Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'

• URL: http://arxiv.org/abs/2607.07764v1
• Date: Wed, 08 Jul 2026 15:32:43 GMT
• Title: Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'
• Abstract summary: This quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it.<n>We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the $p$-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.

Related papers

• Boltzmann counting in Hilbert space [39.146761527401424]
  We introduce a geometric entropy for quantum preparations, defined as the logarithm of the Hilbert-space volume of pure states compatible with a given set of constraints.<n>We analyze three classes of constraints: restriction to a subspace, fixed expectation values, and coarse-grained subsystem descriptions.<n>The resulting framework provides a geometric measure of quantum ignorance at the level of the preparation.
  arXiv  Detail & Related papers  (2026-08-20T15:01:51Z)
• Hysteretic squashed entanglement in many-body quantum systems [42.085941481155295]
  Entanglement in many-body quantum systems is distributed across spatial regions.<n>We propose hysteretic entanglement $T_sq$, a conditional entanglement that measures the genuine quantum correlations between two subregions.<n>We show that $T_sq$ can detect genuine quantum correlations across both adjacent and long-range subsystems.
  arXiv  Detail & Related papers  (2026-03-10T17:00:49Z)
• Exponential Advantage from One More Replica in Estimating Nonlinear Properties of Quantum States [16.185988658474635]
  We prove that estimation of $mathrmtr(rhok O)$ for a broad class of observables $O$ is exponentially hard for any protocol restricted to $(k-1)$-replica joint measurements.<n>Results establish, for the first time, an exponential separation between $(k-1)$- and $k$-replica protocols for any integer $k>2$.
  arXiv  Detail & Related papers  (2025-09-28T17:46:43Z)
• Distances between pure quantum states induced by a distance matrix [0.0]
  We construct an infinite family of distances $d_p$ on the complex projective space $mathbbP(mathbbCn)$.<n>The paper significantly extends earlier work, resolving an important question about the geometry of quantum state space imposed by the quantum Wasserstein distances.
  arXiv  Detail & Related papers  (2025-09-18T08:23:33Z)
• Geometric monotones of violations of quantum realism [89.99666725996975]
  Quantum realism states that projective measurements in quantum systems establish the reality of physical properties, even in the absence of a revealed outcome.<n>This framework provides a nuanced perspective on the distinction between classical and quantum notions of realism, emphasizing the contextuality and complementarity inherent to quantum systems.<n>We derive geometric monotones of quantum realism using trace distance, Hilbert-Schmidt distance, Schatten $p$-distances, Bures, and Hellinger distances.
  arXiv  Detail & Related papers  (2024-12-16T10:22:28Z)
• Isometries of the qubit state space with respect to quantum Wasserstein distances [0.0]
  We study isometries of quantum Wasserstein distances and divergences on the quantum bit state space.<n>We describe isometries with respect to the quantum symmetric Wasserstein divergence $d_sym$, the divergence induced by all of the Pauli matrices.
  arXiv  Detail & Related papers  (2024-08-19T10:41:32Z)
• Quantum Lower Bounds by Sample-to-Query Lifting [33.82353457014144]
  We propose a new method for proving quantum query lower bounds by a quantum sample-to-query lifting theorem.<n>We provide unified proofs for some known lower bounds, including those for phase/amplitude estimation and Hamiltonian simulation.
  arXiv  Detail & Related papers  (2023-08-03T14:41:49Z)
• A vertical gate-defined double quantum dot in a strained germanium double quantum well [48.7576911714538]
  Gate-defined quantum dots in silicon-germanium heterostructures have become a compelling platform for quantum computation and simulation. We demonstrate the operation of a gate-defined vertical double quantum dot in a strained germanium double quantum well. We discuss challenges and opportunities and outline potential applications in quantum computing and quantum simulation.
  arXiv  Detail & Related papers  (2023-05-23T13:42:36Z)
• On the monotonicity of a quantum optimal transport cost [91.3755431537592]
  We show that the generalization of the $2$-Wasserstein distance proposed by Chakrabarti et al. is not monotone under partial traces. We propose a stabilized version of the original definition, which we show to be monotone under the application of general quantum channels.
  arXiv  Detail & Related papers  (2022-11-21T18:33:50Z)
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices [13.452510519858995]
  We show a generalization of the Wasserstein distance of order 1 to quantum spin systems on the lattice $mathbbZd$. We also prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality.
  arXiv  Detail & Related papers  (2022-10-20T17:46:18Z)
• Monotonicity of the quantum 2-Wasserstein distance [0.0]
  We show that for $N=2$ dimensional Hilbert space the quantum 2-Wasserstein distance is monotonous with respect to any single-qubit quantum operation. We conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps in any dimension $N$.
  arXiv  Detail & Related papers  (2022-04-15T09:57:39Z)
• Quantum Monge-Kantorovich problem and transport distance between density matrices [0.0]
  We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between $rhoA$ and $rhoB$. We introduce an associated measure of proximity of quantum states, called SWAP-fidelity, and discuss its properties and applications in quantum machine learning.
  arXiv  Detail & Related papers  (2021-02-15T19:00:08Z)
• The Quantum Wasserstein Distance of Order 1 [16.029406401970167]
  We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit. We also propose a generalization of the Lipschitz constant to quantum observables.
  arXiv  Detail & Related papers  (2020-09-09T18:00:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.