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Distances between pure quantum states induced by a distance matrix

Tomasz Miller, Rafał Bistroń

TL;DR

Problem: construct meaningful distances on the space of pure quantum states from an arbitrary distance matrix and study their metric properties. Approach: define $d_p$ via the bivector $x\wedge y$ and a cost operator $E$ on $\Lambda^2\mathbb{C}^n$, then prove the triangle inequality for all $p\ge 2$ using convexity and exterior-algebra techniques. Key contributions: (i) $d_p$ is a true metric on $\mathbb{P}(\mathbb{C}^n)$ for any $n\ge 2$ and $p\ge 2$; (ii) any finite metric space embeds isometrically into pure states under $d_p$; (iii) the $n=3$ case reduces to a spectral condition $2\rho(E)\le \operatorname{tr}(E)$; (iv) an auxiliary convexity theorem drives the general proof. Significance: this solidifies the foundation for quantum Wasserstein-type distances and supports applications in quantum information science.

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 complex projective space $\mathbb{P}(\mathbb{C}^n)$ modelling the space of pure states of an $n$-level 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 nontrivial auxiliary 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.

Distances between pure quantum states induced by a distance matrix

TL;DR

Problem: construct meaningful distances on the space of pure quantum states from an arbitrary distance matrix and study their metric properties. Approach: define via the bivector and a cost operator on , then prove the triangle inequality for all using convexity and exterior-algebra techniques. Key contributions: (i) is a true metric on for any and ; (ii) any finite metric space embeds isometrically into pure states under ; (iii) the case reduces to a spectral condition ; (iv) an auxiliary convexity theorem drives the general proof. Significance: this solidifies the foundation for quantum Wasserstein-type distances and supports applications in quantum information science.

Abstract

With the help of a given distance matrix of size , we construct an infinite family of distances (where ) on the complex projective space modelling the space of pure states of an -level 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 are indeed distance functions -- in particular, that they satisfy the triangle inequality -- we employ methods of analysis, multilinear algebra and convex geometry, obtaining a nontrivial auxiliary 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 in quantum information science.

Paper Structure

This paper contains 3 sections, 6 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. The $n=3$ case and its consequences
  3. An auxiliary convexity result and the proof of the triangle inequality

Key Result

Theorem 1

Let $n \geq 2$. For any $n \times n$ distance matrix $(E_{ij})$ and any $p \geq 2$, the map $d_p: \mathbb{P}(\mathbb{C}^n) \times \mathbb{P}(\mathbb{C}^n) \rightarrow {\mathbb R}$ given by dp is a distance function. In particular, it satisfies for any $\textup{x}, \textup{y}, \textup{z} \in \mathbb{P}(\mathbb{C}^n)$.

Figures (1)

  • Figure 1: An illustration of Remark \ref{['remint']} (for $n=6$). Given any $n$-element metric space $(A,\rho)$, we can isometrically embed it into $(\mathbb{P}(\mathbb{C}^n),d_p)$ with the distance function $d_p$ defined via \ref{['dp']} for any chosen $p \geq 2$. In doing so, the elements $a_i$ of $A$ are promoted to the orthonormal basis states $\mathbf{e}_i$ (defined up to a phase factor). Loosely speaking, the above construction provides a general way to 'quantize' any finite metric space.

Theorems & Definitions (15)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 5 more