Quantum Wasserstein distance and its relation to several types of fidelities

This paper establishes connections between various definitions of quantum Wasserstein distance and quantum fidelities by proving that optimizing over separable bipartite states yields quantities equal to the Uhlmann-Jozsa fidelity (specifically for qubits) and the superfidelity, while also demonstrating the triangle inequality for certain cases involving pure states.

The Gist

Imagine you are trying to measure the "distance" between two different quantum states. In the classical world, if you have two piles of sand (representing two different distributions of matter), the "Wasserstein distance" is like the minimum amount of work needed to move the sand from one pile to the other. It's a very useful way to say how different two things are.

In the quantum world, things get tricky. Quantum states are like clouds of probability rather than solid piles of sand. Scientists have invented several different ways to measure the "distance" between these quantum clouds, but they often use complicated math that treats the clouds as if they are made of a single, inseparable whole.

This paper, written by G´eza T´oth and J´ozsef Pitrik, asks a simple but profound question: What happens if we stop treating these quantum clouds as inseparable wholes and instead look at them as collections of simple, separate pieces?

Here is a breakdown of their findings using everyday analogies:

1. The Two Main Approaches: The "Whole Cake" vs. The "Separate Slices"

The authors looked at existing definitions of quantum distance.

• The "Whole Cake" Approach: Some definitions assume the two quantum states are linked in a complex, "entangled" way (like a cake that cannot be cut). This is the standard, complex way of doing things.
• The "Separate Slices" Approach: The authors asked, "What if we force the distance calculation to only use 'separable' states?" Think of separable states as two cakes that are sitting next to each other but are not glued together. They are just simple mixtures of independent slices.

2. The Big Discovery: Connecting the Dots

The authors found that when you force the math to use these "separate slices," many of the complicated, different-looking distance formulas actually turn out to be the same thing.

• The Analogy: Imagine you have three different recipes for making a cake: one calls for "flour," one for "wheat powder," and one for "ground grain." They sound different. But if you realize that flour, wheat powder, and ground grain are all just different names for the same ingredient, you realize all three recipes are actually making the exact same cake.
• The Result: The paper proves that several distinct quantum distance formulas, when simplified to "separable" states, are mathematically identical. This connects different branches of quantum physics that previously seemed unrelated.

3. The "Self-Distance" Mystery

In classical physics, the distance between an object and itself is always zero. If you measure the distance from your house to your house, it's 0 miles.

However, in some quantum definitions, the distance from a state to itself is not zero. It's like saying your house is 5 miles away from itself.

• The paper shows that if you use the "separate slices" method, you can get two types of results:
  1. Non-zero self-distance: The state is "far" from itself (this relates to something called "Quantum Fisher Information," which measures how sensitive a system is to change).
  2. Zero self-distance: The state is perfectly close to itself (this relates to the "Trace Distance" and "SWAP-fidelity").

The authors showed that these two different outcomes come from two slightly different ways of setting up the "separate slices" math.

4. The "Magic Mirror" (Fidelity)

One of the most famous tools in quantum physics is called Fidelity. It's like a "similarity score" between two quantum states. A score of 1 means they are identical; 0 means they are completely different.

The authors discovered a surprising new way to calculate this score. They proved that the "similarity score" (specifically, the square root of the Uhlmann-Jozsa fidelity) can be calculated by looking at all possible ways to break the states down into "separate slices" and finding the best match.

• The Analogy: Imagine you want to know how similar two complex paintings are. Instead of looking at the whole canvas, you break both paintings down into thousands of tiny, separate brushstrokes. You then try to pair up the brushstrokes from Painting A with the best matching brushstrokes from Painting B. The authors proved that if you do this perfectly, you get the exact same similarity score as the most complex, high-level method.

5. The Triangle Rule

In geometry, the "Triangle Inequality" says that if you go from Point A to Point B, and then from B to C, the total distance cannot be shorter than going directly from A to C. (You can't take a shortcut by stopping at a third point).

The authors proved that for some of these new "separable" distance measures, this rule holds true if one of the states is "pure" (a simple, non-mixed state, like a single, clear note on a piano). If the states are messy mixtures, the rule is harder to prove, but they found strong evidence that it likely holds there too.

6. The Special Case of Qubits (Two-Level Systems)

For the simplest quantum systems (called qubits, which are like coins that can be heads, tails, or a mix of both), the authors found a perfect match.

• They showed that for qubits, the "separable" distance measure is exactly equal to the standard "similarity score" (Fidelity).
• The Analogy: It's like discovering that for small, simple objects, the complicated "work needed to move sand" formula is exactly the same as the simple "how much they look alike" formula.

Summary

The paper is essentially a unification project. It takes several complicated, high-level definitions of "quantum distance" and shows that if you look at them through the lens of "separable states" (simple, non-entangled pieces), they collapse into a few basic, identical concepts.

• They connected the Quantum Wasserstein Distance (a transport cost) to Quantum Fidelity (a similarity score).
• They showed that for simple systems (qubits), these concepts are mathematically identical.
• They provided a new, simpler way to calculate these distances by breaking complex quantum states into simpler, separable parts.

