Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements
aa r X i v : . [ m a t h - ph ] D ec Convergence Conditions of Mixed Statesand their von Neumann Entropyin Continuous Quantum Measurements
Toru Fuda ∗ Department of MathematicsHokkaido UniversitySapporo 060-0810Japan
Abstract
By carrying out appropriate continuous quantum measurementswith a family of projection operators, a unitary channel can be ap-proximated in an arbitrary precision in the trace norm sense. In par-ticular, the quantum Zeno effect is described as an application. In thecase of an infinite dimension, although the von Neumann entropy isnot necessarily continuous, the difference of the entropies between thestates, as mentioned above, can be arbitrarily made small under someconditions.
The quantum Zeno effect (QZE) is a quantum effect which was shown byMisra and Sudarshan in [5]. This effect demonstrates that, in quantummechanics, continuous measurements can freeze a state. Of course, thiseffect is peculiar to quantum mechanics. Such an effect is not observed inclassical mechanics. The QZE has been extensively investigated by manyresearchers since its discovery.Recently, some general mathematical aspects of quantum Zeno effectwere investigated in [2]. In particular, continuous measurements of a statealong a certain curve in a Hilbert space were considered. Roughly speaking, ∗ E-mail: [email protected]
Let H be a separable Hilbert space of state vectors of a quantum system S .We denote the inner product and the norm of H by h · , · i (anti-linear in thefirst variable and linear in the second) and k · k , respectively. Let d ( ≤ ∞ ) bethe dimension of H . We denote all bounded linear operators, all compactoperators, all trace-class operators, all density operators, and all unitaryoperators on H by B ( H ) , C ( H ) , T ( H ) , S ( H ) , and U ( H ), respectively. Amixed state of S is represented as an element of S ( H ). We denote the tracenorm by k · k := Tr | · | . The Hamiltonian of the quantum system S is givenby a self-adjoint operator H which is time independent. The domain of H is denoted as D ( H ).Let us consider the following two maps on S ( H ):2. ( Unitary channel )Let U be a unitary operator on H and E U be a map on S ( H ) whichis given by E U ρ := U ρU ∗ , ∀ ρ ∈ S ( H ) . In particular, in the case U = e − itH ( t ∈ R ), we denote E e − itH by E t .2. ( Projection channel )Let P := { P n } n be a family of projection operators on H with P m ⊥ P n ( m = n ) , I = P n P n , and E P be a map on S ( H ) which is given by E P ρ := X n P n ρP n , ∀ ρ ∈ S ( H ) . Now, consider a state ρ ∈ S ( H ) fixed and suppose that one of theSchatten decompositions is given by ρ = d X n =1 λ n | Ψ n ih Ψ n | , (2.1)where, for all Ψ , Φ ∈ H , we denote the operator h Ψ , · i Φ by | Φ ih Ψ | . In (2.1),we allow λ n = 0 to take Ψ n such that { Ψ n } dn =1 is a complete orthonormalsystem (CONS). We remark that it is not necessarily λ n ≥ λ n +1 in thisrepresentation.Let us consider a time interval [0 , τ ] with τ >
