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Approach to optimal quantum transport via states over time

Matt Hoogsteder-Riera, John Calsamiglia, Andreas Winter

TL;DR

This work introduces a quantum analogue of optimal transport by formulating transport costs via states over time, defined as $Q = \rho \star J$ where the Jordan product couples an input state with a CPTP map’s Jamiołkowski representation. The cost $\mathcal{K}(\rho,\sigma)$ is computed through an SDP over Jamiołkowski matrices, and the authors develop a detailed unitary-invariant cost analysis, deriving explicit expressions for commuting and pure states and establishing limit behavior as the dimension grows. They reveal that the quantum transport cost can exhibit asymmetry and discontinuities, distinguishing it qualitatively from Monge’s classical theory, and they identify major open questions about the geometry of the state-over-time cone and its dual. Overall, the paper provides a concrete primal formulation and substantive structural results, enabling finite-dimensional SDP studies of quantum transport with potential physical cost interpretations.

Abstract

We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. These two data are naturally integrated into their Jordan product, which is called state over time (``stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.

Approach to optimal quantum transport via states over time

TL;DR

This work introduces a quantum analogue of optimal transport by formulating transport costs via states over time, defined as where the Jordan product couples an input state with a CPTP map’s Jamiołkowski representation. The cost is computed through an SDP over Jamiołkowski matrices, and the authors develop a detailed unitary-invariant cost analysis, deriving explicit expressions for commuting and pure states and establishing limit behavior as the dimension grows. They reveal that the quantum transport cost can exhibit asymmetry and discontinuities, distinguishing it qualitatively from Monge’s classical theory, and they identify major open questions about the geometry of the state-over-time cone and its dual. Overall, the paper provides a concrete primal formulation and substantive structural results, enabling finite-dimensional SDP studies of quantum transport with potential physical cost interpretations.

Abstract

We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where the mass distribution is generalised to a density matrix, and the transport plan to a completely positive and trace preserving map. These two data are naturally integrated into their Jordan product, which is called state over time (``stote''), and the transport cost is postulated to be a linear function of it. We explore the properties of this transport cost, as well as the optimal transport cost between two given states (simply the minimum cost over all suitable transport plans). After that, we analyse in considerable detail the case of unitary invariant cost, for which we can calculate many costs analytically. These findings suggest that our quantum transport cost is qualitatively different from Monge's classical transport.

Paper Structure

This paper contains 17 sections, 25 theorems, 134 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Jordan product motivation and properties
  4. States over two times
  5. States over multiple times
  6. Quantum transport cost and optimal transport
  7. Results on properties
  8. Cost of identity
  9. Positivity
  10. Symmetry
  11. Triangle inequality
  12. General properties
  13. Unitary invariant cost
  14. Commuting density matrices
  15. Limit $d\rightarrow \infty$
  16. ...and 2 more sections

Key Result

Theorem 2.3

Let $\omega$ be a Hermitian operator on $\mathcal{B}(\mathcal{H}_A\otimes\mathcal{H}_B)$ such that $\rho=\Tr_B\left[\omega\right]\geq0$. Then let $B=\{\ket{ik}\}$ be a product basis of $\mathcal{H}_A\otimes H_B$ such that $\{\ket{i}\}$ is a diagonal basis of $\rho$ with associated eigenvalues $\{p_i Then, $\omega=\rho\star J$. Moreover, if $\rho$ is faithful then $J$ is Hermitian and unique with t

Figures (4)

  • Figure 1: Stoat (also stote in old spelling), mustela erminea; not to be confused with the common weasel, mustela nivalis.
  • Figure 2: Proofs of the expressions in \ref{['lem:PTstrongDuality']} using tensor network notation.
  • Figure 3: Plot comparing the global optimal cost with the result of optimising only over unitaries for various values of $p_\rho$, where the states are $\rho=p_\rho\ketbra{0}+(1-p_\rho)\frac{1}{d}{\hbox{\small1}\mkern-5.5mu1}$ and $\sigma=p_\rho\ketbra{\varphi}+(1-p_\rho)\frac{1}{d}{\hbox{\small1}\mkern-5.5mu1}$.
  • Figure 4: Symmetry gap between for $\tilde{K}_0={\hbox{\small1}\mkern-5.5mu1}-\frac{1}{d}\mathcal{S}$, $\rho=\ketbra{0}$ and $\sigma=(1-p_\sigma)\rho+p_\sigma{\hbox{\small1}\mkern-5.5mu1}/d$.

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 56 more