arXiv subjects

József Pitrik

Publications and source records attributed to József Pitrik.

13 recordsLinked to original sources

arXiv2026

Relations between different definitions of the quantum Wasserstein distance

The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the square of the self-distance equals the Wigner-Yanase skew information in this case. Moreover, for finite-dimensional systems of dimension $d\ge2$ and any number of operators, we show that the Golse-Mouhot-Paul-Caglioti self-distance is bounded from below by the De Palma-Trevisan self-distance.

quant-ph↗

arXiv2026

General method for obtaining the energy minimum of spin Hamiltonians for separable states

We present a general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. For ferromagnetic Ising and Ising-like models with nearest-neighbor interactions on lattices of any dimension and on a fully connected graph in an external field, this minimum is given by a compact analytic formula involving the quantum Fisher information. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the direct extraction of both the quantum Fisher information and the fidelity from correlation measurements on the ground states of suitably engineered spin models.

quant-ph↗

arXiv2026

Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

math-ph↗

arXiv2026

Wasserstein distances and divergences of order $p$ by quantum channels

We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.

math-ph↗

arXiv2026

Quantum Wasserstein distance and its relation to several types of fidelities

We consider several definitions of the quantum Wasserstein distance based on an optimization over general bipartite quantum states with given marginals. Then, we examine the quantities obtained after the optimization is carried out over bipartite separable states instead. We prove that several of these quantities are equal to each other. Thus, we connect several approaches in the literature. We prove the triangle inequality for some of these quantities for the case of one of the three states being pure. As a byproduct, we show that the square root of the Uhlmann-Jozsa quantum fidelity can also be written as an optimization over separable states with given marginals. We use this to prove that some of these quantities equal the Uhlmann-Jozsa quantum fidelity for qubits. We also find relations with the superfidelity.

quant-ph↗

arXiv2025

The swap transpose on couplings translates to Petz' recovery map on quantum channels

In [Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234], De Palma and Trevisan described a one-to-one correspondence between quantum couplings and quantum channels realizing transport between states. The aim of this short note is to demonstrate that taking the Petz recovery map for a given channel and initial state is precisely the counterpart of the swap transpose operation on couplings. That is, the swap transpose of the coupling $Π_Φ$ corresponding to the channel $Φ$ and initial state $ρ$ is the coupling $Π_{rec}$ corresponding to the Petz recovery map $Φ_{rec}.$

math-ph↗

arXiv2025

On the metric property of quantum Wasserstein divergences

Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels, and they are conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure, and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general, for an arbitrary choice of states.

math-ph↗

arXiv2023

Quantum Wasserstein distance based on an optimization over separable states

We define the quantum Wasserstein distance such that the optimization of the coupling is carried out over bipartite separable states rather than bipartite quantum states in general, and examine its properties. Surprisingly, we find that the self-distance is related to the quantum Fisher information. We present a transport map corresponding to an optimal bipartite separable state. We discuss how the quantum Wasserstein distance introduced is connected to criteria detecting quantum entanglement. We define variance-like quantities that can be obtained from the quantum Wasserstein distance by replacing the minimization over quantum states by a maximization. We extend our results to a family of generalized quantum Fisher information quantities.

quant-ph↗

arXiv2022

Quantum Wasserstein isometries on the qubit state space

We describe Wasserstein isometries of the quantum bit state space with respect to distinguished cost operators. We derive a Wigner-type result for the cost operator involving all the Pauli matrices: in this case, the isometry group consists of unitary or anti-unitary conjugations. In the Bloch sphere model, this means that the isometry group coincides with the classical symmetry group $\mathbf{O}(3).$ On the other hand, for the cost generated by the qubit "clock" and "shift" operators, we discovered non-surjective and non-injective isometries as well, beyond the regular ones. This phenomenon mirrors certain surprising properties of the quantum Wasserstein distance.

math-ph↗

arXiv2020

A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means

It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted $\mathcal{A} \# \mathcal{H}$-mean introduced by Kim, Lawson, and Lim.

math.FA↗

arXiv2020

Quantum Hellinger distances revisited

This short note aims to study quantum Hellinger distances investigated recently by Bhatia et al. [Lett. Math. Phys. 109 (2019), 1777-1804] with a particular emphasis on barycenters. We introduce the family of generalized quantum Hellinger divergences, that are of the form $ϕ(A,B)=\mathrm{Tr} \left((1-c)A + c B - A σB \right),$ where $σ$ is an arbitrary Kubo-Ando mean, and $c \in (0,1)$ is the weight of $σ.$ We note that these divergences belong to the family of maximal quantum $f$-divergences, and hence are jointly convex and satisfy the data processing inequality (DPI). We derive a characterization of the barycenter of finitely many positive definite operators for these generalized quantum Hellinger divergences. We note that the characterization of the barycenter as the weighted multivariate $1/2$-power mean, that was claimed in the work of Bhatia et al. mentioned above, is true in the case of commuting operators, but it is not correct in the general case.

math-ph↗

arXiv2015

Maps on positive definite matrices preserving Bregman and Jensen divergences

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.

math.FA↗

arXiv2015

On the joint convexity of the Bregman divergence of matrices

We characterize the functions for which the corresponding Bregman divergence is jointly convex on matrices. As an application of this characterization, we derive a sharp inequality for the quantum Tsallis entropy of a tripartite state, which can be considered as a generalization of the strong subadditivity of the von Neumann entropy. (In general, the strong subadditivity of the Tsallis entropy fails for quantum states, but it holds for classical states.) Furthermore, we show that the joint convexity of the Bregman divergence does not imply the monotonicity under stochastic maps, but every monotone Bregman divergence is jointly convex.

math-ph↗