Introduction to Coherent States and Quantum Information Theory

Transcription

1 Introduction to Coherent States and Quantum Information Theory arxiv:quant-ph/0090v 9 Jan 00 Kazuyuki FUJII Department of Mathematical Sciences Yokohama City University Yokohama JAPAN Abstract The purpose of this paper is to introduce several basic theorems of coherent states and generalized coherent states based on Lie algebras su() and su(,), and to give some applications of them to quantum information theory for graduate students or non experts who are interested in both Geometry and Quantum Information Theory. In the first half we make a general review of coherent states and generalized coherent states based on Lie algebras su() and su(,) from the geometric point of view and, in particular, prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space. We also make a short review of Holonomic Quantum Computation (Computer) and show a geometric construction of the well known Bell states by making use of generalized coherent states. address : fujii@yokohama-cu.ac.jp Home-page :

2 In the latter half we apply a method of generalized coherent states to some important topics in Quantum Information Theory, in particular, swap of coherent states and cloning of coherent ones. We construct the swap operator of coherent states by making use of a generalized coherent operator based on su() and show an imperfect cloning of coherent states, and moreover present some related problems. We also present a problem on a possibility of calculation or approximation of coherent state path integrals on Holonomic Quantum Computer. In conclusion we state our dream, namely, a construction of Geometric Quantum Information Theory.

3 Introduction This paper is the pair to the preceding one [4] and the aim is to introduce geometric aspects of coherent states and generalized coherent ones based on Lie algebras su(, ) and su() and to apply them to quantum information theory for graduate students or non experts (in this field) who are interested in both Geometry and Quantum Information Theory. Coherent states or generalized coherent states play a crucial role in quantum physics, in particular, quantum optics, see [] and its references or []. They also play an important one in mathematical physics, see [3]. For example, they are very useful in performing stationary phase approximations to path integral, see [4], [5] and [6]. In the theory of coherent states or generalized coherent ones the resolution of unity is just a key concept, see []. Is it possible to understand this fact from the geometric point of view? For a set of coherent or generalized coherent states we can define a projector from the manifold consisting of parameters of them to infinite dimensional Grassmann manifold (called classifying spaces in K Theory). Making use of this projector we can calculate several geometric quantities such as Chern characters, see for example [9]. In particular, we prove that each resolution of unity can be obtained by the curvature form of some bundle on the parameter space. Let us turn to Quantum Information Theory (QIT). The main subjects in QIT are (i) Quantum Computation (ii) Quantum Cryptgraphy (iii) Quantum Teleportation As for general introduction to QIT see [], [] and [3], [4]. The aim of this paper is to apply geometric methods to QIT, or more directly A Geometric Construction of Quantum Information Theory.

4 We are developing the theory of geometric quantum computation called Holonomic Quantum Computation, see [0], [], [] and [5] [9], and we are also studying geometric construction of the Bell states or the generalized Bell ones, see [40], [4]. We are interested in geometric method of Homodyne Tomography [4], [5] or geometric one of Quantum Cryptgraphy [6], [7]. On the other hand, the method of path integral plays a very important role in Quantum Mechanics or Quantum Field Theory. However it is not easy to calculate complicated path integrals with classical computers. We are interested in it from the quantum information theory s point of view. That is, can we calculate or approximate some path integral in polynomial times with Quantum Computers (Holonomic Quantum Computer especially)? Unfortunately we cannot answer this question, however we believe this problem becomes crucial for Quantum Computers. By the way it seems to the author that our calculations suggest some profound relation to recent non commutative differential geometry or non commutative field theory, see [8] or [9]. This is very interesting, but beyond the scope of this paper. We show the relation diagramatically Classical Information Theory Classical Geometry Century Quantum Information Theory Quantum Geometry We expect that some readers would develop this subject. In the latter half of this paper we treat special topics in Quantum Information Theory, namely, swap of coherent states and cloning of coherent states. It is not difficult to construct a universal swap operator (see Appendix), however for coherent states we can construct a special and better one by making use of a generalized coherent operator based on su(). On the other hand, to construct a cloning operator is of course not easy by the no cloning theorem [43]. However for coherent states we can make an approximate cloning 3

5 ( imperfect cloning in our terminology) by making use of the same coherent operator based on su(). This and some method in [37] may develop a better approximate cloning method. We also present some related problems on these topics. We have so many problems to be solved in the near future. The author expects strongly that young mathematical physicists or information theorists will take part in this fruitful field. The contents of this paper are as follows : Introduction Coherent States 3 Generalized Coherent States Based on su(, ) 3. General Theory 3. Some Formulas 3.3 A Supplement 3.4 Barut Girardello Coherent States 4 Generalized Coherent States Based on su() 4. General Theory 4. Some Formulas 4.3 A Supplement 5 Schwinger s Boson Method 6 Universal Bundles and Chern Characters 7 Calculations of Curvature Forms 7. Coherent States 7. Generalized Coherent States Based on su(, ) 4

6 7.3 Generalized Coherent States Based on su() 8 Holonomic Quantum Computation 8. One Qubit Case 4. Two Qubit Case 9 Geometric Construction of Bell States 9. Review on General Theory 9. Review on Projective Spaces 9.3 Bell States Revisited 0 Topics in Quantum Information Theory 0. Some Useful Formulas 0. Swap of Coherent States 0.3 Imperfect Cloning of Coherent States 0.4 Swap of Squeezed like States? 0.5 A Comment Path Integral on a Quantum Computer Discussion and Dream Appendix A Proof of Disentangling Formulas B Universal Swap Operator C Calculation of Path Integral D Representation from SU() to SO(3) 5

7 Coherent States We make a review of some basic properties of displacement (coherent) operators within our necessity. For the proofs see [3] or [] Let a(a ) be the annihilation (creation) operator of the harmonic oscillator. If we set N a a (: number operator), then [N, a ] = a, [N, a] = a, [a, a] =. () Let H be a Fock space generated by a and a, and { n n N {0}} be its basis. The actions of a and a on H are given by a n = n n, a n = n + n +, N n = n n () where 0 is a normalized vacuum (a 0 = 0 and 0 0 = ). From () state n for n are given by n = (a ) n n! 0. (3) These states satisfy the orthogonality and completeness conditions m n = δ mn, n n =. (4) n=0 Let us state coherent states. For the normalized state z H for z C the following three conditions are equivalent : (i) a z = z z and z z = (5) (ii) z = e z / z n n = e z / e za 0 (6) n! n=0 (iii) z = e za za 0. (7) In the process from (6) to (7) we use the famous elementary Baker-Campbell-Hausdorff formula e A+B = e [A,B] e A e B (8) whenever [A, [A, B]] = [B, [A, B]] = 0, see [] or [3]. This is the key formula. 6

8 Definition The state z that satisfies one of (i) or (ii) or (iii) above is called the coherent state. The important feature of coherent states is the following resolution (partition) of unity. C [d z] π z z = n=0 n n =, (9) where we have put [d z] = d(rez)d(imz) for simplicity. We note that z w = e z w + zw = z w = e z w, w z = z w, (0) so z w < if z w and z w if z and w are separated enough. We will use this fact in the following. Since the operator D(z) = e za za for z C () is unitary, we call this a displacement (coherent) operator. For these operators the following properties are crucial. For z, w C D(z)D(w) = e z w zw D(w)D(z), () D(z + w) = e (z w zw) D(z)D(w). (3) Here we list some basic properties of this operator. (a) Matrix Elements The matrix elements of D(z) are (i) n m n D(z) m = e z n! m! ( z)m n L n (m n) ( z ), (4) (ii) n m n D(z) m = e z m! n! zn m L m (n m) ( z ), (5) where L n (α) is the associated Laguerre s polynomial defined by ( ) n n + α x L (α) n (x) = ( ) j j j=0 n j j!. (6) In particular L n (0) is the usual Laguerre s polynomial and these are related to diagonal elements of D(z). Here let us list the generating function and orthogonality condition of 7

