C. W. Helstrom, "Quantum Detection and Estimation Theory," Chap. VIII.4, Academic, New York, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of more general positive operator valued measures (POVMs) Generalized measurements on a system s can be thought of as orthogonal measurements on an extended system , which may not be orthogonal in s alone. These have applications in a broad number of areas including precision measurement [79], quantum communication in the context of entanglement 111 purification [80] and quantum error correction [22] The existence of POVMs that optimize information retrieval and state processing in various contexts have been discovered, but it remains a problem to implement them in real physical ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....a nice mathematical device, but in fact acquires direct physical signi cance. For instance, Gregoratti and Werner [11] have exploited this correspondence in a scheme for recovery of classical and quantum information from noise by making a generalized quantum measurement (described by a POVM [13]) on the environment Hilbert space of a noisy quantum channel [the Hilbert space E in the ancilla form (6) 5 Characterization of quantum operations by positive operators The correspondence between linear maps from a matrix algebra Mm into a matrix algebra M n and linear functionals on M ....

C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....certain basic blocks can serve as channels. We take our cue from quantum theory of open systems [31] and say that any physically acceptable channel can be realized as a sequence of the following steps: a) adjunction of an auxiliary system (called the ancilla in the terminology of Helstrom [58]) in some xed initial state, b) unitarily implemented evolution of the enlarged system, and (c) restriction to the original subsystem. In other words, any channel must be of the form described in Example 2.2.9. Luckily it turns out Latin for housemaid; we choose not to dwell on the ....

....is unknown, but we are told that it is drawn from some known set f m g m=1 according to the probability distribution fp m g m=1 . Our task is to devise a measurement that would maximize the probability of correctly identifying the state. This is known as the M ary quantum detection problem [58]. Any measurement we would perform will be described by a POVM Fm on the M element set f1; Mg. Given the state , the probability of identifying as m is equal to tr ( Fm ) Thus the average probability of correct decision using the POVM F : fFm g is given by P c [F ] p m tr ....

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C.W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....and information theory, quantum tomography, quantum optics, and quantum measurement theory. Also many conceptual problems, like the problem of joint measurability of noncommutative quantities, or the problem of classical limit of quantum mechanics have greatly advanced by this tool. The monographs [1, 2, 3, 4, 5, 6, 7] exhibit various aspects of these developments. Any positive trace one operator T (a state) defines a phase space observable Q T according to the rule Q T (E) 1 2# # E e i(qP pQ) T e i(qP pQ) dq dp, where E is a Borel subset of the (two dimensional) phase space. It is well known that all ....

C.W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976.

.... He called the matrix L S t (t) symmetrized logarithmic derivative (SLD) because SLD is introduced as a quantum counterpart of a logarithmic derivative in the classical estimation theory (throughout this paper, the term classical estimation means estimation of probability distributions) 1] [2]. Our starting point is the following queries: Why SLD plays such an important role both in the quantum estimation theory and in Uhlmann s parallelity Is this just a coincidence 2 Quantum estimation theory In this section, conventional theory of quantum estimation is reviewed briefly. In the ....

....2 Bj g =Trae( M(B) 2) 2 when the true value of the parameter is . Here, M , which is called measurement, is a mapping from subsets B ae 4 to non negative Hermitian matrices in H, such that M(OE) O;M(4) I; M( 1 [ i=1 B i ) 1 X i=1 M(B i ) B i B j = OE; i 6= j) 3) see Ref. [2],p.53 and Ref. 3] p.50. Conversely, some apparatus corresponds to any measurement M [6] 5] A pair ( M;4) is called an estimator. An estimator ( M;4) is said to be locally unbiased at if E [ jM;4] i E [ j ( jM;4] ffi j i (i; j =1; m) 4) hold at , where ....

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C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....of Engineering, UniversityofTokyo, Bunkyo ku, Tokyo 113, Japan. 1 1 Introduction A quantum statistical model is a family of density operators ae defined on a certain separable Hilbert space H with finite dimensional real parameters = i ) n i=1 which are to be estimated statistically [1][2] For one parameter case, it is rather easy to find the most informativelower bound of the variance of measurement, since the Cram er Rao bound with respect to the SLD is always locally attainable (global feature is another problem) 3] 4] On the other hand, for multi parameter case, it has ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). 17

....lies in B 2 oe( Omega writes Prf 2 Bjaeg =traeM(B) 1) by use of the map M from oe( Omega to nonnegative Hermitian operator which satisfies M(OE) O;M( Omega = I; M( 1 [ i=1 B i ) 1 X i=1 M(B i ) B i B j = OE; i 6= j) 2) so that (1) define a probability measure (see Ref. [8], p.53 and Ref. 9] p.50) We call the map M the measurement, because there always exist an physical experiment corresponds to the map M which satisfies (2) 21] 16] The purpose of the quantum estimation is to identify the density operator of the given physical system from the data obtained by ....

