Braunstein, S. L. and C. M. Caves (1994). Statistical distance and the geometry of quantum states. Phys. Review Letters 72, 3439--3443.  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

....many other measurements Alice could make, but they all have the same property, of only providing a small, random, amount of information about the original state, and destroying it in the process. In fact it is a result from the theory of quantum statistical inference due to Helstrom (1967, 1976) Braunstein and Caves (1994), that whatever measurement is carried out by Alice, the Fisher information matrix based on the probability distribution of the outcome of the experiment, concerning the unknown parameters #, #, has a strictly positive lower bound. The famous no cloning theorem could also be invoked here: it is ....

.... of the observable #(#) If the parameter # is actually a vector, then one defines quantum scores component wise, and finally defines the quantum information matrix elementwise by I Q (#) ij = trace( 1 2 #(#) #(#) i #(#) j #(#) j #(#) i ) The following quantum information inequality due to Braunstein and Caves (1994) is crucial: I(#; M) # I Q (#) for all measurements M . From this inequality one immediately has the quantum Cramer Rao inequality, Helstrom (1967) for all measurements M , and any unbiased estimator # # based on the outcome of that measurement, Var( # #) # I Q (#) 1 . To prove the ....

Braunstein, S. L. and C. M. Caves (1994). Statistical distance and the geometry of quantum states. Phys. Review Letters 72, 3439--3443.

....phase are identified, we have a finite dimensional manifold known as projective Hilbert space, and an associated metric known as the Fubini Study metric, which is the infinitesimal form of the distance d projective (ji; j i) 1 Gamma min (Rejhje iOE j i) 2 : 3. 1) Braunstein and Caves [25] defined a statistical distance which is a Riemannian metric on the space of quantum mechanical density operators. This metric may be defined in terms of the structure of correlations between measurements on systems described by such density operators [26, 27] or it may be defined in a way ....

....[26, 27] or it may be defined in a way originally due to Wootters [28] who considered only pure states) in terms of the number of distinguishable states lying between two states described by density operators. The form of this metric for infinitesimally separated states is known, and is given by [25] ds 2 S = 8[1 Gamma tr q ae 1=2 (ae dae)ae 1=2 ] 3.2) The definition in terms of distinguishability is best understood by assuming ae and ae dae to lie along a path parametrized by, say, Then the problem of distinguish52 ing between the two density operators may identified with ....

[Article contains additional citation context not shown here]

Samuel L. Braunstein and Carlton M. Caves, "Statistical distance and the geometry of quantum states," Physical Review Letters, vol. 72, pp. 3439--3443, 1994.

.... of the so called dualistic geometry, a generalization of the Riemannian geometry [1] It is natural to ask whether some geometrical methods are also useful in quantum estimation theory [2] 3] Indeed, many authors have tried to find geometrical aspects of quantum estimation theory [4] 5] 6] 7][8]. We should notice, however, that there exist a variety of manners to define quantum counterparts of geometrical notions which played essential roles in the classical estimation theory, such as logarithmic derivative, Fisher information, Cram er Rao inequality, etc. Moreover, only a few of them ....

S. L. Braunstein and C. M. Caves, "Statistical distance and the geometry of quantum states," Phys. Rev. Lett. 72, 3439--3443 (1994).

....full exponential model. Various pleasant properties of such quantum exponential models then follow from standard properties of the full exponential models. The classical Cram er Rao bound for the variance of an unbiased estimator x of is Var(x) i( M ) Gamma1 : 4.3) Combining (4. 3) with Braunstein and Caves (1994) quantum information bound (6.3) which we state in Sect. 6.2 yields Helstrom s (1976) quantum Cram er Rao bound Var(x) I( Gamma1 (4.4) whenever x is the result of a quantum measurement. It is a classical result that, under certain regularity conditions, the following are equivalent: i) ....

....depend on the version (in case more than one exists) of the symmetric logarithmic derivative of ae. 6.2 Relation to Classical Expected Information Suppose that is one dimensional. There is an important relationship between expected quantum information and classical expected information, due to Braunstein and Caves (1994), namely that for any measurement M with density m with respect to a oe finite measure on X , i( M ) I( 6.3) The proof is based on the Cauchy Schwarz inequality. A necessary and sufficient condition for equality in (6.3) is that for almost all x in fx : p(x; 0g, m(x) 1=2 ae ....

Braunstein, S.L. and Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Review Letters 72 (1994) 3439--3443.

....use of the van Trees inequality (see Gill and Levit, 1995) a Bayesian Cram erRao bound, which will allow us to make asymptotic optimality statements without assuming or proving local asymptotic normality. Another useful ingredient will be the recent derivation of the quantum Cram er Rao bound by Braunstein and Caves (1994), linking quantum information to classical expected Fisher information in a particularly neat way. We will show that for certain problems, a new Cram er Rao type inequality of Gill and Massar (1998) does provide an asymptotically achievable bound to the quality of an estimator of unknown ....

.... Define I M ( to be the Fisher information matrix for the parameter in the distribution of the outcome of a measurement M on the quantum system ae( Then (with respect to the usual ordering of symmetric positive semi definite matrices) I M ( I Q ( The result in this form was proved by Braunstein and Caves (1994) for a one dimensional parameter, but the general result is an easy consequence of this by considering the information for arbitrary linear combinations. As a corollary one obtains Helstrom s original form of the theorem as a lower bound to the variance of an unbiased estimator of based on the ....

Braunstein, S.L. and Caves, C.M. (1994). Statistical distance and the geometry of quantum states. Physical Review Letters 72, 3439--3443.

....as statistical applications of this geometric approach. To the best of our knowledge, Riemannian metric on quantum states was first considered by Helstrom in connection with state estimation theory [13] Since Helstrom s work, several other metrics appeared in the literature, see for example [6] [7], 12] 23] and Uhlmann approached Helstrom s metric in a different way ( 25] 26] The present paper is organized as follows. In Section II we survey the work of Chentsov both in the probabilistic and in the quantum case. We explain how he arrived at the study of invariant metrics on the ....

....metric ( 10] For example this space is not locally symmetric and all sectional curvatures are greater than 1. Braunstein and Caves obtained recenly the same metric by optimizing over all generalized quantum measurements that can be used to distinguish neighboring quantum states D and D dD ([7]) IV. Monotone metrics If a distance between density matrices expresses statistical distinguishability then this distance must decrease under coarse graining. A good example of coarse graining arises when a density matrix is partitioned in the form of a 2 Theta 2 block matrix, and the ....

S.L. Braunstein, C.M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72(1994), 3439--3443

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC