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One-parameter pure state estimation based on the symmetric logarithmic derivative Akio Fujiwara METR 94-8 July

1994 One-parameter pure state estimation based on the symmetric logarithmic derivative Akio Fujiwara, University of Tokyo

3 , Abstract A statistical parameter estimation theory for quantum pure state models is presented.

We rst investigate a general frame- work of the pure state estimation theory and derive quantum counterparts of the Fisher metric. Then we formulate a one pa- rameter estimation theory, based on the symmetric logarithmic derivatives, and clarify the dierences between pure state models and strictly positive models. Keywords : quantum estimation theory, pure state, symmet- ric logarithmic derivative, quantum Fisher metric 3Department of Mathematical Engineering and Information Physics, Faculty of Engi- neering, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan.

1 1 Introduction A quantum statistical model is a family of density operators dened on a certain separable Hilbert space H with nite-dimensional real parameters = (i)n i=1 which are to be estimated statistically. In order to avoid sin- gularities, the conventional quantum estimation theory [1][2] has been often restricted to models that are composed of strictly positive density operators. ItwasHelstrom [3]who successfullyintroduced the symmetrizedlogarithmic derivative for the one parameter estimation theory as a quantum counter- part of the logarithmic derivative in the classical estimation theory. The right logarithmic derivative is another successful counterpart introduced by Yuen and Lax [4] in the expectation parameter estimation theory for quan- tum gaussian models, which provided a theoretical background of optical communication theory. Quantum information theorists have also kept away from degenerated states, such as pure states, for mathematical convenience [5]. Indeed, the von Neumann entropy cannot distinguish the pure states, and the relative entropies diverge. In this paper, however, we try to construct an estimation theory for pure state models, and clarify the dierences between the pure state case and the strictly positive state case. In Sec. 2, we prove some crucial lemmas which will provide fudamentals of the pure state estimation theory. In Sec. 3, we study the quantum counterpart of the logarithmic derivative and the Fisher information which played important roles in the classical estimation theory. The quantum statistical signicance of the Fubini{Study metric is also clar- ied. In Sec. 4, we provide one parameter pure state estimation theory based on the symmetric logarithmic derivative. This is rather analogous to the conventional quantum estimation theory, but reveals the essential dier- ence between the pure state models and the strictly positive models. Some examples are also given in Sec. 5.

2 Preliminaries Let H be a Hilbert space with inner product h j'

i for every ;

'

2 H. Further, let L and Lsa are, respectively, the set of all the (bounded) linear operators and all the self-adjoint operators on H. Given a possibly degen- erated density operator , we dene sesquilinear forms on L: (A;

B) = TrBA3;

(1)

2 hA;

Bi =

1 2Tr(BA3 +A3B);

(2) where A;

B

2 L. These are pre-inner product on L, i.e., possessing all properties of inner product except that (K;

K) and hK;

Ki may be equal to zero for a nonzero K

2 L. Note that the Schwarz inequality also holds for pre-inner product. The forms (1;

1) and h1;

1i become inner products if and only if >

0. If rank =

1 or equivalently