Chapter IIObservables and States in TensorProducts of Hilbert Spaces15 Positive deﬁnite kernels and tensor products of Hilbert spacesSuppose (i , Ᏺi ), 1 i n are sample spaces describing the elementary out-comes and events concerning n different statistical systems in classical proba-bility. To integrate them into a uniﬁed picture under the umbrella of a singlesample space one takes their cartesian product (, Ᏺ) where = 1 2 1 1 1 2 n ,Ᏺ = Ᏺ1 2 1 1 1 2 Ᏺn , the smallest -algebra containing all rectangles of the formF1 2 F2 21 1 12 Fn , Fj 2 Ᏺj for each j . Now we wish to search for an analogue ofthis description in quantum probability when we have n systems where the eventsconcerning the j -th system are described by the set P (Ᏼj ) of all projections ina Hilbert space Ᏼj , j = 1, 2, . . . , n. Such an attempt leads us to consider tensorproducts of Hilbert spaces. We shall present a somewhat statistically oriented ap-proach to the deﬁnition of tensor products which is at the same time coordinatefree in character. To this end we introduce the notion of a positive deﬁnite kernel. X Let ᐄ be any set and let K : ᐄ 2 ᐄ ! ރbe a map satisfying the following: i j K (xi , xj ) 0 i,jfor all i 2 ,ރxi 2 ᐄ, i = 1, 2, . . . , n. Such a map K is called a positive deﬁnitekernel or simply a kernel on ᐄ. We denote by K(ᐄ) the set of all such kernels onᐄ. If ᐄ = f1, 2, . . . , ng, a kernel on ᐄ is just a positive (semi) deﬁnite matrix.If Ᏼ is a Hilbert space the scalar product K (x, y ) = hx, y i is a kernel on Ᏼ. If Gis a group and g ! Ug is a homomorphism from G into the unitary group ᐁ(Ᏼ)of a Hilbert space Ᏼ then K (g , h) = hu, Ug01 h ui is a kernel on G for every u inᏴ. If is a state in Ᏼ then K (X , Y ) = tr X 3 Y is a kernel on Ꮾ(Ᏼ).Proposition 15.1: Let ((aij )), ((bij )), 1 i, j n be two positive deﬁnitematrices. Then ((aij bij )) is positive deﬁnite.Proof: Let A = ((aij )). Choose any matrix C of order n such that CC 3 = A. Let1 , . . . , n ,
92 Chapter II: Observables and States in Tensor Products of Hilbert Spacesfor 1 i, j n. Let j = j j . Then ޅj = 0, ޅi j = aij bij . Thus ((aij bij ))is the covariance matrix of the complex random vector and hence positivedeﬁnite.Corollary 15.2: The space K (ᐄ) of all kernels on ᐄ is closed under pointwisemultiplication.Proof: This follows immediately from Proposition 15.1 and the fact that K isa kernel on ᐄ if and only if for any ﬁnite set fx1 , x2 , . . . , xn g ᐄ the matrix((aij )) where aij = K (xi , xj ) is positive deﬁnite.Corollary 15.3: Let ᐄi , 1 i n be sets and let Ki 2 K (ᐄ ) for each i. Deﬁne iᐄ = ᐄ1 2 1 1 1 2 ᐄn and ( ) = 5 =1 K (x , y ), x = (x1 , . . . , x ), y = (y1 , . . . , y ) K x, y n i i i i n nwhere x , y 2 ᐄ . Then K 2 K (ᐄ). i i iProof: Let x( ) = (x1 , x2 , . . . , x ) 2 ᐄ, 1 r m. Putting a = r r r nr i rsK (x , x ) we observe that i ir is K (x , x ) = 5 =1 a . ( ) ( ) r s n i i rsSince ((a )), 1 r, s m is positive deﬁnite for each ﬁxed i the required result i rsfollows from Proposition 15.1.Proposition 15.4: Let ᐄ be any set and let K 2 K (ᐄ). Then there exists a(not necessarily separable) Hilbert space Ᏼ and a map : ᐄ ! Ᏼ satisfying thefollowing: (i) the set f(x), x 2 ᐄg is total in Ᏼ; (ii) K (x, y ) = h(x), (y )i forall x, y in ᐄ. If Ᏼ0 is another Hilbert space and 0 : ᐄ ! Ᏼ0 is another map satisfy-ing (i) and (ii) with Ᏼ, replaced by Ᏼ0 , 0 respectively then there is a unitaryisomorphism U : Ᏼ ! Ᏼ0 such that U (x) = 0 (x) for all x in ᐄ.Proof: For any ﬁnite set F = fx1 , x2 , . . . , xn g ᐄ it follows from the argumentin the proof of Proposition 15.1 that there exists a complex Gaussian randomvector (1 , 2 , . . . , n ) such that F F F = K (x , x ), 1 ,0 = ޅi, j n. F F ޅi j i j F iIf G = fx1 , x2 , . . . , x , x +1 g F then the marginal distribution of (1 , . . . , ) n n G G nderived from that of (1 , . . . , , +1 ) is the same as the distribution of (1 , G G n G F. . . , ). Hence by Kolmogorov’s Consistency Theorem there exists a Gaussian n F nfamily f , x 2 ᐄg of complex valued random variables on a probability space x(, Ᏺ, P ) such that = ޅ ,0 = ޅK (x, y ) for all x, y 2 ᐄ. x x yIf Ᏼ is the closed linear span of f , x 2 ᐄg in L2 (P ) and (x) = then the x xﬁrst part of the proposition holds.
15 Positive deﬁnite kernels and tensor products of Hilbert spaces 93 To prove the second part consider S = f(x)jx 2 ᐄg, S 0 = f0 (x)jx 2 ᐄgand the map U : (x) ! 0 (x) from S onto S 0 . Then U is scalar productpreserving and S and S 0 are total in Ᏼ and Ᏼ0 respectively. By the obviousgeneralisation of Proposition 7.2 for not necessarily separable Hilbert spaces, Uextends uniquely to a unitary isomorphism from Ᏼ onto Ᏼ0 . The pair (Ᏼ, ) determined uniquely up to a unitary isomorphism by thekernel K on ᐄ is called a Gelfand pair associated with K . We are now ready to introduce the notion of tensor products of Hilbertspaces using Proposition 15.4. Let Ᏼi , 1 i n be Hilbert spaces and let ᐄ =Ᏼ1 , 2 1 1 1 2 Ᏼn be their cartesian product as a set. Then the function Ki (u, v ) =hu, v i, u, v 2 Ᏼi is a kernel on Ᏼi for each i. By Corollary 15.3 the function K (u, v ) = 5n=1 hui , vi i, u = (u1 , . . . , un ), v = (v1 , . . . , vn ) iwhere ui , vi 2 Ᏼi for each i, is a kernel on ᐄ. Consider any Gelfand pair (Ᏼ, )associated with K and satisfying (i) and (ii) of Proposition 15.4. Then Ᏼ is calleda tensor product of Ᏼi , i = 1, 2, . . . , n. We write Ᏼ = Ᏼ1 Ᏼ2 1 1 1 Ᏼn O = n Ᏼi , (15.1) O = i 1 n (u) = u1 u2 1 1 1 u = u n i (15.2) i =1and call (u) the tensor product of the vectors ui , 1 i n. If Ᏼi = h for all ithen Ᏼ is called the n-fold tensor product of h and denoted by h . If, in addition, nui = u for all i in (15.2) then (u) is denoted by u and called the n-th power nof u. (Since (Ᏼ, ) is determined uniquely upto a Hilbert space isomorphism wetake the liberty of calling Ᏼ the tensor product of Ᏼi , 1 i n in (15.1)).Proposition 15.5: The map (u1 , u2 , . . . , un ) ! u1 u2 1 1 1 un from Ᏼ1 2Ᏼ2 2 1 1 1 2 Ᏼn into Ᏼ1 Ᏼ2 1 1 1 Ᏼn deﬁned by (15.1) and (15.2) is multilinear:for all scalars ,
u1 1 1 1 u 01 v n i i ui+1 1 1 1 un (15.4)Furthermore h Ou Ov n , n i = 5n 1 hui , vi i. i= (15.5) N= u u N= i i =1 i =1 i 2 Ᏼi , i = 1, 2, . . . , ng is total in n nThe set f i 1 ij i i 1 Ᏼi .
