Entanglement Relativity in the Foundations of The Open Quantum Systems Theory
aa r X i v : . [ qu a n t - ph ] A p r Entanglement Relativity in Open Quantum Systems Theory 1
Chapter ? E NTANGLEMENT R E L ATIVITY I N T HE F OUNDATIONS OF T HE O PE N Q UANTUM S YST E MS T HE ORY
M. Arsenijevi´c a ∗ , J. Jekni´c-Dugi´c b , D. Todorovi´c a , M. Dugi´c aa Department of Physics, Faculty of Science, 34000 Kragujevac, Serbia b Department of Physics, Faculty of Science and Mathematics, 18000 Niˇs, Serbia A BSTRACT
Realistic many-particle systems dynamically exchange particles with their environments.In classical physics, small variations in the number of constituent particles are commonlyconsidered practically irrelevant. However, in the quantum mechanical context, such andsimilar structural variations are generically taxed due to the so-called Entanglement Rela-tivity. In this paper we point out difficulties in deriving master equation for a subsystem ofan alternative partition of the closed quantum system. We find that the Nakajima-Zwanzigprojection method cannot be straightforwardly used to solve the problem. The emergingtasks and prospects for the consistent foundations are examined.
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M. Arsenijevi´c et al I NTRODUCTION
Realistic many-particle systems dynamically exchange particles with their environ-ments and with other systems. This trivial observation is still largely intact in the founda-tions of quantum theory, including quantum measurement, decoherence and the open quan-tum systems theory. While intuitively exchange of particles is completely clear, it sets ablurred separating line between a system and its environment. Classical physics straightfor-wardly tackles such situations e.g. in motion with variable mass or in the Grand CanonicalEnsemble for systems in thermal equilibrium. However, systematic quantum-mechanicaldescription of such processes is lacking.In quantum measurement or decoherence [1,2,3] or more generally in open quantumsystems [4,5,6] theory, the border line between ’system’ and ’environment’ is rarely ana-lyzed in depth. We construe this fact as a symptom of a subtle and hard problem that is thesubject of this paper.The aim of this paper is to diagnose the problem that is still largely unrecognized butnevertheless of the fundamental importance in the field of open quantum systems and ap-plications. We hope that, this first step in noticing and identifying the problem can help insetting outlines of further progress in the field.In Section 2 we carefully define the problem. In Section 3 we generalize our findings inthe context of the so-called projection-method approach. Section 4 is an illustration of ourconsiderations and their subtlety. Section 5 is conclusion. T HE PROBLEM
Planet Earth is constantly bombarded from the outer space. Provided that the captured-projectile mass is much less than the mass of the Earth, the variations in the Earth’s orbitaround the Sun are practically negligible. Classical statistical physics routinely accounts forhuge stochastic change in the number of particles in the many-particle systems in thermalequilibrium. The Grand Canonical Ensemble of the standard classical statistical mechanicsdescribes a composite system of N particles with the variations of the number of particlesfrom the set { , , , , ..., N } . There is also no problem with description of non-equilibriunmesoscopic systems, such as Brownian particle, which is dressed by the water moleculesthat constantly stick to and come off the particle’s surface–the particle’s ’hydration shell’known also for large molecules (e.g. protein molecules and other biopolymers) in a solution.Except for the ’chaotic’ systems, it seems that classical physics embraces the follow-ing rule: Knowledge of dynamics of an N -particle classical system allows straightforwarddeduction of dynamics for the ( N ± n )-particle systems under the same physical condi-tions (external fields and interactions), as long as n ≪ N . More intuitively, it seems thatindividuality of many-particle classical systems is not threatened by tiny changes in the sys-tem’s separation from the rest of the world–e.g. Brownian particle with n attached watermolecules is typically regarded as practically the same system as with n ′ ( = n ) attachedwater molecules.However, in the quantum mechanical context, the things stand differently. To see this,let us consider the Hamiltonian of interest in the quantum Brownian model, in which theenvironment E monitors the particle’s center of mass position [4] (and references therein).ntanglement Relativity in Open Quantum Systems Theory 3For simplicity, we refer to one dimensional system and the environment