How to detect a possible correlation from the information of a sub-system in quantum mechanical systems
aa r X i v : . [ qu a n t - ph ] M a y How to detect a possible correlation from the information of a sub-system inquantum mechanical systems
Gen Kimura a , ∗ Hiromichi Ohno b , † and Hiroyuki Hayashi c a Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan b Graduate School of Mathematics, Kyushu University,1-10-6 Hakozaki, Fukuoka 812-8581, Japan and c Department of Physics, Waseda University, Tokyo 169-8555, Japan (Dated: October 30, 2018)A possibility to detect correlations between two quantum mechanical systems only from theinformation of a subsystem is investigated. For generic cases, we prove that there exist correlationsbetween two quantum systems if the time-derivative of the reduced purity is not zero. Therefore, anexperimentalist can conclude non-zero correlations between his/her system and some environmentif he/she finds the time-derivative of the reduced purity is not zero. A quantitative estimation ofa time-derivative of the reduced purity with respect to correlations is also given. This clarifies therole of correlations in the mechanism of decoherence in open quantum systems.
PACS numbers: 03.65.Yz, 03.65.Ta
I. INTRODUCTION
In many contexts in physics, it is important toknow the existence (or absence) of correlations [1] ofa system of interest S and its environment E (an-other unknown system). For example, in order toachieve a successful quantum information process-ing, quantum communication or quantum mechani-cal control, one has to manage system-environmentcorrelations which may enhance the decoherence ofthe states of the system. However, in many cases,we know neither the structure of the environmentnor the nature of the interaction of the system. Un-der these circumstances, one has to detect possiblecorrelations between S and E , only from the mea-surements of the system S , not from those of thetotal system S + E . To do this, if an ensemble ofindependently identical systems is available, the fol-lowing well-known criterion [2] for quantum systemscan be applied: (A) If the system S is in a pure state, then S has no correlations with any other environ-ment E . From this statement, an experimentalist can safelyconclude no correlations with any environment ifhe/she found his/her (reduced) state in a pure state.Indeed, some of the unconditional security proofsof quantum cryptography partially rely on this fact[3], where an unknown eavesdropper is assumed toprepare any environment and do anything which isphysically allowed.Unfortunately, statement (A) is unavailable whenthe reduced state is in a mixed state. Indeed, then,any static property of a subsystem cannot provide ∗ Electronic address: [email protected] † Electronic address: [email protected] the information of the correlation, since the samereduced mixed states can be generated from the to-tal states with and without the system-environmentcorrelations [4]. Therefore, in such cases, we wouldneed to use dynamical information as well. Herewhat we would like to discuss and try to show is thefollowing statement: (B) If the time derivative of the purity of S is not zero at time t = t , there exist non-zero correlations with a certain environmentat that time .If this statement is universally true, this makes anexperimentalist possible to confirm non-zero corre-lations with some environment if he/she found thetime derivative of the purity is not zero [5]. The pur-pose of this paper is to investigate statement (B) forarbitrary quantum mechanical systems [6] under theusual postulates for (open) quantum mechanics (see,for instance [7, 8, 9]), which include the followings:(i) [State space] For any quantum mechanicalsystem S , there exists a separable Hilbert space H S .Any state of S is represented by a density operator ρ S — a positive trace class operator on H S with unittrace. The purity P S for ρ S is defined by P S = Tr S { ρ S } . (1)(ii) [Composite system] Let S and E be quan-tum mechanical systems with Hilbert spaces H S and H E . The composite system S + E is associated withthe tensor product Hilbert space H S ⊗ H E . For a total density operator ρ tot on H S ⊗ H E , thereduced states ρ S and ρ E for S and E are given by ρ S = Tr E { ρ tot } and ρ E = Tr S { ρ tot } where Tr S andTr E are the partial traces with respect to S and E ,respectively. (In the following, ρ S and ρ E alwaysrepresent the reduced density operators on S and E from the total density operator ρ tot .) No correla-tions in a density operator ρ tot on S + E equivalentlyTypeset by REVTEXmeans that ρ tot is given by a tensor product of thereduced density operators of the two subsystems: ρ tot = ρ S ⊗ ρ E . (2)(iii) [Evolution] A quantum system S is dynami-cally isolated or open, and without or with a certainenvironment E , the dynamics of S is eventually de-scribed by the von Neumann equation (Schr¨odingerequation) on the total system. Namely, there exists aself-adjoint Hamiltonian H on H S ⊗ H E with whichthe von Neumann equation holds: i ~ ddt ρ tot ( t ) = [ H, ρ tot ( t )] , (3) where ρ tot ( t ) is a density operator on H S ⊗ H E attime t . (In the following, we set Planck’s constant ~ to be .) Notice, however, that there appears a domain-problem when H is an unbounded operator [10]. Toavoid the problem, it is generally adopted in the ax-iomatic approach of quantum mechanics that the dy-namics is governed by a unitary time evolution: ρ tot ( t ) = U t ρ tot U † t , (4)where ρ tot is an initial density operator at t = 0 and U t is a unitary operator given by U t = e − iHt (fora time-independent Hamiltonian H ). Then, for anydensity operator ρ tot , the dynamics (4) is appliedwithout any problem such as a domain-problem. Inthis paper, we assume a unitary dynamics (4) for anisolated quantum system.In a formal analysis, statement (B) for quantummechanical systems can be proved in the followingway: Let the time-derivative of the purity of a quan-tum system S at t = t is not zero. Since the pu-rity does not change in an isolated system, S shouldbe an open system interacting with some environ-ment E . Let H be a self-adjoint Hamiltonian on H S ⊗ H E which reads the von Neumann equation(3). Assume that there are no correlations at t = t ,namely the initial density operator takes a productform ρ tot = ρ S ⊗ ρ E . Then, from the von Neumannequation, we observe, P ′ S ( t ) ≡ ddt P S ( t ) (cid:12)(cid:12)(cid:12) t = t = 2 Tr S (cid:26) ρ S ( t ) ddt ρ S ( t ) (cid:12)(cid:12)(cid:12) t = t (cid:27) = − i Tr S { ρ S Tr E [ H, ρ S ⊗ ρ E ] } = − i Tr SE { ρ S ⊗ I E [ H, ρ S ⊗ ρ E ] } = 0 , (5)where the cyclic property [11] of the trace Tr SE hasbeen used to estimate the last equality. Therefore,by contradiction, we conclude that ρ tot has non-zero correlations at t = t . (In the following, the no-tation of the Newton’s difference quotient such as P ′ S ( t ) ≡ ddt P S ( t ) (cid:12)(cid:12)(cid:12) t = t will be used.) It is worthyto notice that, although use has been made of aHamiltonian in the proof, experimentalists do nothave to know anything about environments includ-ing the way how they are interacting with their sys-tems. Instead, only thing they have to believe is thepostulates of quantum mechanics such as postulates(i),(ii), and (iii).Estimation (5), however, is still rough without suf-ficient mathematical rigor, especially for the case ofinfinite dimensional Hilbert spaces. Moreover, if theHamiltonian is described by an unbounded opera-tor, we have to deal with the domain carefully, whichmakes the statement quite non-trivial. In the follow-ing, we discuss the validity of statement (B) includ-ing infinite dimensional Hilbert spaces in a carefulmanner. In Sec. II, we provide a rigorous versionof statement (B) and show more general statement(Theorem 1) in the case of bounded Hamiltonians,which quantitatively generalize statement (B). Thisshows how purity changes under the existence of cor-relations, and hence clarifies