94 Chapter II: Observables and States in Tensor Products of Hilbert SpacesProof: Only (15.3) remains to be proved. Straightforward computation using O(15.4) and the sesquilinearity of scalar products show that for each ﬁxed i n ku1 1 1 1 ui01 (ui +
vi i)g N= = 0. n For any ui 2 Ᏼi , 1 i n the product vector i 1 ui may be interpreted Oas the multi-antilinear functional n ( ui )(v1 , v2 , . . . , vn ) = 5n 1 hvi , ui i. i= =1 iSuch multi-antilinear functionals generate a linear manifold M to which the scalarNproduct (15.4) can be extended by sesquilinearity to make it a pre-Hilbert space.M is the usual algebraic tensor product of the vector spaces Ᏼi , 1 i n and n i=1 Ᏼi is its completion.Exercise 15.6: (i) Let K be a kernel on ᐄ. A bijective map g : ᐄ ! ᐄ issaid to leave K invariant if K (g (x), g (y )) = K (x, y ) for all x, y in ᐄ. Let GKdenote the group of all such bijective transformations of ᐄ leaving K invariantand let (Ᏼ, ) be a Gelfand pair associated with K . Then there exists a uniquehomomorphism g ! Ug from GK into the unitary group ᐁ(Ᏼ) of Ᏼ satisfyingthe relation Ug (x) = (g (x)) for all x 2 ᐄ, g 2 GK .If (Ᏼ0 , 0 ) is another Gelfand pair associated with K , V : Ᏼ ! Ᏼ0 is the unitaryisomorphism satisfying V (x) = 0 (x) for all x and g ! Ug is the homomor- 0phism from GK into U (Ᏼ0 ) satisfying Ug 0 (x) = 0 (g (x)) for all x 2 ᐄ, g 2 GK 0then V Ug V 01 = Ug for all g . 0 (ii) In (i) let ᐄ be a separable metric space and let K be continuous on ᐄ 2 ᐄ.Suppose G0 GK is a subgroup which is a topological group acting continuouslyon ᐄ 2 ᐄ. Then in any Gelfand pair (Ᏼ, ) the map : ᐄ ! Ᏼ is continuous, Ᏼis separable and the homomorphism g ! Ug restricted to G0 is continuous. (Hint: Use Proposition 7.2 for (i) and examine k(x) 0 (y )k for (ii)). n NExercise 15.7: (i) Let Ᏼi , 1 i n be Hilbert spaces and let Si Ᏼi be a total nsubset for each i. Then the set f i=1 ui jui 2 Si for each ig is total in i=1 Ᏼi . NᏴ is separable if each Ᏼi is separable.
15 Positive deﬁnite kernels and tensor products of Hilbert spaces 95 (ii) There exist unitary isomorphisms U12,3 : (Ᏼ1 Ᏼ2 ) Ᏼ3 ! Ᏼ1 Ᏼ2 Ᏼ3 , U1,23 : Ᏼ1 (Ᏼ2 Ᏼ3 ) ! Ᏼ1 Ᏼ2 Ᏼ3 ,such that U12,3 (u1 u2 ) u3 = U1,23 u1 (u2 u3 ) = u1 u2 u3for all i 2 Ᏼi , i = 1, 2, 3. (Hint: Use Proposition 7.2.)Exercise 15.8: Let feij jj = 1, 2, . . .g be an orthonormal basis in Ᏼi , i =1, 2, . . . , n respectively. Then the set fe1j e2j 1 1 1 enjn jj1 = 1, 2, . . . , j2 = 1, 2, . . . , jn = 1, 2, . . .g 1 2 Nis an orthonormal basis for i=1 Ᏼi . (Note that when dim Ᏼi = mi 1, ji = n1, 2, . . . , mi ). In particular, Ou n i = X f n 5i=1 he iji , u ig i Oe n iji =1 i j1 ,j2 ,... ,jnwhere the right hand side is a strongly convergent sum in i=1 N n Ᏼi . If dim Ᏼi =mi 1 for every i then i=1 dim OᏴn i = m1 m2 1 1 1 mn . i=1Exercise 15.9: Let Ᏼi = L2 (i , Ᏺi , i ), 1 i n where (i , Ᏺi , i ) is a -ﬁnitemeasure space for each i. If (, Ᏺ, ) = 5n 1 (i , Ᏺi , i ) is the cartesian product i=of these measure spaces there exists a unitary isomorphism U : n i=1 Ᏼi ! NL (, Ᏺ, ) such that 2 (Uu1 1 1 1 un )(!1 , . . . , !n ) = 5n 1 ui (!i ) a.e. . i =Exercise 15.10: Let Ᏼn , n = 1, 2, . . . be a sequence of Hilbert spaces and letf g be a sequence of unit vectors where 2 Ᏼ for each n. Suppose n n n M = fuju = (u1 , u2 , . . .), u 2 Ᏼ , u = for all large ng. j j n n K (u, v) = 51 1 huj , vj i, u, v 2 M .Deﬁne j =Then K is a kernel on M . The Hilbert space Ᏼ in a Gelfand pair (Ᏼ, ) associatedwith K is called the countable tensor product of the sequence fᏴn g with respectto the stabilizing sequence fn g. We write (u) = u1 u2 1 1 1 for u 2 M .Suppose fen0 , en1 , . . .g = Sn is an orthonormal basis in Ᏼn such that en0 = nfor each n. Then the set f(u)ju 2 M , uj 2 Sj for each j g is an orthonormalbasis in Ᏼ.
96 Chapter II: Observables and States in Tensor Products of Hilbert SpacesExercise 15.11: Let Ᏼn = L2 (n , Ᏺn , n ), n = 1, 2, . . . , Ᏼ = L2 (, Ᏺ, )where (n , Ᏺn , n ) is a probability space for each n and (, Ᏺ, ) = 51 1 (n , n=Ᏺn , n ) is the product probability space. Then Ᏼ is the countable tensor productof the sequence fᏴn g with respect to the stabilising sequence fn g where n isthe constant function 1 in n for each n. This may be used to construct exampleswhere different stabilising sequences may lead to different countable tensor prod-ucts. (Hint: If and are two distinct probability measures such that then 2 2 1 1 1 and 2 2 1 1 1 are singular with respect to each other.)Exerecise 15.12: Let Ᏼ = L2 ([0, 1], h) be the Hilbert space deﬁned in Section 2where is the Lebesgue measure in [0, 1]. Then any continuous map f : [0, 1] ! Ᏼis an element of Ᏼ. For any such continuous map f deﬁne the element en (f ) in( 8 ރh)n by en (f ) = n=1 f1 8 n0 21 f ( j )g. j nThen lim hen (f ), en (g )i = exphf , g i. n !1Exercise 15.13: Let Ᏼ1 , Ᏼ2 be Hilbert spaces and let ff1 , f2 , . . .g be an or-thonormal basis in Ᏼ2 . Then there exists a unitary isomorphism U : Ᏼ1 Ᏼ2 !Ᏼ1 8 Ᏼ1 8 1 1 1 satisfying Uu v = 8i hfi , v iu for all u 2 Ᏼ1 , v 2 Ᏼ2 .Exercise 15.14: Let L2 (, h) be as deﬁned in Section 2. Then there exists a uniqueunitary isomorphism U : L2 () h ! L2 (, h) satisfying (Uf u)(! ) = f (! )u.Exercise 15.15: Let Ᏼ be a Hilbert space and let I2 (Ᏼ) be the Hilbert space of 3all Hilbert-Schmidt operators in Ᏼ with scalar product hT1 , T2 i = tr T1 T2 . For anyﬁxed conjugation J there exists a unique unitary isomorphism U : Ᏼ Ᏼ ! I2 (Ᏼ)satisfying Uu v = juihJv j for all u, v in Ᏼ. (See Section 9.)Example 15.16:  Let H be any selfadjoint operator in a Hilbert space Ᏼwith pure point spectrum 6(H ) = S . Then S is a ﬁnite or countable subset of .ޒDenote by G the countable additive group generated by S and endowed with ~the discrete topology. Let G be its compact character group with the normalisedHaar measure. For any bounded operator X on Ᏼ and 2 G deﬁne the boundedoperator Z X = ()(H )X(H )d. ~ GLet , 2 S , u, v 2 Ᏼ, Hu = u, Hv = v. Then Z hu, X v i = f ( 0 0 )dghu, Xvi. ~ GHence hu, Xvi = 0 , hu, X v i = if 0 if 6= 0 .
