A sub-ensemble theory of ideal quantum measurement processes
aa r X i v : . [ qu a n t - ph ] N ov A sub-ensemble theory of ideal quantum measurement processes
Armen E. Allahverdyan , Roger Balian and Theo M. Nieuwenhuizen , Yerevan Physics Institute, Alikhanian Brothers Street 2, Yerevan 375036, Armenia Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette cedex, France Institute for Theoretical Physics, Science Park 904, 1090 GL Amsterdam, The Netherlands International Institute of Physics, Lagoa Nova 59078-970, CP: 1613 - Natal / RN, Brazil
Abstract
In order to elucidate the properties currently attributed to ideal measurements, one must explain how the conceptof an individual event with a well-defined outcome may emerge from quantum theory which deals with statisticalensembles, and how di ff erent runs issued from the same initial state may end up with di ff erent final states. This so-called “measurement problem” is tackled with two guidelines. On the one hand, the dynamics of the macroscopicapparatus A coupled to the tested system S is described mathematically within a standard quantum formalism, where“q-probablities” remain devoid of interpretation. On the other hand, interpretative principles, aimed to be minimal,are introduced to account for the expected features of ideal measurements. Most of the five principles stated here,which relate the quantum formalism to physical reality, are straightforward and refer to macroscopic variables. Theprocess can be identified with a relaxation of S + A to thermodynamic equilibrium, not only for a large ensemble E of runs but even for its sub-ensembles. The di ff erent mechanisms of quantum statistical dynamics that ensure thesetypes of relaxation are exhibited, and the required properties of the Hamiltonian of S + A are indicated. The additionaltheoretical information provided by the study of sub-ensembles remove Schrödinger’s quantum ambiguity of the finaldensity operator for E which hinders its direct interpretation, and bring out a commutative behaviour of the pointerobservable at the final time. The latter property supports the introduction of a last interpretative principle, neededto switch from the statistical ensembles and sub-ensembles described by quantum theory to individual experimentalevents. It amounts to identify some formal “q-probabilities” with ordinary frequencies, but only those which referto the final indications of the pointer. The desired properties of ideal measurements, in particular the uniqueness ofthe result for each individual run of the ensemble and von Neumann’s reduction, are thereby recovered with eco-nomic interpretations. The status of Born’s rule involving both A and S is re-evaluated, and contextuality of quantummeasurements is made obvious.Keywords: quantum measurement problem, system-apparatus dynamics, ensemble and sub-ensembles, q-probability, Born rule, minimalist interpretationPACS: 03.65.-w Quantum mechanics, 03.67.-a Quan-tum information, 05.30.Ch Quantum ensemble theory,64.70.Tg Quantum phase transitions, 67.10.Fj Quantumstatistical theoryDOI: Subject Areas: Quantum Physics
1. Introduction
If one wants to be clear about what is meant by“position of an object”, for example of an electron...,then one has to specify definite experiments bywhich the “position of an electron” can be measured;otherwise this term has no meaning at all.
Werner Heisenberg [1]Measurements constitute our sole contact with mi-croscopic reality, but raise many questions, closely re-lated to the connection between microscopic and macro-scopic concepts. Can one explain theoretically whyidentical measurements performed on several systemsidentically prepared provide di ff erent outcomes? For asingle measurement, how is the occurrence of a well-defined result compatible with the irreducibly proba-bilistic nature of quantum theory? Does measurementtheory require a specific principle of quantum mechan-ics? What is the status of Born’s rule? What is therole of the apparatus? Already raised by the found-ing fathers, these questions have witnessed a revival[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Many answers Preprint submitted to Elsevier November 8, 2016 ave been proposed, relying on various interpretationsor on various extensions of quantum mechanics and of-ten inspired by the solution of models, but no consensushas been reached.Here, we regard as usual a measurement as a jointprocess undergone by the tested system S and a macro-scopic apparatus A; the dynamics of a large statisticalensemble E of similarly prepared runs is represented bycurrent equations of quantum statistical mechanics. Themathematical results thereby obtained must then be in-terpreted, so as to relate them to physical facts. How-ever, we do not wish to adopt any specific interpretationof the quantum formalism. Our purpose is more mod-est, as we will limit ourselves to the interpretation of thesole results relevant to the measurement, keeping asideall other quantum degrees of freedom.Our scope is thus double, technical and conceptual .On the technical side, we wish to sort out how muchcan be told about ideal measurements through a quan-tum approach restricted to a formal skeleton devoid ofany interpretation. To this aim, we will study the dy-namics of S + A, not only in the standard way where thedensity operator describes the whole ensemble E of real-isations of the experiment, but also by introducing moreprecise density operators which describe sub-ensembles of E . Governed by the same conventional equations ofmotion, these operators will provide some useful, moredetailed information.On the conceptual side, in order to explain the var-ious features expected for ideal measurements, we willintroduce interpretative principles that link some formal outcomes of the quantum analysis to the physical factspertaining to the measurement. Discussions about in-terpretation will require a clear distinction between ab-stract quantum “probabilities” (termed q-probabilities )and ordinary probabilities regarded as frequencies ofobserving some macroscopic events in the limit of alarge number of repeated experiments. As we wish tointroduce only the most economic principles (or postu-lates) needed to understand ideal measurements, mostquantum objects will be left without interpretation. Sev-eral points below will appear well known or trivial; weincluded them for completeness and continuity of thereasoning. Let us pose more precisely the problem to be solvedand define the notations. We deal with ideal, non demol-ishing measurements. Their purpose is to test a singleobservable ˆ s = P i s i ˆ π i of S (characterised by its (dis-crete) eigenvalues s i and the associated eigenprojectorsˆ π i ), while perturbing S minimally. For instance, in the historical experiment of Stern and Gerlach (1922), thesystem S is one among the silver atoms of a beam in-coming along the x -direction, it is coupled to the ap-paratus through an inhomogeneous magnetic field inthe z -direction; the tested observable ˆ s is then the z -component of the spin of S, and the projectors ˆ π i referto the directions + z and − z . In EPR settings, ˆ π i denotesthe product of two projectors pertaining to the two cor-related spins.As measurements are required to provide experimen-tal access to microscopic physical quantities, their un-derstanding is an essential element to settle the foun-dations and the interpretation of quantum mechanics.Although the conditions of ideality recalled below arerarely fulfilled in the laboratory, it is natural to focus aswe do here on the simplest case of ideal measurements.Indeed, only ideal measurements are dealt with in quan-tum mechanics textbooks, which postulate their charac-teristic properties (Born’s rule and von Neumann’s re-duction) but skip the analysis of the quantum processof interaction between the tested system and the appara-tus, needed to justify these postulates. Moreover, as anygeneral quantum measurement (POVM) can be repre-sented as a partial trace over an ideal measurement [13],a theoretical elucidation of ideal measurements appearsas a prerequisite for a full understanding of real mea-surements, which should rely on the same ideas.An essential feature is the macroscopic size of the ap-paratus A, which forces us to deal with mixed states andnon-equilibrium quantum statistical mechanics. We de-note by ˆ D ( t ) the joint density operator of S + A for alarge ensemble E of runs, and by ˆ r ( t ) = tr A ˆ D ( t ) andˆ R ( t ) = tr S ˆ D ( t ) the marginal density operators of S andA, respectively. At the initial time t =
0, S and A areuncorrelated, S lies in some state ˆ r (0), pure or not, tobe tested and A lies in a metastable state ˆ R (0), so thatˆ D (0) equals ˆ r (0) ⊗ ˆ R (0).The subsequent evolution of S + A should obey quan-tum statistical dynamics. We expect that the apparatus,triggered by an interaction ˆ H SA with S which is firstswitched on and later o ff , will eventually relax towardsone or another among its stable states ˆ R i . These statesshould have equal entropies and energies so as to avoidbias in the measurement; they can be distinguished fromone another by observing, processing or registering thevalue A i of the pointer variable, identified as the expec-tation value A i = tr A ˆ R i ˆ A in the state ˆ R i of some collec- This initial state of A is often called “ready state”, waiting tobe triggered by S. It must therefore be metastable, and hence can berepresented only by a mixed density operator ˆ R (0), not by a pure state,a property often overlooked. A of A. As the pointer is macroscopic,the spectrum of ˆ A is dense, and many eigenvalues of ˆ A lie in the range of each distribution tr A ˆ R i δ ( A − ˆ A ). More-over, these distributions should not overlap for i , j soas to ensure a neat distinction between the possible out-comes A i . Introducing a width ∆ larger than that of ˆ R i and such that ∆ ≪ | A i − A j | for i , j , we shall denoteas ˆ Π i the projector on the eigenspace characterised byeigenvalues of ˆ A lying between A i − ∆ and A i + ∆ . Wethen have tr A ˆ R i ˆ Π j ≃ δ i j .An ideal measurement of the tested observable ˆ s , per-formed on the initial state ˆ r (0) of S, is currently definedas a thought experiment which is supposed to have thefollowing properties. The experiment involves a largenumber of runs during which S and A interact. Oneassumes that these runs can be sorted out at the finaltime t f according to the macroscopic indication A i ofthe pointer, and that the relative number tr ˆ D ( t f ) ˆ Π i ofruns having yielded the outcome A i is given by Born’srule p i = tr ˆ D ( t f ) ˆ π i = tr S ˆ r (0) ˆ π i . One also admits that theoutcome A i is fully correlated with the eigenvalue s i of ˆ s and with the production of the final state ˆ r i = ˆ π i ˆ r (0) ˆ π i / p i of S (von Neumann’s reduction or so-called collapsepostulate, see Lüders [15] ). These properties are ex-pressed by the surmise that one among the final states of the formˆ D i = ˆ r i ⊗ ˆ R i , ˆ r i = p i ˆ π i ˆ r (0) ˆ π i , (1)with p i = tr S ˆ r (0) ˆ π i , should be assigned to S + A aftereach separate run of the measurement.A major di ffi culty arises when one tries to show thatthe above features traditionally attributed to ideal mea-surements result from the application of quantum the-ory to the dynamics of the compound system S + A. In-deed, the very definition of a measurement relies on theconcept of single run , whereas this concept is foreignto standard quantum mechanics which only deals with large statistical ensembles . One is thus faced with theso-called “measurement problem” [9]. (For pure statesa clear definition is given by Home [17].) To solve it,one must supplement the abstract formalism of quan-tum mechanics with some interpretative principles so asto give way to the concept of individual runs in spiteof the inevitably probabilistic nature of quantum theory. It is essential to distinguish the projector ˆ π i for the system S, as-sociated with the eigenvalue s i of ˆ s , from the projector ˆ Π i for theapparatus A, associated with the eigenvalues of ˆ A located in the range( A i − ∆ , A i + ∆ ). English translation and discussion: K. A. Kirkpatrick [16].