0. For the decomposition(2.1), consider a CONS of H denoted by { Ψ n ( t ) } dn =1 which is parametrizedby t ∈ [0 , τ ] with Ψ n (0) = Ψ n (1 ≤ ∀ n ≤ d ). If n ∈ N is fixed, then Ψ n ( · ) isa map from [0 , τ ] to H .We define P ( t ) := {| Ψ n ( t ) ih Ψ n ( t ) |} dn =1 , ( t ∈ [0 , τ ]) . (2.2)Let ∆ : t , t , · · · , t N ( t j ∈ [0 , τ ] , j = 0 , · · · , N ) be an arbitrary partitionof the interval [0 , τ ]:0 = t < t < · · · < t N − < t N = τ. We set∆ k := t k − t k − , ( k = 1 , · · · , N ) , | ∆ | := max ≤ k ≤ N ∆ k , ρ ∆ ( τ ) := E P ( t N ) ◦ E ∆ N ◦ E P ( t N − ) ◦ E ∆ N − ◦ · · · ◦ E P ( t ) ◦ E ∆ ρ. (2.3)In the context of quantum mechanics where ρ ∆ ( τ ) is interpreted as theposterior state that, in the successive measurements at time t , · · · , t N byusing the family of projection operators P ( t ) , · · · , P ( t N ), respectively. Weremark that ρ ∆ ( τ ) is dependent on the form of decomposition (2.1).If ρ ∆ ( τ ) converges with respect to | ∆ | → ρ ∆ ( τ ) = X k λ ∆ ,k | Ψ k ( τ ) ih Ψ k ( τ ) | (2.4)with λ ∆ ,k := X k , ··· ,k N − λ k N Y j =1 (cid:12)(cid:12) h Ψ k j ( t j ) , e − i ∆ j H Ψ k j − ( t j − ) i (cid:12)(cid:12) , ( k N = k ) . (2.5) Let us consider a convergence condition of λ ∆ ,k in the case | ∆ | → γ ∆ ,k := N Y j =1 (cid:12)(cid:12) h Ψ k ( t j ) , e − i ∆ j H Ψ k ( t j − ) i (cid:12)(cid:12) , (2.6) ǫ ∆ ,k := X k , ··· ,kN − ∃ l ∈{ , ··· ,N − } ,k l = k λ k N Y j =1 (cid:12)(cid:12) h Ψ k j ( t j ) , e − i ∆ j H Ψ k j − ( t j − ) i (cid:12)(cid:12) , (2.7)so that λ ∆ ,k = λ k γ ∆ ,k + ǫ ∆ ,k . (2.8)4 heorem 2.1 Assume that there exists k ∈ N such that the following con-ditions hold: ∀ λ ∈ [0 , τ ] , Ψ k ( λ ) ∈ D ( H ) , (2.9) ξ k := sup ≤ λ ≤ τ k H Ψ k ( λ ) k < ∞ , (2.10) η k := sup λ,ν ∈ [0 ,τ ] λ = ν k Ψ k ( λ ) − Ψ k ( ν ) k| λ − ν | < ∞ , (2.11)lim | ∆ |→ N X j =1 Re h Ψ k ( t j ) − Ψ k ( t j − ) , Ψ k ( t j − ) i = 0 . (2.12) Then we have lim | ∆ |→ λ ∆ ,k = λ k . (2.13) Remark 2.2