the role of correlationsin the mechanism of decoherence in open quantumsystems. In Sec. III, we discuss statement (B) in thecase of unbounded Hamiltonians and show a cer-tain counter example. Finally, we slightly modifythe statement (B) to be correct (Theorem 3) for thecase of unbounded Hamiltonians. This is done byadding an assumption of a finite variance of a totalenergy, and hence we conclude that statement (B)is universally valid for all the generic cases. Sec. IVcloses the paper with some concluding remarks anddiscussion. II. THE CASE OF BOUNDEDHAMILTONIANS — QUANTITATIVEESTIMATION OF STATEMENT (B)
In this section, we discuss statement (B) includinginfinite dimensional cases with mathematical rigor,but for the case of bounded Hamiltonians. We ob-tain a useful theorem which generalizes statement(B) in a quantitative manner (Theorem 1). As usualwhen discussing open quantum systems [8], we shalldivide a total Hamiltonian H into the sum of freeHamiltonians H S and H E for systems S and E andan interaction Hamiltonian H int : H = H S ⊗ I E + H int + I S ⊗ H E . (6)We assume H S , H E and H int are bounded self-adjoint operators on H S , H E , and H S ⊗ H E , respec-tively, and hence H is also a bounded self-adjointoperator on H S ⊗ H E .In order to quantify correlations between S and E in a state ρ tot , we use quantum mutual information[12, 13]: I ( ρ tot ) ≡ Tr SE { ρ tot log ρ tot − ρ tot log ρ S ⊗ ρ E } , where ρ S and ρ E are reduced density operators on S and E , respectively. Notice that I ( ρ tot ) ≥
0, and I ( ρ tot ) = 0 iff ρ tot has no correlations. Notice alsothat [15] || ρ tot − ρ S ⊗ ρ E || ≤ I ( ρ tot ) , (7)where || · || is the trace norm || W || ≡ Tr SE n √ W † W o [11].For any density operator ρ tot on H S ⊗ H E , we de-fine the correlation operator ρ cor [16] by ρ cor ≡ ρ tot − ρ S ⊗ ρ E , (8)which is a trace class operator on H S ⊗ H E . By def-inition, it holds that ρ cor = 0 iff ρ tot has no correla-tions with a product form (2). Since Tr E { ρ S ⊗ ρ E } = ρ S , it follows Tr E { ρ cor } = 0 . (9)We have the following quantitative estimation of atime-derivative of the reduced purity: Theorem 1.
Let S and E be quantum mechanicalsystems where the total system S + E is a closedsystem. Let H be a total Hamiltonian and ρ tot bea density operator at t = t . If H is bounded withthe form (6) , then the reduced purity P S ( t ) is time-differentiable at t = t and P ′ S ( t ) = − i Tr SE { ρ S ⊗ I E [ H int , ρ cor ] } . (10) The absolute value of the time-derivative is boundedfrom above by | P ′ S ( t ) | ≤ || ρ S || || [ H int , ρ cor ] || , (11a) ≤ || H int || || ρ cor || , (11b) ≤ || H int || I ( ρ tot ) , (11c) where || · || denotes the operator norm [11].Proof . Notice that [ H, ρ tot ( t )] is a trace class op-erator due to an ideal property of trace class opera-tors [17] and the von Neumann equation (3) holds [9]for any density operator where the time derivative isdefined with respect to the trace norm. Therefore,by observing the inequalities [11]: | Tr { Aρ }| ≤ || Aρ || ≤ || A || || ρ || , ( ∀ A ∈ B ( H ) , ρ ∈ T ( H )) , (12)and || ρ S ( t ) ⊗ I E || ≤ P S ( t ) is differentiable forany time t and we have P ′ S (0) = − i Tr SE { ρ S ⊗ I E [ H, ρ tot ] } . By the cyclic property of the trace [19],it follows Tr SE { ρ S ⊗ I E [ H, ρ S ⊗ ρ E ] } =Tr SE { [ ρ S ⊗ ρ E , ρ S ⊗ I E ] H } = 0, and therefore, wehave P ′ S (0) = − i Tr SE { ρ S ⊗ I E [ H, ρ cor ] } . Moreover, since Tr SE { ρ S ⊗ I E [ H S ⊗ I E , ρ cor ] } =Tr S { ρ S [ H S , Tr E ρ cor ] } = 0 from (9), and Tr SE { ρ S ⊗ I E [ I S ⊗ H E , ρ cor ] } = Tr SE { [ ρ S ⊗ I E , I S ⊗ H E ] ρ cor } =0 again by the cyclic property of the trace, we ob-tain (10). From (12), [ H int , ρ tot ] ∈ T ( H S ⊗ H E ) and || ρ S ⊗ I E || = || ρ S || , we have | P ′ S (0) | ≤ || ρ S || || [ H int , ρ cor ] || . The second inequality (11b) follows from the triangleinequality for the trace norm, || ρ S || ≤
1, and again(12). The third inequality (11c) follows from (7).