The main principle that we propose (Principle 5, Sec.6) will concern only the macroscopic (quantum) appa-ratus, not the microscopic tested system. Afterwards,one may be in position ( i ) to understand why each in-dividual run produces a well-defined outcome A i , ( ii ) toelucidate how di ff erent runs issued from the same initialstate ˆ D (0) = ˆ r (0) ⊗ ˆ R (0) through deterministic quan-tum equations of motion may end up in di ff erent finalstates having the reduced form ˆ D i , and ( iii ) to demon-strate why the frequencies of the pointer values A i con-verge for a large ensemble of runs to Born’s formal q-probabilities p i = tr S ˆ r (0) ˆ π i which refer only to the sys-tem and to its initial state, irrespective of the apparatusand of the evolution. The explanation we wish to give to the desired prop-erties of ideal measurements has been subjected to adouble constraint. We tried to describe the dynam-ics of S + A by extracting as much mathematical re-sults as possible from a standard quantum formalism,and at the same time to interpret the formal outcomesthus obtained in terms of physical reality by introducingthe weakest possible postulates (or principles) needed.The two di ff erent types of ingredients that enter theapproach, formal and conceptual, will be intertwined.In order to distinguish them, we exhibit all along thetext five “interpretative principles” which relate somemathematical objects to physical properties. Most ofthese principles concern only macroscopic variables through which we have access to reality.The mathematical formalism on which we rely, re-called in Sec. 2, is completed by the first three princi-ples, which we state for completeness but may in factbe regarded as natural or evident. The principle 1 (Sub-sec. 2.2) identifies the formal q-expectation value tr ˆ D ˆ O of a macroscopic observable ˆ O in state ˆ D with the cor-responding physical quantity, in case the correspondingq-variance of ˆ O is negligible. The principle 2 (Subsec.2.3), relevant for the dynamics of S + A, allows us to relyon approximations of quantum statistical mechanics thathave negligible e ff ects upon the physical outcomes ow-ing to the large size of A. The principle 3 (Subsec. 3.2),which determines the density operator that should be as-signed to a system in a situation characterised by somedata, is used here to interpret the expressions of the ini-tial and final states of S + A.The quantum equations of motion refer to a largeensemble E of compound systems S + A, but also ap-ply to sub-ensembles of E (Subsec. 2.4). We there-fore proceed in three steps, which involve successively3 i ) the full set of runs of the measurement, ( ii ) its sub-ensembles and ( iii ) the individual runs. Step (i): Full ensemble . This step has commonlybeen worked out in the literature. The density operatorˆ D ( t ) of the compound system S + A encodes the prop-erties at the time t of a large statistical ensemble E ofrealisations of the measurement. All elements of E areinitially prepared in an identical manner; the result isencoded by the state ˆ D (0) = ˆ r (0) ⊗ ˆ R (0) and the dynam-ics of ˆ D ( t ) is governed by the Liouville–von Neumannequation. One first needs to prove that, at the final time t f of the process, ˆ D ( t ) reachesˆ D ( t f ) = X i p i ˆ D i , ˆ D i = ˆ r i ⊗ ˆ R i , (2)which is a requirement needed for the desired result (1).We will first identify ˆ D ( t f ) as a generalised Gibbsstate (Subsec. 3.3), so that the dynamics which leadsfrom ˆ D (0) to ˆ D ( t f ) can merely be regarded as a relax-ation towards a thermodynamic equilibrium state . Wethen show how this relaxation can be ensured dynami-cally within a purely formal approach (Sec. 4), by deriv-ing (2) through current methods of quantum statisticalmechanics and by relying on some suitable propertiesof the Hamiltonian of S + A (Subsecs. 3.1 and 3.4).What we wish to eventually demonstrate is that, afterthe final time t f , the ensemble E can be split into sub-ensembles E i characterised by the macroscopic outcome A i of the pointer; for each sub-ensemble E i , the com-pound system S + A should lie in the state ˆ D i given by(1), and E i should contain a proportion p i of runs. Theresult (2) is a necessary condition for these propertiesto be satisfied, but one cannot ensure the converse forquantum reasons. If density operators did behave as dis-tributions of classical statistical mechanics, one wouldbe allowed to readily interpret each operator ˆ D i that en-ters (2) as a physical state and its coe ffi cient p i as anordinary probability. However, Schrödinger’s quantumambiguity [18, 19, 20], implies that the operator ˆ D ( t f )can be decomposed not only into a weighted sum of op-erators ˆ D i as in (2), but also into very many other sumsinvolving di ff erent terms. We shall recall (Subsec. 5.1)how contradictions arise when one attempts to interpretthe separate terms of two di ff erent decompositions. Asnothing privileges a priori the decomposition suggestedby the form of (2), the sole establishment of this expres-sion is not su ffi cient to ensure that each of its separateterms is physically meaningful. Other theoretical ingre-dients will help us to find a natural interpretation for thecomponents ˆ D i and p i of ˆ D ( t f ). Step (ii): Sub-ensembles . In order to draw furtherconclusions within the abstract formulation of quantumtheory, we will take advantage of the fact that quan-tum dynamics governs not only ensembles, but alsosub-ensembles. We will therefore make an intermedi-ate step, when going from the full ensemble E towardsindividual runs. We consider an arbitrary sub-ensemble E ( k )sub of runs extracted from E , which includes a propor-tion q ( k ) i of runs having yielded the outcome A i . We needto prove that the state of S + A which describes E ( k )sub endsup in the formˆ D ( k )sub ( t f ) = X i q ( k ) i ˆ D i , ˆ D i = ˆ r i ⊗ ˆ R i , (3)with 0 ≤ q ( k ) i ≤ P i q ( k ) i =
1. Contrary to whatwould happen in classical statistical physics, this ex-pression (3) is not a consequence of (2), as discussedin Subsec. 5.1. It is a further necessary condition, muchstronger than (2), and it must really be demonstrated.Here again as for (2), the desired density operator (3)expresses thermodynamic equilibrium (Subsec. 3.3). Inorder to give, within the standard quantum formalism,a dynamical proof of the relaxation of S + A towardsthis expression (3) for the sub-ensemble E ( k )sub , we in-troduce in Subsec. 5.2 the principle 4 , which allows under some conditions to describe S + A in a more pre-cise way than with ˆ D ( t ) by associating with the vari-ous sub-ensembles of E di ff erent quantum states. Wethereby assume that, at least after some time t ′ f slightlyearlier than t f , the dynamics of a physical sub-ensemble E ( k )sub of E is generated by ordinary quantum equations,even though our available information is not su ffi cientto fully specify the state ˆ D ( k )sub ( t ′ f ) of S + A describing E ( k )sub at the time t ′ f . Then, making use of a specific dynami-cal mechanism, the “poly-microcanonical relaxation” (introduced in [10] under the name of “sub-ensemblerelaxation”) which involves only the (large) apparatus ,we can establish for any physical sub-ensemble the ex-pected result (3), thus removing the quantum ambiguity (Subsec. 5.4). Step (iii): Individual runs.
The result (3), muchstronger than (2), is the most detailed property of idealmeasurements that conventional quantum theory can af-ford. It is a necessary condition, but its mere deriva-tion is not su ffi cient to entail (1), because individualruns lie beyond the realm of the standard formulationof quantum mechanics, and because the ingredients ˆ D i , p i and q ( k ) i of (2) and (3) are still formal quantum quan-tities. Since no interpretation has yet been given to4-probabilities, the numbers q ( k ) i entering (3) are onlymathematical objects, which we indeed would like tointerpret as ordinary probabilities.We will therefore supplement (Subsec. 6.1) the ab-stract formulation of quantum mechanics with a last principle 5 . Its introduction is made natural by theclassical-like properties of the projectors ˆ Π i in the finalstate (Subsec. 5.3), which result from the macroscopicsize of the pointer and from the dynamics. It amounts tointerpret, for any sub-ensemble E ( k )sub , each formal coef-ficient q ( k ) i as the proportion of runs of E ( k )sub that providethe indication A i of the pointer . Equivalently, it amountsto acknowledge the existence of the sub-ensembles E i characterised by the value A i (for which q ( k ) i = q ( k ) i ′ = i ′ , i ). Accordingly the building block ˆ D i of the formal expressions (2) and (3) is identified withthe final state associated with the sub-ensemble E i , sothat it can be assigned to S + A for any individual runof E i . Statements can thus be made about experimentalfacts, and all expected properties of ideal measurementscome out (Sec. 6).We will stress in the conclusion (Sec. 7) that the fea-tures of ideal measurements emerge owing to the macro-scopic size of the apparatus , which plays a major rolein the interpretation. Accordingly, results of measure-ments involving di ff erent settings of apparatuses shouldnot be put together (Subsec. 7.4). We will also recon-sider Born’s rule as a property of the apparatus in thefinal state after its interaction with S (Subsec. 6.4).The formal aspects of the theory lie in the derivationof Eqs. (2) and (3). Such derivations have been achievedat least partly for many specific models [8, 9, 10, 11].As we consider below general ideal measurements, wewill simply sketch how the solution arises from somenecessary properties of the Hamiltonian of S + A, anddemonstrate its technical feasibility by recalling in foot-notes the main features of the detailed dynamical study[10, 21] of the Curie–Weiss (CW) model of quantummeasurement .Moreover, since the derivation of Eqs. (2) and (3)merely amounts to a proof, in the microscopic frame- In the CW model (see ref. [10], sect. 3), S is a spin , the mea-sured observable being its z -component ˆ s z , with outcomes i = ↑ or ↓ .The apparatus simulates a magnetic dot, including N ≫ ˆ σ ( n ) ,which interact through the Ising coupling J , and a phonon thermalbath at temperature T < J ; these spins and the phonons are coupledthrough a dimensionless weak coupling γ . Initially prepared in itsmetastable paramagnetic state, A may switch to one or the other sta-ble ferromagnetic state. The pointer observable ˆ A = N ˆ m = P Nn = ˆ σ ( n ) z is the total magnetisation in the z -direction of the N Ising spins. Thecoupling between S and A is ˆ H SA = − P Nn = g ˆ s z ˆ σ ( n ) z , while ˆ H S = work of quantum statistical dynamics, of the relaxationof S + A towards thermodynamic equilibrium (Subsec.3.4), the reader willing to admit this thermalization mayskip Secs. 4 and 5.