16 Operators in tensor products of Hilbert spaces 97Thus, for any non-zero bounded operator X there exists a 2 S0S = S Xf 0 j , 2 g such that 6= 0 and on the linear manifold D generatedby all the eigenvectors of H [H , [H , X ]] = 2 X , Xu = X0u + X (X0 + X )u , u 2 D. 0 2S0SIfX X X is selfadjoint ( )3 = 0 and X is a “superposition” of bounded “har- X X X S S monic” observables 0 , f 0 + j 2 0 , 0g with respect to . (See H XY tr X 3 Y = tr X0 Y0 + 3 XExample 6.2.) Whenever , are Hilbert-Schmidt operators tr X Y 3 2S0S X (X + X 0and X = X0 + 0 ) 2S0S 0converges in Hilbert-Schmidt norm (i.e., the norm in I2 (Ᏼ)). See Example 6.2.NotesThe presentation here is based on the notes of Parthasarathy and Schmidt .Proposition 15.1 is known as Schur’s Lemma. Proposition 15.4 and Exercise 15.6constitute what may be called a probability theorist’s translation of the famousGelfand-Neumark-Segal or G.N.S. Theorem. Its origin may be traced back to thetheory of second order stationary stochastic processes developed by A.N. Kol-mogorov, N. Wiener, A.I. Khinchine and K. Karhunen . Proposition 7.2, 15.4and 19.4 more or less constitute the basic principles around which the fabric ofour exposition in this volume is woven. e n h It is an interesting idea of Journ´ and Meyer  that ( ) 8 ރin Exercise e f15.12 may be looked upon as a toy Fock space where n ( ) may be imagined asa toy exponential or coherent vector which in the limit becomes the boson Fock L h efspace 0( 2 [0, 1] ) where ( ) is the true exponential or coherent vector. SeeExercise 29.12, 29.13, Parthasarathy , Lindsay and Parthasarathy .16 Operators in tensor products of Hilbert spaces n NWe shall now deﬁne tensor products of operators. To begin with let Ᏼi be a ﬁnite m idimensional Hilbert space of dimension i , = 1, 2, . . . , and Ᏼ = i=1 Ᏼi . n T j mSuppose i is a selfadjoint operator in Ᏼi with eigenvalues f ij , 1 i g e j mand corresponding orthnormal set of eigenvectors f ij , 1 i g so that Tieij = ij eij , 1 j mi, i = 1, 2, . . . , n.Using Exercise 15.8 deﬁne a selfadjoint operator T on Ᏼ by putting On On T eiji = 5n=1iji eiji , 1 ji mi i i=1 i=1
98 Chapter II: Observables and States in Tensor Products of Hilbert Spaces T O =Oand extending linearly on Ᏼ. The operator T has eigenvalues 5n 1 iji and satisﬁes n ui i= n Ti ui for all ui 2 Ᏼi , 1 i n. =1 i =1 iFurthermore kT k = max (j5n=1 ij j, 1 ji mi ) i i = 5i=1 max(jij j, 1 j mi ) = 5n=1 kTi k. n i X XO XOIn particular, we have the identity N n N n j aj k 5 =1 huij , Ti uikn i ij = jh j uij , T j uij ij 1 j ,k N = j = 5= k kk n i XO k 1 = 1 = Ti i 1 N j j 1 n i uij 2 1 (16.1) =1 =1 N= j i 2 Ᏼi , 1 j N , 1 i n, N = 1, 2, . . . . We writefor all scalars j , uijT = 1 Ti = T1 1 1 1 Tn and call it the tensor product of operators Ti , n i1 i n. The next proposition extends this elementary notion to all boundedoperators on Hilbert spaces.Proposition 16.1: Let Ᏼi , 1 i n be Hilbert spaces and let Ti be a bounded Noperator in Ᏼi for each i. Then there exists a unique bounded operator T inᏴ = i=1 Ᏼi satisfying O =O n n n T 2 ui Ti ui for all ui Ᏼi , 1 i n. (16.2) = = = fN = j 2 i 1 i 1Furthermore T k k=5k k Ti . N = 2 g iProof: Let S n i 1 i u ui Ᏼi , 1 i n . For all scalars j , 1 j N 1 uij S , 1 j N we have n X O k = X 5= hand product vectors i N n k j Ti uij j k n uij , Ti3 Ti uik i X 5= h 2 i 1 j= 1 = i 1 1 j ,k N = j k n i 1 uij , Pi Ti3 Ti Pi uik i (16.3) 1 j ,k Nwhere Pi is the projection on the ﬁnite dimensional subspace Mi spanned byfuij , 1 j N g in Ᏼi for each i. Since Pi Ti3 Ti Pi is a positive operator in Mi , k XOit follows from (16.1) and (16.3) that N j n Ti uij k2 5n=1 kPi Ti3 Ti Pi k k i XO N j n uij k2 j =1 i =1 5n=1 kTi3 Ti i kk XO k = = N j j 1 n i uij 2 . 1 (16.4) j =1 i=1
16 Operators in tensor products of Hilbert spaces 99Deﬁne T on the set S by (16.2). Then (16.4) shows that T extends uniquely as alinear map on the linear manifold generated by S . Since S is total in Ᏼ this linearextension can be closed to deﬁne a bounded operator T on Ᏼ satisfying kT k 5n kT 3 T k1=2 = 5n kT k.=1 i i i =1 i i(1 0 )kT k for each i, assuming T 6= 0. Then N =1 uLet now 0 1 be arbitrary. Choose unit vectors ui 2 Ᏼi such that kTi ui k n i i i is a unit vector in Ᏼ and Ou k = k OT u k = 5 i n n kT =1 kTi ui k (1 0 ) 5 =1 k T k . n n n i i i i i i =1 i i=1Thus kT k 5i=1 kTi k. n The operator T determined by Proposition 16.1 is called the tensor productNof the operators Ti , 1 i n. We write T = i=1 Ti = T1 1 1 1 Tn . If Ᏼi = h, nTi = S for all i = 1, 2, . . . , n we write T = S and call it the n-th tensor power nof the operator S . NProposition 16.2: Let Ᏼi , 1 i n be Hilbert spaces and let Si , Ti be boundedoperators in Ᏼi for each i. Let S = i=1 Si , T = i=1 Ti . Then the following n n Nrelations hold:(i) The mapping (T1 , T2 , . . . , Tn ) ! T is multilinear from B (Ᏼ1 ) 21 1 12B (Ᏼn ) into B (Ᏼ1 1 1 1 Ᏼn ); N(ii) ST = i=1 Si Ti , T 3 = i=1 Ti3 ; n n N(iii) If each Ti has a bounded inverse then T has a bounded inverse and T 01 = i Ti01 ;(iv) T is a selfadjoint, unitary, normal or projection operator according to whether each Ti is a selfadjoint, unitary, normal or projection operator;(v) T is positive if each Ti is positive;(vi) If Ti = ju ihv j where u , v 2 Ᏼ for each i then i i i i i T = ju1 u2 1 1 1 u ihv1 v2 1 1 1 v n n j.Proof: This is straightforward from deﬁnitions and we omit the proof.N n i=1 n NProposition 16.3: Let Ti be a compact operator in Ᏼi , i = 1, 2, . . . , n, Ᏼ = Ᏼi , T = T . If Ti has the canonical decomposition in the sense of i=1 iProposition 9.6 Ti = X s (T )jv j i ij ihuij j, i = 1, 2, . . . , n (16.5) jthen T is a compact operator with canonical decomposition T = X sj1 (T1 ) 1 1 1 sjn (Tn )jv1j1 1 1 1 vnjn ihu1j1 1 1 1 unjn j. (16.6) j1 ,j2 ,... ,jn