Condition (2.11) implies that k Ψ k ( λ ) − Ψ k ( ν ) k ≤ η k | λ − µ | , ∀ λ, µ ∈ [0 , τ ] (Lipschitz continuity). In particular, Ψ k ( · ) is strongly con-tinuous, so that the mapping Ψ k ( · ) : [0 , t ] → H is a curve in H .Proof . By using [2, THEOREM 4.2], the assumptions (2.9)–(2.12) implythat lim | ∆ |→ γ ∆ ,k = 1 . (2.14)On the other hand, we can estimate ǫ ∆ ,k as follows. ǫ ∆ ,k = N − X l =0 X k , ··· ,kN − ∀ i>l,k i = k,k l = k λ k N Y j =1 (cid:12)(cid:12) h Ψ k j ( t j ) , e − i ∆ j H Ψ k j − ( t j − ) i (cid:12)(cid:12) (2.15)= N − X l =0 N Y j = l +2 (cid:12)(cid:12) h Ψ k ( t j ) , e − i ∆ j H Ψ k ( t j − ) i (cid:12)(cid:12) X k l ,k l = k (cid:12)(cid:12) h Ψ k ( t l +1 ) , e − i ∆ l +1 H Ψ k l ( t l ) i (cid:12)(cid:12) × X k l − (cid:12)(cid:12) h Ψ k l ( t l ) , e − i ∆ l H Ψ k l − ( t l − ) i (cid:12)(cid:12) × · · · × X k (cid:12)(cid:12) h Ψ k ( t ) , e − i ∆ H Ψ k ( t ) i (cid:12)(cid:12) λ k , (2.16)5n the case where l = 0 , N − { · · · } in (2.16) is given by N Y j =2 (cid:12)(cid:12) h Ψ k ( t j ) , e − i ∆ j H Ψ k ( t j − ) i (cid:12)(cid:12) X k ,k = k (cid:12)(cid:12) h Ψ k ( t ) , e − i ∆ H Ψ k ( t ) i (cid:12)(cid:12) λ k , (2.17) X k N − ,k N − = k (cid:12)(cid:12) h Ψ k ( t l +1 ) , e − i ∆ l +1 H Ψ k l ( t l ) i (cid:12)(cid:12) X k N − (cid:12)(cid:12) h Ψ k N − ( t N − ) , e − i ∆ N − H Ψ k l − ( t l − ) i (cid:12)(cid:12) · · · × X k (cid:12)(cid:12) h Ψ k ( t ) , e − i ∆ H Ψ k ( t ) i (cid:12)(cid:12) λ k , (2.18)respectively.By the Schwarz inequality, we have N Y j = l +2 (cid:12)(cid:12) h Ψ k ( t j ) , e − i ∆ j H Ψ k ( t j − ) i (cid:12)(cid:12) ≤ N Y j = l +2 k Ψ k ( t j ) k · k e − i ∆ j H Ψ k ( t j − ) k ≤ , ∀ l ∈ { , · · · , N − } . For all l ≥ X k l − (cid:12)(cid:12) h Ψ k l ( t l ) , e − i ∆ l H Ψ k l − ( t l − ) i (cid:12)(cid:12) × · · · × X k (cid:12)(cid:12) h Ψ k ( t ) , e − i ∆ H Ψ k ( t ) i (cid:12)(cid:12) λ k ≤ X k l − (cid:12)(cid:12) h Ψ k l ( t l ) , e − i ∆ l H Ψ k l − ( t l − ) i (cid:12)(cid:12) × · · · × X k (cid:12)(cid:12) h e i ∆ H Ψ k ( t ) , Ψ k ( t ) i (cid:12)(cid:12) ≤ X k l − (cid:12)(cid:12) h Ψ k l ( t l ) , e − i ∆ l H Ψ k l − ( t l − ) i (cid:12)(cid:12) × · · · × k e i ∆ H Ψ k ( t ) k ≤ · · · ≤ . Thus (2.16) implies that ǫ ∆ ,k ≤ N − X l =0 X k l ,k l = k (cid:12)(cid:12) h Ψ k ( t l +1 ) , e − i ∆ l +1 H Ψ k l ( t l ) i (cid:12)(cid:12) . (2.19)In the case where k l = k , we have h Ψ k ( t l ) , Ψ k l ( t l ) i = 0. Hence X k l ,k l = k (cid:12)(cid:12) h Ψ k ( t l +1 ) , e − i ∆ l +1 H Ψ k l ( t l ) i (cid:12)(cid:12) = X k l ,k l = k (cid:12)(cid:12) h Ψ k ( t l +1 ) , ( e − i ∆ l +1 H − k l ( t l ) i + h Ψ k ( t l +1 ) − Ψ k ( t l ) , Ψ k l ( t l ) i (cid:12)(cid:12) ≤ X k l ,k l = k n(cid:12)(cid:12) h ( e i ∆ l +1 H − k ( t l +1 ) , Ψ k l ( t l ) i (cid:12)(cid:12) + |h Ψ k ( t l +1 ) − Ψ k ( t l ) , Ψ k l ( t l ) i| o ≤ (cid:8) k ( e i ∆ l +1 H − k ( t l +1 ) k + k Ψ k ( t l +1 ) − Ψ k ( t l ) k (cid:9) . (2.20)6et E