2. Formal principles of quantum mechanics
We tackle the measurement problem within a formu-lation of quantum mechanics which deals only with sta-tistical ensembles. Indeed, this idea underlies most cur-rent interpretations of quantum mechanics, and repeatedexperiments constitute an exploration of the consideredensemble. Individual systems are not directly describedin this framework, which is irreducibly probabilistic, sothat statistical ensembles and sub-ensembles will be es-sential in our approach . The spirit of this formal de-scription is the same as in the C ∗ -algebraic approach[11, 22, 23], although we deal here with finite non rel-ativistic systems. Its principles recalled below do notprejudge any specific interpretation of quantum oddities[13], and it is suited to both microscopic and macro-scopic systems. In fact, S is microscopic and the macro-scopic apparatus A is treated as a finite (though large)object so as to keep control of the time scales character-izing the evolution of S + A. Physical quantities pertaining to a system are repre-sented by “observables” expressed as Hermitean matri-ces in a Hilbert space. Observables behave as randomobjects, but, unlike ordinary random variables, their ran-domness, which arises from their non commutative na-ture, is inherent to the quantum formalism.In the present formal scope, we regard a “quan-tum state” , whether pure or not, merely as a theoreti-cal tool for making probabilistic statements or predic-tions about experiments . It is characterised by a corre-spondence that associates with any observable ˆ O a realnumber h ˆ O i . This correspondence is implemented as We do not allude here to “statistical interpretation” [5] nor to “en-semble interpretation”, terms which depend on the authors, but simplyto “formulation” because interpretation will come out only in the endas a result of a measurement process. We shall abbreviate throughoutby “ensemble” the expression “statistical ensemble”. We subscribe to van Kampen’s theorem IV on quantum measure-ments [24], generalised from pure states ψ to general mixed states ˆ D :“Whoever endows ˆ D with more meaning than is needed for comput-ing observable phenomena is responsible for the consequences”. In this algebraic approach, the observables ˆ O are regarded as el-ements of a vector space, while a state, defined as a linear correspon-dence ˆ O
7→ h ˆ O i , such that h ˆ O i is real and h ˆ O i is non-negative, is anelement of its dual vector space; the q-expectation values h ˆ O i appear O
7→ h ˆ O i = tr ˆ D ˆ O by means of a Hermitean, normalisedand non-negative density operator ˆ D .Such definitions of observables and states look anal-ogous to the corresponding ones in classical statisticalmechanics, where physical quantities are represented byfunctions of the (random) position and momentum vari-ables, where a state is encoded by a density in phasespace, and where expectation values are expressed asintegrals over their product. However, this similitude isonly formal, since the numbers h ˆ O i violate some prop-erties of ordinary expectation values, for instance Bell’sinequalities. Our knowledge is limited by the operatornature of the quantum physical quantities (and not onlyby some ignorance about their values as in classical sta-tistical mechanics). In particular, q-bits represented bytwo-by-two density matrices di ff er from ordinary bits.They can be manipulated only blindly, since the “quan-tum information” (q-information) that they carry is notfully available: Reading a q-bit so as to extract fromit ordinary information in the form of a bit requires ameasurement process which destroys it in part. Simi-larly, for a general density operator, the numbers h ˆ O i may become physically available in the form of ordi-nary expectation values solely in special circumstancesand solely in part, through measurements.One should therefore, as done for q-bits, distinguish h ˆ O i = tr ˆ D ˆ O from an ordinary expectation value by de-nominating it as a “q-expectation value” . Likewise, a “q-correlation” , the q-expectation value of a product oftwo observables, should not be confused with an ordi-nary correlation. Also, the q-expectation value h ˆ π i ofa projection operator ˆ π is not an ordinary probability,but a formal object which we will call “q-probability” rather than “probability” . Born’s rule is not postulatedhere, it will come out (Subsec. 6.4) as a property of theapparatus at the issue of an ideal measurement. We want to extract from the abstract q-informationembedded in density operators some ordinary informa- as scalar products. The representation of states by density matricesarises when one chooses a set of dyadics | η ih η ′ | as basis in the vec-tor space of observables ˆ O , which then appear as linear combinationsof operators | η ih η ′ | with coe ffi cients h η | ˆ O | η ′ i . The matrix element h η ′ | ˆ D| η i of ˆ D is then defined as the q-expectation value of | η ih η ′ | .Other so-called Liouville representations of states, such as the Wignerrepresentation for a particle or the polarisation representation for aspin , are defined through other choices of bases in the dual vectorspaces of observables and states (the basis of Pauli operators for thepolarisation representation of a spin) [25]. The term pre-probability has also been proposed to indicate thatthe formal quantum object p i = h ˆ π i i = tr S ˆ r (0)ˆ π i may be interpreted asa true probability only after achievement of an ideal measurement ofˆ s . tion a ff ording predictions about real events. To this aim,physical interpretations should emerge at the macro-scopic scale, in experimental contexts. Let us alreadypoint out, for a macroscopic quantum system, a simplesituation in which an interpretation is readily providedby a first, trivial principle. Interpretative principle 1 . If the q-variance of a macroscopic observable is negligible in relative size ,its q-expectation value is identified with the value of thecorresponding macroscopic physical variable, even foran individual system.Accordingly, the q-expectation value of ˆ A in thequantum state ˆ R i is identified with the macroscopicpointer value A i . Nevertheless, in the state ˆ D ( t f ) (Eq.(2)), the q-variance of ˆ A is in general large because itspossible values A i are di ff erent, and the interpretation oftr ˆ D ( t f ) ˆ A as an ordinary expectation value will only arisefrom the analysis of the ideal measurement process of ˆ s and from the additional interpretative principle 5 (Sec.6).In spite of the macroscopic nature of the above prin-ciple, it can be used to provide a (somewhat round-about) interpretation of q-expectation values, even formicroscopic systems (Appendix A and Ref. [29]). Letus associate with the system S under study a macro-scopic thought super-system S = { S [1] , S [2] , · · · , S [ N ] } .It is a single compound system obtained by puttingtogether a large number N of subsystems S [ n ] ( n = , , · · · , N ) similar to S. All these subsystems lie in thesame marginal state ˆ D , obtained by tracing out the N -1 other subsystems from the state D ˆ of S .With each ob-servable ˆ O of S we associate the average observable O ˆ = N − P n ˆ O [ n ] of S . It is shown in Appendix A that, whilethe q-expectation values tr ˆ D ˆ O for S and Tr D ˆ O ˆ for S are the same, the q-variance of O ˆ is N times smallerthan the q-variance of ˆ O (provided the subsystems S [ n ] are su ffi ciently weakly q-correlated). The above prin-ciple thus holds for the macroscopic observable O ˆ , sothat the formal q-expectation value h ˆ O i for the (possiblysmall) system S can be identified with the macroscopicvalue of the corresponding average observable O ˆ forthe super-system S . However, q-expectation values willremain without direct interpretation in terms of S itself. This principle does not at all mean “microscopic definiteness”where the system is close to an eigenstate [26, 27, 28]; we refrain frominterpreting microscopic properties. Its use may in particular requirethe assignment of a lower bound to the q-variance of the consideredmacroscopic observable. See footnote 31 in Appendix A and ref. [29]. .3. Dynamics The formalism is completed, for the time-dependenceof the density operator of an isolated system,by the Liouville–von Neumann equation of motion i ~ d ˆ D ( t ) / d t = [ ˆ H , ˆ D ( t )]. Mathematically, this fun-damental dynamic equation is deterministic and re-versible, whereas a measurement process leading fromˆ D (0) to ˆ D ( t f ) is irreversible. We thus have to face inthis context the old paradox of irreversibility , like inclassical statistical mechanics, within replacement ofthe Liouville theorem in phase space by the unitarityin Hilbert space, and to solve it in the same way.As usual in statistical mechanics, it is legitimate inpractice for finite but large systems to disregard eventsthat might occur with an extremely small probability ,to forget about recurrences that might take place afterlarge, unattainable times, and to neglect physically ir-relevant correlations between a macroscopic number ofdegrees of freedom. This view is consistent with theidea that a quantum state is regarded only as a catalogueof knowledge intended for physical predictions. Itsevolution appears as a transfer of q-information amongthe various observables, the most complicated of whichcannot be reached experimentally. A part of the cata-logue thus becomes useless and may be discarded (dis-sipation). Such a coarse graining breaks the constancyof entropy, replacing the conserved von Neumann en-tropy by an increasing relevant entropy [30]. We are ledto the following prescription. Interpretative principle 2 . One may perform a coarsegraining on a density operator ˆ D if this operation has noe ff ect on the physical predictions a ff orded by ˆ D .Standard procedures in quantum statistical mechan-ics are thereby justified. For instance, correlations witha bath or an environment which develop during the re-laxation process are inaccessible and ine ff ective; theymay be discarded. Such approximations, although notmathematically rigorous, are fully justified when theiroutcome is physically indistinguishable from the exactsolution. Moreover, they are necessary to explain irre-versible phenomena, including measurement processes. As an ordinary probability distribution, a quantumstate gathering q-information refers, implicitly or not,to a statistical ensemble E , which is a large collection ofsystems produced under the same conditions and char-acterised by the same available knowledge. However,while ordinary probabilities are defined in terms of the individual events embedded in E , q-probabilities are ab-stract numbers which do not arise from the considera-tion of individual systems. A “state” does not “belong toa system”, it is not an intrinsic property but rather a cat-alogue of knowledge about an ensemble [18, 19]. If onewishes to consider a single system, one should intro-duce a virtual ensemble E encompassing many mentalcopies of the studied system. However, a measurementgathers a large set of runs, and involves a real ensemble E of systems S + A, similarly prepared and evolving inrepeated experiments.Note that di ff erent density operators may simultane-ously be ascribed to the same system, depending on theensemble in which it is embedded , that is, on the infor-mation available about it. This is standard in probabilitytheory: When a dice is repeatedly thrown, the probabil-ity of the outcome “3” is for the full set of runs; it is when only the odd outcomes (“1”, “3”, “5”) are selectedand the even ones discarded; it is for a selection of themiddle ones (“3” or “4”), and 1 for the sub-ensemblecontaining only the outcome “3”. Gaining knowledgeabout an individual system which is originally part of E leads to regard it as member of a sub-ensemble of E , andto modify its probabilistic description by assigning to ita new, more informative state. Such an occurrence ofdi ff erent probability distributions for the same system, depending on the q-information retained about it, whichis trivial in the dice example, may look odd for quantumstates, but it takes place as soon as some non-random se-lection is made among measurement outcomes. Henceit should enter theoretical treatments; indeed, it will becrucial in Sec. 5. Once the existence of di ff erent sub-ensembles is granted, the corresponding states evolve inparallel.However, a specifically quantum di ffi culty arises(Subsec. 5.1). Knowing solely a mixed state such asˆ D ( t f ) does not allow to recognise theoretically within itstates that might describe the sub-ensembles of E , nora fortiori states that might describe its individual sam-ples (although these are evidently distinguished exper-imentally in repeated processes) . The occurrence ofˆ D i within the expression (2) of ˆ D ( t f ) that describes the In classical probability theory, the selection of the elements of asub-ensemble E ( k )sub of E is mathematically implemented [31, 32] bynumbering the events of E with an index n and introducing a function f ( k ) ( n ) that may take two values, 0 if the element n is discarded, 1 if itis selected. The only general condition imposed on the function f isthat the sub-ensemble should become infinite, whenever the ensembledoes. The actual construction of f can (but need not) be related todistinguishing theoretically the individual events, which is of courseexperimentally performed in quantum measurements, but which is al-lowed in quantum theory only in special cases, such as at the issue of ameasurement. This will be discussed in Secs. 5 and 6. Then, the sub- E is not su ffi cient to ensure that this op-erator ˆ D i can be interpreted as final state assigned toan individual run, and we shall need both technical andconceptual developments to reach this conclusion.Indeed, the consideration of sub-ensembles, inspiredfrom the frequency approach to the classical probabilitytheory [31], will be an essential ingredient of the presentapproach to quantum measurements. While the densityoperator (2) of the compound system S + A encompassesq-information about the final state of a large set E ofruns, the final states (3) generated by some specific dy-namics (Subsec. 5.4) will account for the more detailedq-information associated with the sub-ensembles E ( k )sub .As for the final state ˆ D i of the form (1), understandingits occurrence requires solving the measurement prob-lem, as it is assigned to the individual runs of the sub-ensemble E i (Subsec. 6.1). Switching from ˆ D ( t f ) toˆ D i will appear as an updating of information, similarto an updating associated with a gain of information,analogous to an updating of ordinary probabilities afterselection of events characterised by some piece of infor-mation.
