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Whenever a Quantum Environment Emerges As a Classical System, It Behaves Like a Measuring Apparatus

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Whenever a Quantum Environment Emerges As a Classical System, It Behaves Like a Measuring Apparatus Whenever a quantum environment emerges as a classical system, it behaves like a measuring apparatus Caterina Foti1,2, Teiko Heinosaari3, Sabrina Maniscalco3, and Paola Verrucchi4,1,2 1Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy 2INFN, Sezione di Firenze, I-50019, Sesto Fiorentino (FI), Italy 3QTF Centre of Excellence, Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FIN-20014, Turku, Finland 4ISC-CNR, at Dipartimento di Fisica e Astronomia, Università di Firenze, I-50019, Sesto Fiorentino (FI), Italy August 6, 2019 We study the dynamics of a quantum malizes the qualitative argument that the system Γ with an environment Ξ made of reason why we do not observe state su- N elementary quantum components. We perpositions is the continual measurement aim at answering the following questions: performed by the environment. can the evolution of Γ be characterized by some general features when N becomes very large, regardless of the specific form 1 Introduction of its interaction with each and every com- ponent of Ξ? In other terms: should we There exist two closely-related questions about expect all quantum systems with a macro- the quantum mechanical nature of our uni- scopic environment to undergo a somehow verse that keep being intriguing after decades of similar evolution? And if yes, of what thought processing: how is it that we do not ex- type? In order to answer these questions perience state superpositions, and why we cannot we use well established results from large- even see them when observing quantum systems. N quantum field theories, particularly re- As for the latter question, it is somehow assumed ferring to the conditions ensuring a large- that this is due to the continual measurement pro- N quantum model to be effectively de- cess acted upon by the environment. However, scribed by a classical theory. We demon- despite often being considered as an acceptable strate that the fulfillment of these con- answer, this argument is not a formal result, and ditions, when properly imported into the attempts to make it such have been only recently framework of the open quantum systems proposed [1–3]. In fact, the current analysis of dynamics, guarantees that the evolution of the quantum measurement process [4], its Hamil- Γ is always of the same type of that ex- tonian description [5,6], as well as its characteri- pected if Ξ were a measuring apparatus, zation in the framework of the open quantum sys- no matter the details of the actual inter- tems (OQS) dynamics [7] has revealed the qual- action. On the other hand, such details itative nature of the above argument, thus mak- are found to determine the specific basis ing it ever more urgent to develop a rigorous ap- arXiv:1810.10261v3 [quant-ph] 5 Aug 2019 w.r.t. which Γ undergoes the decoherence proach to the original question. This is the main dictated by the dynamical description of goal of our work. the quantum measurement process. This Getting back to the first question, the answer result wears two hats: on the one hand it offered by the statement that microscopic sys- clarifies the physical origin of the formal tems obey quantum rules while macroscopic ob- statement that, under certain conditions, jects follow the classical ones, is by now con- sidered unsatisfactory. Macroscopic objects, in- any channel from ρΓ to ρΞ takes the form of a measure-and-prepare map, as recently deed, may exhibit a distinctive quantum be- shown in Ref. [1]; on the other hand, it for- haviour (as seen for instance in superconductiv- ity, Bose-Einstein condensation, magnetic prop- Caterina Foti: caterina.foti@unifi.it erties of large molecules with S = 1/2), meaning 1 that the large-N condition is not sufficient per- plement such limit in Sec.5, being finally able sé for a system made of N quantum particles to to show what we were looking for. In Sec.6 we behave classically. In fact, there exist assump- comment on the assumptions made, while the re- tions which single out the minimal structure any sults obtained are summed up in the concluding quantum theory should possess if it is to have a section. classical limit [8]. Although variously expressed depending on the approach adopted by different authors (see the thorough discussion on the re- 2 Schmidt decomposition and dynam- lation between large-N limits and classical the- ical maps ories developed in Sec.VII of