consisting of non-interacting linear harmonic oscillators: ˆ H = ˆ H S + ˆ H E + ˆ H SE (1)where ˆ H S = N S X i =1 ˆ p i m i + N S X i = j =1 V ( | ˆ x i − ˆ x j | )ˆ H E = N E X α =1 ˆ p α m α + 12 m α ω α ˆ x α ! ˆ H SE = ˆ X CM ⊗ N E X α =1 κ α ˆ x α (2)where the Latin indices refer to the S system and the Greek indices to the E system, withthe numbers N S and N E of particles in the system and the environment, respectively, withthe pair interactions V , while ˆ X CM = P N S i =1 m i ˆ x i /M and M = P N S i =1 m i .Consider now that the i ◦ th particle of the S system becomes a part of the environment.This is, instead of the S system and the environment E there is the new open system S ′ = S \ i ◦ and the new environment E ′ = E ∪ i ◦ which symbolically reads as a structural (notnecessarily dynamical) transformation: { S, E } → { S ′ , E ′ } , (3)while the composite system C as a whole remains intact by the transformation: S + E = C = S ′ + E ′ .Needless to say, the composite system’s Hamiltonian eq.(1) remains intact by the trans-formation, while it takes another form: ˆ H = ˆ H S ′ + ˆ H E ′ + ˆ H S ′ E ′ (4)where: ˆ H S ′ = N S − X i =1 ˆ p i m i + N S − X i = j =1 V ( | ˆ x i − ˆ x j | )ˆ H E ′ = ˆ H E + ˆ p i ◦ m i ◦ + 1 M ˆ x i ◦ ⊗ N E X α =1 κ α ˆ x α ˆ H S ′ E ′ = 1 M N S − X i =1 m i ˆ x i ⊗ N E X α =1 κ α ˆ x α + N S − X j =1 V ( | ˆ x i ◦ − ˆ x j | ) . (5)The following simplifications can make the two models eq.(2) and eq.(5) similar to eachother: (i) for N S ≫ , the total mass M ≈ M ′ = P N S − i =1 m i , (ii) for large M (i.e. M ′ ), M. Arsenijevi´c et alone can neglect the last term in ˆ H E ′ , or more generally to introduce the normal coordinatesfor the new environment E ′ such that it consists of mutually non-interacting quasi-particles[7,8] and (iii) the pair interactions in ˆ H S ′ E ′ to consider as a weak perturbation. Then,according to the classical intuition, the two open systems S and S ′ and their dynamicsshould appear essentially mutually indistinguishable.However, formal similarity of the quantum Hamiltonians does not in general guaranteethat reduced dynamics of the S system straightforwardly implies the reduced dynamics ofthe S ′ system. This subtle point requires careful examination.Central to derivation of master equations in the density matrix theory of open quantumsystems is the tensor product: ˆ ρ S ( t ) ⊗ ˆ ρ E , (6)which is an ansatz known as Born approximation [4,5,6] and follows in a systematic wayfrom the projection of the composite system’s state ˆ ρ ( t ) [9,10]: P ˆ ρ ( t ) = ˆ ρ S ( t ) ⊗ ˆ ρ E , (7)with P = P and Q = I − P such that Q = Q , while ˆ ρ S ( t ) = tr E ˆ ρ ( t ) carries allinformation about the open system S . Linearity of P excludes the choice ˆ ρ E = tr S ˆ ρ ( t ) .Let us suppose that the reduced dynamics for the S system is well described by a propermaster equation, which assumes validity of eq.(6), i.e. eq.(7). The main observation of thispaper is as follows:( O ) As distinct from the classical counterpart, reduced dynamics of the S system cannot ingeneral be used to derive or deduce dynamics of the S ′ system for the same time interval. This observation asserts that a master equation for the S system cannot be used todeduce/derive master equation for the S ′ system for the same time interval, [0 , t ] , with thefixed initial state, ˆ ρ ( t = 0) , of the total C system.In order to justify the ( O ), we first emphasize the formal yet substantial distinctionbetween the classical and quantum state spaces. In classical physics, the ’phase space’ of N particles is Cartesian product of the N ’phase spaces’ for individual particles. However,in quantum mechanics, the particles state spaces are not in Cartesian but in tensor product.For the composite system C decomposed (structured) as S + E , the Hilbert state space H is the tensor product of the state spaces for the subsystems: H = H S ⊗ H E . (8)The structural transformation eq. (3) gives rise to re-factorization: H = H S ′ ⊗ H E ′ . (9)Invariants of the transformation eq. (3) are: (a) the Hilbert state space H , (b) the com-posite system’s Hamiltonian ˆ H and (c) the composite system’s state in every instant of time.However, the transformation eq.(3) induces a change in the form of the Hamiltonian (e.g.eqs. (2) and (5)) as well as re-factorization eq.(8) → eq.(9). The transformation eq.