3. Preliminaries
Various measurement models have been worked out[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 21, 24, 33]. Inall cases, if we include in A a thermal bath or a pos-sible environment, the compound system S + A is iso-lated, and therefore governed by a Hamiltonian ˆ H = ˆ H S + ˆ H A + ˆ H SA , which should ensure that the state ofS + A evolves unitarily from ˆ D (0) to ˆ D ( t f ). (The sameconclusion holds if the environment is left outside A,in which case the Liouville – von Neumann evolutionshould be replaced by an equation justified by quantumstatistical mechanics.) The most general Hamiltonian ensembles E ( k )sub of theoretical interest for the present argument will bethose for which the coe ffi cients q ( k ) i in (3) di ff er from p i . However,the selections of all such sub-ensembles within the full ensemble E have zero measure (in the sense of Lebesgue measure in the space ofselections [32] when the number of elements of E becomes infinite).Nearly all subsets of E , in particular those obtained by extracting sys-tems at random from E , would be described by the same state ˆ D asthe full set E . It will therefore be essential for our purpose to consider all sub-ensembles E ( k )sub of E . Note also that, if ˆ r i is a mixed state, theruns described by (3) are picked up at random within E i . Note finallythat, if we step away from measurements (for which the state of A isnecessarily mixed) and consider a pure state ˆ D = | ψ ih ψ | , this samestate ˆ D should be assigned to any sub-ensemble and to any individualsystem of E . that may describe an ideal quantum measurement pro-cess should satisfy the following properties.The part ˆ H A associated with the macroscopic appa-ratus A alone must have specific features. It shouldproduce an initial metastable state ˆ R (0), with lifetimelonger than the duration of the measurement, and sev-eral equilibrium states ˆ R i , the possible expected finalstates. A typical example is given by spontaneouslybroken discrete invariance, the macroscopic pointervariable A i being the order parameter which may taketwo or more discrete values. These properties imply inparticular the presence of a bath or an environment, cou-pled to the active part of A including the pointer, whichwill drive it to thermodynamic equilibrium. The weak-ness of such a coupling allows to solve models by meansof standard equations of quantum statistical mechanicswhich eliminate the environment from the Hamiltoniandynamics.As we wish to deal with ideal measurements, the pro-cess should perturb S as little as possible: any observ-able of S compatible with ˆ s , i.e., commuting with itseigenprojectors ˆ π i , should remain una ff ected. The con-servation of all these observables [15] is expressed bythe fact that ˆ H depends on S only through the projectorsˆ π i . Accordingly, the coupling between S and A musthave the form ˆ H SA = P i ˆ π i ⊗ ˆ h i , where ˆ h i are operatorsof A. This form will ensure that the “preferred basis” isindeed the eigenbasis of the projectors ˆ π i . Moreover, if ˆ s takes the value s i , that is, ˆ π i the value 1, the apparatus Ashould end up in its stable state ˆ R i , the pointer variablebeing close to A i and ˆ Π i also taking the value 1. Thiscan be achieved if each ˆ h i behaves as a source that en-ergetically favours relaxation towards ˆ R i , thus breakingexplicitly the equivalence between the various possibleoutcomes A i . (In case the pointer variable A i is an orderparameter, the invariance is explicitly broken by ˆ H SA .)Likewise, ˆ H S must reduce to a linear combination ofprojectors ˆ π i , which only produces trivial phase factors. This form of interaction can allow to describe not only ideal mea-surements involving well separated eigenvalues s i of ˆ s , but also moregeneral measurements for which the projectors ˆ π i , still associatedthrough ˆ h i with the pointer indications A i , are no longer in one-to-onecorrespondence with the eigenvalues of ˆ s . For instance, if some ˆ π i encompasses the eigenspaces of several di ff erent neighbouring eigen-values, selecting the outcome A i will not discriminate them, and thefinal state ˆ r i = ˆ π i ˆ r (0)ˆ π i / p i of S will not be associated with a singleeigenvalue of ˆ s as in an i deal measurement. As another example,consider two orthogonal rank-one projectors ˆ π and ˆ π , coupled withsources ˆ h and ˆ h that produce di ff erent outcomes A and A , and as-sume that ˆ π + ˆ π spans the two-dimensional eigenspace associatedwith a degenerate eigenvalue of ˆ s ; reading the outcome A (or A )then provides more information than this eigenvalue. .2. Assignment of a state to an ensemble of systems The analysis of the measurement process requires theassignment of a density operator to the initial state ofA and the recognition of the nature of the final statesof S + A for the ensemble E of runs and for its sub-ensembles. To this aim one may rely on the following maximum von Neumann entropy criterion [34, 35, 30]. Interpretative principle 3 . Among the states compat-ible with the data available on an ensemble of systems,the least biased predictions are a ff orded by assigning toit the density operator which maximises the von Neu-mann entropy S ( ˆ D ) = − tr ˆ D ln ˆ D .This maximum entropy criterion is most often re-garded as a postulate, issued from the interpretation ofvon Neumann’s entropy as a measure of the informa-tion missing when only ˆ D is known. However, it can bedirectly derived (Appendix A and Ref. [29]) from theintuitive indi ff erence or equiprobability principle (thatLaplace introduced under the name of principle of in-su ffi cient reason), by relying on the identification be-tween q-expectation values of observables and macro-scopic values of the corresponding average observables(Subsec. 2.2).The data are implemented in the form of constraintson the q-expectation values of some observables ˆ O p (1 ≤ p ≤ p max ). As usual, introduction of Lagrangemultipliers provides for the maximum entropy state ˆ D a Boltzmann – Gibbs expression, namely, the exponen-tial of a linear combination of the operators ˆ O p . Forthe apparatus alone, if the equivalence between pointervalues is explicitly broken by adding to the Hamilto-nian ˆ H A the source term ˆ h i , and if the only constraintis about the macroscopic energy h ˆ H A + ˆ h i i , the crite-rion produces the canonical equilibrium density opera-tor ˆ R hi ∝ exp[ − β ( ˆ H A + ˆ h i )]. If the Hamiltonian reducesto ˆ H A , a second constraint, fixing the macroscopic value A i of the pointer, should be introduced to determine thedensity operator ˆ R i which occurs in the expected finalstate (1). A second constraint is also needed to write theexpression of the initial metastable state ˆ R (0) (fixing thepointer value at m = Before analysing the dynamics of the measurementprocess (Secs. 4 and 5), we determine for orientationthe general form ˆ D eq of the possible thermodynamicequilibrium states associated with the Hamiltonian ˆ H ofS + A. Thermodynamic equilibrium is characterised byfixing the values of all the conserved quantities . Besides the constraint on energy, we must account here for theother constants of the motion, to wit, the q-expectationvalues of all the observables of S that commute withthe projectors ˆ π i . Apart from β , the additional Lagrangemultipliers are coe ffi cients which multiply the latter ob-servables.Any equilibrium state ˆ D eq of S + A has therefore ageneralised Gibbsian form, with an exponent contain-ing (apart from − β ˆ H ) an arbitrary operator that com-mutes with all the ˆ π i . Including the Lagrange multi-pliers, such an operator can be written as a sum P i ˆ y i ,where ˆ y i is any operator of S acting inside the diag-onal block i (so that ˆ y i = ˆ π i ˆ y i ˆ π i ). We find thereforeˆ D eq ∝ exp( − β ˆ H + P i ˆ y i ). Noting now that the full ex-ponent, which commutes with the projections ˆ π i , has ablock diagonal structure in a basis where ˆ s is diagonal,we can rewrite ˆ D eq by exhibiting its related block diag-onal structure. Finally, after separation of the variousterms of ˆ H = ˆ H S + ˆ H A + P i ˆ π i ⊗ ˆ h i , we obtain for thethermodynamic equilibrium states of S + A the generalexpressionˆ D eq = X i q i ˆ x i ⊗ ˆ R hi , X i q i = . (4)Each factor q i ˆ x i , which arises from exp( − β ˆ H S + ˆ y i ), isan arbitrary non negative block diagonal operator of S,where ˆ x i = ˆ π i ˆ x i ˆ π i , tr S ˆ x i = s i is non degenerate, ˆ x i re-duces to ˆ π i .) Each factor ˆ R hi ∝ exp[ − β ( ˆ H A + ˆ h i )] in (4)has been interpreted in Subsec. 3.2 as a canonical equi-librium density operator in the space of A, the sourceterm ˆ h i arising now from ˆ H SA . The equilibrium states(4) of S + A are thus parametrized by the temperature forA, by the coe ffi cients q i and the matrices ˆ x i for S.Distinguishing the states ˆ R hi at the macroscopic scalerequires them to be characterized by di ff erent values ofthe pointer, close to A i , with small variances. The op-erators ˆ h i should therefore be su ffi ciently di ff erent fromone another so that the distributions tr A ˆ R hi δ ( A − ˆ A ) ofˆ A (the spectrum of which is dense) have single nar-row peaks, well-separated for di ff erent values of i . Thesame condition will also ensure that, at the beginningof the dynamical process, the apparatus moves out fromits metastable state ˆ R (0) towards one of the equilibriumstates ˆ R h i so as to ensure a proper registration. The valueof ˆ h i should also be su ffi ciently small so that the peakof the distribution associated with ˆ R h i lies close to A i .These properties are easy to satisfy for a macroscopicapparatus . Thermodynamic equilibrium (4) thus en- In the CW model , the factors ˆ h ↓ = − ˆ h ↑ = P Nn = g ˆ σ ( n ) z that occur s i of ˆ s and the macroscopic value of the pointer variable. The states (1), (2) and (3) expected to occur af-ter achievement of an ideal measurement process, fordi ff erent ensembles, all have the equilibrium form (4)within replacement of ˆ R hi by ˆ R i . In fact, the couplingˆ H SA is switched o ff at a time t decoup earlier than the endof the process. Thus, ˆ R hi can relax smoothly and reachˆ R i at the final time t f , provided the Hamiltonian ˆ H A of Adoes not allow direct transitions between di ff erent val-ues A i (this also ensures that the states ˆ R i have a verylong lifetime), and provided ˆ h i is su ffi ciently small sothat ˆ R hi lies in the basin of attraction of ˆ R i .We have stressed (Subsec. 1.2 (i) ) that it is neces-sary (but not su ffi cient) to prove, by studying the dy-namics of a large statistical ensemble E of runs issuedfrom the initial state ˆ D (0) = ˆ r (0) ⊗ ˆ R (0), that it endsup in the state ˆ D ( t f ) expressed by (2). We can iden-tify (2) with a generalised thermodynamic equilibriumstate (4), for which ˆ R hi has evolved towards ˆ R i afterswitching o ff ˆ H SA . The free parameters of ˆ D eq are de-termined from the initial condition ˆ D (0), since the dy-namics keeps track of the conserved quantities; throughthe identification q i ˆ x i = ˆ π i ˆ r (0) ˆ π i ≡ p i ˆ r i we get q i = p i and ˆ x i = ˆ r i .We also need to prove a stronger result, still neces-sary and not su ffi cient (Subsec. 1.2 (ii) ). For a subset E ( k )sub having yielded a proportion q ( k ) i of runs with out-comes A i , the corresponding final state ˆ D ( k )sub should havethe form (3). This final state is again recognised as ageneralised thermodynamic equilibrium state (4), with q i = q ( k ) i , ˆ x i = ˆ r i . (The property q i =
1, ˆ x i = ˆ r i charac-terises the specific sub-ensemble E ( k )sub = E i ).Thus, an ideal measurement process appears as amere relaxation of S + A to generalised thermodynamicequilibrium , for the full ensemble E of runs as wellas for all its sub-ensembles E ( k )sub . In quantum me-chanics, relaxation of ˆ D ( t ) and ˆ D ( k )sub ( t ) towards Gibb-sian generalised thermodynamic equilibrium states (2)and (3) is not granted [36]. For a complete theory ofideal measurement processes, we must therefore justifythese properties within the quantum statistical dynamics in the coupling ˆ H SA behave as a magnetic field applied to A. Theconditions for ˆ h i are satisfied if N ≫ T / g (which lets the probabilityof the states with m < s z = g < T (see ref. [10],sect. 9.4). In the CW model the condition g < T ensures this relaxation (seeref. [10], sect. 7.2). framework. We sketch the main steps of such a tech-nical proof in Secs. 4 and 5, as a prerequisite to therequired consideration of individual runs.If however one admits, in a thermodynamic scope,that the state of S + A relaxes at the final time to theequilibrium forms (2) for the ensemble E and (3) forits sub-ensembles E ( k )sub , one may jump to Sec. 6 whereintroduction of a last, minimalist interpretative principlewill allow us to draw, from the expressions (2) and (3),the desired conclusions about individual measurements.