9 associated Laguerre s polynomials : e xt/( t) ( t) = L (α) α+ n (x)t α for t <, (7) 0 j=0 e x x α L n (α) (x)l m (α) (x)dx = Γ(α + n + ) δ nm for Re(α) >. (8) n! As an interesting application of this formula see the recent [49], or forthcoming [50]. (b) Trace Formula We have This is just a fundamental property. TrU(z) = πδ (z) πδ(x)δ(y) if z = x + iy. (9) (c) Glauber Formula Let A be any observable. Then we have A = C [d z] π Tr[AD (z)]d(z) (0) This formula plays an important role in the field of homodyne tomography, [4] and [5]. (d) Projection on Coherent State The projection on coherent state z is given by z z. But this projection has an interesting expression : z z =: e (a z) (a z) : () where the notation : : means normal ordering. This formula has been used in the field of quantum cryptgraphy, [6] and [7]. We note that z w : e (a z) (a w) : for z, w C with z w. 3 Generalized Coherent States Based on su(,) In this section we introduce some basic properties of generalized coherent operators based on su(, ), see [4] or [3]. 8

10 3. General Theory We consider a spin K (> 0) representation of su(, ) sl(,c) and set its generators {K +, K, K 3 } ((K + ) = K ), [K 3, K + ] = K +, [K 3, K ] = K, [K +, K ] = K 3. () We note that this (unitary) representation is necessarily infinite dimensional. The Fock space on which {K +, K, K 3 } act is H K { K, n n N {0}} and whose actions are K + K, n = (n + )(K + n) K, n +, K K, n = n(k + n ) K, n, K 3 K, n = (K + n) K, n, (3) where K, 0 is a normalized vacuum (K K, 0 = 0 and K, 0 K, 0 = ). We have written K, 0 instead of 0 to emphasize the spin K representation, see [4]. We also denote by K the unit operator on H K. From (3), states K, n are given by K, n = (K +) n n!(k) n K, 0, (4) where (a) n is the Pochammer s notation (a) n a(a + ) (a + n ). These states satisfy the orthogonality and completeness conditions K, m K, n = δ mn, K, n K, n = K. (5) n=0 Now let us consider a generalized version of coherent states : Definition We call a state w e wk + wk K, 0 for w C. (6) the generalized coherent state based on su(, ), [3]. We note that this is the extension of (7) not (5), see [3]. For this the following disentangling formula is well known : e wk + wk = e ζk + e log( ζ )K 3 e ζk or = e ζk e log( ζ )K 3 e ζk +. (7) 9

11 where ζ = ζ(w) wtanh( w ) w (= ζ < ). (8) This is the key formula for generalized coherent operators. Therefore from (3) w = ( ζ ) K e ζk + K, 0 ζ. (9) This corresponds to the right hand side of (6). Moreover since e ζk + K, 0 = we have n=0 ζ n n! Kn + K, 0 = n=0 (K) n n! ζ n K+ (K) n K, 0 = n n! (K) n ζ n K, n n! w = ( ζ ) K (K)n ζ n K, n ζ. (30) n! n=0 This corresponds to the left hand side of (6). Then the resolution of unity corresponding to (9) is C K π tanh( w )[d w] ( tanh ( w ) ) w w w = n=0 C K π n=0 sinh( w )[d w] w w w K [d ζ] = D π ( ζ ) ζ ζ = K, n K, n = K, (3) where C D : w ζ = ζ(w) and D is the Poincare disk in C, see [30]. Here let us construct an example of spin K representations. If we set K + ( ) a, K a, K 3 ( a a + ), (3) then we have [K 3, K + ] = K +, [K 3, K ] = K, [K +, K ] = K 3. (33) That is, the set {K +, K, K 3 } gives a unitary representation of su(, ) with spin K = /4 and 3/4. Now we also call an operator S(w) = e {w(a ) wa } for w C (34) the squeezed operator. 0

12 3. Some Formulas We make some preliminaries for the following section. For that we list some useful formulas on generalized coherent states based ob su(, ). Since the proofs are not so difficult, we leave them to the readers. Formulas For w, w we have { ( ζ )( ζ } K ) (i) w w = ( ζ, (35) ζ ) (ii) w K + w = w w K ζ ζ ζ, (36) (iii) w K w = w w Kζ ζ ζ, (37) where (iv) w K K + w = w w K + 4K ζ ζ ( ζ ζ ). (38) ζ j = w jtanh( w j ) w j When w = w w, then w w =, so we have w K + w = for j =,. (39) K ζ ζ, w K w Kζ ζ, (40) w K K + w = K + 4K ζ Here let us make a comment. From (35) so we want to know the property of ( ζ ). (4) { ( w w ζ )( ζ } K ) = ζ ζ, It is easy to see that ( ζ )( ζ ) ζ ζ. ( ζ )( ζ ) ζ ζ = ζ ζ ζ ζ 0 (4)

13 and (4) = 0 if and only if (iff) ζ = ζ. Therefore { ( w w ζ )( ζ } K ) = ζ ζ (43) because K > (K > 0 from (3)). Of course w w = iff ζ = ζ iff w = w. (44) by (39). 3.3 A Supplement Before ending this section let us make a brief comment on generalized coherent states (6). The coherent states z has been defined by (5) : a z = z z. Why do we define the generalized coherent states w as K w = w w because K is an annihilation operator corresponding to a? First let us try to calculate K w making use of (9). K w = ( ζ ) K K e ζk + K, 0 = ( ζ ) K e ζk + e ζk + K e ζk + K, 0. Here it is easy to see e ζk + K e ζk + = from the relations (), so that n=0 n! [ ζk +, [ ζk +, [,, [ ζk +, K ] ]]] = K + ζk 3 + ζ K +, K w = ( ζ ) K e ζk + (K + ζk 3 + ζ K + ) K, 0 = ζk( ζ ) K e ζk + K, 0 + ζ K + ( ζ ) K e ζk + K, 0 = (Kζ + ζ K + ) w (45) because K K, 0 = 0. Namely we have the equation or more symmetrically (K ζ K + ) w = Kζ w, where ζ = wtanh( w ), (46) w (ζ K ζk + ) w = K w, where ζ = wtanh( w ). (47) w This equation is completely different from (5).

