PDF《One-parameter pure state》

2 = , is called pure. The following lemmas are fundamental. Lemma

1 Suppose is pure. Then the following

3 conditions for linear operators K

2 L are equivalent. (i) (K;

K) = 0, (ii) K = 0, (iii) TrK =

0 and K +K3 = 0. Proof Let us express as = j ih j where j i is a normalized vector in H. Then the following equivalent sequence (K;

K) =

0 () h jKK3j i =

0 () h jK =

0 () j ih jK = 0;

yield (i),(ii). Further, (ii))(iii) is trivial. (iii))(ii) is shown as follows. Operating h j from the left to the assumption j ih jK +K3j ih j = 0;

and invoking another assumption TrK =

0 , h jK3j i = 0, we have

0 = h j ih jK +h jK3j ih j = h jK: Therefore j ih jK = 0.

4 Lemma

2 Suppose is pure. Then the following

3 conditions for linear operators K

2 L are equivalent. (i) hK;

Ki = 0, (ii) K = K3 = 0, (iii) TrK = 0, K +K3 = 0, and K3 +K = 0.

3 Proof (i),(ii) is shown as follows: hK;

Ki =

0 () h jKK3j i+h jK3Kj i =

0 () h jK = 0;

h jK3 =

0 () K = K3 = 0: (ii),(iii) is a sraightforward consequence of Lemma 1.

4 Lemma

3 Suppose is pure. Then the following

3 conditions for self{ adjoint operators K

2 Lsa are equivalent. (i) hK;

Ki = 0, (ii) K = 0, (iii) K +K = 0. Proof Straightforward by setting K = K3 in Lemma 2.

4 Note that in either lemmas, equivalence of the conditions (i) and (ii) holds for any , whereas the condition (iii) is characteristic of pure states. These lemmas are, therefore, eectively employed in the pure state estima- tion theory. Denote by K() the set of linear operators K

2 L satisfying (K;

K) = 0, which are called the kernel of the pre-inner product (1;

1). Also denote by Ksa() the set of self-adjoint operators K

2 Lsa satisfying hK;

Ki = 0, which are called the kernel of the pre-inner product h1;

1i.

3 Quantum Fisher metric suppose we are given a n parameter pure state model: S = f ;

3 = ;

Tr = 1;

2 = ;

2 2 R ng: (3) We dene a family of quantum analogues of the logarithmic derivative by @ @j =

1 2[L;

j +L3 ;

j];

TrL;

j = 0: (4) For instance, @ @j =

1 2[LS ;

j +LS ;

j];

LS ;

j = LS3 ;

j (5)

4 denes the symmetric logarithmic derivative (SLD) LS ;

j introduced by Hel- strom [3]. Furthermore, since every pure state model is written in the form = U0U3 , whereU is unitary,wehaveanotherusefullogarithmicderiva- tive @ @j =

1 2[LA ;

j 0LA ;

j];

TrLA ;

j = 0;

LA ;

j = 0LA3 ;

j;

(6) which may be called the anti-symmetric logarithmic derivative (ALD). In- deed,theALDiscloselyrelatedtothelocalgeneratorA;

j = 0i(@U=@j)U3 of the unitary U such as LA ;

j = 02iA;

j. Thus, (4) denes a certain family of logarithmic derivatives [6]. Denote by T () all the logarithmic deriva- tives which satisfy (4). Lemma

4 Suppose is pure and an arbitrary linear operator A

2 L is given. Then all the quantities (A;

L;

j) with respect to the common j are identical for every logarithmic derivatives L;

j

2 T (). Proof Take any logarithmic derivatives L;

j and L0 ;

j which correspond to the same j, and denote K = L;

j

0 L0 ;

j. Then, from (4), K satises the condition (iii) of Lemma 1. Therefore (K;

K) =

0 holds. This and the Schwarz inequality j(A;

K) j2 (A;

A) (K;

K) ;

lead us to (A;

K) =

0 for all A

2 L.

4 From Lemma 4, we can dene uniquely the complex Fisher information matrix J for the family of logarithmic derivatives (4) whose (j;

k) entry is (L;

j;

L;

k) . The SLD is also not uniquely determined for pure state models. Denote by T S() all the SLD'