(3) alsoleads to a change in the form of the composite system’s instantaneous state. For a pure statein an instant of time t , this is known as Entanglement Relativity , e.g. the equality:ntanglement Relativity in Open Quantum Systems Theory 5 | φ i S ⊗ | χ i E = X i c i | i i S ′ ⊗ | i i E ′ . (10)Entanglement Relativity (ER) is a recently established (and rediscovered) [11-16] corol-lary of quantum mechanics that asserts: Virtually every structural transformation that in-duces tensor re-factorization also induces a change in amount of quantum entanglement inthe composite system C . If the composite system’s state is tensor-product for one structure( S + E ) it is of the entangled form for practically all the alternative structures ( S ′ + E ′ ) ofthe composite system. ER regards every (pure) state in every instant in time t .On the basis of ER, it can be shown that also the more general non-classical correlations,quantified e.g. by ’quantum discord’, are structure-dependent [17]: A tensor-product mixedstate for one structure ( S + E ) acquires a quantum correlated (entangled or discordant) formfor virtually arbitrary alternative structure ( S ′ + E ′ ) of the composite system, e.g.: ˆ ρ S ⊗ ˆ ρ E = X i λ i ˆ ρ S ′ i ⊗ ˆ ρ E ′ i , X i λ i = 1 . (11)Now the classically unknown conditions eqs. (10) and (11) plausibly justify the ( O )statement as follows; the more rigorous consideration is given in Section 3. The projectioneq.(7), as well as the generalizations given in Section 3, do not possess any quantum cor-relations. Equations (6) and (7) are of exactly the same form as the l.h.s. of eqs. (10) and(11), which refer to the S + E structure. The r.h.s. of eqs. (10) and (11), which refer to the S ′ + E ′ structure, carry the quantum correlations that are not accounted for by any general-ization of eq.(7). Hence derivation of master equation that is based on eq.(6), i.e. on eq.(7),for the S system is practically never a simultaneous derivation of the master equation forthe S ′ system. Consequently, the knowledge of dynamics of the S system does not sufficeto conclude much about dynamics of the S ′ system.For the probably most relevant class of Markovian open systems, eqs. (10) and (11)reveal another layer of consideration. The tensor-product initial state ˆ ρ S ( t = 0) ⊗ ˆ ρ E is anecessary condition for Markovian dynamics [5,6]; typically, the environment is supposedthermal, ˆ ρ E = exp( − β ˆ H E ) /tr E exp( − β ˆ H E ) , on the inverse temperature β = 1 /k B T .Due to eqs. (10) and (11) for t = 0 , as long as eq.(6) is valid for the S + E structure, itis practically never fulfilled regarding the alternative S ′ + E ′ structure. Then Markoviandynamics for the S system is virtually never applicable for the alternative S ′ system. Morespecifically: even if eq.(5) can be reduced to eq.(2), and even if the new environment mayalso be in thermal-equilibrium state, P i λ i ˆ ρ S ′ i = exp( − β ′ ˆ H E ′ ) /tr E (exp( − β ′ ˆ H E ′ )) , thereis initial correlation for the S ′ and E ′ systems. Consequently, the reduced S ′ system’sdynamics is not Markovian [5,6] and is also possibly non-completely positive [18].Hence, typically, derivation of the S ′ system’s dynamics has to be started from thescratch–by setting eq.(6) for the S ′ + E ′ structure in an independent derivation of masterequation for the S ′ system. To this end, the knowledge about dynamics of the S system isnot useful. Exceptions to ER are also known, but do not alter our main point. E.g. the transformation eq.(3) does notchange a tensor-product state. However, the more general structural transformations change even such statesand ER applies [16].
M. Arsenijevi´c et al O N THE USE OF THE N AKAJIMA -Z WANZIG PROJECTION METHOD
One may still wonder if, somehow, dynamics of the S system can be used for drawingconclusions on the S ′ system’s dynamics. After all, it’s just one tiny-particle differencebetween the two structures pertaining to eq. (2) and eq. (5). The correlations present on ther.h.s. of eqs. (10) and (11) are due only to a single particle denoted i ◦ . May it be possibleto approximate the r.h.s. of eqs. (10) and (11) by some tensor-product states?In certain special cases (e.g., when the use of the Born-Oppenheimer approximationis allowed), this may be the case. Nevertheless there still remains open the question as towhether ’tiny correlations’ can be safely discarded from considerations in general. In thispaper we are not interested in such subtle and deep questions. Rather, we consider useful-ness of the Nakajima-Zwanzig projection method [9,10] in regard of the ( O ) statement.Our motivation comes from the fact that the Nakajima-Zwanzig and the related(projection-based) methods provide systematic introduction of eq.