Periodic Reporting for period 1 - GWFP (Geometric study of Wasserstein spaces and free probability)
Okres sprawozdawczy: 2019-10-01 do 2021-09-30
We studied geometric aspects of optimal transportation (OT) and quantum information (QI).
Albeit the first version of the OT problem was formulated in the 18th century by Monge, the theory of OT became one of the central topics of analysis only ca. 20 years ago with intimate links to mathematical physics, theory of PDE's, and probability. The importance of OT theory was highlighted by two Fields-medals (Cedric Villani and Alessio Figalli) in the past decade.
Our main objective concerning OT was the study of the metric structure of classical Wasserstein spaces with a strong emphasis on the classification of distance-preserving maps.
Information theoretical dissimilarity measures (in short: divergences) play a central role in QI theory, and the geometric aspects of QI have been studied with a particular intensity in the last decades. Divergences play an essential role in quantum state tomography and quantum process tomography, quantum error correction, and quantum hypothesis testing. We studied the geometry of quantum states equipped with quantum Jensen-Shannon divergence (QJSD), which is the symmetric and bounded version of the von Neumann relative entropy.
Our main objectives concerning QI were to prove the long-standing conjecture on the metric property of the QJSD, to give a center-of-mass interpretation of Kubo-Ando operator means, and to study quantum Hellinger distances.
Impact for society:
OT is a very natural problem with deep connections to other fields of science: economics, physics, etc.
Our results concerning QI have the potential to be applied in quantum computing.
Albeit the first version of the OT problem was formulated in the 18th century by Monge, the theory of OT became one of the central topics of analysis only ca. 20 years ago with intimate links to mathematical physics, theory of PDE's, and probability. The importance of OT theory was highlighted by two Fields-medals (Cedric Villani and Alessio Figalli) in the past decade.
Our main objective concerning OT was the study of the metric structure of classical Wasserstein spaces with a strong emphasis on the classification of distance-preserving maps.
Information theoretical dissimilarity measures (in short: divergences) play a central role in QI theory, and the geometric aspects of QI have been studied with a particular intensity in the last decades. Divergences play an essential role in quantum state tomography and quantum process tomography, quantum error correction, and quantum hypothesis testing. We studied the geometry of quantum states equipped with quantum Jensen-Shannon divergence (QJSD), which is the symmetric and bounded version of the von Neumann relative entropy.
Our main objectives concerning QI were to prove the long-standing conjecture on the metric property of the QJSD, to give a center-of-mass interpretation of Kubo-Ando operator means, and to study quantum Hellinger distances.
Impact for society:
OT is a very natural problem with deep connections to other fields of science: economics, physics, etc.
Our results concerning QI have the potential to be applied in quantum computing.
I. Geometry of Wasserstein spaces.
We described the group of Wasserstein isometries over the unit interval [0,1] and the real line in [G.P. Gehér, T. Titkos, D. Virosztek, Trans. Amer. Math. Soc. 373 (2020), 5855–5883]. As for the real line, we proved isometric rigidity for all positive parameters p != 2. This is in striking contrast with Kloeckner's result on the quadratic (p=2) Wasserstein space which admits non-trivial and exotic isometries. We found a substantial difference between the real line and the interval as well. Namely, the p-Wasserstein space over the interval is rigid for p>1, but for p=1, it has exotic, moreover, mass-splitting isometries. Using this latter phenomenon we gave affirmative answers to the questions posed by Kloeckner in 2010 regarding the existence of exotic and mass-splitting isometries in quadratic Wasserstein spaces.
Our most recent work in this direction is a study of the isometry structure of Wasserstein-Hilbert spaces [G.P. Gehér, T. Titkos, D. Virosztek, The isometry group of Wasserstein spaces: the Hilbertian case. Preprint, submitted]. The key steps here are a metric characterization of Dirac masses and the use of the Wasserstein potential of measures. We introduced this latter notion to study isometries, however, this concept may be useful in a more general context as well. The main results are the existence of non-trivial isometries in the quadratic case, and isometric rigidity for every p != 2.
