this, physicists have invented several different ways to calculate this quantum distance. Think of these different methods as different teams of cartographers trying to map the same mysterious island. They all use different tools and rules, so they often produce slightly different maps.

This paper is about two specific teams of cartographers:

1. The GMPC Team: Led by Golse, Mouhot, Paul, and Caglioti.
2. The DPT Team: Led by De Palma and Trevisan.

Both teams are trying to measure the distance between two quantum states (let's call them "State A" and "State B"). They both look for a special "bridge" (a mathematical object called a coupling) that connects the two states with the least amount of "cost." However, they define the "cost" slightly differently.

The Big Discovery: They Agree on Single Qubits

The authors of this paper, Géza Tóth and József Pitrik, focused on the simplest possible quantum system: a qubit. You can think of a qubit as a single quantum coin that can be heads, tails, or a fuzzy mix of both.

They asked a simple question: If we are only dealing with one single qubit and we are measuring the distance based on just one specific "rule" (one operator), do these two different teams get the same answer?

The answer is yes.

The paper proves that for a single qubit, if you are using a single rule to measure the distance, the GMPC map and the DPT map are identical. The two different definitions of quantum distance collapse into one.

Why is this surprising? (The "Self-Distance" Puzzle)

In the classical world, the distance from a point to itself is always zero. If you are standing in Paris, the distance from Paris to Paris is zero.

In the quantum world, however, a state can have a non-zero "self-distance." This is like saying that if you try to move a quantum coin from its current fuzzy state to the exact same fuzzy state, it still costs some "work."

The paper highlights a fascinating connection:

• The DPT team had already discovered that this "self-distance" is mathematically equal to a quantity called Wigner-Yanase skew information. Think of this as a measure of how much "quantum uncertainty" or "information" is hidden inside the state regarding that specific rule.
• Because the authors proved the two teams agree on single qubits, they can now say: The GMPC team's "self-distance" is also equal to this Wigner-Yanase skew information.

The Magic Trick: Making Everything Real

How did they prove this? They used a clever mathematical "magic trick."

Imagine the quantum state and the rule (the operator) are written in a complex language involving imaginary numbers. The authors showed that for a single qubit, you can always rotate the entire system (like spinning a globe) so that all the numbers become "real" (no imaginary parts).

Once everything is "real," the two different definitions the teams use turn out to be mathematically identical. It's like realizing that two people describing a building—one using a blueprint and one using a 3D model—are actually describing the exact same structure once you realize they are both looking at the same side of the building.

What does this mean for the rest of the paper?

The authors also point out a practical consequence for physicists studying spin chains (long lines of quantum magnets). Because the two distance definitions are now known to be the same for single qubits, physicists can use the simpler formulas from one team to calculate the energy of these magnetic systems. Specifically, they can relate the minimum energy of a system to the Wigner-Yanase skew information without needing to worry about complex "transpose" operations that usually complicate the math.

Summary

• The Problem: Physicists had two different ways to measure distance in the quantum world, and it wasn't clear if they agreed.
• The Solution: For the simplest quantum object (a single qubit) and a single measurement rule, the two methods are exactly the same.
• The Result: This confirms that the "cost" of a quantum state to move to itself is a fundamental measure of quantum information (Wigner-Yanase skew information), regardless of which mathematical definition you use.
• The Limit: This agreement is proven specifically for single qubits with a single operator. The paper does not claim this holds for complex, multi-qubit systems or multiple operators.

In short, the paper unifies two different languages of quantum transport for the simplest case, showing they are just different ways of saying the same thing.

Technical Summary

Technical Summary: Relations between Different Definitions of the Quantum Wasserstein Distance for Qubits

Problem Statement
The field of quantum optimal transport has generated multiple distinct definitions of the quantum Wasserstein distance, each motivated by different physical interpretations (e.g., semi-classical limits, free probability, dynamical interpretations, or quantum channel formalisms). Two prominent definitions are:

1. The GMPC Distance: Defined by Golse, Mouhot, Paul, and Caglioti (GMPC), based on the optimization of a cost function over bipartite states with fixed marginals.
2. The DPT Distance: Defined by De Palma and Trevisan (DPT), also based on an optimization over bipartite states (couplings) but utilizing a different cost function involving matrix transposes.

While it was known that the self-distance (the distance of a state to itself) for the DPT definition equals the sum of Wigner–Yanase skew information, the relationship between the GMPC and DPT definitions remained unclear. Specifically, it was unknown whether these two definitions coincide for general quantum states, and if so, under what conditions. Previous results established that a separable-based quantum Wasserstein distance serves as an upper bound for both, but a direct equivalence had not been proven.

Methodology
The authors investigate the relationship between the squared GMPC distance, $D_{GMPC}^2(\varrho, \sigma)$, and the squared DPT distance, $D_{DPT}^2(\varrho, \sigma)$, specifically for qubit systems (single-qubit states) and a single operator ($N=1$) in the cost function.

The core of their methodology relies on a unitary transformation argument encapsulated in Lemma 1. The authors prove that for any single-qubit Hermitian operator $H$ and any single-qubit density matrix $\varrho$, there exists a unitary operator $U$ such that both the transformed operator $H' = UHU^\dagger$ and the transformed state $\varrho' = U\varrho U^\dagger$ are real matrices.

Using this lemma, the authors demonstrate that the optimization problems defining the two distances can be mapped onto each other. They utilize the property that for real matrices, the transpose operation is equivalent to the identity operation in the context of the cost function's structure, thereby bridging the gap between the definitions which differ by the presence of transposes in the DPT formulation.

Key Contributions and Results

1. Equivalence for Qubits: The paper's main result (Theorem 1) establishes that for single-qubit states $\varrho$ and $\sigma$ and a single operator $H_1$, the squared distances defined by the two approaches are identical:
   $$D_{DPT}^2(\varrho, \sigma) = D_{GMPC}^2(\varrho, \sigma)$$
   This equality holds generally for qubits, not just for specific subsets of states.

2. Implication for Self-Distance: As a direct consequence of this equivalence, the self-distance for the GMPC definition (when $N=1$) is shown to equal the Wigner–Yanase skew information, $I_\varrho(H_1)$. This confirms that the GMPC distance shares the profound connection to quantum estimation theory found in the DPT definition.

3. Spin Chain Energy Relation: The authors derive a consequence for spin chain calculations. They show that the ground state energy of a ferromagnetic model with a single operator $H_1$ and given marginals can be directly related to the Wigner–Yanase skew information via the formula:
   $$\min_{\varrho_{12}} \text{Tr}[-(H_1 \otimes H_1)\varrho_{12}] = I_\varrho(H_1) - \langle H_1^2 \rangle_\varrho$$
   Notably, this formulation does not require matrix transposes, simplifying the connection between energy minimization and skew information for qubits.

Significance and Claims
The paper claims to resolve the ambiguity between two major definitions of the quantum Wasserstein distance in the specific but fundamental case of qubits with a single cost operator. By proving their equivalence, the work unifies the static interpretation (GMPC) and the quantum channel interpretation (DPT) for this regime.

The authors emphasize that this result connects the quantum Wasserstein distance to fundamental quantities in quantum physics and estimation theory, specifically the Wigner–Yanase skew information. They note that while the self-distance in classical optimal transport is always zero, the non-zero self-distance in the quantum case (equal to the skew information) is a distinct feature of quantum optimal transport. The paper modestly limits its scope to qubits and single operators, acknowledging that for $N \geq 2$ or higher-dimensional systems, the relationship may differ, and previous bounds involving quantum Fisher information remain relevant. The work does not propose new experimental setups but rather clarifies the theoretical landscape of existing definitions.