### Abstract

The free analogue of the transportation cost inequality so far obtained for measures is extended to the noncommutative setting. Our free transportation cost inequality is for traded distributions of noncommutative self-adjoint (also unitary) multi-variables in the framework of tracial C*-probability spaces, and it tells that the Wasserstein distance is dominated by the square root of the relative free entropy with respect to a potential of additive type (corresponding to the free case) with some convexity condition. The proof is based on random matrix approximation procedure.

Original language | English |
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Pages (from-to) | 391-412 |

Number of pages | 22 |

Journal | Infinite Dimensional Analysis, Quantum Probability and Related Topics |

Volume | 9 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Mathematical Physics
- Applied Mathematics

### Cite this

*Infinite Dimensional Analysis, Quantum Probability and Related Topics*,

*9*(3), 391-412. https://doi.org/10.1142/S0219025706002457

**Free transportation cost inequalities for noncommutative multi-variables.** / Hiai, Fumio; Ueda, Yoshimichi.

Research output: Contribution to journal › Article

*Infinite Dimensional Analysis, Quantum Probability and Related Topics*, vol. 9, no. 3, pp. 391-412. https://doi.org/10.1142/S0219025706002457

}

TY - JOUR

T1 - Free transportation cost inequalities for noncommutative multi-variables

AU - Hiai, Fumio

AU - Ueda, Yoshimichi

PY - 2006/9/1

Y1 - 2006/9/1

N2 - The free analogue of the transportation cost inequality so far obtained for measures is extended to the noncommutative setting. Our free transportation cost inequality is for traded distributions of noncommutative self-adjoint (also unitary) multi-variables in the framework of tracial C*-probability spaces, and it tells that the Wasserstein distance is dominated by the square root of the relative free entropy with respect to a potential of additive type (corresponding to the free case) with some convexity condition. The proof is based on random matrix approximation procedure.

AB - The free analogue of the transportation cost inequality so far obtained for measures is extended to the noncommutative setting. Our free transportation cost inequality is for traded distributions of noncommutative self-adjoint (also unitary) multi-variables in the framework of tracial C*-probability spaces, and it tells that the Wasserstein distance is dominated by the square root of the relative free entropy with respect to a potential of additive type (corresponding to the free case) with some convexity condition. The proof is based on random matrix approximation procedure.

UR - http://www.scopus.com/inward/record.url?scp=33748521849&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748521849&partnerID=8YFLogxK

U2 - 10.1142/S0219025706002457

DO - 10.1142/S0219025706002457

M3 - Article

AN - SCOPUS:33748521849

VL - 9

SP - 391

EP - 412

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

SN - 0219-0257

IS - 3

ER -