H ( · ) be the spectral measure of Hamiltonian H . By the spectral the-orem, we have k ( e i ∆ l +1 H − k ( t l +1 ) k = Z R | e i ∆ l +1 x − | d k E H ( x )Ψ k ( t l +1 ) k ≤ Z R ∆ l +1 x d k E H ( x )Ψ k ( t l +1 ) k ≤ ∆ l +1 k H Ψ k ( t l +1 ) k . (2.21)The assumptions (2.9)–(2.11) imply that k H Ψ k ( t l +1 ) k ≤ ξ k , k Ψ k ( t l +1 ) − Ψ k ( t l ) k ≤ ∆ l +1 η k . (2.22)Therefore, (2.19), (2.20), (2.21) and (2.22) implies that ǫ ∆ ,k ≤ N − X l =0 (cid:8) k ( e i ∆ l +1 H − k ( t l +1 ) k + k Ψ k ( t l +1 ) − Ψ k ( t l ) k (cid:9) ≤ N − X l =0 (cid:8) ∆ l +1 k H Ψ k ( t l +1 ) k + k Ψ k ( t l +1 ) − Ψ k ( t l ) k (cid:9) ≤ ξ k + η k ) N X l =1 ∆ l . (2.23)By [2, LEMMA 2.2], lim | ∆ |→ N X l =1 ∆ l = 0 . Thus (2.23) implies that lim | ∆ |→ ǫ ∆ ,k = 0. Hence, by (2.8) and (2.14), weobtain (2.13) Remark 2.3
Assume that the conditions of Theorem 2.1 hold. Let a > be a constant and take | ∆ | such that ( ξ k + 2 ξ k η k ) | ∆ | + 2 η k | ∆ | ≤ log aa . (2.24) By the proof of [2, THEOREM 4.2], exp " − a ( ( ξ k + 2 ξ k η k ) N X l =1 ∆ l − N X l =1 Re h Ψ k ( t l ) − Ψ k ( t l − ) , Ψ k ( t l − ) i ) ≤ γ ∆ ,k ≤ . (2.25)7 hen, by (2.23) and (2.25), we have | λ ∆ ,k − λ k | = | λ k ( γ ∆ ,k −
1) + ǫ ∆ ,k | ≤ λ k (1 − γ ∆ ,k ) + ǫ ∆ ,k ≤ λ k − exp " − a ( ( ξ k + 2 ξ k η k ) N X l =1 ∆ l − N X l =1 Re h Ψ k ( t l ) − Ψ k ( t l − ) , Ψ k ( t l − ) i ) +2( ξ k + η k ) N X l =1 ∆ l . (2.26)The following corollary can be easily proven by using [2, COROLLARY4.4]. Corollary 2.4
Assume that there exists k ∈ N such that the following con-ditions hold: Ψ k ( · ) : [0 , τ ] → H is a strongly differentiable mapping , (2.27) ∀ λ ∈ [0 , τ ] , Ψ k ( λ ) ∈ D ( H ) , (2.28) ξ k < ∞ , (2.29)sup ≤ λ ≤ τ k Ψ ′ k ( λ ) k < ∞ , (2.30) where Ψ ′ k ( · ) denotes the strong derivative of Ψ k ( · ) . Then (2.9)–(2.12) hold. Therefore, by Theorem 2.1, (2.13) holds.
Example 2.5
Let A be a self-adjoint operator on H . Assume that thereexists k ∈ N such that the following conditions hold: Ψ k ∈ D ( A ) ∩ \ ≤ λ ≤ τ D ( He − iλA ) , (2.31)sup ≤ λ ≤ τ k He − iλA Ψ k k < ∞ , (2.32) ∀ λ ∈ [ o, τ ] , Ψ k ( λ ) = e − iλA Ψ k . (2.33) In this case, by [2, EXAMPLE 4.5], (2.27)–(2.30) hold. Then by usingCorollary 2.4, (2.9)–(2.13) hold. .3 Trace norm convergence For the decomposition (2.1), we define ρ ( t ) := X n λ n | Ψ n ( t ) ih Ψ n ( t ) | , ∀ t ∈ [0 , τ ] . (2.34)Let us consider conditions of convergence from ρ ∆ ( τ ) to ρ ( τ ) in the tracenorm sense. Theorem 2.6
Assume that the conditions (2.9)–(2.12) hold for all k ∈ N satisfying λ k > .Then we have lim | ∆ |→ k ρ ∆ ( τ ) − ρ ( τ ) k = 0 . (2.35) Proof . By definition of ρ ∆ ( τ ), ρ ( τ ), and equation (2.8), we have k ρ ∆ ( τ ) − ρ ( τ ) k = X k h Ψ k ( τ ) , | ρ ∆ ( τ ) − ρ ( τ ) | Ψ k ( τ ) i = X k | λ ∆ ,k − λ k | = X k | λ k ( γ ∆ ,k −
