Distances between pure quantum states induced by a distance matrix (2509.14727v1)

Abstract: With the help of a given distance matrix of size $n$, we construct an infinite family of distances $d_p$ (where $p \geq 2$) on the the complex projective space $\mathbb{P}(\mathbb{C}^n)$, modelling the space of pure states of an $n$-dimensional quantum system. The construction can be seen as providing a natural way to isometrically embed any given finite metric space into the space of pure quantum states 'spanned' upon it. In order to show that the maps $d_p$ are indeed distance functions -- in particular, that they satisfy the triangle inequality -- we employ methods of analysis, multilinear algebra and convex geometry, obtaining a non-trivial convexity result in the process. The paper significantly extends earlier work, resolving an important question about the geometry of quantum state space imposed by the quantum Wasserstein distances and solidifying the foundation for applications of distances $d_p$ in quantum information science.

• The paper proves that the new dₚ metric for p ≥ 2 satisfies the triangle inequality, firmly establishing it as a valid metric on pure quantum state space.
• It provides a canonical method for isometrically embedding any finite metric space into the quantum state space using weighted Plücker coordinates.
• The approach leverages convexity and multilinear algebra to bridge traditional Hilbert–Schmidt metrics with novel quantum Wasserstein-type distances.

Distances between Pure Quantum States Induced by a Distance Matrix

Introduction and Motivation

The geometry of the space of pure quantum states, formalized as the complex projective space $\mathbb{P}(\mathbb{C}^n)$, is central to quantum information theory, quantum metrology, and quantum machine learning. The standard Hilbert–Schmidt (HS) metric, while mathematically natural, is limited in its ability to capture physically meaningful distinctions between quantum states, particularly in contexts where the geometry of the underlying system or cost structure is nontrivial. Recent developments in quantum optimal transport have motivated the search for more general metrics, notably quantum analogues of Wasserstein distances, that can encode arbitrary cost structures via distance matrices.

This paper introduces and rigorously analyzes a broad family of metrics $d_p$ on $\mathbb{P}(\mathbb{C}^n)$, parameterized by $p \geq 2$ and induced by an arbitrary $n \times n$ distance matrix $(E_{ij})$. The construction generalizes both the HS metric and previously proposed quantum Wasserstein distances, and provides a canonical method for isometrically embedding any finite metric space into the quantum state space. The main technical contribution is a proof that $d_p$ is a bona fide metric for all $p \geq 2$ and any distance matrix, resolving a conjecture from prior work and establishing a robust foundation for applications in quantum information science.

Construction of the $d_p$ Metrics

Given a distance matrix $(E_{ij})$ of size $n$, the authors define, for any $p \geq 2$, the following family of metrics on pure states $|\psi\rangle, |\phi\rangle \in \mathbb{C}^n$ (modulo global phase):

$$d_p(\psi, \phi) = \left( \sum_{i<j} E_{ij}^p |x_i y_j - x_j y_i|^2 \right)^{1/p}$$

where $x_i$ and $y_j$ are the components of $|\psi\rangle$ and $|\phi\rangle$ in the canonical basis. This can be interpreted as the $L^p$ norm of the vector of weighted Plücker coordinates of the simple bivector $|\psi\rangle \wedge |\phi\rangle$, with weights determined by the distance matrix.

For $p=2$ and $E_{ij} = 1 - \delta_{ij}$, this reduces to the HS metric. The construction is motivated by quantum optimal transport, where $(E_{ij})$ plays the role of a cost matrix, and the metric quantifies the "cost" of transforming one pure state into another under this structure.

Main Results

The central theorem establishes that $d_p$ is a metric on $\mathbb{P}(\mathbb{C}^n)$ for all $p \geq 2$ and any distance matrix $(E_{ij})$:

• Triangle Inequality: For all pure states $|\psi\rangle, |\phi\rangle, |\chi\rangle$,

$$d_p(\psi, \phi) \leq d_p(\psi, \chi) + d_p(\chi, \phi)$$

holds for all $p \geq 2$.

• Necessity of $p \geq 2$: The triangle inequality fails for $p < 2$, regardless of the choice of $n$ or $(E_{ij})$.
• Isometric Embedding: Any $n$-point metric space $(A, \rho)$ can be isometrically embedded into $(\mathbb{P}(\mathbb{C}^n), d_p)$ by mapping $a_i \mapsto e_i$ (basis vector), with $d_p(e_i, e_j) = \rho(a_i, a_j)$.
• Extension to Quantum Wasserstein: The $d_p$ family generalizes the pure-state quantum Wasserstein distances of order $p$, providing a unified framework for such metrics.

