Monotone metrics on matrix spaces

Petz, Dénes

doi:10.1016/0024-3795(94)00211-8

Cited by 480 publications

(584 citation statements)

References 7 publications

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“…In general, the length of a vector (matrix) B ij may be position dependent; at a diagonal point A = diag(a 1 , a 2 , ..., a N ) the squared length of B is given by [1] that operator monotonicity implies that f maps R + → R + and is concave. The result of Morozova and Chentsov [32], completed and extended by Petz and Sudár [1,33], states that every monotone metric may be written in the form (2.16) up a proportionality constant. Conversely, this equation determines a monotone Riemannian metric, if the function f (t) = 1/c(t, 1) is: a) operator monotone, and b) self-transposed, f (1/t) = f (t)/t.…”

Section: Bures Metricmentioning

confidence: 99%

Bures volume of the set of mixed quantum states

Sommers

Życzkowski

2003

J. Phys. A: Math. Gen.

164

296

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We compute the volume of the N 2 − 1 dimensional set MN of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N 2 − 1 dimensional hyperhemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, 1 2 S 3 ⊂ R 4 . We find also the area of the boundary of the set MN and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures-Hall normalization constants is derived for an arbitrary N .

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Section: Bures Metricmentioning

confidence: 99%

Bures volume of the set of mixed quantum states

Sommers

Życzkowski

2003

J. Phys. A: Math. Gen.

164

296

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“…Following the work of Petz [44] (see also [3,4]) let us recall that a Riemannian metric on the manifold D of the density matrices ρ can be written in the form…”

Section: Monotone Riemannian Metrics: Generic Formulamentioning

confidence: 99%

“…In the geometrical approach to statistics proposed by Morozova andCencov [43] the monotone metrics can be introduced and studied from a unified point of view due to the work of Petz [44].…”

Section: Introductionmentioning

confidence: 99%

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

Tonchev

2016

Journal of Mathematical Physics

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An analytical approach is developed to the problem of computation of monotone Riemannian metrics (e.g. Bogoliubov-Kubo-Mori, Bures, Chernoff, etc.) on the set of quantum states. The obtained expressions originate from the Morozova,Cencov and Petz correspondence of monotone metrics to operator monotone functions. The used mathematical technique provides analytical expansions in terms of the thermodynamic mean values of iterated (nested) commutators of a model Hamiltonian T with the operator S involved through the control parameter h. Due to the sum rules for the frequency moments of the dynamic structure factor new presentations for the monotone Riemannian metrics are obtained. Particularly, relations between any monotone Riemannian metric and the usual thermodynamic susceptibility or the variance of the operator S are discussed. If the symmetry properties of the Hamiltonian are given in terms of generators of some Lie algebra, the obtained expansions may be evaluated in a closed form. These issues are tested on a class of model systems studied in condensed matter physics.2

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“…The requirement that the distance between density matrices expresses quantum statistical distinguishability implies that this distance must decrease under coarsegraining (stochastic maps). Unlike the classical case, it turns out that there are infinitely many monotone Riemannian metrics satisfying this requirement [24][25][26].…”

Section: On Information Geometry and Statistical Distinguishabilitymentioning

confidence: 99%

On Grover’s search algorithm from a quantum information geometry viewpoint

Cafaro

Mancini

2012

Physica A: Statistical Mechanics and its Applications

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We present an information geometric characterization of Grover's quantum search algorithm. First, we quantify the notion of quantum distinguishability between parametric density operators by means of the Wigner-Yanase quantum information metric. We then show that the quantum searching problem can be recast in an information geometric framework where Grover's dynamics is characterized by a geodesic on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in Grover's algorithm. We also discuss possible deviations from Grover's algorithm within this quantum information geometric setting.

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Monotone metrics on matrix spaces

Cited by 480 publications

References 7 publications

Bures volume of the set of mixed quantum states

Bures volume of the set of mixed quantum states

Monotone Riemannian metrics and dynamic structure factor in condensed matter physics

On Grover’s search algorithm from a quantum information geometry viewpoint

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