PDF《One-parameter pure state》

s which satisfy (5). Lemma

5 Suppose is pure and an arbitrary self-adjoint operator A

2 Lsa is given. Then all the quantities hA;

LS ;

ji with respect to the common j are identical for every SLD LS ;

j

2 T S(). Proof By using Lemma 3, it is proved in the same way as Lemma 4.

4 FromLemma5, wecandeneuniquelytherealFisherinformationmatrix JS for the family of SLD (5) whose (j;

k) entry is hLS ;

j;

LS ;

ki , which is called the SLD{Fisher information matrix. The above results are summarized by the following theorem.

5 Theorem

1 Suppose is pure. Then the complex Fisher information matrix J = [(L;

j;

L;

k) ] and the SLD{Fisher information matrix JS = h hLS ;

j;

LS ;

ki i are uniquely determined on the quotient spaces T ()=K() and T S()=Ksa(), respectively. They are related by JS = ReJ. The (j;

k) entry of JS becomes (JS )jk = 2Tr(@j)(@k);

(7) where @j = @=@j. This metric is identical, up to a constant factor, to the Fubini{Study metric. Proof We only need to prove (7). Dierentiating =

2 , @j = (@j) +(@j): (8) This relation indicates that 2@j is a representative of the SLD. Then (JS )jk = h2@j;

2@ki = 2Tr[(@j)(@k)+(@k)(@j)]: (9) Further, multiplying to (8), we have (@j) = 0: (10) Therefore, by using (8) and (10), (@j)(@k) = [(@j) +(@j)][(@k) +(@k)] = (@j)(@k)+(@j)(@k): This, along with (9), leads to the relation (7). Denoting = j ih j Tr(@j)(@k) = 2[Reh@j j@k i+h j@j ih j@k i];

which is identical to the Fubini{Study metric [8][9].

4 The Fubini{Study metric is known as a gauge invariant metric on a pro- jective Hilbert space [10]. Theorem

1 gives another meaning of the Fubini{ Study metric, i.e., the statistical distance. Wootters [11] also investigated from a statistical viewpoint the distance between two rays, and obtained d( ;

'

) = cos01 jh j'

ij. This is identical, up to a constant factor, to the geodesic distance as measured by the Fubini{Study metric [12]. Theorem 1, together with the following Theorem 2, reveals a deeper connection between them.

6 4 Paremeter estimation of pure states In this section, we give a parameter estimation theory of pure state models based on the SLD. Given a n parameter pure state model (3). In order to handle simultaneous probability distributions of possibly mutually non- commuting observables, an extended framework of measurement theory is needed [1, p. 53] [2, p. 50]. An estimator for is identied to a generalized measurement which takes values on 2. The expectation vector with respect to the measurement M at the state is dened as E[M] = Z ^ PM (d^ ): The measurement M is called unbiased if E[M] = holds for all

2 2, i.e., Z ^ jPM (d^ ) = j;

(j = 1;

111;

n): (11) Dierentiation yields Z ^ j @ @k PM (d^ ) = j k;

(j;

k = 1;

111;

n): (12) If (11) and (12) hold at a certain , M is called locally unbiased at . Ob- viously, M is unbiased i M is locally unbiased at every

2 2. Letting M be a locally unbiased measurement at , we dene the covariance matrix V[M] = [vjk ]

2 Rn2n with respect to M at the state by vjk = Z (^ j 0j)(^ k 0k)PM (d^ ): (13) A lower bound for V[M] is given by the following theorem, which is a quantum version of Cram er{Rao theorem. Theorem

2 Given a pure state model , the following inequality holds for any locally unbiased measurement M: V[M] JS

01 : (14) Proof It is proved almost in the same way as the strictly positive case [2, p. 274], except that h1;

1i is a pre-inner product now.

4 7 When the model is one dimensional, the measurement M is identied with a certain self-adjoint operator T, and the inequalities in the theorem become scalar, i.e., V[T]