(6) and set the basis forthe up-to-date the most general methodological basis of the open systems field [4,5,6]. If the( O ) statement remains valid in the context of the projection-based methods, then it presentsa serious limitation not only to our classical intuition but also to operational procedures indescribing dynamics of the alternate open systems. Bearing in mind the classical intuitionof Sections 1 and 2, in such a case we face yet another non-trivial task in the context of theproblem of transition from quantum to classical.Below, we assume arbitrary bipartitions of a composite system C , S + E = C = S ′ + E ′ , i.e. arbitrary linear canonical transformations (LCTs) that induce the tensor-productstructures (i.e. tensor re-factorization) of the composite system’s Hilbert state space.The key idea behind the Nakajima-Zwanzig projection method [9,10] is presented byeq.(7). It consists of the introduction of a linear projection operator, P , which acts on theoperators of the state space of the composite system ’system+environment’ ( S + E ). If ˆ ρ is the density matrix of the composite system, the projection P ˆ ρ (the ’relevant part’ of thecomposite density matrix) serves to represent a simplified effective description through areduced state of the composite system. The complementary part (the ’irrelevant part’ ofthe composite density matrix), Q ˆ ρ = ( I − P )ˆ ρ . For the ’relevant part’, P ˆ ρ ( t ) , one derivesclosed (’autonomous’) equations of motion in the form of integro-differential equation. Theopen system’s density matrix ˆ ρ S ( t ) = tr E P ˆ ρ ( t ) is required to carry all information aboutthe open system S , equivalently tr E Q ˆ ρ = 0 .The Nakajima-Zwanzig projection method assumes a concrete, in advance chosen andfixed for all time-instants, system-environment split (a ’structure’), S + E . This splitis uniquely defined by the associated tensor product structure of the composite system’sHilbert space, H = H S ⊗ H E . Division of the composite system into ’system’ and ’envi-ronment’ is practically motivated. In principle, the projection method can equally describearbitrary system-environment split i.e. arbitrary factorization of the composite system’sHilbert space. By definition, different factorizations introduce different projectors, denoted P for the S + E structure, and P ′ for some alternative S ′ + E ′ structure of the compositesystem, such that ˆ ρ S = tr E P ˆ ρ ( t ) carries all information about the S system (equivalently tr E Q ˆ ρ ( t ) = 0 ), and ˆ ρ S ′ = tr E ′ P ′ ˆ ρ ( t ) carries all information about the S ′ system (equiva-lently tr E ′ Q ′ ˆ ρ ( t ) = 0 ).The linear projections can be defined [1,19]: (i) P ˆ ρ ( t ) = ( tr E ˆ ρ ( t )) ⊗ ˆ ρ E [for somentanglement Relativity in Open Quantum Systems Theory 7 ˆ ρ E = tr S ˆ ρ ], which is eq.(7), (ii) P ˆ ρ ( t ) = P n ( tr E ˆ P Sn ˆ ρ ( t )) ⊗ ˆ ρ En [with arbitrary or-thogonal supports for the ˆ ρ E s], and (iii) P ˆ ρ ( t ) = P i ( tr E ˆ P Ei ˆ ρ ( t )) ⊗ ˆ P Ei [with arbitraryorthogonal projectors for the E system]; by ˆ P , we denote the projectors on the respectiveHilbert state (factor) spaces. The physical context fixes the choice of the projection–e.g. byan assumption about the initial state. In this paper we stick to the projection (i), which isby far of the largest interest in foundations and applications of the open systems theory. Asit can be easily shown, all the projections (i)-(iii) are free from the quantum correlations(entanglement or discord).Now we provide the main results of this section that are borrowed from [20] with theproofs placed in the appendices. Lemma 1.
For the most part of the composite system’s dynamics, validity of tr E Q ρ ( t ) = tr E (ˆ ρ ( t ) − P ˆ ρ ( t )) = tr E (ˆ ρ ( t ) − ˆ ρ S ( t ) ⊗ ˆ ρ E ) = 0 , ∀ t. (12) implies non-validity of tr E ′ Q ˆ ρ ( t ) = tr E ′ (ˆ ρ ( t ) − ρ S ( t ) ⊗ ˆ ρ E ) = 0 , ∀ t, (13) and vice versa .Lemma 1 reveals that the information ’irrelevant part’ of a projected state for onestructure contains some relevant information regarding an alternative structure of the com-posite system for the most of time instants t . In formal terms: for the most part of thecomposite system’s dynamics, the projection Q ˆ ρ ( Q ′ ˆ ρ ) brings some information aboutthe open system S ′ ( S )–at variance with the Nakajima-Zwanzig projection idea. Hence ∂ P ˆ ρ ( t ) /∂t allows ’tracing out’ regarding only one structure. If that structure is S + E , then tr E ′ ∂ P ˆ ρ ( t ) /∂t = ∂ ˆ ρ S ′ ( t ) /∂t [as long as ˆ ρ S ′ ( t ) = tr E ′ ˆ ρ ( t ) ]. This can be seen also fromthe following argument, which is not restricted to the projection-based methods. Tracingout the E system is dependent on, but not equal to, the tracing out the E ′ system, and viceversa . This dependence follows from the fact that the S and E degrees of freedom areintertwined with the S ′ and E ′ degrees of freedom. Intuitively: ’ tr E ’ (e.g. integrating overthe E ’s degrees of freedom) partly encompasses both the S ′ and the E ′ degrees of freedom. Lemma 2.