QED
Theorem 1 provides a quantitative estimation of atime-derivative of the reduced purity in terms of theamount of correlations I ( ρ tot ) and the strength of in-teraction || H int || [20]. It is worth to notice that theinequalities (11)s include the following well-knownfact [5]: the purity of system does not change with-out an interaction with an environment. Indeed, ex-perimentalists usually confirm the existence of aninteraction between the system and some environ-ment, if they find the reduced purity not to be con-stant. However, not only that, (11)s imply that cor-relations play an essential role in changing the purityeven in the existence of an interaction. Moreover,Eq. (10) implies that the commutator between theinteraction Hamiltonian and the correlation opera-tor is essential for the changes of purity, or decoher-ence.From Theorem 1, we obtain a rigorous version ofstatement (B): Theorem 2.
With the same assumptions as in The-orem 1, if there are no correlations at t = t : ρ tot = ρ S ⊗ ρ E , then P S ( t ) is time-differentiableat t = t and P ′ S ( t ) = 0 . In other words, if thetime-derivative of the reduced purity is not zero, thenthere exists a non-zero correlation between S and E at that time.Proof . Since ρ tot = ρ S ⊗ ρ E implies ρ cor = 0, wehave P ′ S ( t ) = 0 from inequality (11a). QED
It should be noticed that the opposite statementdoes not generally true. (For instance, if H int = 0,we have P ′ S ( t ) = 0 even in the presence of correla-tions.) Therefore, it is incorrect to infer no correla-tions when the time-derivative of the reduced purityis zero. Notice also that the above theorems do notcontradict with the results in Ref. [21] where we haveshown that an effect of an initial correlation does notappear in van Hove’s limit (the weak coupling limit)and therefore system S behaves as if the total sys-tem started from the factorized initial state. Indeed,this is true only for the van Hove time scale τ = λ t where λ ≪ τ , we can find a differencebetween no correlations and non-zero correlations aswe have seen in the above Theorems. (See also [16]for an effect of an initial correlation.) III. THE CASE OF UNBOUNDEDHAMILTONIANS — COUNTER EXAMPLESOF STATEMENT (B)
In the previous section, we have confirmed thatstatement (B) is universally true for any boundedHamiltonian. However, Hamiltonians are generallyunbounded, especially from above, like that of theharmonic oscillator. Notice that, although the quan-titative estimation (11) in Theorem 1 turns out tobe trivial when || H int || = ∞ , we may still expectthe validity of Theorem 2, i.e., statement (B). Inthis section, we discuss statement (B) in the caseof unbounded Hamiltonians. However, as we shallsee below, the statement itself can be generally bro-ken down. In the following, we provide a counterexample of statement (B).[ Counter Example of statement (B) ]Let our system be described by H S = H S ⊗ H S where H S is a separable Hilbert space with an infi-nite dimension, and H S is a 2 dimensional Hilbertspace, H S ≃ C . (For instance, it is a system ofa non-relativistic electron with spin 1 / H E ≃ C , which is also a 2 dimensional Hilbertspace.Assume that initially the total system is in a state ρ tot = ρ S ⊗ ρ E which has no correlations, where ρ S = ∞ X n =1 p n | φ n ih φ n | ⊗ | s ih s | , ρ E = | e ih e | , (13)with p n ≥ , P ∞ n =1 p n = 1, and orthonormal bases {| φ n i} ∞ n =1 , {| s n i} n =1 , and {| e n i} n =1 of H S , H S ,and H E , respectively.We