arXiv:2203.10857 · International Journal of Geometric Methods in Modern Physics

Monotone metric tensors in quantum information geometry

Florio M. Ciaglia, Fabio Di Cosmo, Fabio Di Nocera, Patrizia Vitale

Published Oct 14, 2023• 6 citations

Abstract

In this paper, we review some geometrical aspects pertaining to the world of monotone quantum metrics in finite dimensions. Particular emphasis is given to an unfolded perspective for quantum states that is built out of the spectral theorem and is naturally suited to investigate the comparison with the classical case of probability distributions.

DOI: 10.1142/s0219887824400048

Related papers

Unfolding of Relative g-Entropies and Monotone Metrics †

Fabio Di Nocera

We discuss the geometric aspects of a recently described unfolding procedure and show the form of objects relevant in the field of quantum information geometry in the unfolding space. In particular, we show the form of the quantum monotone metric tensors characterized by Petz and retrace in this unfolded perspective a recently introduced procedure of extracting a covariant tensor from a relative g-entropy.

arXiv:2207.08118International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt)Dec 15, 20220 citations

Monotone metrics and measures

Ingemar Bengtsson, Karol Życzkowski

Probability theory is a measure theory – with a soul. Mark Kac Section 2.6 was devoted to classical ensembles , that is to say ensembles defined by probability measures on the set of classical probability distributions over N events. In this chapter quantum ensembles are defined by choosing probability measures on the set of density matrices of size N . A warning should be issued first: there is no single, naturally distinguished measure in M ( N ) , so we have to analyse several measures, each of them with different physical motivations, advantages and drawbacks. This is in contrast to the set of pure quantum states, where the Fubini–Study measure is the only natural choice for a measure that defines ‘random states’. A simple way to define a probability measure goes through a metric. Hence we will start this chapter with a review of the metrics defined on the set of mixed quantum states. Monotone metrics In Section 2.5 we explained how the Fisher metric holds its distinguished position due to the theorem of Čencov, which states that the Fisher metric is the unique monotone metric on the probability simplex Δ N –1 . Now that the latter has been replaced by the space of quantum states M ( N ) we must look again at the question of metrics. Since the uniqueness in the classical case came from the behaviour under stochastic maps, we turn our attention to stochastic quantum maps – the completely positive, trace preserving maps discussed in Chapter 10. A distance D in the space of quantum states M ( N ) is called monotone if it does not grow under the action of a stochastic map Φ, If a monotone distance is geodesic the corresponding metric on M ( N ) is called monotone.

Cambridge University Press eBooksMay 11, 20060 citations

On metric of classical channel spaces: non-asymptotic theory

Keiji Matsumoto

The aim of the manuscript is to characterize monotone `metric' in the space of Markov map. Here, `metric' means the square of the norm defined on the tangent space, and not necessarily induced from an inner product (this property hereafter will be called inner-product-assumption), different from usual metric used in differential geometry. As for metrics in So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov proved the monotone metric in probability distribution space is unique up to constant multiple, and identical to Fisher information metric. Petz characterized all the monotone metrics in the quantum state space using operator mean. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. (Notably, we do not use the inner-product-assumption.) To obtain this result, `resource conversion' technique, which is widely used in quantum information, is used. We consider distillation from and formation to a family of channels. Under these axioms, we identify the largest and the smallest `metrics'. Interestingly, they are not induced from any inner product, i.e., not a metric. Indeed, one can prove that any `metric' satisfying our axioms can not be a metric. This result has some impact on the axiomatic study of the monotone metric in the space of classical and quantum states, since both conventional theory relies on the inner-product-assumption. Also, we compute the lower and the upper bound for some concrete examples.

arXiv:1001.3543Jan 20, 20102 citations

Tomographic reconstruction of quantum metrics

Marco Laudato, Giuseppe Marmo, Fabio M. Mele +2 more

Abstract In the framework of quantum information geometry we investigate the relationship between monotone metric tensors uniquely defined on the space of quantum tomograms, once the tomographic scheme is chosen, and monotone quantum metrics on the space of quantum states, classified by operator monotone functions, according to the Petz classification theorem. We show that different metrics can be related through a change in the tomographic map and prove that there exists a bijective relation between monotone quantum metrics associated with different operator monotone functions. Such a bijective relation is uniquely defined in terms of solutions of a first order second degree differential equation for the parameters of the involved tomographic maps. We first exhibit an example of a non-linear tomographic map that connects a monotone metric with a new one, which is not monotone. Then we provide a second example where two monotone metrics are uniquely related through their tomographic parameters.

arXiv:1704.01334Journal of Physics A Mathematical and TheoreticalNov 30, 20179 citations

Geometrical Statistics Classical and Quantum

Bengtsson, I

This is a review of the ideas behind the Fisher--Rao metric on classical probability distributions, and how they generalize to metrics on density matrices. As is well known, the unique Fisher--Rao metric then becomes a large family of monotone metrics. Finally I focus on the Bures--Uhlmann metric, and discuss a recent result that connects the geometric operator mean to a geodesic billiard on the set of density matrices.

Sep 2, 20053 citations

Geometrical Statistics—Classical and Quantum

Ingemar Bengtsson

This is a review of the ideas behind the Fisher‐Rao metric on classical probability distributions, and how they generalize to metrics on density matrices. As is well known, the unique Fisher‐Rao metric then becomes a large family of monotone metrics. Finally I focus on the Bures‐Uhlmann metric, and discuss a recent result that connects the geometric operator mean to a geodesic billiard on the set of density matrices.

arXiv:quant-ph/0509017AIP conference proceedingsJan 1, 20060 citations