The two structure-adapted projectors P and P ′ do not mutually commute [ P , P ′ ]ˆ ρ ( t ) = 0 (14) for the most of the time instants t .Very much like noncomutativity of quantum observables, Lemma 2 asserts that theprojection-based information contents regarding different structures of a composite systemare mutually exclusive for the most of the time instants t . Formally, there is no state ˆ ρ ( t ) of a composite system for which the equality P ˆ ρ ( t ) = ˆ ρ ( t ) = P ′ ˆ ρ ( t ) can be fulfilled forarbitrary instant of time t .Lemma 1 and lemma 2 refer to all projection-based methods and exclude acquisitionof information about an open system S ′ from the master equation known for the alternative(albeit possibly similar) open system S in the same instant (or interval) of time, and viceversa . In effect, the ( O ) statement is justified and leads to the conclusion that derivation ofmaster equations has to be performed for every set of the degrees of freedom (i.e. for everyopen system, S , S ′ etc.) separately, in accord with equations (10) and (11). M. Arsenijevi´c et al A NALYSIS OF THE QUANTUM B ROWNIAN MOTION
In order to illustrate subtlety of our considerations, we stick to eq.(2) as the Caldeira-Leggett model of quantum Brownian motion (QBM) [4,21].In the Schr¨odinger picture the QBM master equation for the initial separable state( ˆ ρ ( t = 0) = ˆ ρ S ( t = 0) ⊗ ˆ ρ E ) with the environment on temperature T : d ˆ ρ S ( t ) dt = − ı ¯ h [ ˆ H S , ˆ ρ S ( t )] − ıγ ¯ h [ˆ x S , { ˆ p S , ˆ ρ S ( t ) } ] − mγk B T ¯ h [ˆ x S , [ˆ x S , ˆ ρ S ( t )]] . (15)The curly brackets denote the ’anticommutator’, m is the mass while ˆ x S and ˆ p S are theposition and momentum of the particle and γ is the semi-empirical friction coefficient.Eq.(15) is not of the Lindblad form and hence by definition [5,6] is not Markovian.Interestingly enough, eq.(15) applies even for initially correlated state and for arbitrarystrength of interaction in the composite system as well as for arbitrary ’spectral density’(which defines the friction coefficient γ ) . Non-Markovianity of eq.(15) is behind its ’ro-bustness’, which is the ultimate basis of the observation of QBM effect for an alternativestructure of the total system C [7].Now we emphasize the variations offered by eq.(15). First, it is known that eq.(15) canbe transformed in a Lindblad form for sufficiently high temperature T [4]. Second, for themassive particle, the second term proportional to γ can be neglected. This constitutes the’recoilless’ variant of the Caldeira-Leggett model and provides the Lindblad-form masterequation: d ˆ ρ S ( t ) dt = − ı ¯ h [ ˆ H S , ˆ ρ S ( t )] − mγk B T ¯ h [ˆ x S , [ˆ x S , ˆ ρ S ( t )]] , (16)which is similar with the scattering-decoherence master equation–see eq.(3.66) in [1].It is remarkable that already at the level of fixed structure , C = S + E , we can see non-trivial variations in the form of master equation and consequently regarding the S system’sdynamics.Now we refer to certain structural variations of the dynamics described by eq.(16). Weare aiming at the cases in which the classical intuition can be justified; see e.g. commentsbelow eq.(5). In all other cases we do not expect the classical intuition to be very useful.Concretely, we are interested in eq.(3) as well as in the opposite case, i.e. whenan environmental particle, denoted α ◦ , is joined the S system thus providing a newopen system, S ′′ = S ∪ α ◦ , and new environment, E ′′ = E \ α ◦ ; needless to say, C = S + E = S ′ + E ′ = S ′′ + E ′′ . This situation also describes the ’Schr¨odinger’scat’–the cat represented by the S system while the α ◦ th environmental particle flows out ofthe radioactive source.Regarding the structural transformation ( S, E ) → ( S ′′ , E ′′ ) , (17) See e.g. eq.(4.226) in Ref. [4]. For examples of the more general structural transformations see e.g. Refs. [7,16,22,23]. ntanglement Relativity in Open Quantum Systems Theory 9that is accompanied by tensor re-factorization, H S ⊗ H E → H S ′′ ⊗ H E ′′ , the Hamiltonianeq.(2) i.e. eq.(5) takes the form: ˆ H = ˆ H S ′′ + ˆ H E ′′ + ˆ H S ′′ E ′′ . (18)In eq.(18): ˆ H S ′′ = ˆ H S + ˆ p α ◦ m α ◦ + ˆ X CM ⊗ κ α ◦ ˆ x α ◦ ˆ H E ′ = N E − X α =1 ˆ p α m α + 12 m α ω α ˆ x α ! ˆ H S ′′ E ′′ = ˆ X CM ⊗ N E − X α =1 κ α ˆ x α . (19)As distinct from eq.(5), eq.(19) is already of the form of eq.(2): there only appears anew interaction in the system’s (in the S ′′ system’s) self Hamiltonian . Therefore, the twostructure variations, eq.(3) and eq.(17), are not mutually equivalent.Our starting model is eq.