14 3.4 Barut Girardello Coherent States Now let us make a brief comment on Barut Girardello coherent states, [33]. The states w (w C) defined by K w = w w (48) are called the Barut Girardello coherent states. This definition is a natural genelization of (5) because K is an annihilation operator. In the preceding section we denoted by a capital letter K a level of the representation of su(, ). But to avoid some confusion in this subsection we use a small letter k instead of K. The solution is easy to find and given by w = n=0 w n n!(k) n k, n (49) up to the normalization factor. Compare this with (30). Let us determine the inner product. Noting that we have w w = (k) n = n=0 Γ(k + n) Γ(k) ( ww ) n n!(k) n = n=0 = (k) n = ( ww ) n n!(k) n Γ(k) Γ(k + n) w w = Γ(k)( ww ) k+ I k ( ww ), where I ν (z) is the modified Bessel function of the first kind : Therefore ( z ν I ν (z) = ) (z/) n n!γ(ν + n + ). n=0 w = w w / = { Γ(k) w k+ I k ( w ) } /. (50) This gives the normalization factor of (49). Therefore the normalized solution of (48) corresponding to (6) is given by w = { Γ(k) w k+ I k ( w ) } / 3 n=0 w n n!(k) n k, n. (5)

15 Next we show the resolution of unity. C dµ( w, w) w w C K k ( w ) [d w] w w = k, (5) πγ(k) where K ν (z) is the modified Bessel function whose integral representation is given by ( ) π z ν K ν (z) = dye zy (y ) ν /, ν > Γ(ν + /). The proof of (5) is not so easy, so we give it. C dµ( w, w) w w = = = n=0 m=0 n=0 n=0 0 C w n w m dµ( w, w) k, n k, m (53) n!(k) n m!(k) m r n dµ(r) k, n k, n n!(k) n { n!(k) n 0 dµ(r)r n } k, n k, n where we have integrated on θ making use of w = re iθ. Here 0 dµ(r)r n = 4 drr k+n K k (r) Γ(k) 0 = 4 + n + + k k + n + k + Γ(k )Γ( ) Γ(k) 4 Γ(k + n) = Γ(n + ) = (k) n n!, Γ(k) where we have used the famous formula 0 dxx µ K ν (ax) = 4 ( a) For the proof see [34]; Appendix B. C dµ( w, w) w w = n=0 Γ( µ + ν )Γ( µ ν ) a > 0, Reµ > Reν. Therefore we have n!(k) n (k) n n! k, n k, n = k, n k, n = k. n=0 Their states have several interesting structures, but we don t consider them in this paper. See [34], [35] and [36] as for further developments and applications. A comment is in order. Here let us compare two types of coherent states based on Lie algebra su(, ) Perelomov type (section 3.) and Barut Girardello one (section 3.4). The measures satisfying resolution of unity must be positive, so we have 4

16 () Perelomov type K > ( = (3)) () Barut Girardello type K > 0 ( = (5)) 4 Generalized Coherent States Based on su() In this section we introduce some basic properties of generalized coherent operators based on su(), see [4] or [3]. 4. General Theory We consider a spin J (> 0) representation of su() sl(,c) and set its generators {J +, J, J 3 } ((J + ) = J ), [J 3, J + ] = J +, [J 3, J ] = J, [J +, J ] = J 3. (54) We note that this (unitary) representation is necessarily finite dimensional. The Fock space on which {J +, J, J 3 } act is H J { J, m 0 m J} and whose actions are J + J, m = (m + )(J m) J, m +, J J, m = m(j m + ) J, m, J 3 J, m = ( J + m) J, m, (55) where J, 0 is a normalized vacuum (J J, 0 = 0 and J, 0 J, 0 = ). We have written J, 0 instead of 0 to emphasize the spin J representation, see [4]. We also denote by J the unit operator on H J. From (3), states J, m are given by J, m = (J +) m m!j P m J, 0, (56) where J P m = (J)(J ) (J m + ). These states satisfy the orthogonality and completeness conditions J, m J, n = δ mn, J m=0 J, m J, m = J. (57) 5

17 Now let us consider a generalized version of coherent states : Definition We call a state v e vj + vj J, 0 for v C. (58) the generalized coherent state based on su(), [3]. We note that this is the extension of (7) not (5), see [3]. For this the following disentangling formula is well known : e vj + vj = e ηj + e log(+ η )J 3 e ηj or = e ηj e log(+ η )J 3 e ηj +. (59) where η = η(v) vtan( v ) v. (60) This is the key formula for generalized coherent operators. Therefore from (55) v = ( + η ) J eηj + J, 0 η. (6) This corresponds to the right hand side of (6). Moreover since we have e ηj + J, 0 = = m=0 J m=0 η m m! Jm + J, 0 = m=0 JP m ηm J+ m J m! JP m m! J, 0 = m=0 JP m η m J, m m! JC m η m J, m (6) v = ( + η ) J J m=0 JC m η m J, m η. (63) This corresponds to the left hand side of (6). Then the resolution of unity corresponding to (9) is = C C J + π J + π tan( v )[d v] ( + tan ( v )) v C v v = [d η] ( + η ) η η = J m=0 where C C CP : v η = η(v), see [30]. 6 J + π sin( v )[d v] v v v J, m J, m = J, (64)

18 4. Some Formulas We make some preliminaries for the following section. For that we list some useful formulas on generalized coherent states based on su(). Since the proofs are not so difficult, we leave them to the readers. Formulas For v, v we have { ( + η η ) } J (i) v v = ( + η )( + η, (65) ) J η (ii) v J + v = v v, (66) + η η Jη (iii) v J v = v v, (67) + η η where (iv) v J J + v = v v J + 4J η η ( + η η ). (68) η j = v jtan( v j ) v j When v = v v, then v v =, so we have v J + v = for j =,. (69) J η + η, v J Jη v + η, (70) v J J + v = J + 4J η Here let us make a comment. From (65) v v = so we want to know the property of It is easy to see that { ( + η ). (7) + η η ( + η )( + η ) + η η ( + η )( + η ). } J, + η η ( + η )( + η ) = η η ( + η )( + η ) 0 (7) 7

19 and (7) = 0 if and only if (iff) η = η. Therefore { v v + η η } J = ( + η )( + η ) (73) because J > (from (64)). Of course v v = iff η = η iff v = v. (74) by (69). 4.3 A Supplement Before ending this section let us make a brief comment on generalized coherent states (58). The coherent states z has been defined by (5) : a z = z z. Why do we define the generalized coherent states w as J v = v v because J is an annihilation operator corresponding to a? First let us try to calculate J v making use of (6). J v = ( + η ) J J e ηj + J, 0 = ( + η ) J e ηj + e ηj + J e ηj + J, 0. Here it is easy to see e ηj + J e ηj + = from the relations (54), so that m=0 m! [ ηj +, [ ηj +, [,, [ ηj +, J ] ]]] = J ηj 3 η J +, J v = ( + ζ ) J e ηj + (J ηj 3 η J + ) J, 0 = ηj( + η ) J e ηj + J, 0 η J + ( + ζ ) J e ηj + J, 0 = (Jη η J + ) v (75) because J J, 0 = 0 and J 3 J, 0 = J J, 0. Namely we have the equation or more symmetrically (J + η J + ) v = Jη v, where η = vtan( v ), (76) v (η J + ηj + ) v = J v, where η = vtan( v ). (77) v 8

20 This equation is completely different from (5). 5 Schwinger s Boson Method Here let us construct the spin J and K representations making use of Schwinger s boson method. We consider the system of two-harmonic oscillators. If we set a = a, a = a ; a = a, a = a, (78) then it is easy to see [a i, a j ] = [a i, a j ] = 0, [a i, a j ] = δ ij, i, j =,. (79) We also denote by N i = a i a i number operators. Now we can construct representation of Lie algebras su() and su(, ) by making use of Schwinger s boson method, see [4], [5]. Namely if we set su() : J + = a a, J = a a, J 3 = ( ) a a a a, (80) su(, ) : K + = a a, K = a a, K 3 = ( a a + a a + ), (8) then we have su() : [J 3, J + ] = J +, [J 3, J ] = J, [J +, J ] = J 3, (8) su(, ) : [K 3, K + ] = K +, [K 3, K ] = K, [K +, K ] = K 3. (83) In the following we define (unitary) generalized coherent operators based on Lie algebras su() and su(, ). Definition We set su() : U J (v) = e vj + vj for v C, (84) su(, ) : U K (w) = e wk + wk for w C. (85) 9

