arXiv math-ph/03050602003-05-29math-phmath.DGmath.MP

Monotone Riemannian metrics on density matrices with non-monotone scalar curvature

Attila Andai

备注：: 15 pages, to be published in Journal of Mathematical Physics
发表信息：: Journal of Mathematical Physics, 44, 3675-3688, 2003.
DOI：: 10.1063/1.1592874

英文摘要：

The theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant - monotone - metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on the positive real line satisfying g(1) = 0. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in s…

1998-08-30math-phmath.MPmath.OA

Geodesic distances on density matrices

We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.

2003-12-17math-phmath.DGmath.MP

On the Curvature of Monotone Metrics and a Conjecture Concerning the Kubo-Mori Metric

It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its expected quantum statistical meaning. The scalar curvature is explained in more detail for three examples, the Bures metric, the largest monotone metric and the Kubo-Mori metric. In particular, we show an important conjecture of Petz concerning the Kubo-Mori metric up to a formal proof of the concavity of a certain function on R_+^3. This concavity seems to …

1999-06-02math-phmath.DGmath.MP

Characterization of symmetric monotone metrics on the state space of quantum systems

The quantum Fisher information is a Riemannian metric, defined on the state space of a quantum system, which is symmetric and decreasing under stochastic mappings. Contrary to the classical case such a metric is not unique. We complete the characterization, initiated by Morozova, Chentsov and Petz, of these metrics by providing a closed and tractable formula for the set of Morozova-Chentsov functions. In addition, we provide a continuously increasing bridge between the smallest and largest symmetric monotone metrics.

2006-03-29math-phmath.MP

Geometry of quantum state space and quantum correlations

Quantum state space is endowed with a metric structure and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical considerations on the quantum state space. In this article, considering the quantum state space being spanned by 2x2 density matrices, we determine a particular Riemannian metric for a state \r{ho} and show that if \r{ho} gets entangled with another quantum state, the negativity of the generated entangled state is, upto a constant factor, equals to square root …

2015-05-30math-phmath.MPquant-ph

Riemannian metrics on positive definite matrices related to means

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $ϕ$ in the form $K_D^ϕ(H,K)=\sum_{i,j}ϕ(λ_i,λ_j)^{-1} Tr P_iHP_jK$ when $\sum_iλ_iP_i$ is the spectral decomposition of the foot point $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $D\mapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distribu…

2008-11-08math-phmath.FAmath.MP

更多相关论文收起更多论文

Volume of the quantum mechanical state space

The volume of the quantum mechanical state space over $n$-dimensional real, complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean measure is computed, and explicit formulas are presented for the expected value of the determinant in the general setting too. The case when the state space is endowed with a monotone metric or a pull-back metric is considered too, we give formulas to compute the volume of the state space with respect to the given Riemannian metric. We present the volume of the space of qubits with respect to various monotone metrics. It turns out that the …

2006-04-27math-phmath.DGmath.MP

Gaussian distributions on Riemannian symmetric spaces, random matrices, and planar Feynman diagrams

Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as partition functions, are complicated, intractable integrals in general that depend in a highly non-linear way on the mean of the given distribution. Nonetheless, on Riemannian symmetric spaces, the partition functions are independent of the mean and reduce to integrals over finite dimensional vector spaces. These are generally still hard to compute numerica…

2021-06-02math-phmath.MPmath.PR

Communication tasks in operational theories

We investigate the question which communication tasks can be accomplished within a given operational theory. The concrete task is to find out which communication matrices have a prepare-and-measure implementation with states and measurement from a given theory. To set a general framework for this question we develop the ultraweak matrix majorization in the set of communication matrices. This preorder gives us means to determine when one communication task is more difficult than another. Furthermore, we introduce several monotones which can be used to compare and characterize the communication …

2020-03-11math-phmath.MPquant-ph

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in [F. D. Cunden, F. Mezzadri, N. O'Connell and N. J. Simm, arXiv:1805.08760] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the gener…

2018-10-18math-phmath.COmath.MP

分级色散谐振器阵列中的频率排序：基于体积-相互作用矩阵

Frequency sorting in graded dispersive resonator arrays via volume-interaction matrices

École Polytechnique（巴黎综合理工学院）

本研究通过体积-势矩阵约化，证明单调材料分级可使色散谐振器阵列实现解析可控的频率排序，并给出逆设计方法及数值验证。

2026-09-25math-phmath.APmath.MP

Contraction coefficients for noisy quantum channels

Generalized relative entropy, monotone Riemannian metrics, geodesic distance, and trace distance are all known to decrease under the action of quantum channels. We give some new bounds on, and relationships between, the maximal contraction for these quantities.

2015-08-14math-phmath.MPmath.OA

偶数次单位根万花筒 Yang-Baxter 代数中的精确链词阈值与单调路径结构

Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra

Peking University（北京大学）；Lanzhou University（兰州大学）

本文证明了偶数阶万花筒 Yang-Baxter 代数中链词消失的精确阈值，刻画了非零词的秩，并揭示了其单调路径代数结构及根的性质。

2026-10-19math-phmath.MPnlin.SI