4. Dynamics of system and apparatus for the full setof runs
As indicated in Sec. 1.2, the first step in the analy-sis of an ideal measurement process consists in derivingthe form (2) for the final state ˆ D ( t f ) of S + A associatedwith the ensemble E , by solving the dynamical equa-tions with the initial condition ˆ D (0) = ˆ r (0) ⊗ ˆ R (0). Ini-tiated long ago on a model [33], such a task has beenachieved for many other specific models [10]. We onlysurvey here the formal features of the solution in thegeneral case, postponing any interpretation.Since the measurement problem is related to thefoundations of physics, a theoretical analysis shouldrely on the most fundamental dynamical law, that is,the Liouville–von Neumann equation i ~ d ˆ D ( t ) / d t = [ ˆ H , ˆ D ( t )] which governs an isolated, large but finite sys-tem. It is therefore preferable (but not compulsory) toconsider that A includes the needed thermal bath or en-vironment so that S + A is isolated . Taking then intoaccount the above form of ˆ H including the interactionˆ H SA = P i ˆ π i ⊗ ˆ h i , and the approximate commutation[ ˆ H S , ˆ r (0)] ≃ r ( t )of S is perturbed only by the interaction ˆ H SA during theprocess, we check that ˆ D ( t ) can be parameterised asˆ D ( t ) = X i , j ˆ π i ˆ r (0) ˆ π j ⊗ ˆ R i j ( t ) (5)in terms of a set ˆ R i j ( t ) = ˆ R † ji ( t ) of operators in theHilbert space of A. The latter operators must be foundby solving the equations of motion i ~ d ˆ R i j ( t )d t = ( ˆ H A + ˆ h i ) ˆ R i j ( t ) − ˆ R i j ( t )( ˆ H A + ˆ h j ) , (6)with the initial conditions ˆ R i j (0) = ˆ R (0).The dynamics thus involves solely the apparatus. Its coupling with the tested system occurs in (6) only10hrough ˆ h i and ˆ h j , a specific property of ideal measure-ments. Ideality involves separation of S from A uponachievement of the measurement, meaning that each ˆ h i is switched o ff at the last stage of the process. Moreover, the dynamics of each block ˆ R i j of the density matrix ˆ D ,whether i = j or i > j , is decoupled from the dynamicsof the other blocks . (The i < j blocks follow by Her-miticity.)If the environment is regarded as external to the ap-paratus, with weak interactions, its elimination fromthe equations of motion, achieved by standard methodsof quantum statistical mechanics, produces additionalterms in (6). However the decoupling still takes place.In any case, the evolution of ˆ D ( t ) towards the equilib-rium state ˆ D ( t f ) is an irreversible process, during whichthe coarse grained entropy increases. The compatibilityof this feature with the reversibility of the di ff erentialequations (6) is ensured by the principle 2 of Subsec.2.3, which allows us to disregard physically irrelevantelements issued from the exact equations, and thus tojustify approximations of quantum statistical mechan-ics. The macroscopic number of degrees of freedom forthe bath and for the pointer included in A, and a suit-able choice of parameters in ˆ H A and ˆ H SA will thereforebe needed, for each model, to explain the required re-laxations and to estimate their time scales, as illustratedby the CW model . In decoherence approaches, whichfocus on the disappearance of the o ff -diagonal blocksˆ R i j for i , j , irreversibility is ensured by the large sizeof an external environment [7, 9].Two types of relaxation, with di ff erent time scales,arise independently from the dynamical equations (6) .( i ) “Truncation”: For i , j , the coherent contribu-tions ˆ R i j ( t ) decay for all practical purposes owing to thedi ff erence between ˆ h i and ˆ h j , and rather quickly vanish.The o ff -diagonal blocks of the density matrix ˆ D ( t ) arethus truncated as regards the physically attainable ob-servables . Depending on the model, this decay may Authors do not always give the same meaning to the variouswords used. We term as truncation the disappearance of the o ff -diagonal blocks of the density matrix of S + A under the e ff ect of anarbitrary mechanism (including dephasing), and specialise decoher-ence to the production of this e ff ect by interaction with an environ-ment or a thermal bath. We term as registration the process whichleads each diagonal block to the correlated state ˆ r i ⊗ ˆ R i , and as reduc-tion the transition from ˆ r (0) to some ˆ r i for an individual run. The matrix elements of ˆ R ij ( t ) with i , j contain rapidly oscillat-ing phase factors. As for any irreversible process, physical quantitiesinvolve sums over very many of them, which cancel out for times lessthan the huge recurrence time. So for all practical purposes they canbe omitted after the relaxation time owing to the macroscopic size ofthe apparatus, in spite of the constant value of the sum tr A ˆ R ij ( t ) ˆ R † ij ( t ) be governed by di ff erent mechanisms .( ii ) “Registration”: For i = j , the evolution of ˆ R ii ( t )governed by (6) is a mere relaxation from the metastablestate ˆ R (0) to the equilibrium state ˆ R hi in the presenceof the source ˆ h i , and then to ˆ R i after ˆ H SA is switchedo ff . The correlation between s i and A i needed to registerthe outcome is thereby established . Since registrationrequires a dumping of free energy into the bath, it itstypically slower than truncation.These two irreversible processes are unrelated andshould not be confused. The registration consists in the establishment of correlations between the pointer andthe tested observable , that we generally denoted as ˆ s (for the CW model, this tested variable is the componentˆ s z of the spin), whereas the truncation proceeds throughgradual creation and subsequent vanishing of correla-tions between the pointer and observables that do notcommute with ˆ s (for the CW model, these observablesare the transverse components ˆ s x and ˆ s y ). Both are es- of the modulus square of the matrix elements of ˆ R ij ( t ). However,would one wish to calculate mathematical objects, for instance tocheck that the exact von Neumann entropy (without coarse graining)remains constant, they would definitely contribute. This “truncation” process has abundantly been studied in the lit-erature on measurements. It is often supposed to be the result of adecoherence produced by a coupling with an external environment. Inthe case of measurements, the o ff -diagonal blocks to be suppressed bythe dynamics (6) are those which relate di ff erent eigenvalues s i of ˆ s in a basis diagonalizing ˆ s . However, as discussed in Subsec. 3.1, Smust be coupled to A (including the environment) by an interactionof the form ˆ H SA = P i ˆ π i ⊗ ˆ h i , where each operator ˆ h i should ensurerelaxation towards the equilibrium state ˆ R hi of A. Thus, explaining thetruncation by a decoherence process may be satisfactory only if thecoupling with an external environment has a particular form depend-ing both on the tested system and on the pointer observable, so thatS + A is piloted by a potential ˆ H SA of the above type (see ref. [10],sect. 2.7). In the CW model , several mechanisms occur, involvingor not a thermal bath. Over the short time scale ~ / g √ N , truncationresults (see ref. [10], sect. 5) from the dephasing between the os-cillations yielded by the factor exp 2 it ~ − P Nn = g ˆ σ ( n ) z entering ˆ R ↑↓ ( t ),which have di ff erent frequencies (due to the randomness of σ ( n ) z in theinitial paramagnetic state of A). Information is thereby lost through acascade of correlations of higher and higher order, less and less acces-sible, between ˆ s x or ˆ s y and the spins of A, in such a way that ˆ R ↑↓ ( t )practically tends to zero as regards the accessible observables. Recur-rences are wiped out (see ref. [10], sect. 6), either by the coupling γ with the phonon bath (provided T / J ≫ γ ≫ g / NT ), or by a spread δ g in the couplings g of ˆ H SA (provided δ g ≫ g / √ N ). While much attention has been paid to the vanishing of the o ff -diagonal blocks, the relaxation of the diagonal blocks is too often dis-regarded, although it produces the correlations that ensure the pos-sibility of reading the outcome. In the CW model (see ref. [10],sect. 7), this process is triggered by ˆ h i which makes ˆ R (0) unstable andshould be su ffi ciently large to exclude false registrations ( g ≫ J / √ N ).Later on, the relaxation of ˆ R ii ( t ) to ˆ R hi , and finally to ˆ R i after ˆ H SA isswitched o ff , is governed by the dumping of free energy from the mag-net to the phonon bath; its characteristic duration is the registrationtime ~ /γ ( J − T ). .Thus, microscopic dynamics confirms the surmise ofrelaxation towards the generalised thermodynamic equi-librium state (2) for S + A in the ensemble E . As S and Ahave been decoupled at some time t decoup before t f , theremainder of our discussion will involve only the appa-ratus .