II. Quantum information theory.
The main result of [J. Pitrik, D. Virosztek, Lett. Math. Phys. 110 (2020), 2039–2052] is a characterization of the barycenter of finitely many positive matrices with respect to quantum Hellinger divergences by a fixed-point equation. As a byproduct, we pointed out that the fixed-point equation given previously by Bhatia et al. is incorrect.
Continuing this line of study we showed that (general) symmetric Kubo-Ando means admit a barycenter (divergence center) interpretation [J. Pitrik, D. Virosztek, Linear Algebra Appl. 609 (2021), 203–217]. This result immediately led to a definition of weighted and multivariate extensions of these operator means.
The quantum Jensen-Shannon divergence (QJSD) is the channel capacity (Holevo capacity) of a binary classical-quantum channel with respect to the (Umegaki) quantum relative entropy on the one hand, and the Jensen gap for the von Neumann entropy on the other hand. It was conjectured more than a decade ago by Lamberti et al. that the square root of the QJSD is a genuine metric. We proved this conjecture in [ D. Virosztek, Adv. Math. 380
(2021), 107595.].
We described the group of Wasserstein isometries over the unit interval [0,1] and the real line in [G.P. Gehér, T. Titkos, D. Virosztek, Trans. Amer. Math. Soc. 373 (2020), 5855–5883]. As for the real line, we proved isometric rigidity for all positive parameters p != 2. This is in striking contrast with Kloeckner's result on the quadratic (p=2) Wasserstein space which admits non-trivial and exotic isometries. We found a substantial difference between the real line and the interval as well. Namely, the p-Wasserstein space over the interval is rigid for p>1, but for p=1, it has exotic, moreover, mass-splitting isometries. Using this latter phenomenon we gave affirmative answers to the questions posed by Kloeckner in 2010 regarding the existence of exotic and mass-splitting isometries in quadratic Wasserstein spaces.
Our most recent work in this direction is a study of the isometry structure of Wasserstein-Hilbert spaces [G.P. Gehér, T. Titkos, D. Virosztek, The isometry group of Wasserstein spaces: the Hilbertian case. Preprint, submitted]. The key steps here are a metric characterization of Dirac masses and the use of the Wasserstein potential of measures. We introduced this latter notion to study isometries, however, this concept may be useful in a more general context as well. The main results are the existence of non-trivial isometries in the quadratic case, and isometric rigidity for every p != 2.
II. Quantum information theory.
The main result of [J. Pitrik, D. Virosztek, Lett. Math. Phys. 110 (2020), 2039–2052] is a characterization of the barycenter of finitely many positive matrices with respect to quantum Hellinger divergences by a fixed-point equation. As a byproduct, we pointed out that the fixed-point equation given previously by Bhatia et al. is incorrect.
Continuing this line of study we showed that (general) symmetric Kubo-Ando means admit a barycenter (divergence center) interpretation [J. Pitrik, D. Virosztek, Linear Algebra Appl. 609 (2021), 203–217]. This result immediately led to a definition of weighted and multivariate extensions of these operator means.
The quantum Jensen-Shannon divergence (QJSD) is the channel capacity (Holevo capacity) of a binary classical-quantum channel with respect to the (Umegaki) quantum relative entropy on the one hand, and the Jensen gap for the von Neumann entropy on the other hand. It was conjectured more than a decade ago by Lamberti et al. that the square root of the QJSD is a genuine metric. We proved this conjecture in [ D. Virosztek, Adv. Math. 380
(2021), 107595.].
It is safe to say that the results described in the previous sections (isometry groups of classical Wasserstein spaces discovered, a long-standing conjecture about the metric property of the quantum Jensen-Shannon divergence proved, connections between Kubo-Ando operator means and barycenters established) are beyond the state of the art.