21 For the latter convenience let us list well-known disentangling formulas once more. We have su() : U J (v) = e ηj + e log(+ η )J 3 e ηj, where η = vtan( v ), v (86) su(, ) : U K (w) = e ζk + e log( ζ )K 3 e ζk, where ζ = wtanh( w ). w (87) For the proof see Appendix A. As for a generalization of these formulas see [3]. Now let us make some mathematical preliminaries for the latter sections. We have easily U J (t)a U J (t) = cos( t )a tsin( t ) a, t (88) U J (t)a U J (t) = cos( t )a + tsin( t ) a, t (89) so the map (a, a ) (U J (t)a U J (t), U J (t)a U J (t) ) is (U J (t)a U J (t), U J (t)a U J (t) ) = (a, a ) cos( t ) tsin( t ) t tsin( t ) t cos( t ). We note that On the other hand we have easily cos( t ) tsin( t ) t tsin( t ) t cos( t ) SU(). U K (t)a U K (t) = cosh( t )a tsinh( t ) a t, (90) U K (t)a U K (t) = cosh( t )a tsinh( t ) a, t (9) so the map (a, a ) (U K (t)a U K (t), U K (t)a U K (t) ) is (U K (t)a U K (t), U K (t)a U K (t) ) = (a, a ) cosh( t ) tsinh( t ) t tsinh( t ) t cosh( t ). We note that cosh( t ) tsinh( t ) t tsinh( t ) t cosh( t ) SU(, ). 0

22 Before ending this section let us ask a question. What is a relation between (85) and (34) of generalized coherent operators based on su(.)? The answer is given by : Formula We have U J ( π 4 )S (w)s ( w)u J ( π 4 ) = U K (w), (9) where S j (w) = (34) with a j instead of a, see [5]. Namely, U K (w) is given by rotating the product S (w)s ( w) by U J ( π 4 ). Proof It is easy to see U J (t)s (w)s ( w)u J (t) = U J (t)e w {(a ) (a ) } w {(a ) (a ) } UJ (t) = e X (93) where X = w w { (UJ (t)a U J (t) ) (U J (t)a U J (t) ) } { (UJ (t)a U J (t) ) (U J (t)a U J (t) ) }. (94) From (88) and (89) we have X = {( w cos ( t ) t sin ) ( ( t ) t (a ) cos ( t ) t sin ) ( t ) t w {( cos ( t ) t sin ) ( ( t ) t a cos ( t ) t sin ) ( t ) t (a ) (t + } t)sin( t ) a t a a (t + } t)sin( t ) a a. t (95) Here we set t = π 4, then X = w (a a ) w (a a ) = wa a wa a = e X = U K (w). Namely, we obtain the formula. Next let us prove the following

23 Formula U J (t)s (α)s (β)u J (t) = U J (t)e { } α (a ) ᾱ (a ) + β (a ) β (a ) U J (t) = e X (96) where From (88) and (89) we have If we set X = α (U J(t)a U J (t) ) ᾱ (U J(t)a U J (t) ) + β (U J(t)a U J (t) ) β (U J(t)a U J (t) ). X = { cos ( t )α + t sin } ( t ) t β (a ) { cos ( t )ᾱ + t sin ( t ) t + { cos ( t )β + t sin } ( t ) t α (a ) { + (βt α t) sin( t ) t then it is easy to check so, in this case, Therefore cos ( t ) β + t sin ( t ) t ᾱ } β a a a ( β t ᾱt) sin( t ) a a. (97) t βt α t = 0 βt = α t, (98) cos ( t )α + t sin ( t ) t β = α, cos ( t )β + t sin ( t ) t α = β, X = α(a ) ᾱa + β(a ) βa. U J (t)s (α)s (β)u J (t) = S (α)s (β). (99) That is, S (α)s (β) commutes with U J (t) under the condition (98). We use this formula in the following. } a

24 6 Universal Bundles and Chern Characters In this section we introduce some basic properties of pull backed ones of universal bundles over the infinite dimensional Grassmann manifolds and Chern characters, see [9]. Let H be a separable Hilbert space over C. For m N, we set St m (H) { V = (v,, v m ) H H V V GL(m;C) }. (00) This is called a (universal) Stiefel manifold. Note that the unitary group U(m) acts on St m (H) from the right : St m (H) U(m) St m (H) : (V, a) V a. (0) Next we define a (universal) Grassmann manifold Gr m (H) { X M(H) X = X, X = X and trx = m }, (0) where M(H) denotes a space of all bounded linear operators on H. Then we have a projection π : St m (H) Gr m (H), π(v ) V (V V ) V, (03) compatible with the action (0) (π(v a) = V a{a (V V ) a}(v a) = π(v )). Now the set {U(m), St m (H), π, Gr m (H)}, (04) is called a (universal) principal U(m) bundle, see [9] and [4]. We set E m (H) {(X, v) Gr m (H) H Xv = v}. (05) Then we have also a projection π : E m (H) Gr m (H), π((x, v)) X. (06) The set {C m, E m (H), π, Gr m (H)}, (07) 3

25 is called a (universal) m th vector bundle. This vector bundle is one associated with the principal U(m) bundle (04). Next let M be a finite or infinite dimensional differentiable manifold and the map P : M Gr m (H) (08) be given (called a projector). Using this P we can make the bundles (04) and (07) pullback over M : { U(m), St, π St, M } P {U(m), St m (H), π, Gr m (H)}, (09) { C m, Ẽ, π Ẽ, M} P {C m, E m (H), π, Gr m (H)}, (0) U(m) U(m) C m C m St St m (H) M P Gr m (H) Ẽ E m (H) M P Gr m (H) see [9]. (0) is of course a vector bundle associated with (09). For this bundle the (global) curvature ( ) form Ω is given by Ω = PdP dp () making use of (08), where d is the usual differential form on Ω. For the bundles Chern characters play an essential role in several geometric properties. In this case Chern characters are given by Ω, Ω,, Ω m/ ; Ω = Ω Ω, etc, () where we have assumed that m = dimm is even. In this paper we don t take the trace of (), so it may be better to call them densities for Chern characters. To calculate these quantities in infinite dimensional cases is not so easy. In the next section let us calculate these ones in the special cases. 4

26 Let us now define our projectors for the latter aim. In the following, for H we treat H = H in section, H = H K in section 3 and H = H J in section 4 at the same time. For u, u,, u m C we consider a set of coherent or generalized coherent states { u, u,, u m } and set V m (u) = ( u, u,, u m ) V m (3) where u = (u, u,, u m ). Since V m V m = ( u i u j ) M(m,C), we define D m {u C m det(v m V m ) 0}. (4) We note that D m is an open set in C m. For example, for m = and m = V V =, det(v a V ) = ā = a 0, where a = u u. So from (0), (44) and (74) we have D = {u C no conditions} = C, (5) D = {(u, u ) C u u }. (6) For D m (m 3) it is not easy for us to give a simple condition like (6). Problem For the case m = 3 make the condition (4) clear like (6). At any rate V m St m (H) for u D m. Now let us define our projector P as follows : P : D m Gr m (H), P(u) = V m (V m V m) V m. (7) In the following we set V = V m for simplicity. Let us calculate (). Since dp = V (V V ) dv { V (V V ) V } + { V (V V ) V }dv (V V ) V (8) where d = m j= ( duj u j + dū j ū j ), we have PdP = V (V V ) dv { V (V V ) V } 5