.... , M; is called locally unbiased at . 2. 2 SLD CR inequality, the attainable CR type bound Analogically to the estimation theory of probability distributions, in the quantum estimation theory,wehave the following SLD CR inequality, which is proved for the exact state model by Helstrom [7][8], and is proved for the pure state model byFujiwara and Nagaoka [5] V [ j M ] i J S ( j 01 ; 5) i.e. V [ j M ] 0 (J S ( 01 is non negative definite. Here V [ j M]isa covariance matrix of an unbiased measurement M , and J S ( is called ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). 15

....by Prf 2 Bj g =trae( M(B) 9) when represents the true value of parameter. Here M is a mapping of subsets B ae R m to non negative Hermitian operators on H, such that M(OE) O;M(R m ) I; 10) M( 1 [ i=1 B i ) 1 X i=1 M(B i ) B i B j = OE; i 6= j) 11) 5 (see Ref. [5],p.53 and Ref. 6] p.50. M is called a generalized measurement or measurement, because there is a corresponding measuring apparatus to any M satisfying (11) 11] 15] A measurement E is said to be simple if E is projection valued. A generalized measurement M is called an unbiasedmeasurement in ....

....Z ( j 0 j ) k 0 k )trae( M(d ) 14) 2. 3 SLD CR bound and the attainable CR type bound Analogically to the classical estimation theory, in the quantum estimation theory, we have the following SLD CR inequality, which is proved for the exact state model by Helstrom [4][5], and is proved for the pure state model byFujiwara and Nagaoka [3] V [M ] J S ( 01 ; 15) i.e. V [M ] 0 (J S ( 01 is non negative definite. Here V [M]isacovariance matrix of an unbiased measurement M , and J S ( is called SLD Fisher information matrix, and is ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....together with in offering some clear cut methodologies. It was also found that it gives an important application 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, ....

.... ( the normalization factor, and the Einstein s summation convention i f i (x) P i i f i (x) is used. In this section, we investigate in detail the quantum counterpart of this notion. We first give a brief summary of the conventional quantum estimation theory. For details, consult [2][3] Suppose we are given a statistical parametric model composed of strictly positive density operators: S = fae ; ae = ae 3 0; Tr ae = 1; 2 2 ae R n g: 1) Here, 1 ; 1 1 1 ; n ) is the parameter to be estimated statistically. An estimator for is identified to a ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....A quantum statistical model is a family of density operators ae defined on a certain separable Hilbert space H with finite dimensional real parameters = i ) n i=1 which are to be estimated statistically. In order to avoid singularities, the conventional quantum estimation theory [1][2] has been often restricted to models that are composed of strictly positive density operators. It was Helstrom [3] who successfully introduced the symmetrized logarithmic derivatives for the one parameter estimation theory as a quantum counterpart of the logarithmic derivative in the classical ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

.... 1 Department of Mathematical Engineering and Information Physics UniversityofTokyo, Bunkyo ku,Tokyo 113, Japan 1 1 Introduction Quantum estimation theory deals with the identification of the density matrix or given system by use of data produced by appropriately desigened experiment [4][5][6] From this quantum estimation theoretical point of view, there are two natural geometridal structures in the space of full rank density matrices. One is Uhlmann s parallelism and the other is Nagaoka s information geometry. Uhlmann s parallelism is generalization of Berry s phase, which, by ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

....2 S and provides multiple copies of the secret state when requested to do so by Player 2. The task for Player 2 is to design an observable (POVM) or sequence of observables which allows him to guess the secret state in as few requests to Player 1 as possible. Similar scenarios have been considered [5]. Definition 1 Let S = fae 1 ; ae N g be a set of states and A = fA 1 ; Am g a POVM. For each ae 2 S we define the elimination set of ae with respect to A as EA (ae) fA i 2 A : tr(aeA i ) 0g and the elimination operator of ae with respect to A as A ae = P A i 2EA (ae) A i ....

Helstrom, Carl W. Quantum Detection and Estimation Theory. Academic Press, 1976.

....using all of the possible code words. Denote an individual code word by j s i i. In order to read Alice s messages, Bob will have to employ a decoding observable to determine which signal j s i i is present. This decoding observable 7 will in general be a positive operator measurement (or POM)[10, 11]. Bob will want to choose his decoding observable so that he will deduce Alice s message with as small a probability of error as possible. Bob s essential problem is to distinguish between a collection of vectors in the Hilbert space H l . Let fj OE k ig be a collection of such vectors ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 74-83.

....property is not itself easy to prove, but it is a property of S(ae) itself, without reference to measurements. 2. We employ a physical model of the measurement process instead of a mathematically defined observable . This model is sufficient to describe any generalized measurement (or POVM) [4], where distinct outcomes are represented by positive operators instead of projections. 3. Because our model of measurement includes the effect of the measurement process on the system Q, we are in fact able to arrive at a stronger result than Equation 1. Our version is much sharper than Kholevo s ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976), pp. 74-83.