5. Through sub-ensembles towards individual runs
The expression ˆ D ( t f ) = P i p i ˆ D i of the final state ofS + A thus derived for the large ensemble E of runs mightsuggest that the task is over. It seems to mean that theset E gathers, as expected, a proportion p i of individ-ual runs having ended up in the state ˆ D i . However, asalready indicated in Subsec. 1.2 ( i ) and explained indetail below (Subsec. 5.1), Schrödinger’s quantum am-biguity makes it fallacious to postulate directly such aninterpretation of the separate terms p i ˆ D i of ˆ D ( t f ). Inorder to justify it, we adopt the following strategy. Wewill first draw (Sec. 5) further information from the dy-namics of sub-ensembles in a formal quantum frame,and this will allow us to introduce afterwards an indis-putable interpretative principle (Sec. 6). Classical probabilities presuppose the existence of asample space, so that an ordinary probability distribu-tion can be identified with the set of relative frequenciesof occurrence of some property among a large numberof individual events. The construction of classical sub-ensembles relies on the possibility of distinguishing in-dividual events so as to select part of them . One canthus readily infer from a classical probability distribu-tion a unique set of sub-ensembles E ( k )sub and their associ-ated distributions ˆ D ( k )sub .The situation is di ff erent in quantum mechanics.There also, the assignment of a density operator to anensemble of systems is a means of making statementsabout experimental facts pertaining to this ensemble,but experiments performed with di ff erent apparatusescan provide results (such as the violation of Bell’s in-equalities) which are not compatible with the existenceof a sample space describing individual systems.Formally, this di ffi culty is expressed by Schrödinger’s quantum ambiguity of the decompositions of a density In an analogy to Nuclear Magnetic Resonance, truncation is sim-ilar to T processes and generally much faster than registration, whichbears some analogy to T processes. operator [18, 19, 20], which we illustrate by the sim-ple, well known example of an unpolarised ensemble E of spins in the state ˆ I . The decomposition ˆ I = | z ih z | + |− z ih− z | for this state seems to mean that halfof the spins are polarised along | + z i , the other half along |− z i . However, the same argument applied to the alter-native decomposition ˆ I = | x ih x | + |− x ih− x | would implythat E might be split into four sub-ensembles, each ofwhich would gather spins polarised simultaneously intwo orthogonal directions, which is nonsensical. Andthere exist many other decompositions, suggesting in-terpretations contradictory to each other, hence mean-ingless. Likewise, it is obviously inconsistent to inter-pret the state of an unpolarised spin extracted from asinglet pair as a mixture of completely polarised spins .The above argument is general. Any mixed state ˆ D can be decomposed in an infinity of ways as a weightedsum of projectors onto pure states, which need not beorthogonal and which cannot be decomposed further .Attempting to interpret simultaneously two di ff erent de-compositions would lead to contradictions. Due tothis quantum ambiguity, the irreducible nature of q-probabilities forbids the recognition of a sample spacethat would refer to individual systems and would un-derlie density operators. Hence, if nothing else than ˆ D is known, the logical incompatibility between arbitrarymathematical decompositions prevents us from giving aphysical meaning to the separate terms of such a decom-position.For measurements, once the expression (2) of ˆ D ( t f )has been globally derived as in Sec. 4, the existenceof mathematical decompositions of ˆ D ( t f ) incompatiblewith the particular one ˆ D ( t f ) = P i p i ˆ D i makes it unjus-tified to bluntly infer (as is often done) that each indi-vidual run ends up in one or another of the states ˆ D i .There, extra information will be searched by noting thatthe runs are expected to be tagged after measurement by the indication of the pointer, allowing the considera-tion of sub-ensembles. The idea that the same dynami-cal equations govern both E and its sub-ensembles willhelp us to pursue within the quantum formalism as far One could argue that an interpretation might arise from theknowledge of the preparation of the ensemble E . If E is built byputting together two equal-sized sub-ensembles of spins polarised inthe directions | z i and |− z i , respectively, and provided we keep track ofthe origin of each sample, it is legitimate to interpret separately eachterm of ˆ I = | z ih z | + |− z ih− z | . However, if the two sub-ensembleshave merged at random, only ˆ I is meaningful. No experiment canallow to distinguish two di ff erent preparations of the ensemble E hav-ing led to the same mixed state, or to distinguish di ff erent populationswithin E , if no other information than this state is available. All other decompositions, involving mixed states, are built bygrouping terms of the decompositions in terms of pure states
12s possible, postponing interpretation so as to introduceweakest possible interpretative principles. Our next task(Subsec. 5.4) will therefore consist in proving that, for all possible physical sub-ensembles E ( k )sub of E , S + Aends up in a state of the form (3), ˆ D ( k )sub = P i q ( k ) i ˆ D i . We remind (Subsec. 2.4) that a given individual sys-tem can statistically be described by di ff erent quantumstates, depending on our information about the phys-ical sub-ensemble in which it is embedded. Thesestates, usually mixed, are related to one another. Whentwo disjoint sub-ensembles, E ( k )sub (with N ( k )sub elements)and E ( k ′ )sub (with N ( k ′ )sub elements), described by ˆ D ( k )sub andˆ D ( k ′ )sub , respectively, merge to constitute an ensemble E described by ˆ D (with N = N ( k )sub + N ( k ′ )sub elements),the q-expectation values defined by the correspondenceˆ O
7→ h ˆ O i for E , E ( k )sub and E ( k ′ )sub have the same additivityproperty as ordinary averages. This is expressed at eachtime by ˆ D ( t ) = λ ˆ D ( k )sub ( t ) + (1 − λ ) ˆ D ( k ′ )sub ( t ) , (7)with the weight λ = N ( k )sub / N . All three states ˆ D ( t ),ˆ D ( k )sub ( t ) and ˆ D ( k ′ )sub ( t ) are governed by the same dynamicalequations, involving the Hamiltonian that characterisesthe considered system.However, conversely, due to the matrix nature of ˆ D ,there exist many operators ˆ D dec issued from decompo-sitions of the type (7) which cannot be associated withsub-ensembles and have no physical meaning. In fact,the mathematical decompositions of ˆ D depend on con-tinuous parameters, so that the number of states ˆ D dec is infinite. In contrast, if the ensemble E has N ele-ments, the number 2 N − N , the numberof physical sub-ensembles E ( k )sub described by states ˆ D ( k )sub is still smaller, growing polynomially with N , becauseeach E ( k )sub should contain many elements. Thus, only atiny proportion of decompositions of the type (7) maydescribe a splitting of E into physical sub-ensembles. If an ensemble E is constructed by putting together two ensem-bles E ( k )sub and E ( k ′ )sub , the ingredients λ , ˆ D ( k )sub and ˆ D ( k ′ )sub of the decom-position (7) of the state ˆ D associated with E keep a physical meaningas long as the sub-ensembles E ( k )sub and E ( k ′ )sub can be identified within E . However, if track is lost of these sub-ensembles within E , the twoterms in ˆ D cannot be determined from any experiment performed byextracting samples at random from E . The impossibility of extracting from ˆ D alone in-formation about individual runs and even about sub-ensembles is the form taken here by the measurementproblem . In order to overcome it, we need a crite-rion allowing a quantum description of physical sub-ensembles of runs extracted from E . After achievement of the measurement process, observation and selectionof the pointer indications would a ff ord identification ofthe sub-ensembles E ( k )sub characterised by the proportions q ( k ) i of runs that have produced the outcome A i . We ex-pect a state ˆ D ( k )sub ( t f ) of the form (3) to be assigned toeach one, and we also expect the same state to describethe sub-ensemble at the final time just before reading .It is then natural to postulate that, at least during thevery last stage t ′ f < t < t f of the process, one can asso-ciate quantum states ˆ D ( k )sub ( t ) to the sub-ensembles E ( k )sub although the latter cannot yet be identified. Hence westate: Interpretative principle 4 . Density operators whichobey the probabilistic and dynamic rules of quantummechanics may be assigned not only to a large statisticalensemble of systems, but also to any one of its physicalsub-ensembles. Such a simultaneous assignment of sev-eral sub-ensemble dependent states to similar systemscan be done during a short delay preceding the timewhen the sub-ensembles will be identified through somemacroscopic property.This principle implies that the evolution of the densityoperators ˆ D ( k )sub ( t ) during the time lapse t ′ f < t < t f isgoverned by the same equations as for ˆ D ( t ). (This isconsistent with the fact that, in the Heisenberg picture,the dynamical equations do not depend on the state.) Wewish to work out these dynamical equations from t ′ f to t f so as to prove that ˆ D ( k )sub ( t ) relaxes to the expected form(3). However we have a priori no information aboutˆ D ( k )sub ( t ′ f ) at the new initial time t ′ f , except for the factthat ˆ D ( k )sub ( t ′ f ) is an element of some decomposition (7)of ˆ D ( t ′ f ), a property which will yield constraints on thisinitial state. Note that, if the state ˆ D ( k )sub entering (7) wereassociated with a sub-ensemble picked at random from E , it would be for large E the same as ˆ D itself. The sub-ensembles E ( k )sub of interest are therefore scarce within E . Individual runs and sub-ensembles evidently existexperimentally at all times; they can be tagged andfollowed during the whole process (“waiting for theoutcome”). Nevertheless, we cannot consider theo-retically the physical sub-ensembles E ( k )sub at arbitrarytimes. We can acquire information about the initial state13 D ( k )sub ( t ′ f ) only from Eq. (7), through knowledge pre-viously obtained about the state ˆ D ( t ′ f ) associated withthe full ensemble E . We therefore choose t ′ f su ffi cientlylate so that the interaction ˆ H SA between A and S hasbeen switched o ff and that this state ˆ D ( t ′ f ) has alreadyreached the final form (2). We also take t ′ f su ffi cientlyearly so that the