100 Chapter II: Observables and States in Tensor Products of Hilbert SpacesIf Ti 2 I1 (Ᏼi ) for each i then T 2 I1 (Ᏼ) and kT k1 = 5n=1 kTi k1 , tr T i = 5i=1 tr Ti . nIn particular, if each Ti is a state so is T .Proof: Since fuij , j = 1, 2, . . .g, fvij , j = 1, 2, . . .g are orthonormal sets in Ᏼi foreach i it follows that fu1j1 1 1 1 unjn g, fv1j1 1 1 1 vnjn g are orthonormal sets inᏴ. Equation (16.5), the second part of Proposition 16.1 and (vi) in Proposition 16.2 Pyield (16.6) as an operator norm convergent sum over the indices (j1 , j2 , . . . , jn ). XIf kTi k1 = j jsj (Ti )j 1 for each i then we have kT k 1 = jsj (T1 ) 1 1 1 sjn (Tn )j = 5n 1 kTi k1 . 1 i = j1 ,j2 ,... ,jnIf feij , j = 1, 2, . . .g is an orthonormal basis in Ᏼi for each i it follows that fe1j 1 1 1 enjn g X he 1 1 1 e T 1 1 1 T e 1is an orthonormal basis in Ᏼ and tr T = njn , 1 1 1 enjn i X he T e i 1j1 1 n 1j1 j1 ,... ,jn n = 5i=1 , Xhe T e i iji i iji j1 ,... ,jn n = 5i=1 ij , i ij j = 5i=1 tr Ti , ndue to the absolute convergence of all the sums involved. Finally, if each Ti ispositive so is T . If tr Ti = 1 for each i then tr T = 1.then putting Ᏼ = n i=1 N n i=1 i N If (Ᏼi , P (Ᏼi ), i ) is a quantum probability space for each i = 1, 2, . . . , n Ᏼi , = we obtain a new quantum probabilityspace (Ᏼ, P (Ᏼ), ) called the product of the quantum probability spaces (Ᏼi , P (Ᏼi ), i ), i = 1, 2, . . . , n.If Xi is a bounded selfadjoint operator in Ᏼi or, equivalently, a bounded realvalued observable in Ᏼi then X = X1 1 1 1 Xn is a bounded real valuedobservable in Ᏼ and the expectation of X in the product state is equal to5n 1 tr i Xi , the product of the expectations of Xi in the state i , i = 1, 2, . . . , n. i= Now the stage is set for achieving our goal of combining the description ofevents concerning several statistical experiments into those of a single experiment.Suppose that the events of the i-th experiment are described by the elements ofP (Ᏼi ) where Ᏼi is a Hilbert space, i = 1, 2, . . . , n. Let Ᏼ = i Ᏼi . If Pi 2 P (Ᏼi )then we view the event Pi as the element Pi = 1 1 1 1 1 Pi 1 1 1 1, ^
16 Operators in tensor products of Hilbert spaces 101(called the ampliation of Pi in Ᏼ) in ᏼ(Ᏼ) and interpret P = i=1 Pi = 5n 1 Pi 2i= ^ N nᏼ(Ᏼ) (see Proposition 16.2) as the event signifying the simultaneous occurrenceof Pi in the i-th experiment for i = 1, 2, . . . , n. These are the quantum probabilis-tic equivalents of measurable rectangles in products of sample spaces. The nextproposition indicates how ᏼ(Ᏼ) is “generated” by P (Ᏼi ), i = 1, 2, . . . , n.Proposition 16.4: Every element P in ᏼ(Ᏼ) can be obtained as a strong limitof linear combinations of projections of the form i=1 Pi , Pi 2 P (Ᏼi ). n NProof: Choose an increasing sequence fQik , k = 1, 2, . . .g of ﬁnite dimen-Nsional projections so that s.limk!1 Qik = 1 in Ᏼi for each i. Then Qk = n Q increases to the identity strongly in Ᏼ and for any bounded opera- i=1 iktor T in Ᏼ, s.limK !1 Qk T Qk = T . Thus it sufﬁces to show that for ﬁxed k the Noperator Qk T Qk can be expressed as a linear combination of product projectionsof the form i=1 Pi , Pi 2 P (Ᏼi ). Replacing Ᏼi by the range of Qik if necessary nwe may assume that each Ᏼi has dimension mi 1. Let feir , 1 r mi g bean orthonormal basis in Ᏼi and let Ers = jeis iheir j. Then every operator T on iᏴ can be expressed as a linear combination of operators of the form i=1 Erj sj . n j NEach Ers can be expressed as a linear combination of selfadjoint operators: j j3 1 j 1 j 3 Ers = (Ers + Ers ) + if (Ers 0 Ers )g. j j 2 2iSince every selfadjoint operator in Ᏼi admits a spectral decomposition in termsof eigen projections the proof is complete.Proposition 16.5: Let (i , Ᏺi ) be measurable spaces and let i : Ᏺi ! P (Hi ) n i=be the product measurable space and Ᏼ = i=1 Ᏼi . Then there exists a unique Nbe an i -valued observable in Ᏼi for i = 1, 2, . . . , n. Let (, Ᏺ) = 5n 1 (i , Ᏺi )-valued observable : Ᏺ ! ᏼ(Ᏼ) such that (F1 2 1 1 1 2 Fn ) = O n Fi ) for all Fi 2 Ᏺi , i = 1, 2, . . . , n. i( (16.7) =1 iProof: By Proposition 7.4 there exists a ﬁnite or countable family of -ﬁnitemeasures fij , j = 1, 2, . . . , g and a unitary isomorphism Ui : Ᏼi ! j L2 (ij ) Lsuch that Ui i (E )Ui01 = M ij E ) for all E 2 Ᏺi , ( 1 i n. j Of8 L MLWe have n j 2 ij )g = ( 2 2 2j 2 1 1 1 2 njn ). ( 1j 1 2 =1i j1 ,... ,jnDeﬁne U = U1 1 1 1 Un and (F ) = U 01 f M 1j1 21112 njn F gU , F 2 Ᏺ. ( ) j1 ,... ,jnThen (16.7) is obtained.