27 after some calculation. Therefore we obtain PdP dp = V (V V ) [dv { V (V V ) V }dv ](V V ) V. (9) Our main calculation is dv { V (V V ) V }dv, which is rewritten as dv { V (V V ) V }dv = [{ V (V V ) V }dv ] [{ V (V V ) V }dv ] (0) since Q V (V V ) V is also a projector (Q = Q and Q = Q). Therefore the first step for us is to calculate the term { V (V V ) V }dv. () Let us summarize our process of calculations : st step { V (V V ) V }dv (), nd step dv { V (V V ) V }dv (0), 3 rd step V (V V ) [dv { V (V V ) V }dv ](V V ) V (9). 7 Calculations of Curvature Forms In this section we only calculate the curvature forms (m = ). The calculations even for the case m = are complicated enough, see [7] and [8]. For m 3 calculations may become miserable. 7. Coherent States In this case z z =, so our projector is very simple to be P(z) = z z. () In this case the calculation of curvature is relatively simple. From (9) we have PdP dp = z {d z ( z z )d z } z = z z {d z ( z z )d z }. (3) 6

28 Since z = exp( z )exp(za ) 0 by (6), d z = {(a z )dz z } d z z = {a dz } { ( zdz + zd z) z = a dz } d( z ) z, so that ( z z )d z = ( z z )a z dz = (a z a z ) z dz = (a z) dz z because ( z z ) z = 0. Similarly d z ( z z ) = z (a z)d z. Let us summarize : ( z z )d z = (a z) dz z, d z ( z z ) = z (a z)d z. (4) Now we are in a position to determine the curvature form (3). d z ( z z )d z = z (a z)(a z) z d z dz = d z dz after some algebra. Therefore Ω = PdP dp = z z d z dz. (5) From this result we know Ω πi = z z dx dy π when z = x + iy. This just gives the resolution of unity in (9). 7. Generalized Coherent States Based on su(, ) In this case w w =, so our projector is very simple to be P(w) = w w. (6) In this case the calculation of curvature is relatively simple. From (9) we have PdP dp = w {d w ( K w w )d w } w = w w {d w ( K w w )d w }, (7) where d = dw w + d w w. Since w = ( ζ ) K exp(ζk + ) K, 0 by (9), d w = { dζk + + Kdlog( ζ ) } w (8) 7

29 by some calculation, so that ( K w w )d w = ( K w w )K + w dζ = (K + w K + w ) w dζ ( = K + K ζ ) ζ dζ w (9) because ( K w w ) w = 0. Similarly we have d w ( K w w ) = w ( K Now we are in a position to determine the curvature form (7). Kζ ) ζ d ζ (30) d w ( K w w )d w ( = w K Kζ ) ( ζ K + { = = K ζ ) ζ w d ζ dζ w K K + w K ζ ζ w K w Kζ ζ w K + w + 4K ζ ( ζ ) d ζ dζ K ( ζ ) d ζ dζ (3) after some algebra with (40) and (4). Therefore From this result we know Ω πi = K π Ω = PdP dp = w w Kd ζ dζ ( ζ ). (3) dζ dζ ( ζ ) w w = K π by (9) when ζ = ζ + ζ. If we define a constant then we have dζ dζ ( ζ ) ζ ζ C K = K K, (33) Ω C K πi = K dζ dζ π ( ζ ζ ζ. (34) ) This gives the resolution of unity in (3). But the situation is a bit different from [7] in which the constant corresponding to C K is just one. Problem What is a (deep) meaning of C K? } 8

30 7.3 Generalized Coherent States Based on su() In this case v v =, so our projector is very simple to be P(v) = v v. (35) In this case the calculation of curvature is relatively simple. From (9) we have PdP dp = v {d v ( J v v )d v } v = v v {d v ( J v v )d v }, (36) where d = dv v + d v v. Since v = ( + η ) J exp(ηj + ) J, 0 by (6), by some calculation, so that d v = { dηj + Jdlog( + η ) } v (37) ( J v v )d v = ( J v v )J + v dη = (J + v J + v ) v dη ( = J + J η ) + η dη v (38) because ( J v v ) v = 0. Similarly we have ( d v ( J v v ) = v J Now we are in a position to determine the curvature form (36). d v ( J v v )d v ( = v J Jη ) ( + η J + { = = v J J + v J η ) + η v d η dη Jη ) + η d η (39) J η + η v J v Jη + η v J + v + 4J η ( + η ) } d η dη J ( + η dη (40) ) d η after some algebra with (70) and (7). Therefore From this result we know Ω πi = J π Jd η dη Ω = PdP dp = v v ( + η ). (4) dη dη ( + η ) v v = J π 9 dη dη ( + η ) η η

31 by (6) when η = η + η. If we define a constant then we have C J = J + J, (4) Ω C J πi = J + dη dη π ( + η η η. (43) ) This gives the resolution of unity in (64). But the situation is a bit different from [7] in which the constant corresponding to C J is just one. Problem What is a (deep) meaning of C J? 8 Holonomic Quantum Computation In this section we introduce the concept of Holonomic Quantum Computation, see [9]. Let M be a parameter space and we denote by λ its element. Let λ 0 be a fixed reference point of M. Let H λ be a family of Hamiltonians parameterized by M which act on a Fock space H. We set H 0 = H λ0 for simplicity and assume that this has a m-fold degenerate vacuum : H 0 v j = 0, j = m. (44) These v j s form a m-dimensional vector space. We may assume that v i v j = δ ij. Then (v,, v m ) St m (H) and m F 0 x j v j x j C = C m. Namely, F 0 is a vector space associated with o.n.basis (v,, v m ). j= Next we assume for simplicity that a family of unitary operators parameterized by M W : M U(H), W(λ 0 ) = id. (45) is given and H λ above is given by the following isospectral family H λ W(λ)H 0 W(λ). (46) 30

32 In this case there is no level crossing of eigenvalues. Making use of W(λ) we can define a projector m P : M Gr m (H), P(λ) W(λ) v j v j W(λ) (47) and have the pullback bundles over M from (09) and (0) j= { U(m), St, π St, M }, { C m, Ẽ, π Ẽ, M}. (48) For the latter we set vac = (v,, v m ). (49) In this case a canonical connection form A of { U(m), St, π St, M } is given by A = vac W(λ) dw(λ) vac, (50) where d is a usual differential form on M, and its curvature form by F da + A A, (5) see [0] and [9]. Let γ be a loop in M at λ 0., γ : [0, ] M, γ(0) = γ(). For this γ a holonomy operator Γ A is defined as the path ordered integral of A along γ : { } Γ A (γ) = Pexp A γ U(m), (5) where P means path-ordered. See [9]. This acts on the fiber F 0 at λ 0 of the vector bundle { C m, Ẽ, π, Ẽ M} as follows : x Γ A (γ)x. The holonomy group Hol(A) is in general subgroup of U(m). In the case of Hol(A) = U(m), A is called irreducible, see [9]. In the Holonomic Quantum Computation we take Encoding of Information = x F 0, Processing of Information = Γ A (γ) : x Γ A (γ)x. (53) 3

