169 research outputs found
Wigner-Yanase information on quantum state space:the geometric approach
In the search of appropriate riemannian metrics on quantum state space the
concept of statistical monotonicity, or contraction under coarse graining, has
been proposed by Chentsov. The metrics with this property have been classified
by Petz. All the elements of this family of geometries can be seen as quantum
analogues of Fisher information. Although there exists a number of general
theorems sheding light on this subject, many natural questions, also stemming
from applications, are still open. In this paper we discuss a particular member
of the family, the Wigner-Yanase information.
Using a well-known approach that mimics the classical pull-back approach to
Fisher information, we are able to give explicit formulae for the geodesic
distance, the geodesic path, the sectional and scalar curvatures associated to
Wigner-Yanase information. Moreover we show that this is the only monotone
metric for which such an approach is possible
On the monotonicity of scalar curvature in classical and quantum information geometry
We study the statistical monotonicity of the scalar curvature for the
alpha-geometries on the simplex of probability vectors. From the results
obtained and from numerical data we are led to some conjectures about quantum
alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this
last conjecture implies the truth of the Petz conjecture about the monotonicity
of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte
On the characterisation of paired monotone metrics
Hasegawa and Petz introduced the notion of dual statistically monotone
metrics. They also gave a characterisation theorem showing that
Wigner-Yanase-Dyson metrics are the only members of the dual family. In this
paper we show that the characterisation theorem holds true under more general
hypotheses.Comment: 12 pages, to appear on Ann. Inst. Stat. Math.; v2: changes made to
conform to accepted version, title changed as wel
A volume inequality for quantum Fisher information and the uncertainty principle
Let be complex self-adjoint matrices and let be a
density matrix. The Robertson uncertainty principle gives a bound for the quantum
generalized covariance in terms of the commutators . The right side
matrix is antisymmetric and therefore the bound is trivial (equal to zero) in
the odd case .
Let be an arbitrary normalized symmetric operator monotone function and
let be the associated quantum Fisher information. In
this paper we conjecture the inequality that gives a
non-trivial bound for any natural number using the commutators . The inequality has been proved in the cases by the joint efforts
of many authors. In this paper we prove the case N=3 for real matrices
Fisher information and means: some questions in the classical and quantum settings
The algebraic and geometric properties of Fisher Information, its relations with the theory of operator means have been very active fields in the last decades. In this paper I try to shed some light on recent results and to address some open questions which could in principle give some new directions to the field
A characterisation of Wigner-Yanase skew information among statistically monotone metrics
Let M-n = M-n(C) be the space of n x n complex matrices endowed with the Hilbert-Schmidt scalar product, let S-n be the unit sphere of M-n and let D-n subset of M-n be the space of strictly positive density matrices. We show that the scalar product over D-n introduced by Gibilisco and Isola(3) (that is the scalar product induced by the map D-n There Exists rho --> rootrho is an element of S-n) coincides with the Wigner-Yanase monotone metric
A dynamical uncertainty principle in von Neumann algebras by operator monotone functions
Suppose that A(1),..., A(N) are observables (selfadjoint matrices) and rho is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det{Cov(rho) (A(j), A(k) )}, using the commutators [A(j), A(k)]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [rho, A(j)] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones
Fisher information and Stam inequality on a finite group
We prove a discrete version of the Stam inequality for random variables taking values on a finite group
Volume of the quantum mechanical state space
The volume of the quantum mechanical state space over -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 volume of the space of qubits
can be infinite too. We characterize those monotone metrics which generates
infinite volume.Comment: 17 page
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