(2), for which we assume the initial tensor product state andthermal environment E : ˆ ρ ( t = 0) = ˆ ρ S ⊗ ˆ ρ E = ˆ ρ S ⊗ α exp( − β ˆ H α ) Z α , (20)with the one-particle ’statistical sum’ Z α = tr α exp( − β ˆ H α ) .The interactions generated by the V pair-interactions in the S system suggest correla-tions in the initial S system’s state, ˆ ρ S ( t = 0) = P µ m ˆ ρ S ′ m ⊗ ˆ ρ i ◦ m , which gives correlatedinitial state for the S ′ + E ′ structure: ˆ ρ ( t = 0) = X m µ m ˆ ρ S ′ m ⊗ ˆ ρ E ′ m ≡ X m µ m ˆ ρ S ′ m ⊗ (cid:16) ˆ ρ i ◦ m ⊗ ˆ ρ E (cid:17) , X m µ m = 1 . (21)However, at variance with eq.(11) , eq.(20) directly provides tensor-product initial statefor the S ′′ + E ′′ structure: ˆ ρ ( t = 0) = ˆ ρ S ′′ ⊗ ˆ ρ E ′′ ≡ ˆ ρ S ⊗ exp( − β ˆ H α ◦ ) Z α ◦ ! ⊗ α = α ◦ exp( − β ˆ H α ) Z α . (22)Further we focus on the S ′′ + E ′′ structure.The projection is defined: Rigorously, the new structure ( S + α ◦ ) + E ′′ , where the α ◦ particle is not in interaction with E ′′ . If weassume that the mass M ′′ of S ′′ is approximately M , the two models eq.(2) and eq.(19) become practicallyindistinguishable. This nicely exhibits the subtlety of ’ quantum correlations relativity’, eq.(10) and eq.(11): for some specialstates (here: tensor-product states) and for a special pair of structures (here: S + E and S ′′ + E ′′ ) one shouldnot worry about the quantum correlations relativity. However, the worry remains for almost all other kinds ofre-structuring even for the initial tensor-product state regarding the starting S + E structure of the total system. P ˆ ρ ( t ) = ˆ ρ S ( t ) ⊗ α exp( − β ˆ H α ) Z α , (23)providing that ⊗ α exp( − β ˆ H α ) /Z α = tr S ˆ ρ ( t ) (and analogously for the other structures).For the S ′′ + E ′′ structure, the projection reads: P ′′ ˆ ρ ( t ) = (cid:16) tr E \ α ◦ ˆ ρ ( t ) (cid:17) ⊗ ˆ ρ E ′′ . (24)In an instant of time t > , we expect correlations in the S ′′ system, e.g. ˆ ρ S ′′ ( t ) := tr E \ α ◦ ˆ ρ ( t ) = P i λ i ( t )ˆ ρ Si ( t ) ⊗ ˆ ρ α ◦ i ( t ) . Hence with the use of eq.(24): tr E ′′ ∂ P ′′ ˆ ρ ( t ) ∂t = ∂∂t X i λ i ( t )ˆ ρ Si ( t ) ⊗ ˆ ρ α ◦ i ( t ) . (25)On the other hand, from eq.(23): tr E ′′ P ˆ ρ ( t ) = ˆ ρ S ( t ) ⊗ exp( − β ˆ H α ◦ ) Z α ◦ , (26)which instead of eq.(25) gives: tr E ′′ ∂ P ˆ ρ ( t ) ∂t = ∂∂t ˆ ρ S ( t ) ⊗ exp( − β ˆ H α ◦ ) Z α ◦ . (27)The absence of the exact correlations appearing in eq.(25) clearly illustrates Lemma 1,i.e. implies that tr E ′′ Q ˆ ρ ( t ) = 0 , even in this case in which eq.(11) is not applicable.Hence despite the classical similarity, the open systems S , S ′ and S ′′ are subjected todifferent dynamics. While the S system, as well possibly as the S ′′ system, undergoes africtionless Markovian dynamics eq.(16), the S ′ system may be expected to be subjected toneither Markovian [5,6] nor frictionless [4,21] and possibly non-completely-positive [18]dynamics. Exact master equations for the S ′ and S ′′ systems can follow from the inde-pendent e.g. projection-based analysis. Regarding the S ′ system, derivation of the masterequation should start from the projection P ′ ˆ ρ ( t ) = ˆ ρ S ′ ( t ) ⊗ ˆ ρ E ′ with the initial state ofthe form of eq.(21), while for the S ′′ system, the derivation should regard eq.(25) with theinitial state of the form of eq.(22)–that does not bring any substantial new observation thatis of interest for the present paper. C ONCLUSION
Recently, it was shown [7] that non-similar subsystems of a composite system can havesimilar dynamics. In contrast to the classical intuition of Sections 1 and 2, in this paper weshow that classically indistinguishable many-particle systems can undergo non-equivalentdynamics. Moreover, the shortcuts in describing dynamics of the alternative open systemsmay not even exist. It is our conjecture that the classical intuition fits with some specialstructures of many-particle systems [24] that can still require certain assumptions, such ase.g. the Born-Oppenheimer adiabatic approximation. Those results set a new layer in the See eq.(22). ntanglement Relativity in Open Quantum Systems Theory 11long standing problem of the transition from quantum to classical [1] that will be discussedelsewhere.Perhaps not surprisingly, our findings are implied by the classically unknown Quan-tum Correlations Relativity [11-17]–a not-yet-fully-appreciated rule of the universally validquantum theory–and some other, classically surprising, findings may be expected.Currently it appears that description of the alternative subsystems dynamics should beperformed for every alternate open system separately. The classical intuition, that similarsystems should bear similar dynamics, appears unreliable. A CKNOWLEDGEMENT
The work on this paper is financially supported by Ministry of education, science andtechnological development, Serbia, grant no 171028 and in part for MD by the ICTP-SEENET-MTP grant PRJ-09 ”Strings and Cosmology” in frame of the SEENET-MTP Net-work. R EFERENCES [1] Joos, E.; Zeh, H. D.; Kiefer, C.; Giulini, D.; Kupsch, J.; Stamatescu, I.-O.