1) + ǫ ∆ ,k |≤ X k λ k (1 − γ ∆ ,k ) + X k ǫ ∆ ,k = X k λ k (1 − γ ∆ ,k ) + X k ( λ ∆ ,k − λ k γ ∆ ,k )= 2 − X k λ k γ ∆ ,k . (2.36)Note that | λ k γ ∆ ,k | ≤ λ k ( ∀ k ∈ N ) , X k λ k = 1 . The assumptions (2.9)–(2.12) imply thatlim | ∆ |→ λ k γ ∆ ,k = λ k ( ∀ k ∈ N ) . Hence, by using Lebesgue’s dominated convergence theorem, we havelim | ∆ |→ X k λ k γ ∆ ,k = 1 . Remark 2.7
Assume that the conditions of Theorem 2.6 hold and that sup k,λ k =0 ξ k < ∞ and sup k,λ k =0 η k < ∞ hold. Then, for a > , we cantake | ∆ | such that (2.24) holds for all k with λ k = 0 . Then we have (2.25)for all k ∈ N with λ k = 0 . Hence, by (2.36), for all k ∈ N , we obtain thefollowing estimation: | λ ∆ ,k − λ k | ≤ k ρ ∆ ( τ ) − ρ ( τ ) k ≤ − X k λ k exp " − a ( ( ξ k + 2 ξ k η k ) N X l =1 ∆ l − N X l =1 Re h Ψ k ( t l ) − Ψ k ( t l − ) , Ψ k ( t l − ) i ) . (2.37)The following corollary and example can be easily proven by using Corol-lary 2.4, Example 2.5, and Theorem 2.6. Corollary 2.8
Assume that the conditions (2.27)–(2.30) hold for all k ∈ N with λ k > . Then we have (2.35). Example 2.9
Let A be a self-adjoint operator on H . Assume that the con-ditions (2.31)–(2.33) hold for all k ∈ N with λ k > . Then we have (2.35). In Example 2.9, let us consider the case of d < ∞ . It is easy to see thatthe assumptions (2.31)–(2.32) are satisfied. On the other hand, by Stone’stheorem, for all U ∈ U ( H ), there exists a self-adjoint operator A such that U = e − iτA . Since ρ ( τ ) = U ρU ∗ , we have lim | ∆ |→ k ρ ∆ ( τ ) − U ρU ∗ k = 0 . This fact shows that, in the case d < ∞ , an arbitrary state in { U ρU ∗ | U ∈ U ( H ) } can be approximated (in the trace norm sense) by states obtainedafter an appropriate continuous measurements. In other words, in this case,we can approximate an arbitrary unitary channel by continuous quantummeasurements. Let Ψ k ∈ D ( H ) and Ψ k ( λ ) = Ψ k ( ∀ λ ∈ [0 , τ ]) holds for all k ∈ N with λ k > A = 0 in Example 2.9. Then (2.9)–(2.12) hold forall k ∈ N with λ k >
0. Hence, we have (2.35).10his means that, by the series of measurement with respect to the familyof the projection operators {| Ψ k ih Ψ k |} k , transitions to states different fromthe initial state are hindered. This can be interpreted as a quantum Zenoeffect for mixed states. Let ϕ : [0 , ∞ ) ∋ λ