The authors did not discuss medical applications or future technologies in this paper; their goal was purely to clarify the mathematical relationships between these different ways of measuring quantum differences.

Technical Summary

Technical Summary: Quantum Wasserstein Distance and its Relation to Several Types of Fidelities

Problem Statement
The paper addresses the proliferation of definitions for the quantum Wasserstein distance, a central concept in quantum optimal transport. Various approaches exist in the literature, including those based on quantum channels (De Palma and Trevisan), static interpretations (Golse, Mouhot, Paul, Caglioti), and cost functions involving the SWAP operator or antisymmetric costs. A key challenge is determining whether these distinct definitions are fundamentally related or if they can be unified. Specifically, the authors investigate the quantities obtained when the optimization in these distance definitions is restricted from the set of all physical bipartite quantum states to the subset of separable states. The paper seeks to connect these approaches, determine if they reduce to basic types, and explore their relations to established quantum information measures such as fidelities and the quantum Fisher information.

Methodology
The authors systematically analyze several definitions of the quantum Wasserstein distance by performing the optimization over separable states rather than general states. The methodology involves:

1. Reformulation of Definitions: Taking existing definitions (e.g., De Palma-Trevisan type, SWAP-fidelity based) and modifying the constraint set to include only separable bipartite states ($\varrho_{12} \in \text{Sep}$).
2. Convex Roof Analysis: Expressing the resulting quantities as optimizations over decompositions of mixed states into ensembles of pure product states. This leverages the mathematical structure of convex and concave roofs.
3. Analytical Proofs: Proving equalities and inequalities between the new separable-state quantities and standard measures like the Uhlmann-Jozsa fidelity, Bures distance, and superfidelity.
4. Special Cases: Analyzing specific scenarios, such as when one of the states is pure, when a full set of observables (generators of $SU(d)$) is used, and for qubit systems ($d=2$).
5. Numerical Verification: Using semidefinite programming (SDP) to compute these distances for random states, utilizing the fact that for two qubits, the set of Positive Partial Transpose (PPT) states coincides with the set of separable states, allowing for exact numerical solutions.

Key Contributions and Results

• Unification of Approaches: The paper demonstrates that optimizing over separable states leads to two primary categories of quantities:

  1. Those based on the standard De Palma-Trevisan cost function, which can have a non-zero self-distance.
  2. Those based on a modified cost function (subtracting self-distances), which yields a zero self-distance.
     The authors prove that for a full set of observables, the modified De Palma-Trevisan distance optimized over separable states equals the distance based on the trace distance and the SWAP-fidelity optimized over separable states.

• Relation to Fidelities:

  • Uhlmann-Jozsa Fidelity: The square root of the Uhlmann-Jozsa fidelity, $\sqrt{F(\varrho, \sigma)}$, is proven to be expressible as an optimization over separable states (Theorem 1). Specifically, it is the convex roof of the overlap of pure states in the decomposition.
  • SWAP-fidelity: The SWAP-fidelity optimized over separable states, $F_{S,sep}$, is shown to be the convex roof of the squared overlap of pure states.
  • Qubit Equality: For qubits ($d=2$), the authors prove that the SWAP-fidelity optimized over separable states is exactly equal to the Uhlmann-Jozsa fidelity ($F_{S,sep} = F$). This is a consequence of the equality between the Uhlmann-Jozsa fidelity and the superfidelity for qubits.

• Triangle Inequality: The paper proves that the triangle inequality holds for the modified quantum Wasserstein distance (based on separable state optimization) in the specific case where one of the three states involved is pure. This is established by utilizing the fact that the distance for pure states satisfies the triangle inequality and extending this via the convex roof structure.

• Connection to Quantum Fisher Information: The self-distance of the separable-state optimized De Palma-Trevisan distance is shown to be related to the quantum Fisher information (QFI). For a single operator, the self-distance equals $F_Q/4$. The paper establishes bounds relating the Wigner-Yanase skew information, the QFI, and the self-distances.

• Relations to Other Divergences: The authors derive bounds connecting the new separable-state distances to the Bures distance, the superfidelity, and various quantum divergences (Jensen, Bregman, Tsallis), showing that these distances often serve as upper or lower bounds for these other quantities.

Significance and Claims
The paper claims to provide a unified framework for understanding various definitions of the quantum Wasserstein distance by showing that they often reduce to the same quantities when restricted to separable states.

• It establishes a direct link between the quantum Wasserstein distance and the quantum Fisher information, connecting optimal transport theory with quantum metrology.
• It demonstrates that the Uhlmann-Jozsa fidelity can be formulated as an optimization over separable states, offering a new perspective on this fundamental measure.
• The result that for qubits, the separable-state SWAP-fidelity equals the Uhlmann-Jozsa fidelity simplifies the calculation of these distances for two-level systems.
• The proof of the triangle inequality for cases involving pure states supports the metric-like properties of these quantities, although the paper notes that the general triangle inequality for arbitrary mixed states remains an open question (supported by numerical evidence but not rigorously proven for all cases).

The authors conclude that these findings connect seemingly unrelated fields of quantum physics, such as optimal transport, entanglement theory, and quantum metrology, by revealing that many distance measures are fundamentally linked through the geometry of separable states.