The quantum analog of the Fisher information metric of a probability simplex is searched and seve... more The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.
Let Mn be the space of all positive definite n × n complex matrices of trace 1. A Riemannian metr... more Let Mn be the space of all positive definite n × n complex matrices of trace 1. A Riemannian metric k on Mn is said to be monotone if for every stochastic map T the following holds: kD(T (X), T (Y )) ≤ kD(X,Y ) D ∈MnX, Y ∈ TDMn (Recall that a linear map T is stochastic if it is completely positive and T (Mn) ⊂Mn ) On the base of Morozova and Chentsov’s work [7] D. Petz showed in [8] that for every monotone metric k there exists a symmetric and positive operator monotone function f such that (1) kD(X,Y ) = Tr(KD(X)Y ), K−1 D = f(LDR −1 D )RD where LD, RD are the operators of left and right multiplication by D. A function f is positive operator monotone if A ≤ B implies f(A) ≤ f(B) for every positive matrices A,B with order r and for each integer r. Such an f is symmetric if f(x) = xf(x−1), this condition implies that KD(X∗) = KD(X)∗, so KD maps the space of selfadjoint operators into itself. In addition f(x) possesses the following integral representation:
The quantum analog of the Fisher information metric of a probability simplex is searched and seve... more The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.
Let Mn be the space of all positive definite n × n complex matrices of trace 1. A Riemannian metr... more Let Mn be the space of all positive definite n × n complex matrices of trace 1. A Riemannian metric k on Mn is said to be monotone if for every stochastic map T the following holds: kD(T (X), T (Y )) ≤ kD(X,Y ) D ∈MnX, Y ∈ TDMn (Recall that a linear map T is stochastic if it is completely positive and T (Mn) ⊂Mn ) On the base of Morozova and Chentsov’s work [7] D. Petz showed in [8] that for every monotone metric k there exists a symmetric and positive operator monotone function f such that (1) kD(X,Y ) = Tr(KD(X)Y ), K−1 D = f(LDR −1 D )RD where LD, RD are the operators of left and right multiplication by D. A function f is positive operator monotone if A ≤ B implies f(A) ≤ f(B) for every positive matrices A,B with order r and for each integer r. Such an f is symmetric if f(x) = xf(x−1), this condition implies that KD(X∗) = KD(X)∗, so KD maps the space of selfadjoint operators into itself. In addition f(x) possesses the following integral representation:
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Papers by Csaba Sudár