33 F 0 AX 3 X 3 λ 0 3 M Quantum Computational Bundle 8. One Qubit Case Let H 0 be a Hamiltonian with nonlinear interaction produced by a Kerr medium., that is H 0 = hxn(n ), (54) where X is a certain constant, see [] and []. The eigenvectors of H 0 corresponding to 0 is { 0, }, so its eigenspace is Vect { 0, } = C. The vector space Vect { 0, } is called -qubit (quantum bit) space and we set F 0 = Vect { 0, } and vac = ( 0, ). 3

34 Now we consider the following isospectral family of H 0 : H (α,β) = W(α, β)h 0 W(α, β), (55) W(α, β) = D(α)S(β). (56) In this case and we want calculate where M = { (α, β) C } (57) A = vac W dw vac (58) d = dα α + dᾱ ᾱ + dβ β + d β. (59) β Since A is anti hermitian (A = A), we can write A = A α dα + A β dβ A α dᾱ A β d β (60) where A α = vac W W α vac The calculation of A α and A β is as follows ([5]) : A α = ᾱ A β = β( + cosh( β )) 4 β A β = vac W W β vac. βsinh( β ) L + cosh( β )F + E, β (6) (K + ) L, (6) where E = 0, F = 0, K = 0, L =. Then making use of these ones we can show that the holonomy group generated by (58) is irreducible in U(), namely just U(), see [] and [5]. This is very crucial fact to Holonomic Quantum Computation. 33

35 8. Two Qubit Case We consider the system of two particles, so the Hamiltonian that we treat in the following is H 0 = hxn (N ) + hxn (N ). (63) The eigenspace of 0 of this Hamiltonian becomes therefore F 0 = Vect { 0, } Vect { 0, } = Vect { 0, 0, 0,,, 0,, } = C 4. (64) We set vac = ( 0, 0, 0,,, 0,, ). Next we consider the following isospectral family of H 0 : H (α,β,λ,µ,α,β ) = W(α, β, λ, µ, α, β )H 0 W(α, β, λ, µ, α, β ), (65) W(α, β, λ, µ, α, β ) = W (α, β )O (λ, µ)w (α, β ). (66) where O (λ, µ) = U J (λ)u K (µ), W j (α j, β j ) = D j (α j )S j (β j ) for j =,. (67) O 7 7 W W In this case and we have only to calculate the following M = { (α, β, λ, µ, α, β ) C 6} (68) A = vac W dw vac, (69) where d = dα + dᾱ + dβ + d α ᾱ β β β + dλ λ + d λ λ + dµ µ + d µ µ + dα + dᾱ + dβ + d α ᾱ β β β. (70) 34

36 The calculation of (69) is not easy, but we can determine it, see [5], [6] and [8] for the details. But we cannot determine its curvature form which is necessary to look for the holonomy group (Ambrose Singer theorem) due to too complication. Then the essential point is Problem Is the connection form (69) irreducible in U(4)? Our analysis in [8] shows that the holonomy group generated by A may be SU(4) not U(4). To obtain U(4) a sophisticated trick higher dimensional holonomies [44] may be necessary. See also [3]. 9 Geometric Construction of Bell States In this section we introduce the geometric constraction of Bell states by making use of coherent states based on su(), [40]. One of purpose of Quantum Information Theory is to clarify a role of entanglement of states, so that we would like to look for geometric meaning of entanglement. The famous Bell states ([45], [46]) given by ( ), (7) ( 0 0 ), (7) ( ), (73) ( 0 0 ) (74) are typical examples of entanglement. It is very interesting that these play an essential role in Quantum Teleportation, see []. We would like to reconstruct these states in a geometric manner. 35

37 9. Review on General Theory Let us make a review of [4] and rewrite the result with our method. Let G be a compact linear Lie group (for example G = U(n)) and consider a coherent state representation of G whose parameter space is a compact complex manifold S = G/H, where H is a subgroup of G. For example G = U(n) and H = U(k) U(n k), then S = U(n)/U(k) U(n k) = G k (C n ), which is called a complex Grassmann manifold, [3], [6]. Let Z be a local coordinate on S and Z a generalized coherent state in some representation space V ( = C K for some high K N). Then we have, by definition, the measure dµ(z, Z ) that satisfies the resolution of unity dµ(z, Z ) Z Z = V S and S dµ(z, Z ) = dimv. (75) Next we define an anti-automorphism : S S. We call Z Z an antiautomorphism if and only if (i) Z Z induces an automorphism of S, (76) (ii) is an anti-map, namely Z W = W Z. (77) Now let us redefine the generalized Bell state in [4] as follows : Definition The generalized Bell state is defined as Then we have B B = dimv = dimv = dimv = dimv = dimv B = S S S S S S where we have used (75) and (77). S S dimv S dµ(z, Z ) Z Z. (78) dµ(z, Z )dµ(w, W )( Z Z )( W W ) dµ(z, Z )dµ(w, W ) Z W Z W dµ(z, Z )dµ(w, W ) Z W W Z dµ(z, Z ) Z Z dµ(z, Z ) =, 36

38 9. Review on Projective Space We make a review of complex projective spaces, [9], [5] and [8]. For N N the complex projective space CP N is defined as follows : For ζ, µ C N+ {0} ζ is equivalent to µ (ζ µ) if and only if ζ = λ µ for λ C {0}. We show the equivalent relation class as [ζ] and set CP N C N+ {0}/. When ζ = (ζ 0, ζ,, ζ N ) we write usually as [ζ] = [ζ 0 : ζ : : ζ N ]. Then it is well known that CP N has N + local charts, namely Since N CP N = U j, U j = {[ζ 0 : : ζ j : : ζ N ] ζ j 0}. (79) j=0 ( ζ0 (ζ 0,, ζ j,, ζ N ) = ζ j,, ζ j,, ζ j+,, ζ ) N, ζ j ζ j ζ j ζ j we have the local coordinate on U j ( ζ0,, ζ j, ζ j+,, ζ ) N. (80) ζ j ζ j ζ j ζ j However the above definition of CP N is not tractable, so we use the well known expression by projections CP N = G (C N+ ) = {P M(N + ;C) P = P, P = P and trp = } (8) and the correspondence [ζ 0 : ζ : : ζ N ] ζ 0 + ζ + + ζ N ζ 0 ζ 0 ζ ζ 0 ζn ζ ζ0 ζ ζ ζn ζ N ζ0 ζ N ζ ζ N P. (8) If we set ζ = Nj=0 ζ j ζ 0 ζ ζ N, (83) 37

39 then we can write the right hand side of (8) as For example on U (z, z,, zn) = we have where P = ζ ζ and ζ ζ =. (84) ( ζ, ζ,, ζ ) N, ζ 0 ζ 0 ζ 0 z z N z z z z N P(z,, z N ) = + N j= z j z N z N z z N = (z, z,, z N ) (z, z,, z N ), (85) (z, z,, z N ) = + N j= z j Let us give a more detail description for the cases N = and. z z N. (86) (a) N = : P(z) = P(w) = + z where z = w + where w = z z z = z z, + z w w w z = w w, w +, z = ζ ζ 0, on U, (87) w, w = ζ 0 ζ, on U. (88) 38