102 Chapter II: Observables and States in Tensor Products of Hilbert Spaces The unique observable satisfying (16.7) in Proposition 16.5 is called thetensor product of the observables i , i = 1, 2, . . . , n. We write = 1 1 1 1 n .Proposition 16.6: Let Ᏼ1 , Ᏼ2 be Hilbert spaces and let T be a trace class operatorin Ᏼ = Ᏼ1 Ᏼ2 . Then there exists a unique trace class operator T1 in Ᏼ1 suchthat tr T1 X = tr T (X 1) for all X in B(Ᏼ1 ). (16.8)If T is a state in Ᏼ then T1 is a state in Ᏼ1 .Proof: For any compact operator X in Ᏼ1 deﬁne (X ) = tr T (X 1). ByProposition 9.12, (iii) we have j(X )j kT k1 kX k. In particular, is a continuouslinear functional on I1 (Ᏼ1 ). By Schatten’s Theorem (Theorem 9.17) there existsa T1 in I1 (Ᏼ1 ) such that tr T1 X = tr T (X 1) for all X in I1 (Ᏼ1 ). Since themaps X ! tr T1 X and X ! tr T (X 1) are strongly continuous in B(Ᏼ1 ) weobtain (16.8). If T is positive we have hu, T1 ui = tr T1 juihuj = tr T (juihuj 1) 0for all u in Ᏼ1 . Thus T1 0. Putting X = 1 in (16.8) we get tr T1 = tr T . Thisproves that T1 is a state if T is a state. The operator T1 in Proposition 16.6 is called the relative trace of T in Ᏼ1 .If T is a state then the relative trace of T1 is the analogue of marginal distributionin classical probability.Exercise 16.7: Suppose Ᏼ1 , Ᏼ2 are two real ﬁnite dimensional Hilbert spacesof dimensions m1 , m2 respectively and Ᏼ = Ᏼ1 Ᏼ2 . Let ᏻ(Ᏼ1 ), ᏻ(Ᏼ2 ) andᏻ(Ᏼ1 Ᏼ2 ) be the real linear spaces of all selfadjoint operators in Ᏼ1 , Ᏼ2 andᏴ1 Ᏼ2 respectively. Then we have dim ᏻ(Ᏼi ) = 2 mi (mi + 1), i = 1, 2 and 1dim ᏻ(Ᏼ1 Ᏼ2 ) = 2 m1 m2 (m1 m2 + 1). In particular, 1 dim ᏻ(Ᏼ1 Ᏼ2 ) dim ᏻ(Ᏼ1 ) 1 dim ᏻ(Ᏼ2 ) if mi 1, i = 1, 2, .On the other hand, if Ᏼ1 , Ᏼ2 are complex Hilbert spaces dim ᏻ(Ᏼi ) = mi and 2 dim ᏻ(Ᏼ1 Ᏼ2 ) = dim ᏻ(Ᏼ1 ) dim ᏻ(Ᏼ2 ).(In the light of Proposition 16.4 this indicates the advantage of working withcomplex Hilbert spaces in dealing with observables concerning several quantumstatistical experiments).Exercise 16.8: (i) Let Tj be (a not necessarily bounded) selfadjoint operator inᏴj with spectral representation Z Tj = ( ) xj dx , j = 1, 2, . . . , n, ޒj being a real valued observable in Ᏼj for each j . Deﬁne the selfadjoint operator Z T1 111 T = n x1 x2 111x n 1 2 1 1 1 (dx1 dx2 1 1 1 dx ) n n ޒn
16 Operators in tensor products of Hilbert spaces 103 D = D(T ) is a core for T for eachj T 111 T then the linear manifold Mby Proposition 12.1 and Theorem 12.2. If generated by fu 111 unjuj 2 Dj for each jg j 1 j jis a core for 1 n and T 111 Tn u 111 un = T u 111 Tn un 1 1 1 1for all uj 2 D(Tj ), 1 j n. (ii) Let Ti = 1 111 1 Ti 1 111 1 1 (be the i-th ampliation of Ti ) where ^Ti is in the i-th position. Then T + 111 + Tn is essentially selfadjoint on M with its ^ ^ (x + 111 + x ) 111 (dx 111 dx ). 1 Rclosure being the selfadjoint operator ޒn 1 n 1 n 1 n T j then 1T 111 T n has ﬁnite expectation in the product state = 1 111 (iii) If j is a state in Ᏼj and j has ﬁnite expectation in the state j for each n hT 111 T i hT iand n 1 111n = 5i=1 i i n 1in the notation of Proposition 13.6.Exercise 1 orthonormal basis in thenffj g is an16.9: Let T 2 I (Ᏼ Ᏼ Ᏼ ) and let T be its relative trace in Ᏼ . If 1 2 2 1 1 hu, T vi = j hu fj , Tv fj i for all u, v in Ᏼ 1 X 1where the right hand side converges absolutely.Exercise 16.10: Let Ᏼ1 , Ᏼ2 be Hilbert spaces and Ᏼ = Ᏼ1 Ᏼ2 . Then the*-algebra generated byᏮ(Ᏼ). 1fX X jX 2 B 2 i i (Ᏼi ), = 1, 2 is strongly dense in g in Ᏼ2 there exists a unique linear map : ޅᏮ(Ᏼ) !B (ii) For any trace class operator (Ᏼ1 ) satisfying hu, (ޅX )vi = tr X (jvihuj ) for all u, v 2 Ᏼ , X 2 Ꮾ(Ᏼ) 1 k(ޅX )k kk kX k.where 1 B (iii) If is a state then ޅis called the -conditional expectation map fromᏮ(Ᏼ) into (Ᏼ1 ). The conditional expectation map satisﬁes the following prop-erties:(1) ޅ ,1 = 1 ޅ X 3 = ( ޅX )3, kޅX k kX k;(2) ( ޅA X B 1) = A( ޅX )B, for all A, B 2 B(Ᏼ1), X 2 Ꮾ(Ᏼ); 1) ( 1i,j k Yi ( ޅXi Xj )Yj 0 for all Xi 2 Ꮾ(Ᏼ), Yi 2 B (Ᏼ1 ). In particu- P 3 3 lar, ޅX 0 whenever X 0.(3)
104 Chapter II: Observables and States in Tensor Products of Hilbert SpacesExercise 16.11: Let fᏴn , n 0g be a sequence of Hilbert spaces and fn , n 1gbe unit vectors, n 2 Ᏼn . Deﬁne Ᏼ[n+1 = Ᏼn+1 Ᏼn+2 1 1 1 with respect tothe stabilising sequence n+1 , n+2 , . . . and Ᏼn] = Ᏼ0 Ᏼ1 1 1 1 Ᏼn . In theHilbert space ~ Ᏼ = Ᏼ0 Ᏼ[1 = Ᏼn ] Ᏼ[n+1 , n = 1, 2, . . .consider the increasing sequence of *-algebras = fX 1 n 1 jX 2 B(Ᏼn )g, n = 0, 1, 2 . . . Bn] [ + ]Deﬁne Bn = B (Ᏼn ) and 1n to be the identity in Bn . There exists a unique linear ] ]map ޅn : B1 ! Bn satisfying ] ] hu, ޅn (X )v i = hu n 1 , Xv n 1 i for all u, v 2 Ᏼn , X 2 B1 ] [ + [ + ] [ + + ~where n 1 = n 1 n 2 1 1 1 and B1 = B (Ᏼ). Indeed, + ޅn (X ) = ޅn n (X ) 1 n 1 . ] j [ +1 ih [ +1 j [ +The maps fޅn] gn0 satisfy the following properties:(i) ޅn] 1 = 1, ޅn] X 3 = (ޅn] X )3 , kޅn] X k kX k;(ii) ޅn] AXB = Aޅn] (X )B whenever A, B 2 Bn] ; ޅm] ޅn] = ޅn] ޅm] = ޅm] whenever m n;(iii)(iv) P 1i,j k Yi ޅn] (Xi Xj )Yj 0 for all Yi 2 Bn] , 3 3 Xi 2 B . In particular 1 ޅn] X 0 whenever X 0;(v) s.limn!1 ޅn] X = X for all X in B1 .Exercise 16.12: (i) In the notations of Exercise 16.11 a sequence fXn g in B1is said to be adapted if Xn 2 Bn] for every n. It is called a martingale if ޅn01] Xn = Xn 01 for all n1Suppose A = fAn gn 1 is any sequence of operators where An 2 B(Ᏼn ), hn , An n i = 0, n = 1, 2, . . . ~Deﬁne An = 1n An 1[n+1 , 01] Mn (A) = A + A + 1 1 1 + A if n = 0, 0 ~1 ~2 ~n if n 1.Then fMn (A)gn0 is a martingale. For any two sequences A, B of operatorswhere An , Bn 2 B (Ᏼn ), hn , An n i = hn , Bn n i = 0 for each n ޅm] fMn (A) 0 Mm (A)g3 fMn (B ) 0 Mm (B )g Xn = hAj j , Bj j i for all n m 0. j =m+1
17 Symmetric and antisymmetric tensor products 105 (ii) For any sequence E = (E1 , E2 , . . .) of operators such that En 2 Bn01 , n = 1, 2, . . . ]deﬁne 0 In (A, E ) = Pn EnfMn(A) 0 Mn0 (A)g 1 1 if if n = 0, n 1.Then fIn (A, E )gn0 is a martingale. Furthermore ޅn01] In (A, E )3 In (B , F ) = In01 (A, E )3 In01 (B , F ) 3 + hAn n , Bn n iEn Fnfor all n 1.Exercise 16.13: For any selfadjoint operator X in the Hilbert space Ᏼ deﬁnethe operator S (X ) in 2ރᏴ by S (X ) = (1 1) exp[i2 X ] where j ,1 j 3 are the Pauli spin matrices. Then S (X ) = S (X )01 = S (X )3 and2 [S (X ) + S (0X )] = 1 cos X . Thus S (X ) and S (0X ) are spin observables1with two-point spectrum f01, 1g but their average can have arbitrary spectrum inthe interval [01, 1]. (See also Exercise 4.4, 13.11.)NotesThe role of tensor products of Hilbert spaces and operators in the construction ofobservables concerning multiple quantum systems is explained in Mackey .For a discussion of conditional expectation in non-commutative probability theory,see Accardi and Cecchini . Exercise 16.13 arose from discussions with B.V. R.Bhat.17 Symmetric and antisymmetric