Technical Approach

The proof strategy is multifaceted, combining analysis, multilinear algebra, and convex geometry:

• Reduction to Orthonormal Triples: The triangle inequality for arbitrary states is reduced to the case of orthonormal triples, leveraging the structure of the exterior algebra and the properties of the induced operator on $\Lambda^2 \mathbb{C}^n$.
• Convexity and Symmetry: A key technical result is a convexity theorem for functions of the squared moduli of minors of order 2 and 3, which underpins the triangle inequality for the $d_p$ metrics.
• Spectral Conditions: For $n=3$, the triangle inequality is shown to be equivalent to a spectral condition on the induced operator, specifically $2 \rho(M) \leq \operatorname{tr} M$, where $M$ is the positive-definite operator associated with the distance matrix.
• Generalization to $n > 3$: The argument is extended to higher dimensions by considering the restriction to 3-dimensional subspaces and applying the convexity result.

Implications and Applications

Theoretical Implications

• Metric Geometry of Quantum State Space: The result provides a systematic method for endowing $\mathbb{P}(\mathbb{C}^n)$ with a rich family of metrics tailored to arbitrary cost structures, significantly generalizing the standard HS geometry.
• Quantum Optimal Transport: The $d_p$ metrics are directly connected to quantum optimal transport problems, offering a rigorous foundation for the use of quantum Wasserstein distances in both pure and mixed state settings.
• Embedding of Metric Spaces: The isometric embedding result implies that any finite metric space can be "quantized" as a subset of pure quantum states, preserving all pairwise distances. This has potential implications for the paper of quantum analogues of classical metric geometry and for the design of quantum algorithms sensitive to underlying metric structures.

Practical Implications

• Quantum Machine Learning: The ability to encode arbitrary cost structures into the geometry of quantum state space is relevant for quantum kernel methods, quantum clustering, and other machine learning tasks where the notion of distance is central.
• Quantum Sensing and Metrology: The flexibility in defining distances may allow for the design of state spaces optimized for specific sensing tasks, where the cost of distinguishing between states is non-uniform.
• Quantum Information Processing: The $d_p$ metrics can be used to analyze the robustness of quantum protocols under noise models that are not isotropic, or to design error-correcting codes adapted to nontrivial geometries.

Numerical and Structural Observations

• The triangle inequality is strictly violated for $p < 2$, as demonstrated by explicit counterexamples, establishing the sharpness of the $p \geq 2$ threshold.
• The construction is agnostic to the choice of distance matrix, allowing for arbitrary (finite) metric spaces to be embedded, which is not possible with the standard HS metric.
• The proof techniques rely on convexity properties of symmetric functions and the structure of the exterior algebra, suggesting possible generalizations to other settings where similar algebraic structures are present.

Limitations and Open Questions

• The construction is restricted to pure states; extension to mixed states or to infinite-dimensional Hilbert spaces is nontrivial and remains open.
• The metrics $d_p$ are not Riemannian for general $(E_{ij})$, and their geodesic structure is not analyzed in this work.
• The operational meaning of $d_p$ for arbitrary $p > 2$ and arbitrary cost matrices in concrete quantum protocols warrants further investigation.

Future Directions

• Extension to Mixed States: Developing analogous metrics for mixed quantum states, possibly via convex roof constructions or by leveraging the geometry of density matrices.
• Quantum Algorithm Design: Exploiting the flexibility of $d_p$ metrics in quantum algorithms that require custom notions of distance, such as quantum nearest neighbor search or quantum clustering.
• Optimal Transport on Quantum Graphs: Applying the embedding results to paper quantum transport on graphs or networks, where the underlying metric is non-Euclidean.
• Resource Theories: Investigating the role of $d_p$ metrics in resource theories where the cost of state conversion is not uniform.

Conclusion

This work rigorously establishes a broad family of metrics on the space of pure quantum states, parameterized by an arbitrary distance matrix and $p \geq 2$, and proves that these metrics satisfy the triangle inequality. The results provide a robust mathematical foundation for the use of quantum Wasserstein-type distances in quantum information science, enable the isometric embedding of arbitrary finite metric spaces into quantum state space, and open new avenues for both theoretical and applied research in quantum geometry, optimal transport, and quantum machine learning.