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Int. J. Theor. Phys. , 2215-2225.[16] Jekni´c-Dugi´c, J.; Arsenijevi´c, M.; Dugi´c, M. Quantum Structures: A View of theQuantum World ; LAP Lambert Acad Publ: Saarbr¨ucken, 2013.[17] Dugi´c, M.; Arsenijevi´c, M.; Jekni´c-Dugi´c, J.
Sci. China Phys. Mech. Astron. , 732-736.[18] Brodutch, A.; Datta, D.; Modi, K.; Rivas, ´A.; Rodriguez-Rosario, C. A. Phys. Rev.A , 042301.[19] Gemmer, J.; Michel, M.; Mahler, G. Quantum Thermodynamics, Lecture Notes inPhysics ; Springer-Verlag: Berlin, 2004; Vol. 657.2 M. Arsenijevi´c et al[20] Arsenijevi´c, M.; Jekni´c-Dugi´c, J.; Dugi´c, M. (2013). A Limitation of theNakajima-Zwanzig projection method. http://arxiv.org/abs/1301.1005.[21] Caldeira, A. O.; Leggett, A.
Physica A , 587.[22] Stokes, A.; Kurcz, A.; Spiller, T. P.; Beige, A.
Phys. Rev. A , 053805.[23] Arsenijevi´c, M.; Jekni´c-Dugi´c, J.; Dugi´c, M. Chin Phys B , 020302.[24] Arsenijevi´c, M.; Jekni´c-Dugi´c, J.; Dugi´c, M. (2013). Zero Discord for MarkovianBipartite Systems. http://arxiv.org/abs/1204.2789. A PPENDIX
A: P
ROOF OF L EMMA Given eq.(12), i.e. tr E Q ˆ ρ ( t ) = 0 , ∀ t , we investigate the conditions that should befulfilled in order for eq.(13), i.e. tr E ′ Q ˆ ρ ( t ) = 0 , ∀ t , to be fulfilled. The Q projector refersto the S + E , not to the S ′ + E ′ structure. Therefore, in order to calculate tr E ′ Q ˆ ρ ( t ) , weuse ER. We refer to the projection (i), Section 3, in an instant of time: P ˆ ρ = ( tr E ˆ ρ ) ⊗ ˆ ρ E . (28)A) Pure state ˆ ρ = | Ψ ih Ψ | , while, due to eq.(12), tr E Q| Ψ ih Ψ | = 0 .We consider the pure state presented in its (not necessarily unique) Schmidt form | Ψ i = X i c i | i i S | i i E , (29)where ˆ ρ S = tr E | Ψ ih Ψ | = P i p i | i i S h i | , p i = | c i | and for arbitrary ˆ ρ E = tr S | Ψ ih Ψ i .Given ˆ ρ E = P α π α | α i E h α | , we decompose | Ψ i as: | Ψ i = X i,α c i C iα | i i S | α i E , (30)with the constraints: X i | c i | = 1 = X α π α , X α | C iα | = 1 , ∀ i, (31)Then Q| Ψ ih Ψ | = | Ψ ih Ψ | − X i,α p i π α | i i S h i | ⊗ | α i E h α | . (32)We use ER: | i i S | α i E = X m,n D iαmn | m i S ′ | n i E ′ (33)with the constraints: X m,n D iαmn D i ′ α ′ ∗ mn = δ ii ′ δ αα ′ . (34)With the use of eqs.(30) and (33), eq.(32) reads:ntanglement Relativity in Open Quantum Systems Theory 13 X m,m ′ n,n ′ [ X i,i ′ ,α,α ′ c i C iα c ∗ i ′ C ∗ i ′ α ′ D iαmn D i ′ α ′ ∗ m ′ n ′ − X i,α p i π α D iαmn D iα ∗ m ′ n ′ ] | m i S ′ h m ′ | ⊗ | n i E ′ h n ′ | . (35)After tracing out, tr E ′ : X m,m ′ { X i,α,n X i ′ ,α ′ c i C iα c ∗ i ′ C ∗ i ′ α ′ D iαmn D i ′ α ′ ∗ m ′ n − p i π α D iαmn D iα ∗ m ′ n }| m i S ′ h m ′ | (36)Hence tr E ′ Q| Ψ ih Ψ | = 0 ⇔ X i,α,n [ X i ′ ,α ′ c i C iα c ∗ i ′ C ∗ i ′ α ′ D iαmn D i ′ α ′ ∗ m ′ n − p i π α D iαmn D iα ∗ m ′ n ] = 0 , ∀ m, m ′ . (37)Introducing notation, Λ mn ≡ P i,α c i C iα D iαmn , one obtains: tr E ′ Q| Ψ ih Ψ | = 0 ⇔ A mm ′ ≡ X n [Λ mn Λ m ′ ∗ n − X i,α p i π α D iαmn D iα ∗ m ′ n ] = 0 , ∀ m, m ′ . (38)Notice: X m A mm = 0 . (39)which is equivalent to tr Q| Ψ ih Ψ | = 0 , see eq.