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.
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.
Abstract In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential calculus. The cases N = 2 , N = 3 are discussed in detail and notable limits are analyzed. The radial limit procedure has been used to recover quantum metrics for lower rank states, such as pure states. By using the tomographic picture of quantum mechanics we have obtained the Fisher–Rao metric for the space of quantum tomograms and derived a reconstruction formula of the quantum metric of density states out of the tomographic one. A new inequality obtained for probabilities of three spin-1/2 projections in three perpendicular directions is proposed to be checked in experiments with superconducting circuits.
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.
Quantum monotone metric was introduced by Petz [Linear Algebra Appl. 244, 81–96 (1996)], and it was proved that quantum monotone metrics on the set of quantum states with trace one were characterized by operator monotone functions. Later, these were extended to monotone metrics on the set of positive operators whose traces are not always the one based on completely positive, trace preserving maps. It was shown that these extended monotone metrics were characterized by operator monotone functions continuously parameterized by traces of positive operators and did not have some ideal properties such as monotonicity and convexity with respect to the positive operators. In this paper, we introduce another extension of quantum monotone metrics that have monotonicity under completely positive, trace non-increasing maps and additive noise. We prove that our extended monotone metrics can be characterized only by static operator monotone functions from few assumptions without assuming continuities of metrics. We show that our monotone metrics have some natural properties such as additivity of direct sum, convexity, and monotonicity with respect to positive operators.
The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.
The proposed in J. Math. Phys. v.57,071903 (2016) analytical expansion of monotone (contractive) Riemannian metrics (called also quantum Fisher information(s)) in terms of moments of the dynamical structure factor (DSF) relative to an original intensive observable is reconsidered and extended. The new approach through the DSF which characterizes fully the set of monotone Riemannian metrics on the space of Gibbs thermal states is utilized to obtain an extension of the spectral presentation obtained for the Bogoliubov-Kubo-Mori metric (the generalized isothermal susceptibility) on the entire class of monotone Riemannian metrics. The obtained spectral presentation is the main point of our consideration. The last allows to present the one to one correspondence between monotone Riemannian metrics and operator monotone functions (which is a statement of the Petz theorem in the quantum information theory) in terms of the linear response theory. We show that monotone Riemannian metrics can be determined from the analysis of the infinite chain of equations of motion of the retarded Green's functions. Inequalities between the different metrics have been obtained as well. It is a demonstration that the analysis of information-theoretic problems has benefited from concepts of statistical mechanics and might cross-fertilize or extend both directions, and vice versa. We illustrate the presented approach on the calculation of the entire class of monotone (contractive) Riemannian metrics on the examples of some simple but instructive systems employed in various physical problems.
Some non-linear generalizations of classical Radon tomography were recently introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the straight lines of the standard Radon map are replaced by quadratic curves (ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids, hyperboloids, spheres). We consider here the quantum version of this novel non-linear approach and obtain, by systematic use of the Weyl map, a tomographic encoding approach to quantum states. Non-linear quantum tomograms admit a simple formulation within the framework of the star-product quantization scheme and the reconstruction formulae of the density operators are explicitly given in a closed form, with an explicit construction of quantizers and dequantizers. The role of symmetry groups behind the generalized tomographic maps is analyzed in some detail. We also introduce new generalizations of the standard singular dequantizers of the symplectic tomographic schemes, where the Dirac delta-distributions of operator-valued arguments are replaced by smooth window functions, giving rise to the new concept of "thick" quantum tomography. Applications for quantum state measurements of photons and matter waves are discussed.