....1) R for 0 R 0 ( 1) where ( 1) ln Tr S 2 = ln a X i;j=1 i j j i j j j 2 : For 0 ( 1) R C the function E r ( R) is a [ convex and is given by E r ( R) s R ) s RR; where s R is the root of the equation 0 ( s R ) R (see Fig. 2) 3. The expurgated lower bound When we choose codewords randomly there is certain probability that some codewords will coincide that makes error probability for such code equal to 1. It turns out that probability to choose such a bad code does not in uence essentially the average code error ....

.... codewords, for which at least M codewords satisfy k [2E r k ] 1=r ; 12) for arbitrary 0 r 1 (without loss of generality we can assume that (12) holds for k = 1; M ) Then we can use an estimate from [2] to evaluate the righthand side of (12) By using the inequality p 3 2 1 2 2 for 0, one obtains k 2[1 u k jG 1 2 (u 1 ; u M 0 ) u k ] 2 3 u k jG(u 1 ; u M 0 ) u k u k jG 2 (u 1 ; u M 0 ) u k = X i6=k j u i j u k j 2 ; where the summation ....

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C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, NY, 1976.

....A quantum statistical model is a family of density operators ae defined on a certain separable Hilbert space H with finite dimensional real parameters = i ) n i=1 which are to be estimated statistically. In order to avoid singularities, the conventional quantum estimation theory [1][2] has been often restricted to models that are composed of strictly positive density operators. It was Helstrom [3] who successfully introduced the symmetrized logarithmic derivative for the one parameter estimation theory as a quantum counterpart of the logarithmic derivative in the classical ....

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976). 10

....aspect when the unknown state belongs to a finite dimensional Hilbert space. Thus the setting of the problem is that we may dispose of a finite number N of copies of an unknown quantum state ae (pure or mixed) Our task is to determine ae as well as possible. This is by now a classical problem[9][10] A common approach is first to specify a cost function which numerically quantifies the deviation of the estimate from the true state. One then tries to devise a measurement and estimation strategy which minimizes the mean cost. Since the mean cost typically depends on the unknown state ....

....estimation procedures In particular, what does the boundary of this set of attainable m.q.e. matrices look like In the case when the parameter is one dimensional (p = 1) the problem has been solved: a bound on the variance of unbiased estimators the quantum Cram er Rao bound was given in [9], and a strategy for attaining the bound in the large N limit was proposed in [15] This justifies taking the bound to induce a distinguishability metric on the space of states [13] 14] In the case of a multidimensional parameter however, though different bounds for the matrix W have been ....

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C. W. Helstrom, Quantum Detection and Estimation Theory, Academic, New York, 1976.

....factor G(W ) Gamma1=2 is a major source of analytical difficulties in the noncommutative case. Note that the vectors w 1 ; w M need not be linearly independent; in the case of linearly independent coherent state vectors (5) is related to the suboptimal receiver described in [Helstrom 1976], Sec. VI.3(e) It was shown in [Holevo 1978] that by using this decision rule one obtains the upper bound min X (X; W ) 2 M Sp i E Gamma Gamma(W ) 1=2 j = 1 M Sp i E Gamma Gamma(W ) 1=2 j 2 ; 6) where E is the unit M Theta M matrix and Sp is the trace of M Theta ....

.... can be calculated explicitly, 2 M E Sp i E Gamma Gamma(W ) 1=2 j = Tr f( S Omega n ) where f(z) 2 M [1 Gamma q 1 (M Gamma 1)z (M Gamma 1) 1 Gamma p 1 Gamma z) This function strangely resembles the expression for the Bayes error in the equiangular case [Helstrom 1976] rel. VI.2.10) although does not coincide with it. The function f(z) admits standard estimates f(z) z min f(M Gamma 1)z; 2g 2(M Gamma 1) s z 1 s ; 0 s 1 ; allowing to prove the following result [Burnashev and Holevo 1997] Theorem 1. For all M;n and 0 s 1 E min X (W; X) 2(M ....

C. W. Helstrom, Quantum Detection and Estimation Theory. New York: Academic Press 1976.

....S : fae 2 S(H)j = 1 ; d ) 2 Theta ae R d g; where the set S(H) denotes the set of densities on H. The most general description of a quantum measurement probability is given by the mathematical concept of a positive operator valued measure (POM) on the system state space [7, 8]. Generally speaking, if Omega is a measurable space, a measurement M satisfies the following: M(B) M(B) 0; M( 0; M( Omega Gamma = Id on H; for any Borel B ae Omega : M( i B i ) X i M(B i ) for B i B j = i 6= j) fB i g is a countable subsets of Omega : In the quantum ....

C. W. Helstrom, Quantum Detection and Estimation Theory, (Academic Press, New York, 1976).

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C. W. Helstrom, "Quantum Detection and Estimation Theory," Chap. VIII.4, Academic, New York, 1976.

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C. W. Helstrom. Quantum Detection and Estimation Theory. Academic Press, New York, 1976.

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C.W. Helstrom. Quantum Detection and Estimation Theory. Academic Press, New York, 1976.

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Carl W. Helstrom. Quantum Detection and Estimation Theory. Academic Press, 1976.

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C. W. Helstrom, \Quantum Detection and Estimation Theory," Chap. VIII.4, Academic, New York, 1976.

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C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

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