40 (b) N = : P(z, z ) = z z + z + z z z z z = (z, z ) (z, z ), z z z z where ( ζ (z, z ) = + z + z z, (z, z ) =, ζ ) ζ 0 ζ 0 on U, (89) z P(w, w ) = w w w w w + + w w w = (w, w ) (w, w ), w w w w where w ( ζ0 (w, w ) = w + + w, (w, w ) =, ζ ) ζ ζ on U (90), w P(v, v ) = v v v v v + v + v v v v = (v, v ) (v, v ), v v where v ( ζ0 (v, v ) = v + v v, (v, v ) =, ζ ) + ζ ζ on U 3. (9) 9.3 Bell States Revisited In this subsection we show that (78) coinsides with the Bell states (7) (74) by choosing anti-automorphism suitably. We treat first of all the case of spin. From here we identify 0 = ( ) 0 and = ( ) 0, 39

41 so we have η = + η ( 0 + η ) =. (9) + η η In this case we consider the following four anti-automorphisms (76) and (77) : () η = η () η = η (3) η = η (4) η = η. (93) Now by making use of these we define Definition where we have put for simplicity () B = () B = (3) B = (4) B = It is easy to see from (87) and (88) Then making use of elementary facts C C C C dµ(η, η) = π dµ(η, η) η η, (94) dµ(η, η) η η, (95) dµ(η, η) η / η, (96) dµ(η, η) η / η, (97) [d η] ( + η ). () η = η = + η ), + η ( 0 (98) () η = η = η ), + η ( 0 (99) (3) η = / η = + ), + η ( η 0 (00) (4) η = / η = + ), + η ( η 0 (0) [d η] π C ( + η ) + η = π π C [d η] ( + η ) η + η = π 40 C C [d η] η ( + η ) + η =, [d η] ( + η ) η + η = 0,

42 we obtain easily () B = ( ), (0) () B = ( 0 0 ), (03) (3) B = ( ), (04) (4) B = ( 0 0 ). (05) We just recovered the Bell states (7) (74)!! We can say that four B in Definition are overcomplete expression (making use of generalized coherent states) of the Bell states. This is an important point of view. Since we consider the case of higher spin J, we write η as η J = ( + η ) J J k=0 JC k η k k (06) to emphasize the dependence of spin J. Here we have set k = J, k for simplicity. From the above result it is very natural to define Bell states with spin J as follows because the parameter space is the same CP : Definition where () B = () B = (3) B = (4) B = J + J + J + J + C C C C dµ(η, η) = J + π dµ(η, η) η J η J, (07) dµ(η, η) η J η J, (08) dµ(η, η) η J / η J, (09) dµ(η, η) η J / η J, (0) [d η] ( + η ). Let us calculate η J, η J, / η J and / η J. It is easy to see () η J = ( + η ) J J k=0 JC k η k k, () 4

43 () η J = (3) / η J = (4) / η J = ( + η ) J ( + η ) J J k=0 ( + η ) J From this lemma and the elementary facts J + π C J k=0 JC k ( ) k η k k, () JC k η k J k, (3) J k=0 JC k ( ) k η k J k. (4) [d η] η k ( + η ) ( + η ) = for 0 k J, J JC k we can give explicit forms to the Bell states with spin J : () B = J + J k=0 k k, (5) J () B = ( ) k k k, (6) J + (3) B = k=0 J + J k=0 k J k, (7) J (4) B = ( ) k k J k. (8) J + k=0 We obtained the Bell states with spin J which are a natural extension of usual ones (J = /). A comment is in order. For the case J = : () () (3) (4) 3 ( ), 3 ( ), 3 ( ), 3 ( ). It is easy to see that they are not linearly independent, so that only this case is very special (peculiar). Comment We cannot give a geometric construction of Bell states by making use of generalized coherent states based on su(, ) (Lie algebra of non compact Lie group). 4

44 Because the parameter space in this case is a Poincare disk D = {ζ C ζ < } and the measure on it is given by see (3). Therefore we have D dµ(ζ, ζ) = K π dµ(ζ, ζ) = (K ) 0 [d ζ] ( ζ ), dr ( r) =! Compare this with (75). This is a reason why we cannot determine a normalization. 0 Topics in Quantum Information Theory In this section we don t introduce a general theory of quantum information theory (see for example []), but focus our attension to special topics of it, that is, swap of coherent states cloning of coherent states Because this is just a good one as examples of applications of coherent and generalized coherent states and our method developed in the following may open a new possibility. First let us define a swap operator : S : H H H H, S(a b) = b a for any a, b H (9) where H is the Fock space in Section. It is not difficult to construct this operator in a universal manner, see Appendix B. But for coherent states we can construct a better one by making use of generalized coherent operators in the preceding section. Next let us introduce no cloning theorem, [43]. For that we define a cloning (copying) operator C which is unitary C : H H H H, C(h 0 ) = h h for any h H. (0) 43

45 It is very known that there is no cloning theorem No Cloning Theorem We have no C above. The proof is very easy (almost trivial). Because h = h + h H and C is a linear operator, so C(h 0 ) = C(h 0 ). () The LHS of () is C(h 0 ) = h h = 4(h h), while the RHS of () C(h 0 ) = (h h). This is a contradiction. This is called no cloning theorem. Let us return to the case of coherent states. For coherent states α and β the superposition α + β is no longer a coherent state, so that coherent states may not suffer from the theorem above. Problem Is it possible to clone coherent states? At this stage it is not easy, so we will make do with approximating it (imperfect cloning in our terminology) instead of making a perfect cloning. We write notations once more. Coherent States α = D(α) 0 for α C Squeezed like States β = S(β) 0 for β C 0. Some Useful Formulas We list and prove some useful formulas in the following. Now we prepare some parameters α, ǫ, κ in which ǫ, κ are free ones, while α is unknown one in the cloning case. Let us 44

46 unify the notations as follows. α : (unknown) α = α e iχ, () ǫ : known ǫ = ǫ e iφ, (3) κ : known κ = κ e iδ, (4) Let us start. (i) First let us calculate S(ǫ)D(a)S(ǫ). (5) For that we show S(ǫ)aS(ǫ) = cosh( ǫ )a e iφ sinh( ǫ )a. (6) Proof is as follows. For X = (/){ǫ(a ) ǫa } we have easily [X, a] = ǫa and [X, a ] = ǫa, so S(ǫ)aS(ǫ) = e X ae X = a + [X, a] +! [X, [X, a]] + [X, [X, [X, a]]] + 3! From this it is easy to check = a ǫa + ǫ! a ǫ ǫ a + 3! } = { + ǫ + a ǫ } { ǫ + ǫ 3 +! ǫ 3! = cosh( ǫ )a ǫsinh( ǫ ) a = cosh( ǫ )a e iφ sinh( ǫ )a. ǫ S(ǫ)D(α)S(ǫ) = D ( αs(ǫ)a S(ǫ) ᾱs(ǫ)as(ǫ) ) = D ( cosh( ǫ )α + e iφ sinh( ǫ )ᾱ ). (7) a Therefore D(e ǫ α) if φ = χ S(ǫ)D(α)S(ǫ) = D(e ǫ α) if φ = χ + π By making use of this formula we can change a scale of α. (8) (ii) Next le us calculate S(ǫ)S(α)S(ǫ). (9) 45