use the following Hamiltonian H , whose spec-tral decomposition reads H = ∞ X n =1 4 X k =1 h nk | φ n ⊗ χ k ih φ n ⊗ χ k | , with eigenvalues (point spectra) h n = 0 , h n = h n = h n , h n = 2 h n with h n ≥ n ∈ N ), where {| χ k i} k =1 is an orthonormal basis of H S ⊗ H E given by | χ i ≡ √ | s ⊗ e i + i | s ⊗ e i ) , | χ i ≡ | s ⊗ e i , | χ i ≡ | s ⊗ e i , | χ i ≡ √ | s ⊗ e i − i | s ⊗ e i ) . By the above spectral decomposition, it is easy tosee that H is a positive self-adjoint operator on H S ⊗ H E , which is unbounded when the sequence { h n } is not bounded from above. The time evolu-tion map U t = exp( − iHt ) is given by U t = ∞ X n =1 | φ n ih φ n | ⊗ X nt , where X nt ≡ | χ ih χ | + e − ih n t ( | χ ih χ | + | χ ih χ | ) + e − i h n t | χ ih χ | . By (13) we have ρ tot ( t ) = U t ρ tot U † t = ∞ X n =1 p n | φ n ih φ n | ⊗ | X nt s ⊗ e ih X nt s ⊗ e | , where | X nt s ⊗ e i = e − ih n t (cos( h n t ) | s ⊗ e i − sin( h n t ) | s ⊗ e i ). By taking a partial traceover E , we have ρ S ( t ) = P ∞ n =1 p n | φ n ih φ n | × (cos ( h n t ) | s ih s | +sin ( h n t ) | s ih s | ). From this, weobtain an analytical form of the reduced purity: P S ( t ) = ∞ X n =1 p n (cos ( h n t ) + sin ( h n t ))= P S (0) − ∞ X n =1 ( p n sin(2 h n t )) = 34 P S (0) + 14 ∞ X n =1 p n cos[4 h n t ] , (14)where P S (0) = P ∞ n =1 p n . Therefore, if the infinitesum in (14) and the time-derivative is commuta-tive, we obtain P ′ S (0) = 0 and statement (B) holds.For instance, let p n = n , and h n = nE with aunit of energy E . Then, since | ddt p n cos[4 h n t ] | = | nE sin[ nE t ]4 n | ≤ nE n and P ∞ n =1 nE n < ∞ , it fol-lows that P ∞ n =1 p n cos[4 h n t ] is differentiable withrespect to t and we have ddt P ∞ n =1 p n cos[4 h n t ] = P ∞ n =1 p n h n sin[4 h n t ]. Hence, this example satisfiesstatement (B) even in the case of unbounded Hamil-tonians. See FIG. 1 (a). (In the following, we set E to be 1. )However, we can construct a counter example ofstatement (B) in the sense that P S ( t ) is not differ-entiable with respect to t at t = 0 even when aninitial state is given in a product form. We providean interesting example that P S ( t ) is continuous but FIG. 1: Time evolution of the reduced purity (14) for(a) p n = n , h n = n and (b) p n = n , h n = n π , witha unit of time ω ≡ ~ /E . Notice that in both cases theHamiltonians are unbounded from above. One sees theflat time derivative at t = 0 in (a) which makes statement(B) to be true, while one sees non-differentiability in (b)which breaks down statement (B). not differentiable at anytime t by connecting the re-duced purity to the so-called Weierstrass function f ( t ; a, b ) [22], defined by f ( t ; a, b ) = ∞ X n =0 a n cos( b n πt ) , with two parameters 0 < a < b satisfying ab > π . It is known that thefunction is continuous everywhere but differentiablenowhere with respect to t . From the form of (14),a proper choice of p n and h n , for instance, p n = n h n = n π , makes P S ( t ) an essentially Weierstrassfunction: P S ( t ) = 14 (1 − cos( πt ) + f ( t ; 14 , , (15)(See FIG. 1 (b).) This provides a counter example ofstatement (B). Namely, even with a product initialstate, a time derivative of the purity is not necessar-ily zero; though this case just provides a case of anon-existence of the time-derivative.Therefore, in the case of unbounded Hamiltonians,we need to modify our statement (B). Indeed, thefollowing weaker statement can be proved to be true: Theorem 3.