tensor productsThere is a special feature of quantum mechanics which necessitates the introductionof symmetric and antisymmetric tensor products of Hilbert spaces. Suppose thata physical system consists of n identical particles which are indistinguishablefrom one another. A transition may occur in the system resulting in merely theinterchange of particles regarding some physical characteristic (like position forexample) and it may not be possible to detect such a change by any observablemeans. Suppose the statistical features of the dynamics of each particle in isolationare described by states in some Hilbert space Ᏼ. According to the procedure Noutlined in Section 15, 16 the events concerning all the n particles are describedby the elements of ᏼ(Ᏼ ). If Pi 2 ᏼ(Ᏼ), 1 i n then i=1 Pi signiﬁes the n nevent that Pi occurs for each i. If the particles i and j (i j ) are interchanged anda change cannot be detected then we should not distinguish between the eventsP1 1 1 1Pn and P1 1 1 1Pi01 Pj Pi+1 1 1 1Pj01 Pi Pj+1 1 1 1Pn , wherein the second product the positions of Pi and Pj are interchanged. This suggeststhat the Hilbert space Ᏼ is too large and therefore admits too many projections n
106 Chapter II: Observables and States in Tensor Products of Hilbert Spacesand does not qualify for the description of events concerning n identical particles.In order to take into account this curtailment in the degrees of freedom, the desiredreduction of the Hilbert space may be achieved by restriction to a suitable subspaceof Ᏼ which is “invariant under permutations”. We shall make this statement nmore precise in the sequel. Let Sn denote the group of all permutations of the set f1, 2, . . . , ng. Thusany 2 Sn is a one-to-one map of f1, 2, . . .n, ng onto itself. For each 2 Sn letU be deﬁned on the product vectors in Ᏼ by U u1 1 1 1 un = u01 (1) 1 1 1 u01 (n) 17.1) (where 01 is the inverse of . Then U is a scalar product preserving the mapof the total set of product vectors in Ᏼ onto itself. Hence by Proposition 7.2, nU extends uniquely to a unitary operator on Ᏼ , which we shall denote by U nitself. Clearly U U0 = U0 for all , 0 2 Sn . (17.2)Thus ! U is a homomorphism from the ﬁnite group Sn into ᐁ(Ᏼ ). The nclosed subspaces Ᏼ fu 2 Ᏼ jU u = u for all s n n = 2 Sn g, (17.3) Ᏼ fu 2 Ᏼ jU u = ( )u for all a n n = 2 Sn g, (17.4)where ( ) = 61 according to whether the permutation is even or odd arecalled respectively the n-fold symmetric and antisymmetric tensor products of Ᏼ.They are left invariant by the unitary operators U , 2 Sn . There do exist othersuch permutation invariant subspaces of Ᏼ but it seems that they do not feature nfrequently in a signiﬁcant form in the physical description of n identical particles.If the statistical features of the dynamics of a single particle are described by stateson ᏼ(Ᏼ) and the dynamics of n such identical particles is described by states inᏼ(Ᏼ ) for n = 2, 3, . . . then such a particle is called boson. Instead, if it is sndescribed by states on ᏼ(Ᏼ ) for every n then such a particle is called fermion. an(This nomenclature is in honour of the physicists S.N. Bose and E. Fermi whopioneered the investigation of statistics of such particles). We shall now present some of the basic properties of Ᏼ and Ᏼ in the s a n nnext few propositions. E and F be operators in Ᏼ deﬁned by n XUProposition 17.1: Let 1 E= , (17.5) n! 2S F = 1 X U n ( ) , (17.6) n! 2Sn
17 Symmetric and antisymmetric tensor products 107where U is the unitary operator satisfying (17.1) and ( ) is the signature of thepermutation . Then E and F are projections onto the subspaces Ᏼ and Ᏼ sn anrespectively. If K Sn is any subgroup E = E #1 XU , (17.7) K F = F #1 X U 2 K ( ) (17.8) K 2 Kwhere K is the cardinality of K . Furthermore, for any ui , vi 2 Ᏼ, 1 i n # = 1 X hEu1 1 1 1 un , Ev1 1 1 1 vn i = hu1 1 1 1 un , Ev1 1 1 1 vn i n 5i=1 hui , v(i) i, (17.9) n! 2S n X hF u1 1 1 1 un , F v1 1 1 1 vn i = hu1 1 1 1 un , F v1 1 1 1 vn i n! 2Sn ()5i=1 hui , v(i) i 1 n = det((hui , vj i)). 1 = (17.10) n!Proof: From (17.2) we have E 3 = n! 1 X U3 X U = 1 01 = E, 2 Sn n! 2Sn F 3 = n! 1 X U3 EU = U E = E 2 , 1 X 01 01 (17.11) 2 Sn ( ) = n! 2Sn ( )U = F, F U = U F = ()F , F 2 = F .Thus E and F are projections. Furthermore (17.3), (17.4) and (17.11) imply thatEu = u if and only if u 2 Ᏼn and F v = v if and only if v 2 Ᏼn . This s aproves the ﬁrst part. Summing up over 2 K and dividing by #K in the secondequation of (17.11) we obtain (17.7). Multiplying by ( ), summing over 2 Kand dividing by #K in the last equation of (17.11) we obtain (17.8). The ﬁrst part of (17.9) is a consequence of the fact that E is a projection.By deﬁnitionhu1 1 1 1 un , Ev1 1 1 1 vn i = n! 1 X hu 1 1 1 u 1 n , v (1) 1 1 1 v(n) i 2 1 = X hu v i Sn n! 2Sn 5i i , (i) .This proves (17.9). (17.10) follows exactly along the same lines.
108 Chapter II: Observables and States in Tensor Products of Hilbert Spaces P It may be mentioned here that the quantity 2Sn 5n 1 hui , v(i) i occurring i=on the right hand side of (17.9) is the permanent of the matrix ((hui , vj i)) incontrast to the determinant occurring in (17.10).Corollary 17.2: Let En and Fn denote respectively the projections E and F ndeﬁned on Ᏼ by (17.5) and (17.6) for each n = 1, 2, . . .. Then for u 2 Ᏼ , `v2Ᏼ m , w 2 Ᏼ n E`+m+n fE`+m (u v ) wg = E`+m+n fu Em+n (v w)g = E`+m+n (u v w), F`+m+n fF`+m (u v ) wg = F`+m+n fu Fm+n (v w)g = F`+m+n (u v w).Proof: This is immediate from (17.7) and (17.8) if we consider the permutationgroups S`+m and Sm+n as subgroups of S`+m+n so that 2 S`+m leaves thelast n elements of f1, 2, . . . , ` + m + ng ﬁxed whereas 2 Sm+n leaves the ﬁrst` elements of f1, 2, . . . , ` + m + ng ﬁxed.Proposition 17.3: Let fei , i = 1, 2, . . .g be an orthonormal basis for Ᏼ. For any k Nui 2 Ᏼ, i = 1, 2, . . . , k and positive integers r1 , r2 , . . . , rk satisfying r1 + 1 1 1 +rk = n denote by i=1 u i the element u1 1 1 1 u1 u2 1 1 1 u2 1 1 1 uk i r1 1 1 uk where ui is repeated ri times for each i. Let E and F be the projectionsdeﬁned respectively by (17.5) and (17.6) in Ᏼ . Then the sets n ( r ! 1n!1 r ! )1=2 E Oe k rj f 1 ij ji1 i2 1 1 1 ik , 1 k j =1 rj 1 for each 1 j k, r1 + r2 + 1 1 1 + rk = n, k = 1, 2, . . . , ng, f (n!)1=2 F ei1 ei2 1 1 1 ein ji1 i2 1 1 1 in gare orthonormal bases in Ᏼ and Ᏼ respectively. In particular, if dim Ᏼ s a n n =N 1 then N +n01 dim Ᏼ = sn , 0N 1 n if n N , dim Ᏼ a = n n 0 otherwise.Proof: The ﬁrst part follows from (17.9) and (17.10). The formula for dim Ᏼ s nis immediate if we identify it as the number of ways in which n indistinguishableballs can be thrown in N cells. Similarly the dimension of Ᏼ can be identiﬁed anwith the number of ways in which n indistinguishable balls can be thrown in Ncells so that no cell has more than one ball.