7→ − λ log λ ∈ [0 , ∞ ), where ϕ (0) := 0. Then ϕ iscontinuous, concave, and subadditive. Let S ( ρ ) be the von Neumann entropyof ρ ∈ S ( H ). i.e. S ( ρ ) := Tr ϕ ( ρ ) . In the case d < ∞ , by Fannes’ inequality, we have for all ρ , ρ ∈ S ( H ) k ρ − ρ k ≤ /e = ⇒ | S ( ρ ) − S ( ρ ) | ≤ k ρ − ρ k log d + ϕ ( k ρ − ρ k ) . Therefore the von Neumann entropy is continuous with respect to the tracenorm.On the other handin the case d = ∞ although the von Neumann entropyis lower semi-continuous with respect to the trace norm i.e. lim n →∞ k ρ n − ρ k = 0 ⇒ S ( ρ ) ≤ lim inf n →∞ S ( ρ n )), it is not necessarily continuous.Moreover, it is known that the set { ρ ∈ S ( H ) | S ( ρ ) < ∞} is of the firstcategory [10].In what follows, we deal with the case where d = ∞ only.For ρ ∆ ( τ ) and ρ considered in the section 2, conditions of the convergence S ( ρ ∆ ( τ )) → S ( ρ ) are given by the following theorem. Theorem 3.1
Assume that the conditions (2.9)–(2.11) hold for all k ∈ N ,and that the condition (2.12) holds for all k ∈ N with λ k > . Suppose thatthe following conditions hold: ξ k → , η k → k → , (3.1) S ( ρ ) < ∞ , (3.2) X k ϕ ( ξ k ) < ∞ , X k ϕ ( η k ) < ∞ . (3.3) Then lim | ∆ |→ S ( ρ ∆ ( τ )) = S ( ρ ( τ )) = S ( ρ ) . (3.4)11 emark 3.2 The function ϕ is monotone increasing on [0 , /e ] and ξ k = sup ≤ λ ≤ τ k H Ψ k ( λ ) k = sup ≤ λ ≤ τ Z R x d k E H ( x )Ψ k ( λ ) k . Hence, ξ k → k → ∞ ) implies that there exists N ∈ N such that, for all k > N , ϕ ( ξ k ) ≥ sup ≤ λ ≤ τ ϕ (cid:18)Z R x d k E H ( x )Ψ k ( λ ) k (cid:19) . By Jensen’s inequality, we have ϕ (cid:18)Z R x d k E H ( x )Ψ k ( λ ) k (cid:19) ≥ Z R ϕ ( x ) d k E H ( x )Ψ k ( λ ) k . Hence, for all k > N , ϕ ( ξ k ) ≥ sup ≤ λ ≤ τ Z R ϕ ( x ) d k E H ( x )Ψ k ( λ ) k . Then, we have ∀ k > N , ∀ λ ∈ [0 , τ ] , Ψ k ( λ ) ∈ D ( p ϕ ( H )) , ϕ ( ξ k ) ≥ sup ≤ λ ≤ τ k p ϕ ( H )Ψ k ( λ ) k . Moreover, using the estimate that X k ϕ ( ξ k ) = N X k =1 ϕ ( ξ k ) + ∞ X k = N +1 ϕ ( ξ k ) ≥ N X k =1 ϕ ( ξ k ) + ∞ X k = N +1 sup ≤ λ ≤ τ k p ϕ ( H )Ψ k ( λ ) k ≥ N X k =1 ϕ ( ξ k ) + sup ≤ λ ≤ τ ∞ X k = N +1 k p ϕ ( H )Ψ k ( λ ) k , we obtain ξ k → k → ∞ ) , X k ϕ ( ξ k ) < ∞ = ⇒ ∃ N ∈ N , sup ≤ λ ≤ τ ∞ X k = N +1 k p ϕ ( H )Ψ k ( λ ) k < ∞ . (3.5) Particularly, in the case H ∈ B ( H ) , we have, for all Φ ∈ H , Z R ϕ ( x ) d k E H ( x )Φ k ≤ sup x ∈ σ ( H ) ϕ ( x ) Z R d k E H ( x )Φ k = sup x ∈ σ ( H ) ϕ ( x ) ·k Φ k < ∞ . ence, we obtain p ϕ ( H ) ∈ B ( H ) . Therefore, by (3.5), we have ξ k → k → ∞ ) , X k ϕ ( ξ k ) < ∞ = ⇒ ϕ ( H ) ∈ T ( H ) . (3.6) We remark that, in this case, if Hamiltonian H is represented as a densityoperator, then ϕ ( H ) ∈ T ( H ) means S ( H ) < ∞ .Proof . The assumption of this theorem and Theorem 2.6 imply thatlim | ∆ |→ k ρ ∆ ( τ ) − ρ ( τ ) k = 0 . Hence we have w- lim | ∆ |→ ρ ∆ ( τ ) = ρ ( τ ),where w- lim means weak limit.By (2.8), (2.23) and γ ∆ ,k ≤