Let H be a self-adjoint Hamiltonianbounded from below, but not necessarily boundedfrom above. Let ρ tot be a density operator at t = t . If the variance of H with respect to ρ tot isfinite , then ρ tot = ρ S ⊗ ρ B ⇒ P ′ S ( t ) = 0 . The assumption of the boundedness of the Hamil-tonian from below is physically required so that thesystem to be stable. Hence, even when the Hamilto-nian H is unbounded, statement (B) is correct pro-vided that the total state has a finite variance of H . In fact, it is easy to see that the variance of H is infinite for the initial state used for the counterexample in (15).To avoid redundant technical difficulties whendealing with unbounded Hamiltonians, in thepresent paper, we do not give a proof of Theorem3. Instead, we just notice the followings: First, afiniteness of the variance of H with respect to apure state ρ tot = | ψ ih ψ | is equivalent to that | ψ i is in the domain of H . Therefore, from the mathe-matical point of view, the assumption of a finitenessof the variance of H allows us to avoid a domain-problem for unbounded operators. Second, the vonNeumann equation holds when the variance of H isfinite, which is the essential reason for the Theorem3 to be correct [23]. We plan to discuss and providea systematic investigation for the case of unboundedHamiltonians in the forthcoming paper, including acomplete proof of Theorem 3. IV. CONCLUDING REMARKS ANDDISCUSSION
We have discussed the problem how one can detectpossible correlations between the system of interest S and an environment from the knowledge (by ob-servations) of the system S only. We conjecturedstatement (B), from which one can conclude non-zero correlations with some environment when thetime derivative of the reduced purity is not zero. Insome sense, it is a counterpart of statement (A); onecan conclude no correlations when the reduced pu-rity is 1 using statement (A), while one can concludecorrelations when the time derivative of the reducedpurity is not zero. For instance, an experimental-ist first can use statement (A), and if his/her stateis in a pure state, he/she can conclude no correla-tions. If the state is in a mixed state, then he/shecan use statement (B) and check the time-derivativeof the purity. If the time-derivative is not zero,he/she can conclude the existence of correlations,provided that statement (B) is universality true. Inthis paper, we have investigated the validity of state-ment (B) for arbitrary quantum mechanical systems.When the total Hamiltonian is bounded, we provedit to be universally correct (Theorem 2), by giving amore general statement (Theorem 1) which quanti-tatively implies statement (B). Theorem 1 also clar-ifies the cause of a purity-change (decoherence) dueto an interaction and correlations. However, whenthe total Hamiltonian is unbounded, we have alsoshown a counter example of statement (B). In theexample, the reduced purity evolves essentially asa Weierstrass function even with a product initialstate, whence the differentiability of the reduced pu-rity has been broken down in statement (B). There-fore, a certain modification is necessary for state-ment (B). If one considers a state with a finite vari-ance of energy as a natural realization in nature,one can conclude the universality of statement (B)for all the generic states in that sense. However,considering our original goal to estimate a possiblecorrelation, especially for the situation where we donot know anything about environment (other thanour theoretical knowledge of quantum theory), it ispreferable to assume nothing additional for an en-vironment [24]. In order for this, another plausibleconjecture will be Conjecture 1. ∃ P ′ S (0) and P ′ S (0) = 0 ⇒ ρ tot = ρ S ⊗ ρ B . If this is correct, it turns out that one can con-clude non-zero correlations if one finds non-zero timederivative (including the differentiability) of the re-duced purity. In this direction, in the forthcom-ing paper, we will discuss statement (B) includinga complete proof of Theorem 3 and an investigationof the above conjecture. Also the case of a quantumfield by using an algebraic formalism of quantumfields [25] will be presented elsewhere.