relaxation time for each sub-ensembleis shorter than the duration t f − t ′ f of the evolution (Sub-sec. 5.4). We wish to prove that all the states ˆ D ( k )sub ( t ) associ-ated with physical subsets of runs end up in the re-quired form ˆ D ( k )sub ( t f ) = P i q ( k ) i ˆ D i . This property mightbe regarded as intuitive, since this is just a relaxationtowards a generalised thermodynamic equilibrium state,often supposed to be ensured by an environment. How-ever, even though the probability distribution ˆ D ( t ′ f ) as-sociated with the full ensemble E has already reachedits equilibrium form ˆ D ( t f ) = P i p i ˆ D i , the distributionsˆ D ( k )sub ( t f ) associated with its sub-ensembles may still beo ff equilibrium. A dynamical derivation is necessary toestablish their relaxation rigorously.Since S and A have been decoupled at the time t decoup before t ′ f , the Hamiltonian reduces for t > t ′ f to ˆ H S + ˆ H A . To simplify the discussion, we assume here that theeigenvalues of ˆ s are non degenerate, hence ˆ r i = ˆ π i = | s i ih s i | , and that ˆ H S = . The dynamics of ˆ D ( k )sub ( t ) istherefore governed for t > t ′ f by the Hamiltonian ˆ H A ofthe apparatus alone .We now need to characterise the initial states ˆ D ( k )sub ( t ′ f ).These operators cannot be fully determined, but theymust arise from some decomposition (7) of ˆ D ( t ′ f ) = ˆ D ( t f ) = P i p i ˆ D i = P i p i ˆ π i ⊗ ˆ R i , a density operatorthat we first analyse. The state ˆ R i describes canoni-cal equilibrium of the apparatus, with moreover a con- At earlier times, the form of ˆ D ( t ) would provide weaker con-straints on the initial state of ˆ D ( k )sub ( t ). Still earlier and especially at thebeginning of the process, no reasonable splitting of ˆ D ( t ) even exists.The very possibility of considering the quantum states ˆ D ( k )sub ( t ) at thetime t ′ f and hence later emerges from the relaxation of the state ˆ D ( t )which occurred at earlier times t . For degenerate eigenvalues s i , the only change in the forthcomingderivation, if the states ˆ r i ≡ | i ih i | are pure, is the replacement of | s i i bythe ket | i i in the eigenspace of ˆ s associated with s i . If the density oper-ator ˆ r i is mixed, we note that this operator of S is not modified by theprocess, while remaining fully coupled with A i for t > t ′ f . We shouldtherefore preserve this property when we consider the decompositions(7) of ˆ D which produce the states ˆ D ( k )sub of physical sub-ensembles E ( k )sub . The poly-microcanonical relaxation of A then produces againthe final state (10). A non-vanishing ˆ H S would generate for each i adi ff erent phase factor, which is ine ff ective. straint on the macroscopic value A i for the pointer (Sub-sec 3.3). As A is macroscopic, the fluctuations of ˆ H A around h ˆ H A i and of the pointer observable ˆ A around A i are small in relative size, and it is legitimate to replacein ˆ D ( t ′ f ) the canonical equilibrium states ˆ R i of A by mi-crocanonical ones, ˆ R µ i , as regards both the energy andthe pointer variable. Thus, within the Hilbert space ofA, we denote as | A i , η i a basis of kets constrained bythe fact that the macroscopic energy lies in some smallrange and that ˆ A also lies between A i − ∆ and A i + ∆ ,where 2 ∆ is larger than the width of ˆ R i (Sec. 1.1).As the spectrum is dense, the index η may take a verylarge number G i of values. We have denoted as ˆ Π i theprojector over the eigenspace of ˆ A associated with theeigenvalues lying between A i − ∆ and A i + ∆ (for arbi-trary energies), hence, ˆ R µ i ˆ Π j = ˆ R µ i δ i j . The equivalencebetween the canonical and microcanonical states ˆ R i andˆ R µ i is expresed by tr A ˆ R i ˆ Π j ≃ tr A ˆ R µ i ˆ Π j = δ i j . The mi-crocanonical equilibrium state of A is then proportionalto a projector:ˆ R µ i = G i X η | A i , η ih A i , η | . (8)Accordingly, the state ˆ D ( t ′ f ) ≃ P i p i ˆ π i ⊗ ˆ R µ i (whereˆ π i = | s i ih s i | ) does not lie in the full Hilbert space H ofS + A, but in its small, shrunken subspace H shr spannedby the kets | s i i| A i , η i . In this subspace, the tested sys-tem and the pointer value are correlated . Since the ini-tial state ˆ D ( k )sub ( t ′ f ) associated with the sub-ensemble E ( k )sub must be an element of some decomposition (7) of ˆ D ( t ′ f ),it is also constrained to lie in the subspace H shr . It musttherefore have the formˆ D ( k )sub ( t ′ f ) = X i , j ,η,η ′ | s i i| A i , η i K ( k ) ( i , η ; j , η ′ ; t ′ f ) h s j |h A j , η ′ | , (9)where K ( k ) is a Hermitean, normalised and nonnegativematrix, which however remains unknown . Strictly speaking, this equality holds only for the microcanonicalstates ˆ R µ i . However, tr A ˆ R i ˆ Π i is close to 1 if ∆ is su ffi ciently largecompared to the width of ˆ R i , and tr A ˆ R i ˆ Π j for i , j is negligible if ∆ is small compared to the distance between the possible outcomes A i . All mathematical decompositions of ˆ D ( t ′ f ) having the form (7)give rise to arbitrary operators of the form (9). According to the prin-ciple 4, some of these operators (but we cannot determine which ones)describe physical sub-ensembles, and we consider only these, whereasthe other ones, much more numerous, are physically meaningless asdiscussed in Subsecs. 5.1 and 5.2. .4. Poly-microcanonical relaxation We now consider the dynamics of ˆ D ( k )sub ( t ) for t > t ′ f ,governed by the Hamiltonian ˆ H A of the sole apparatus and starting from the partially unknown initial condi-tion (9). As ˆ D ( k )sub ( t ) is an element of some decompo-sition (7) of ˆ D ( t ) which is constant, it remains in theshrunken subspace H shr and retains the form (9) where K ( k ) depends on time. Only the part of ˆ H A that lives inthe Hilbert subspace H shr is relevant for the evaluationof the time dependence of K ( k ) ( i , η ; j , η ′ ; t ). We will relyfor the sub-ensembles on a new relaxation mechanism[10], which we term here as “poly-microcanonical” .One can regard it as a generalisation of the standardmicrocanonical relaxation [37, 38, 39, 40] which, forany initial state in the only Hilbert subspace H i of H spanned by the kets | s i i| A i , η i for given i , produces a de-cay towards the state (8) proportional to the projector on H i .Here, we consider the subspace H shr of H , the di-rect sum of several microcanonical subspaces H i . Weassume that weak interactions in the apparatus (includ-ing the environment) induce among the kets | A i , η i rapidtransitions within each subspace H i . Such interactionsare realistic for a macroscopic apparatus; they have littlee ff ect on the processes described in Sec. 4. In each ele-mentary transition, η is modified while both the macro-scopic energy and the macroscopic pointer value arenot a ff ected. The absence of jumps between di ff erentpointer values is needed to ensure the stability of thestates ˆ R i . Owing to this conservation of macroscopicquantities, the process is very rapid .The poly-microcanonical relaxation is thus a “quan-tum collisional process”, irreversible for a large ap-paratus. Acting separately in each sector, on bothsides | A i , η i and h A j , η ′ | of (9), it produces two di ff er-ent e ff ects. ( i ) For i = j , the result is the same asfor the standard microcanonical relaxation. All terms η , η ′ disappear from K ( k ) ( i , η ; i , η ′ ; t ), while the terms K ( k ) ( i , η ; i , η ; t ) all tend to one another, their sum remain-ing constant. Altogether, the coherences disappear and Two di ff erent mechanisms achieving such a process have beenfully worked out for the CW model (see ref. [10], sec. 11.2), and it hasbeen shown that they produce the result (10). In the more realistic one(see ref. [10], Appendices H and I), the transitions that modify η areproduced by an interaction ˆ V between the magnet and the bath whichhas a variance v = tr ˆ V ; an average delay θ separates successive tran-sitions. The poly-microcanonical relaxation may take place even if ˆ V is not macroscopic, with a variance scaling as v ∝ N a ( a <
1) for large N . For a short θ that scales as θ ∝ / N b ( a < b < a ), the characteris-tic time τ sub = ~ / v θ scales as 1 / N c where c = a − b , 0 < c < a < t f be-cause registration involves a macroscopic dumping of energy from themagnet to the bath, in contrast to the poly-microcanonical relaxation. the populations equalise within each sector . ( ii ) For i , j , all contributions to (9) fade out and eventuallyvanish, so that the di ff erent sectors i become uncorre-lated . Both e ff ects occur over the same time scale τ sub ,which (by definition of t ′ f ) is shorter than t f − t ′ f . As themechanism is already e ff ective before t ′ f , the relaxationis likely to have already been e ff ective at t ′ f . Anyhow,ˆ D ( k )sub reaches at the final time t f > t ′ f + τ sub the “poly-microcanonical” equilibrium ˆ D ( k )sub ( t f ) = X i q ( k ) i ˆ r i ⊗ ˆ R µ i , q ( k ) i = X η K ( k ) ( i , η ; i , η ; t ′ f ) = tr ˆ D ( k )sub ( t ′ f ) ˆ Π i (10)This general expression for the final state ˆ D ( k )sub ( t f ) as-sociated with any physical sub-ensemble depends on theinitial condition (9) only through the coe ffi cients q ( k ) i .The distinction between canonical and microcanonicalequilibria being macroscopically irrelevant, we have de-rived within the quantum dynamical formalism the re-laxation to the required equilibrium form (3) for arbi-trary sub-ensembles E ( k )sub . All of these involve at thefinal time the same building blocks ˆ D i , so that the quan-tum ambiguity has been removed . ffi cients q ( k ) i The form (3) or (10) for the final state associated withany sub-ensemble E ( k )sub , together with the property tr A ˆ R i ˆ Π j = δ i j , implytr ˆ D ( k )sub ( t f ) ˆ Π i = q ( k ) i . (11)Each weight q ( k ) i is therefore identified as the q-probability of occurrence of the macroscopic value A i for the pointer in the sub-ensemble E ( k )sub of runs of themeasurement. The narrowness ( ∆ ≪ | A i − A j | ) of thespectrum of the projectors ˆ Π i entails that for any E ( k )sub the q-distribution tr ˆ D ( k )sub ( t f ) δ ( ˆ A − A ) of ˆ A is strongly peakedaround the values A i , with the weights q ( k ) i . These quan-tum properties are still formal and call for an interpreta-tion (Sec. 6).We also note, by using the commutation [ ˆ R i , ˆ Π j ] = The present process should not be confused with those of Sec. 4.On the one hand, in contrast to the latter, it involves only the appara-tus (which includes a bath or an environment). On the other hand, itrequires the achievement of both the truncation and the registration.