17 Symmetric and antisymmetric tensor products 109 Using Proposition 17.3 it is possible to compare the different probabilitydistributions that arise in the “statistics of occupancy”. More precisely, let be a P 1 ih jf j ih jstate in Ᏼ with dim Ᏼ = N . Let ej , j = 1, 2, . . . , N be an orthonormal jbasis of Ᏼ such that = j pj ej ej . Let ej ej signify the event “particle goccupies cell number j ”. Now consider n distinguishable particles whose statis-tics are described by the quantum probability space (Ᏼ , ᏼ(Ᏼ ), ). The n n n j 111 ih 111 jprojection ej1 ejn ej1 ejn signiﬁes the event “particle i occupiescell ji for each i = 1, 2, . . . , n”. Let r = (r1 , . . . , rN ) where rj is the number of E = X je111 111 e n ihe 111 e n jparticles in cell j so that r1 + 0 r + rN = n. Deﬁne j1 j j1 j (17.12)where the summation on the right hand side is over all (j1 , . . . , jn ) such that the fj gcardinality of i ji = j is rj for j = 1, 2, . . . , N . Er is a projection whose range 0has dimension r1 !111!rN ! and it signiﬁes the event that cell j has rj particles for neach j . Then n 0 tr Er = n! 111 pr2 pr2 pN . r1 ! rN ! 1 1 rN 111 (17.13)In this case we say that the particles obey the Maxwell-Boltzmann statistics and nthe probability that there are rj particles in cell j for each j in the state isgiven by (17.13). Suppose that the n particles under consideration are n identical bosons. n n nThen the Hilbert space Ᏼ is replaced by Ᏼ and correspondingly by its srestricton to Ᏼn . To make this restriction a state we put s n = Ᏼn . n j X s s s n n c = tr = tr E = s s sN p11 p22 pN 111 s1 +111+sN =n nand observe that c01 is a state, E being deﬁned by (17.5). The quantum sprobability space (Ᏼ n , ᏼ(Ᏼn ), c01 n ) describes the statistics of n identical s s sbosons. Denote by Er (b) the projection on the one dimensional subspace generated Oby the vector = (r1 , . . . , r ), r1 + 111 + r = n. N rj er =E e , r N N j =1Then the probability of ﬁnding rj particles in cell j for each j = 1, 2, . . . , N is s n (b) tr c01 Er = pr1 pN 1 rN P ps1 pN s1 +111sN =n 1 sN 111 (17.14) 111In this case we say that the particles obey the Bose-Einstein statistics. n n When the particles are n identical fermions Ᏼ is replaced by Ᏼ which a is non-trivial if and only if n N , i.e., the number of particles does not exceed
110 Chapter II: Observables and States in Tensor Products of Hilbert Spacesthe number of cells. The state is replaced by its restriction to Ᏼ . Once n a n = tr n Ᏼn = j a Xagain to make this restriction a state we have to divide it by c0 pi1 pi2 1 1 1 pin . 1 i1 i2 111in NThen the quantum probability space describing the statistics of n identical fermions an an 01 a nis (Ᏼ , ᏼ(Ᏼ ), c0 ) where = n Ᏼn a n a j (f )If Er denotes the one dimensional projection on the subspace generated by thevector = F ei ei fr 1 ein 2 111 where r = (r1 , . . . , rN ), rij = 1, j = 1, 2, . . . , n and the remaining ri are 0 andF is deﬁned by (17.6) then the probability that cell ij is occupied by one particlefor each j = 1, 2, . . . , n and the remaining cells are unoccupied is given by tr c0 01 n E (f ) = a r P j j 111ji Nppij pj pi p 1 2 111 n 111 pjn . (17.15) 1 1 2 n 1 2In this case we say that the particles obey the Fermi-Dirac statistics. For a comparison of the three distributions (17.13)–(17.15) consider the caseof two cells and n particles. Let the state of a single particle be given by e1 = 2 e1 , 1 e2 = 2 e2 1where fe1 , e2 g is an orthonormal basis in Ᏼ. According to Maxwell-Boltzmann n 0nstatistics the number of particles in cell 1 has a binomial distribution given by Pr (cell 1 has k particles) = 2 , 0k n kwhereas Bose-Einstein statistics yield Pr (cell 1 has k particles) = 1 0k n. n+1 , According to the ﬁrst distribution the probability that all the particles occupya particular cell is 20n whereas the second distribution assigns the enhanced 1probability n+1 to the same event. In the case of fermions it is impossible to have more than two of them whenthere are only two cells available and if a cell is occupied by one particle thesecond one has to be occupied by the other. Bosons tend to crowd more thanparticles obeying Maxwell-Boltzmann statistics and fermions tend to avoid eachother.
18 Examples of discrete time quantum stochastic ﬂows 111Exercise 17.4: Let be an irreducible unitary representation of Sn and let( ) = tr ( ) be its character. Suppose (1) = d denotes the dimension of therepresentation . Deﬁne E = d X( )U n! 2S nwhere U is the unitary operator satisfying (17.1). Then E is a projection foreach , E U = U E for all 2 Sn , E1 E2 = 0 if 1 6= 2 and E = 1. P(Hint: Use Schur orthogonality relations).Example 17.5:  (i) The volume of the region 1 = f(x1 , x2 , . . . , xN 01 )jxj 0 for all j , 0 x1 + 1 1 1 + xN 01 1gin ޒ N 01 is [(N 0 1)!]01 . (ii) Let r1 , . . . , rN be non-negative integers such that r1 + 1 1 1 + rN = n.Then Z n! r rN N + n 0 1 01 p11 1 1 1 pN (N 0 1)!dp1 dp2 1 1 1 dpN 01 = 1 r1 ! 1 1 1 rN ! nwhere pN = (1 0 p1 0 p2 0 1 1 1 0 pN 01 ). This identity has the following interpretation. Suppose all the probabilitydistributions (p1 , p2 , . . . , pN ) for the occupancy of the cells (1, 2, . . . , N ) by aparticle are equally likely and for any chosen prior distribution (p1 , p2 , . . . , pN ) theparticle obeys Maxwell-Boltzmann statistics. Then one obtains the Bose-Einsteindistribution (17.4) with pj = N , j = 1, 2, . . . , N . 1NotesRegarding the role of symmetric and antisymmetric tensor products of Hilbertspaces in the statistics of indistinguishable particles, see Dirac . For an inter-esting historical account of indistinguishable particles and Bose-Einstein statistics,see Bach . Example 17.5 linking Bose-Einstein and Maxwell-Boltzmann statis-tics in the context of Bayesian inference is from Kunte .18 Examples of discrete time quantum stochastic ﬂowsUsing the notion of a countable tensor product of a sequence of Hilbert spaceswith respect to a stabilising sequence of unit vectors and properties of conditionalexpectation (see Exercise 16.10, 16.11) we shall now outline an elementary pro-cedure of constructing a “quantum stochastic ﬂow” in discrete time which is ananalogue of a classical Markov chain induced by a transition probability matrix. For a Hilbert space Ᏼ any subalgebra Ꮾ Ꮾ(Ᏼ) which is closed under theinvolution * and weak topology is called a W 3 algebra or a von Neumann algebra.If Ᏼi , i = 1, 2 are Hilbert spaces, Ꮾi Ꮾ(Ᏼi ), i = 1, 2 are von Neumann algebrasdenote by Ꮾ1 Ꮾ2 the smallest von Neumann algebra containing fX1 X2 jXi 2
