Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem
arXiv:1803.02673 · doi:10.1063/5.0068344
Abstract
The earth mover's distance is a measure of the distance between two probabilistic measures. It plays a fundamental role in mathematics and computer science. The Kantorovich-Rubinstein theorem provides a formula for the earth mover's distance on the space of regular probability Borel measures on a compact metric space. In this paper, we investigate the quantum earth mover's distance. We show a no-go Kantorovich-Rubinstein theorem in the quantum setting. More precisely, we show that the trace distance between two quantum states can not be determined by their earth mover's distance. The technique here is to track the bipartite quantum marginal problem. Then we provide inequality to describe the structure of quantum coupling, which can be regarded as quantum generalization of Kantorovich-Rubinstein theorem. After that, we generalize it to obtain into the tripartite version, and build a new class of necessary criteria for the tripartite marginal problem.
arXiv admin note: text overlap with arXiv:quant-ph/0506138 by other authors
References in corpus (12)
- Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
- Area laws in quantum systems: mutual information and correlations
- N-representability is QMA-complete
- The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
- Quantum marginal problem and representations of the symmetric group
- Quantum state transformations and the Schubert calculus
- Proving Expected Sensitivity of Probabilistic Programs
- Quantum Monge-Kantorovich problem and transport distance between density matrices
- Reconstructing quantum states from single-party information
- Consistency of Local Density Matrices is QMA-complete
- Quantum Relational Hoare Logic with Expectations
- Entropic bounds for the quantum marginal problem