47 From the definition { ( S(ǫ)S(α)S(ǫ) = S(ǫ)exp α(a ) ᾱa )} S(ǫ) e Y/ where Y = α ( S(ǫ)a S(ǫ) ) ᾱ ( S(ǫ)aS(ǫ) ). From (6) and after some calculations we have Y = { cosh ( ǫ )α e iφ sinh ( ǫ )ᾱ } (a ) { cosh ( ǫ )ᾱ e iφ sinh ( ǫ )α } a + ( e iφ α + e iφ ᾱ) sinh( ǫ )(a a + aa ) = { cosh ( ǫ )α e iφ sinh ( ǫ )ᾱ } (a ) { cosh ( ǫ )ᾱ e iφ sinh ( ǫ )α } a + ( e iφ α + e iφ ᾱ)sinh( ǫ )(a a + ) ( = [a, a ] = ), or Y = { cosh ( ǫ )α e iφ sinh ( ǫ )ᾱ } K + { cosh ( ǫ )ᾱ e iφ sinh ( ǫ )α } K + ( e iφ α + e iφ ᾱ)sinh( ǫ )K 3 (30) with {K +, K, K 3 } in (3). This is our formula. Now e iφ α + e iφ ᾱ = α ( e i(φ χ) + e i(φ χ) ) = i α sin(φ χ), so if we choose φ = χ, then e iφ ᾱ = e iχ e iχ α = α and cosh ( ǫ )α e iφ sinh ( ǫ )ᾱ = ( cosh ( ǫ ) sinh ( ǫ ) ) α = α, and finally Y = α(a ) ᾱa. That is, S(ǫ)S(α)S(ǫ) = S(α) S(ǫ)S(α) = S(α)S(ǫ). The operators S(ǫ) and S(α) commute if the phases of ǫ and α coincide. 46

48 (iii) Third formula is : For V (t) = e itn where N = a a (a number operator) V (t)d(α)v (t) = D(e it α). (3) The proof is as follows. V (t)d(α)v (t) = exp ( αv (t)a V (t) ᾱv (t)av (t) ). It is easy to see V (t)av (t) = e itn ae itn = a + [itn, a] + [itn, [itn, a]] +! = a + ( it)a + ( it) a +! = e it a. Therefore we obtain V (t)d(α)v (t) = exp ( αe it a ᾱe it a ) = D(e it α). This formula is often used as follows. α V (t) α = V (t)d(α)v (t) V (t) 0 = D(e it α) 0 = e it α, (3) where we have used V (t) 0 = 0 becase N 0 = 0. That is, we can add a phase to α by making use of this formula. 0. Swap of Coherent States The purpose of this section is to construct a swap operator satifying α α α α. (33) Let us remember U J (κ) once more U J (κ) = e κa a κa a for κ C. 47

49 We note an important property of this operator : U J (κ) 0 0 = 0 0. (34) The construction is as follows. U J (κ) α α = U J (κ)d(α ) D(α ) 0 0 = U J (κ)d (α )D (α ) 0 0 = U J (κ)d (α )D (α )U J (κ) U J (κ) 0 0 = U J (κ)d (α )D (α )U J (κ) 0 0 by (34), (35) and U J (κ)d (α )D (α )U J (κ) = U J (κ)exp { α a ᾱ a + α a ᾱ a } UJ (κ) = exp { α (U J (κ)a U J (κ) ) ᾱ U J (κ)a U J (κ) +α (U J (κ)a U J (κ) ) ᾱ U J (κ)a U J (κ) } exp(x). (36) From (88) and (89) we have { X = cos( κ )α + κsin( κ ) { α }a cos( κ )ᾱ + κsin( κ ) ᾱ }a κ κ { + cos( κ )α κsin( κ ) } { α a cos( κ )ᾱ κsin( κ ) ᾱ }a, κ κ so ( exp(x) = D cos( κ )α + κsin( κ ) ) ( α D cos( κ )α κsin( κ ) ) α κ κ ( = D cos( κ )α + κsin( κ ) ) ( α D cos( κ )α κsin( κ ) ) α. κ κ Therefore we have from (36) α α cos( κ )α + κsin( κ ) κ If we write κ as κ e iδ, then the above formula reduces to α cos( κ )α κsin( κ ) α. κ α α cos( κ )α + e iδ sin( κ )α cos( κ )α e iδ sin( κ )α. 48

50 Here if we choose sin( κ ) =, then α α e iδ α e iδ α = e iδ α e i(δ+π) α. Now by operating the operator V = e iδn e i(δ+π)n where N = a a from the left (see (3)) we obtain the swap α α α α. A comment is in order. In the formula we set α = α and α = 0, then the formula reduces to U J (κ)d (α)u J (κ) = D (cos( κ )α)d ( e iδ sin( κ )α). (37) 0.3 Imperfect Cloning of Coherent States We cannot clone coherent states in a perfect manner likely α 0 α α for α C. (38) Then our question is : is it possible to approximate? We show that we can at least make an imperfect cloning in our terminology against the statement of [37]. Let us start. The method is almost same with one in the preceding subsection, but we repeat it once more. Operating the operator U J (κ) on α 0 U J (κ) α 0 = U J (κ) {D(α) } 0 0 = U J (κ)d (α) 0 0 =U J (κ)d (α)u J (κ) U J (κ) 0 0 = U J (κ)d (α)u J (κ) 0 0 by (34) =D (cos( κ )α)d ( e iδ sin( κ )α) 0 0 by (37) =D (cos( κ )α)d (e i(δ+π) sin( κ )α) 0 0 = { D(cos( κ )α) D(e i(δ+π) sin( κ )α) } 0 0. Operating the operator e i(δ+π)n on the last equation D(cos( κ )α) e i(δ+π)n D(e i(δ+π) sin( κ )α)

51 =D(cos( κ )α) e i(δ+π)n D(e i(δ+π) sin( κ )α)e i(δ+π)n e i(δ+π)n 0 0 =D(cos( κ )α) e i(δ+π)n D(e i(δ+π) sin( κ )α)e i(δ+π)n 0 0 =D(cos( κ )α) D(e i(δ+π) sin( κ )αe i(δ+π) ) 0 0 by (3) =D(cos( κ )α) D(sin( κ )α) 0 0 = cos( κ )α sin( κ )α. Namely we have constructed α 0 cos( κ )α sin( κ )α. (39) This is an imperfect cloning what we have called. A comment is in order. The authors in [37] state that the perfect cloning (in their terminology) for coherent states is possible. But it is not correct as shown below. Their method is very interesting, so let us introduce it. Before starting let us prepare a notation for simplicity (7) : S(ǫ)D(α)S(ǫ) = D( α), α cosh( ǫ )α + e iφ sinh( ǫ )ᾱ. Operating the operator S(ǫ) S(e iδ ) from the left S(ǫ) S(e iδ ) α 0 = { S(ǫ) S(e iδ ) } {D(α) } 0 0 = S(ǫ)D(α) S(e iδ ) 0 0 = S(ǫ)D(α)S(ǫ) S(ǫ) S(e iδ ) 0 0 = D( α)s(ǫ) S(e iδ ) 0 0 = {D( α) } { S(ǫ) S(e iδ ) } 0 0 = D ( α) { S(ǫ) S(e iδ ) } 0 0. Operating the operator U J (κ) (remember that κ = κ e iδ ) from the left U J (κ)d ( α) { S(ǫ) S(e iδ ) } 0 0 =U J (κ)d ( α) { S(ǫ) S(e iδ ) } U J (κ) U J (κ)