(32).B) Mixed (e.g. non-entangled) state. ˆ ρ = X i λ i ˆ ρ Si ˆ ρ Ei , ˆ ρ Si = X m p im | χ im i S h χ im | , ˆ ρ Ei = X n π in | φ in i E h φ in | , (40)In eq.(40), having in mind eq.(28), tr E Q ˆ ρ = 0 , while tr E ˆ ρ = P p κ p | ϕ p i S h ϕ p | , and ˆ ρ E = P q ω q | ψ q i E h ψ q | 6 = tr S ˆ ρ .Constraints: X i λ i = 1 = X p κ p = X q ω q , X m p im = 1 = X n π in , ∀ i. (41)Now we make use of ER and, for comparison, we use the same basis {| a i S ′ | b i E ′ }| χ im i S | φ in i E = X a,b C imnab | a i S ′ | b i E ′ , | ϕ p i S | ψ q i E = X a,b D pqab | a i S ′ | b i E . (42)Constraints: X a,b C imnab C im ′ n ′ ∗ ab = δ mm ′ δ nn ′ , X a,b D pqab D p ′ q ′ ∗ ab = δ pp ′ δ qq ′ . (43)So4 M. Arsenijevi´c et al Q ˆ ρ = ˆ ρ − ( tr E ˆ ρ ) ⊗ ˆ ρ E = X a,a ′ ,b,b ′ { X i,m,n λ i p im π in C imnab C imn ∗ a ′ b ′ − X p,q κ p ω q D pqab D pq ∗ a ′ b ′ }| a i S ′ h a ′ | ⊗ | b i E ′ h b ′ | . (44)Hence tr E ′ Q ˆ ρ = 0 ⇔ Λ aa ′ ≡ X i,m,n,b λ i p im π in C imnab C imn ∗ a ′ b − X p,q,b κ p ω q D pqab D pq ∗ a ′ b = 0 , ∀ a, a ′ . (45)Again, for a = a ′ : X a Λ aa = 0 , (46)as being equivalent with tr Q ˆ ρ = 0 , see eq.(44).Validity of eq.(13) assumes validity of eq.(38) for pure and of eq.(45) for mixed states.Both eq.(38) and eq.(45) represent the sets of simultaneously satisfied equations. We donot claim non-existence of the particular solutions to eq.(38) and/or to eq.(45), e.g. for thefinite-dimensional systems. Nevertheless, we want to emphasize that the number of statesthey might refer to is apparently negligible compared to the number of states for which thisis not the case. For instance, already for the fixed a and a ′ , a small change e.g. in κ s (whilebearing eq.(41) in mind) undermines equality in eq.(45).Quantum dynamics is continuous in time. Provided eq.(12) is fulfilled, validity ofeq.(13) might refer only to a special set of the time instants. So we conclude: for the mostpart of the open S ′ -system’s dynamics, eq.(13) is not fulfilled. By exchanging the roles ofeq.(12) and eq.(13) in our analysis, we obtain the reverse conclusion, which completes theproof. Q.E.D. A PPENDIX
B: P
ROOF OF L EMMA The commutation condition, [ P , P ′ ]ˆ ρ ( t ) = 0 , ∀ t . With the notation ˆ ρ P ( t ) ≡ P ˆ ρ ( t ) and ˆ ρ P ′ ( t ) ≡ P ′ ˆ ρ ( t ) , the commutativity reads: P ˆ ρ P ′ ( t ) = P ′ ˆ ρ P ( t ) , ∀ t . Then P ˆ ρ P ′ ( t ) = tr E ˆ ρ P ′ ( t ) ⊗ ˆ ρ E = ˆ ρ S ( t ) ⊗ ˆ ρ E , while P ′ ˆ ρ P ( t ) = tr E ′ ˆ ρ P ( t ) ⊗ = σ S ′ ( t ) ⊗ σ E ′ . So, thecommutativity requires the equality σ S ′ ( t ) ⊗ σ E ′ = ˆ ρ S ( tt
ROOF OF L EMMA The commutation condition, [ P , P ′ ]ˆ ρ ( t ) = 0 , ∀ t . With the notation ˆ ρ P ( t ) ≡ P ˆ ρ ( t ) and ˆ ρ P ′ ( t ) ≡ P ′ ˆ ρ ( t ) , the commutativity reads: P ˆ ρ P ′ ( t ) = P ′ ˆ ρ P ( t ) , ∀ t . Then P ˆ ρ P ′ ( t ) = tr E ˆ ρ P ′ ( t ) ⊗ ˆ ρ E = ˆ ρ S ( t ) ⊗ ˆ ρ E , while P ′ ˆ ρ P ( t ) = tr E ′ ˆ ρ P ( t ) ⊗ = σ S ′ ( t ) ⊗ σ E ′ . So, thecommutativity requires the equality σ S ′ ( t ) ⊗ σ E ′ = ˆ ρ S ( tt ) ⊗ ˆ ρ E , ∀ tt