112 Chapter II: Observables and States in Tensor Products of Hilbert SpacesᏮi , i = 1, 2g in Ꮾ(Ᏼ1 Ᏼ2 ). If is a trace class operator in Ᏼ2 , followingExercise 16.10, deﬁne the operator ( ޅZ ), Z 2 Ꮾ(Ᏼ1 Ᏼ2 ) by the relation hu, ( ޅZ )v i = tr Z jv ihuj , u, v 2 Ᏼ1 (18.1)in Ᏼ1 .Proposition 18.1: ( ޅZ ) 2 Ꮾ1 if Z 2 Ꮾ1 Ꮾ2 .Proof: If Z = X1 X2 then (18.1) implies that ( ޅZ ) = (tr X2 )X1 . Thus theproposition holds for any ﬁnite linear combination of product operators in Ꮾ1 Ꮾ2 .Suppose that = 6pj jej ihej j is a state where pj 0, 6pj = 1 and fej g is anorthonormal set and w.limn!1 Zn = Z in Ꮾ(Ᏼ1 Ᏼ2 ). Then by (18.1) lim hu, ( ޅZn )v i = lim X n!1 !1 p hu e , Z v e i n j j j n j = 6p hu e , Zv e i j j j = hu, ( ޅZ )vi for all u, v 2 Ᏼ1 . In other words ޅis weakly continuous if is a state. Since ޅis linear in the same property follows for any trace class operator. Now the required result isimmediate from the deﬁnition of Ꮾ1 Ꮾ2 . Let Ᏼ0 , Ᏼ be Hilbert spaces where dim Ᏼ = d 1. Let fe0 , e1 , . . . , ed01 gbe a ﬁxed orthonormal basis in Ᏼ and let Ꮾ0 Ꮾ(Ᏼ0 ) be a von Neumann algebrawith identity. Putting Ᏼn = Ᏼ, n = e0 for all n 1 in Exercise 16.11 constructthe Hilbert spaces Ᏼn] , Ᏼ[n+1 for each n 0. Deﬁne the von Neumann algebras Ꮾn] = fX 1 1 jX 2 Ꮾ0 Ꮾ(Ᏼn )g, [n+ n 0, Ꮾ = Ꮾ0 Ꮾ(Ᏼ 1 ). [Property (v) in Exercise 16.11 implies that Ꮾ is the smallest von Neumann algebracontaining all the Ꮾn , n 0. fᏮn] g is increasing in n. By Proposition 18.1 the[n+1 -conditional expectation ޅn] of Exercise 16.11 maps Ꮾ onto Ꮾn] . Any algebra with identity and an involution * is called a *-unital algebra.If Ꮾ1 , Ꮾ2 are *-unital algebras and : Ꮾ1 ! Ꮾ2 is a mapping preserving * andidentity then is called a *-unital map.Proposition 18.2: Let : Ꮾ0 ! Ꮾ0 Ꮾ(Ᏼ) be a *-unital homomorphism. Deﬁnethe linear maps j : Ꮾ0 ! Ꮾ0 by i i j X ( ) = ޅj ih j ((X )), ej ei 0 i, j d 0 1, X 2 Ꮾ0 (18.2)Then the following holds: (1) = , (X 3 ) = (X )3 ; j P 010 (X ) (Y ) for all X , Y i i i(i) j j j i (X Y ) = d(ii) i j k= i k k j 2 Ꮾ0 .
18 Examples of discrete time quantum stochastic ﬂows 113Proof: From (18.1) and (18.2) we have hu, j (X )v i = hu ei , (X )v ej i. iIf X = 1 the right hand side of this equation is hu, v ij . Furthermore i 3 3 hu, j (X )v i = hu ei , (X )v ej i = hv ej , (X )u ei i i 3 = hv , i (X )ui = hu, i (X ) v i. j jThis proves (i). To prove (ii) choose an orthonormal basis fun g in Ᏼo and observethat hu, j (XY )v i = hu ei , (X ) (Y )v ej i i Xu e Xu e u 3 = h (X )u ei , (Y )v ej i = h , ( ) k ih ek , (Y )v ej i Xu Xu u Y v i r r r ,k i k = h , k( ) r ih , j( ) i X X u Y v r k,r i 3 k = h k( ) , j( ) i =h u X X Y v k , i k( ) k j( ) i. kProposition 18.3: Let , Ᏼ0 , Ᏼ1 , Ꮾ0 be as in Proposition 18.2. Deﬁne the mapsjn : Ꮾ0 ! Ꮾn , n = 0, 1, 2, . . . inductively by ] jn (X ) = X j0 (X ) = X 1 1 , j1 (X ) = (X ) 1 2 , [ jn01 (k (X ))1n01 i ] [ jei ihek j 1[n+1 . (18.3) 0 0 i,j d 1Then jn is a *-unital homomorphism for every n. Furthermore ޅn01] jn (X ) = jn01 (0 (X )) for all n 1, X 0 2 Ꮾ0 ,where ޅn01] is the [n -conditional expectation map of Exercise 16.11.Proof: We prove by induction. For n = 0, 1 it is immediate. Let n 2. Then by Xj 0 0(i) in Proposition 18.2 and the induction hypothesis we have jn (1) = i 1 ( j )1n 1] j i ih j j e e 1[n+1 Xe e n i,j = 0 1n 1] j i ih i j 1[n+1 = 1 i
114 Chapter II: Observables and States in Tensor Products of Hilbert Spacesand jn (X )3 = X jn01 (i (X 3 ))1n01] j je ihe j 1 j i [n+1 i,j = jn (X ). 3 X 0By (ii) in Proposition 18.2 and induction hypothesis we have jn (X )jn (Y ) = jn 1 (j (X ))jn01 (` (Y ))1n01] i k je ihe j 1 k X 0X j i ` [n+1 i,j ,k,` = jn 1 k (X )` (Y ))1n01] i k je ihe j 1 X0 i ` [n+1 i,` k = jn 1 (` (XY ))1n01] i je ihe j 1 i ` [n+1 . i,`This proves the ﬁrst part. By the deﬁnition of ޅn] in Exercise 16.11 and the factthat he0 , jei ihej je0 i = 0 j , the second part follows from (18.3). i 0Corollary 18.4: Let fjn , n 0g be the *-unital homomorphisms of Proposition18.3. Write T = 0 . Then for 0 n0 n1 1 1 1 nk 1, Xi 2 Ꮾ0 , 1 i k 0 ޅn0 ] jn1 (X1 )jn2 (X2 ) 1 1 1 jnk (Xk ) = jn0 (T n1 0n0 (X1 T n2 0n1 (X2 1 1 1 (X 01 T k 0 (18.4) k n 0 nk 1 (X )) 1 1 1)Proof: By Exercise 16.11 ޅn0 ] = ޅn0 ] ޅnk01 ] . Since jn1 (X1 ) 1 1 1 jnk01 (Xk01 ) isan element of Ꮾnk01] it follows from the same exercise that ޅn0 ] jn1 (X1 ) 1 1 1 jnk (Xk ) (18.5) = ޅn0] jn1 (X1 ) 1 1 1 jnk01 (Xk01 )ޅnk01 ] (jnk (Xk )).Since ޅnk01 ] = ޅnk01] ޅnk01 +1] 1 1 1 ޅnk 01] it follows from Proposition 18.3 that ޅnk01 ] jnk (Xk ) = jnk01 (T nk 0nk01 (X )).Substituting this in (18.5), using the fact that jnk01 is a homomorphism and re-peating this argument successively we arrive at (18.4).Proposition 18.5: The map T = 0 from Ꮾ0 into itself satisﬁes the following: 0P(i) T is a *-unital linear map on Ꮾ(Ᏼ0 ); (ii) for any Xi , Yi 2 Ꮾ0 , 1 i k, 3 3 1i,j k Yi T (Xi Xj )Yj 0 for every k . In particular, T (X ) 0 wheneverX 0.