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arXiv:2006.05739 · Journal of Mathematical Physics

Quantum monotone metrics induced from trace non-increasing maps and additive noise

Koichi Yamagata

Published May 1, 2020• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.5129058

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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.
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.
In a unital C*-algebra with a faithful trace functional τττ, the set Dτ(A)D_τ(A)Dτ​(A) of positive ρ∈Aρ\in Aρ∈A of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space Dτ(A)D_τ(A)Dτ​(A) in terms of the Bures metric. Linear maps on A that map Dτ(A)D_τ(A)Dτ​(A) back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space Dτ(A)D_τ(A)Dτ​(A), (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space Dτ(A)D_τ(A)Dτ​(A), and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra A∗∗A^{**}A∗∗.
Josza’s definition of fidelity [R. Jozsa, J. Mod. Opt. 41(12), 2315–2323 (1994)] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ. In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positive elements ρ ∈ A for which τ(ρ) = 1. The second setting is more operator theoretic: by fixing a faithful normal semifinite trace τ on a semifinite von Neumann algebra M, we define and consider the fidelity of pairs of positive operators in M of unit trace. The main results of this paper address monotonicity and preservation of fidelity under the action of certain trace-preserving positive linear maps of A or of the predual M∗. Our results in the von Neumann algebra setting are novel in that we focus on the Schrödinger picture rather than the Heisenberg picture, and they also yield a new proof of a theorem of Molnár [Rep. Math. Phys. 48(3), 299–303 (2001)] on the structure of fidelity-preserving quantum channels on the trace-class operators.
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.
In this paper we investigate quantum Hellinger type divergences which were studied by Bhatia-Gaubert-Jain (2019), Pitrik-Virosztek (2020), and Dinh-Lie-Osaka-Phan (2025). In particular, when [Formula: see text] is a convex function defined by the form [Formula: see text] ([Formula: see text]) and [Formula: see text] is an operator monotone function with [Formula: see text], we introduce the quantum quantitative measure [Formula: see text] for positive definite matrices [Formula: see text] and [Formula: see text], and show that [Formula: see text] is a quantum divergence in the sense of Bhatia-Gaubert-Jain and also show that it is jointly convex and satisfies the data processing property by a trace preserving positive unital map [Formula: see text], that is, [Formula: see text].
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.