1, we have λ ∆ ,k ≤ λ k + 2( ξ k + η k ) N X l =1 ∆ l . By lim | ∆ |→ P Nl =1 ∆ l = 0, there exists δ > | ∆ | < δ ⇒ P Nl =1 ∆ l < / . Thus λ ∆ ,k ≤ λ k + ξ k + η k ( | ∆ | < δ ) . (3.7)We set σ := X k ( λ k + ξ k + η k ) | Ψ k ( τ ) ih Ψ k ( τ ) | . (3.8)By the assumption of this theorem, σ ∈ C ( H ). On the other hand, (3.7)implies that ρ ∆ ( τ ) ≤ σ ( | ∆ | < δ ) . (3.9)Moreover, by the assumption of this theorem and subadditivity of ϕ , wehave S ( σ ) = X k ϕ ( λ k + ξ k + η k ) (3.10) ≤ S ( ρ ) + X k ϕ ( ξ k ) + X k ϕ ( η k ) < ∞ . (3.11)Hence, by Simon’s dominated convergence theorem for entropy [4, THEO-REM A.3], we have lim | ∆ |→ S ( ρ ∆ ( τ )) = S ( ρ ( τ )) . It is obvious that S ( ρ ( τ )) = S ( ρ ) holds.13 emark 3.3 In the proof of Theorem 3.1, we used that S ( ρ ) < ∞ , X k ϕ ( ξ k ) < ∞ , X k ϕ ( η k ) < ∞ = ⇒ X k ϕ ( λ k + ξ k + η k ) < ∞ . (3.12) Conversely, we can show that, under condition (3.1), X k ϕ ( λ k + ξ k + η k ) < ∞ = ⇒ S ( ρ ) , X k ϕ ( ξ k ) , X k ϕ ( η k ) < ∞ (3.13) as follows. By λ k + ξ k + η k → k → ∞ ) , we have ∃ N ∈ N , ∀ k > N , max { λ k , ξ k , η k } ≤ λ k + ξ k + η k < /e. Hence, by the fact that ϕ is a monotone increasing function on [0 , /e ] , weobtain max ∞ X k = N +1 ϕ ( λ k ) , ∞ X k = N +1 ϕ ( ξ k ) , ∞ X k = N +1 ϕ ( η k ) ≤ ∞ X k = N +1 ϕ ( λ k + ξ k + η k ) . Therefore, we have (3.13). Thus, in Theorem 3.1, we can replace the con-dition (3.2) and (3.3) with P k ϕ ( λ k + ξ k + η k ) < ∞ . Example 3.4
Let A be a self-adjoint operator on H . Assume that A, H ∈ C ( H ) , and that A and H are strongly commuting. Moreover, we assume that ∀ k ∈ N , ∀ λ ∈ [0 , τ ] , Ψ k ( λ ) = e − iλA Ψ k , (3.14) S ( ρ ) < ∞ , X k ϕ ( k H Ψ k k ) < ∞ , X k ϕ ( k A Ψ k k ) < ∞ . (3.15) Then, the compactness, the strong commutativity of A and H , and (3.14) im-ply that ξ k = k H Ψ k k → , η k = k A Ψ k k → k → ∞ ) . Hence, the assump-tion of Theorem 3.1 is satisfied. Hence, we have S ( ρ ∆ ( τ )) → S ( ρ ) ( | ∆ | → . In Example 3.4, let us consider the case of A = 0. The following factcan be easily seen: H ∈ C ( H ) , Ψ k ( λ ) = Ψ k ( ∀ k ∈ N , ∀ λ ∈ [0 , τ ]) , S ( ρ ) < ∞ , X k ϕ ( k H Ψ k k ) < ∞ = ⇒ lim | ∆ |→ S ( ρ ∆ ( τ )) = S ( ρ ) . (3.16)This is the case of QZE. We remark that, if { Ψ k } k is a sequence of eigen-vectors of H , we have P k ϕ ( k H Ψ k k ) = Tr ϕ ( H ) < ∞ . Then, in (3.16), wecan replace the condition P k ϕ ( k H Ψ k k ) < ∞ with ϕ ( H ) ∈ T ( H ).14 cknowledgments The author would like to thank Professor Asao Arai for valuable comments.
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