Acknowledgement
We are grateful to Profs. S. Pascazio, I. Ohba,S. Tasaki, H. Nakazato, M. Ozawa, and F. Hiai fortheir continued encouragements and helpful advices.We would like to thank Drs. M. Mosonyi, M. Hotta,K. Yuasa, K. Imafuku, and P. Facchi for their fruit-ful comments and useful discussions. In particular,we appreciate Profs. Pascazio, Tasaki, Nakazato,and Dr. Yuasa for their careful readings of themanuscript prior publication and Dr. Mosonyi forthe useful discussion about the validity of the vonNeumann equation. This research is supported bythe Grant-in-Aid for JSPS Research Fellows. [1] In this paper, we mean a correlation as the statisti-cal correlation in a state of composite physical sys-tem S + E : We say there are no correlations between S + E (a statistical independence) iff the joint prob-ability distribution between any two observables O S from S and O E from E is a product of the probabil-ity distributions for O S and O E . Otherwise, we saythat there exist non-zero correlations.[2] See for instance; B. d’Espagnat, Conceptual Foun-dations of quantum Mechanics (Benjamin, Read-ing Mass., 1976); B. d’Espagnat, Lett. Nuovo Ci-mento, (16), 823 (1971). Although the proof thereis given only for finite dimensional cases, we noticethat statement (A) is true even for quantum fields,where quantum states are treated as positive linearfunctionals on the algebras of observables.[3] A. Ekert, Phys. Rev. Lett. 67, 661 (1991).[4] Some researchers might take a stance that all themixedness originates from correlations with someenvironment, and eventually the total system shouldbe always described by a pure state. If this is univer-sally true, then statement (A) is enough to concludethe existence of correlations when system S is in amixed state. However, our capacity is more generaland statement (B) is still useful even if there existsmixedness not originated from correlations.[5] Notice that an experimentalist would usually con-clude the existence of non-zero interaction withsome environment when the time derivative of the purity of S is not zero, since the purity of system S does not change if S is isolated. Compared tothis, statement (B) allows experimentalists to con-firm non-zero correlations with some environment.(See Theorem 1 below.)[6] In a rigorous sense, in this paper, we only treat anenvironment which is a quantum mechanical sys-tem with a fixed separable Hilbert space. Therefore,quantum fields with infinite degrees of freedom arenot included. These cases will be investigated in theforth coming paper.[7] J. von Neumann, Mathematische Grundlagen derQuantenmechanik (Springer, Berlin, 1932) [transl.by E. T. Beyer:
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Classics on Fractals (Addison-Wesley Pub-lishing Company, 1993, 3-9).[23] Indeed, the indifferentiability of (15) is essentiallydue to the fact that the von Neumann equation doesnot hold in this example. However, it is worth notic-ing that we can provide an example which satisfiesstatement (B) even when von Neumann equationdoes not hold.[24] Moreover, in order to include the case where an en-vironment is a quantum field which has infinite de-grees of freedom, the assumption of the environmentto be described by a particular fixed Hilbert spaceshould also be removed.[25] R. Haag,
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