Ref. [8]), these as- sumptions imply precise physical constraints on We consider the unitary evolution of an isolated the quantum theory that describes a macroscopic bipartite system Ψ = Γ + Ξ, with Hilbert space quantum system if this has to behave classically. HΓ ⊗ HΞ ; being Ψ isolated, it is In what follows, these assumptions will formally ˆ characterize the quantum environment, in order |Ψ(t)i = e−iHt |Ψi , (1) to guarantee that the environment, and it alone, behaves classically. The relevance of the sentence where ~ = 1 and Hˆ is any Hamiltonian, describ- "and it alone" must be stressed: indeed, the work ing whatever interaction between Γ and Ξ. The done in the second half of the last century on state |Ψi is assumed separable the N → ∞ limit of quantum theories is quite comprehensive but it neglects the case when the |Ψi = |Γi ⊗ |Ξi , (2) large-N system is the big partner of a principal quantum system, that only indirectly experiences meaning that we begin studying the evolution at such limit. This is, however, an exemplary situ- a time t = 0 when both Γ and Ξ are in pure ation in quantum technologies and OQS, hence states. This is not a neutral assumption, and we the questions asked at the beginning of this In- will get back to it in Sec.6. troduction have recently been formulated in the At any fixed time τ, there exists a Schmidt corresponding framework [1–3,9–16]. decomposition of the state (1), X In this work, we develop an original approach |Ψ(τ)i = cγ |γi |ξγi , (3) which uses results for the large-N limit of quan- γ tum theories in the framework of OQS dynamics. + This allows us to show that details of the interac- with γ = 1, ..., dimHΓ, cγ ∈ R for γ ≤ γmax ≤ P 2 tion between a quantum principal system Γ and dimHΓ, cγ = 0 for γ > γmax, γ cγ = 1, its environment Ξ are irrelevant in determining and the symbol ⊗ understood (as hereafter done the main features of the state of Ξ at any time whenever convenient). The states {|γi}HΓ , and τ in the large-N limit, as long as such limit im- {|ξji}HΞ with j = 1, ... dim HΞ, form what we plies a classical behaviour for Ξ itself. If this is will hereafter call the τ-Schmidt bases, to remind the case, indeed, such state can always be recog- that the Schmidt decomposition is state-specific nized as that of an apparatus that measures some and therefore depends on the time τ appearing observable of the principal system. The relation in the LHS of Eq.(3), in whose RHS we have between our findings and the two questions that instead understood the τ-dependence of cγ, |γi, open this section is evident. and |Rγi, for the sake of a lighter notation. Con- sistently with the idea that Ξ is a macroscopic The paper is structured as follows. In the first system, we take γmax < dimHΞ: therefore, the section we define the dynamical maps character- states {|ξγi}HΞ entering Eq.(3) are a subset of izing the two evolutions that we aim at compar- the pertaining τ-Schmidt basis. Given that |Γi ing. We do so through a parametric representa- is fully generic, the unitary evolution (1) defines, tion introduced in Sec.3. In Sec.4, we focus on a via ρΞ = TrΓ ρΨ , the CPTP linear map (from Γ- peculiar property of generalized coherent states, to Ξ-states) particularly relevant when the large-N limit is considered. As the environment is doomed to be X 2 E: |ΓihΓ| → ρΞ = cγ |ξγihξγ| . (4) macrocopic and behave classically, we then im- γ 2 Being the output ρΞ a convex sum of orthogo- Further using the Schmidt coefficients, we con- nal projectors, Eq.(4) might describe a projec- struct the separable state tive measurement acted upon by Ξ on the prin- M M cipal system Γ, by what is often referred to as |Ψ i = |Γi ⊗ |Ξ i , (8) measure-and-prepare (m&p) map. However, for where |Γi is the same as in Eq.(2), while |ΞMi = this being the case, the probability reproducibil- P c |ξ i , with c and |ξ i as in Eq.(3). Finally ity condition [17] must also hold, meaning that, γ γ γ γ γ we define given X ˆ M |Γi = aγ |γi , (5) |ΨMi = e−iH τ |ΨMi , (9) γ τ 2 2 ˆ ˆ it should also be cγ = |aγ| , ∀γ. This condi- that reads, using OΓ |γi = εγ |γi, OΞ |ξγi = P tion, however, cannot be generally true, if only Eγ |ξγi, and |Γi = γ aγ |γi, for the τ-dependence of the Schmidt coefficients M −iHˆ Mτ X X {cγ} which is not featured by the set {aγ}. In |Ψτ i =e aγ |γi cγ0 |ξγ0 i fact, there exists a dynamical model (the Ozawa’s γ γ0 model [5] for projective von Neumann measure- X −iϕγγ0 = aγ |γi cγ0 e |ξγ0 i , (10) 2 ment described in AppendixA) for which cγ = γ,γ0 2 |aγ| , ∀γ and ∀τ. Such model is defined by a Hamiltonian where the operators acting on Γ with ϕγγ0 ≡ τgεγEγ0 ∈ R.
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