15r ˆ D ( k )sub ( t f )[ ˆ Π i , ˆ O ] = O of S + A. More gener-ally, if the typical dimension G of the projectors ˆ Π i islarge, and if ˆ P , ˆ P ′ and ˆ P ′′ denote arbitrary projectionoperators with finite dimension whereas G ≫
1, onereadily shows, by expansion on the basis | s i i| A i , η i , thattr ˆ D ( k )sub ( t f ) ˆ P ′ [ ˆ Π i , ˆ P ] ˆ P ′′ = O G ! (12b)is small. Any operator of S + A containing as a factor acommutator [ ˆ Π i , ˆ O ] of a pointer observable with an ar-bitrary observable ˆ O (finite for large G ) can be writtenas a weighted sum of terms (12b). Hence, Eqs. (12a-b)express that the q-expectation value, in the final state ,of any operator depending on the projectors ˆ Π i through commutators [ ˆ Π i , ˆ O ] with arbitrary finite observables ˆ O, is negligible for a macroscopic pointer. We shall relyon this property in Subsection 6.1.Finally the coe ffi cients q ( k ) i that characterise the states(10) derived for the whole collection of sub-ensembles E ( k )sub of E possess a hierarchic structure embedded inthe following additivity property. If some sub-ensemble E ( k )sub is split into two smaller sub-ensembles E ( k ′ )sub and E ( k ′′ )sub , containing N ( k ′ ) and N ( k ′′ ) elements, respectively,the corresponding weights q ( k ) i satisfy q ( k ) i = N ( k ′ ) q ( k ′ ) i + N ( k ′′ ) q ( k ′′ ) i N ( k ′ ) + N ( k ′′ ) . (13)This is a consequence of Eq. (7) for ˆ D ( k ′ )sub and ˆ D ( k ′′ )sub with λ = N ( k ′ ) / ( N ( k ′ ) + N ( k ′′ ) ) and of the expression (10)defining the still formal q-probabilities q ( k ) i . Thus, for allpossible sub-ensembles, the various final states ˆ D ( k )sub ( t f )satisfy a hierarchic structure characterised by their form(3) and by the additivity (13) of the q-probabilities q ( k ) i .Such an addition rule is obvious for ordinary probabilitydistributions, and we may suspect that we are beginningto land in standard probability theory , but the resultsproved above, though suggestive, are only formal andstill call for physical interpretation.
6. Emergence of classical features
The expressions (2) and (3) derived above are themost detailed results about ideal measurements pro-vided by a strictly formal quantum statistical frame-work free from any interpretation, where one does not deal with individual systems but only with statisticalensembles – possibly Gedanken, but physically consis-tent (Sec. 2). We have not only shown that the ini-tial state ˆ D (0) of S + A for a run randomly extractedfrom the ensemble E relaxes to ˆ D ( t f ), but also thatthe states associated with all its possible sub-ensembles E ( k )sub reach at the final time t f the equilibrium structureˆ D ( k )sub ( t f ) = P i q ( k ) i ˆ D i involving the same building blocks ˆ D i . However, nothing yet ensures that each operatorˆ D i can be interpreted as a final state (1) assigned tosome sub-ensemble E i yet to uncover, characterised bythe outcome A i .What remains thus to be done is to interpret the q-probabilities q ( k ) i = tr ˆ D ( k )sub ( t f ) ˆ Π i , still mathematical co-e ffi cients, as ordinary probabilities. In the frequencyapproach , ordinary probabilities appear as numbersassociated with a large ensemble and with its sub-ensembles, which have the following properties: theyare non-negative and normalized; they are additive fordisjoint sub-ensembles; they may take any value rang-ing from 0 to 1. Here, although density operators di ff erfrom probabilities because they do not refer to any sam-ple space, the set q ( k ) i of q-probabilities satisfy the aboveproperties of classical probabilities including the hierar-chic additive structure (13), except for the last one, totake any value between 0 and 1. In fact, they came outin Eq. (10) as formal objects; nothing ensured that their range extends down to and up to
1, although nothingin the quantum formalism prevents this. In order to re-late these mathematical objects to physical events, weought to supplement the quantum rules of Sec. 2 bypostulating a last interpretative principle.
Instead of identifying any q-probability with an ordi-nary probability, which would lead to paradoxes (Sub-sec. 2.1), we wish to introduce a much weaker principle,by imposing stringent conditions on the objects that willget an interpretation. We rely on the following heuris-tic argument. The essential feature that distinguishesquantum mechanics from classical statistical mechan-ics is the non-commutative nature of the algebra of ob-servables. The set of projectors ˆ Π i associated with themacroscopic values A i of the pointer present in this re-spect a remarkable feature. Consider their commutators[ ˆ Π i , ˆ O ] with arbitrary observables ˆ O ( ˆ O being boundedwhen the typical dimension G of the projectors becomeslarge). Eqs. (12a-b) imply that, in ˆ D ( t f ) and in anystate ˆ D ( k )sub ( t f ) describing the outcome of a sub-ensemble,all q-expectation values involving commutators [ ˆ Π i , ˆ O ]have become negligible as 1 / G . The observables ˆ Π i (as16ell as their linear combinations which describe proper-ties of the pointer variable) thus behave at the final timet f as if they commuted with the full algebra . The quan-tum nature of these macroscopic variables has becomeconcealed as a result of the dynamics, so that they takein a commutative behaviour at the final stage of the pro-cess. This restrictive, quasi classical property makes thefollowing principle natural. Interpretative principle 5 . Consider a set of macro-scopic orthogonal projectors Π i , a state ˆ D associated ata given time with an ensemble E and the states ˆ D ( k )sub associated with its sub-ensembles E ( k )sub . If the projec-tors have in these states the commutative behaviour ex-pressed by Eqs. (12a-b), their q-expectation values q ( k ) i can be interpreted as physical probabilities for exclusiveevents, i. e., as relative frequencies .In the abstract formulation of quantum mechanics,arbitrary q-probabilities have no reason to be inter-preted as relative frequencies (Subsec. 2.1). This iden-tification, for the specific ones q ( k ) i submitted to theabove conditions on the projectors Π i and on the statesˆ D ( k )sub ( t f ), is imposed by macroscopic experience, whileremaining in harmony with the quantum rules. The above principle implies that the mathematicalstructure (10) of the density operators pertaining to thewhole set of sub-ensembles reflects the physical struc-ture of these sub-ensembles. More precisely, since theweights q ( k ) i are interpreted as standard probabilities inthe sense of frequencies, they can now take values rang-ing from 0 to 1, with 0 and 1 included. By taking q j = δ i j in Eq. (3), we thus theoretically acknowledge the Introduced here for the pointer variable to solve the measurementproblem, this interpretation of q-probabilities for macroscopic quan-tities can be used in other contexts, such as the quantum dynamicsof phase transitions with spontaneously broken invariance . There,ˆ D ( t ) denotes the state of a statistical ensemble E of systems, identi-cally prepared at the macroscopic scale, and ˆ D i the equilibrium statescharacterised by discrete values of the macroscopic order parameter A i . We assume that the initial state ˆ D (0) and the Hamiltonian ˆ H aresu ffi ciently symmetric so as to avoid favouring the occurrence at thefinal time of a single outcome A i . We thus expect ˆ D ( t ) to relax to-wards a state ˆ D ( t f ) of the form (2). Provided time scales are suitable,we also expect, as in Sec. 5, the states ˆ D ( k )sub ( t ) associated with allsub-ensembles to relax to the hierarchical structure (3). The presentprinciple can then be used to explain within quantum mechanics why,in the considered circumstances, the order parameter takes in eachsingle experiment a well-defined value, but not always the same. Im-plicitly assuming the principle 5, the community has rightfully notbeen bothered about this subtlety. (experimentally obvious) existence of sub-ensembles E i characterised by the value A i of the pointer, and to whichthe final state ˆ D i = ˆ r i ⊗ ˆ R i of S + A is assigned. An ar-bitrary sub-ensemble E ( k )sub can now be regarded as themerger of sub-ensembles E i , each q ( k ) i being understoodas the proportion of individual runs tagged by A i in E ( k )sub .The q-probability p i = tr ˆ D ( t f ) ˆ Π i is interpreted as theproportion in E of runs having ended up with the in-dication A i . We thus recover all expected well knownproperties of ideal measurements.Whether the macroscopic pointer is observed or not,the formal quantum dynamics and the above inter-pretative principles ensure the existence of the sub-ensembles E i , but the latter can be explicitly identifiedonly by reading or registering the pointer indication soas to tag the runs. Two steps are thus necessary to gofrom the initial state ˆ D (0) to ˆ D i . First, the irreversibledynamics of the coupled system S + A leads to ˆ D ( t f ) = P i p i ˆ D i for the ensemble E , and to ˆ D ( k )sub = P i q ( k ) i ˆ D i with unknown coe ffi cients q ( k ) i for its sub-ensembles.The second step, leading then to one of the componentsˆ D i , is not a consequence of some evolution , but the re-sult of selecting the particular outcome A i . It merelyamounts to an updating of q-information by switchingfrom the full ensemble E to the sub-ensemble E i (as inthe dice example of Subsec. 2.4).The complete correlation established by the processbetween the pointer indications A i and the final statesˆ D i gives access to some features of microscopic real-ity. After selection of the outcome A i and separation ofthe system S from the apparatus, this system is set intothe quantum state ˆ r i = tr A ˆ D i = ˆ π i ˆ r (0) ˆ π i / p i , for whichthe tested observable ˆ s has the well-defined value s i (ifthe eigenvalue s i is non degenerate, ˆ r i = ˆ π i ). We therebyderive von Neumann’s reduction , which expresses the fi-nal marginal state of S for the sub-ensemble E i in termsof the state ˆ r (0) initially assigned to S. An ideal mea-surement with selection of the outcome A i constitutesa preparation of the state ˆ r i , from which we can predictthe q-expectation values of all observables of S. Repeat-ing the measurement of ˆ s then leaves S unchanged.Consideration of sub-ensembles sheds light on local-ity issues . Take for instance as system S a pair of parti-cles 1 and 2 lying far apart and carrying the spins ˆ s (1) andˆ s (2) , initially prepared in the singlet state ˆ r (0) = ( | ↑↓i− | ↓↑ i )( h↑↓ | − h↓↑ | ). The measured observable is the z -component ˆ s (1) z of the spin 1. The pointer has two pos-sible outcomes A ↑ and A ↓ associated with s (1) z = s (1) z = −
1, and their selection at the time t f produces twosub-ensembles E ↑ and E ↓ of runs for which the reducedstates of S are ˆ r ↑ = | ↑↓i h↑↓| and ˆ r ↓ = | ↓↑i h↓↑| , respec-17ively. The interaction ˆ H SA is localized in the vicinity ofthe particle 1 and is switched on during the time lapse0 < t < t decoup (with t decoup < t f ). The fact that the parti-cle 2 lies beyond the range of the apparatus is consistentwith the time-invariance, for the full ensemble E , of itsmarginal state ˆ r (2) ( t ) = ( | ↑i h↑ | + | ↓i h↓ | ). However,for the sub-ensemble E ↑ of runs, one can assign to S thereduced state ˆ r ↑ = | ↑↓i h↑↓ | after the reading time t f ,but also already after the decoupling time t decoup , sinceS cannot evolve after t decoup . Likewise, one can assign tothe spin 2 the reduced marginal state ˆ r (2) ↑ = | ↓i h↓| at anytime t >
0. The change of state of the spin 2 from ˆ r (2) (for E ) to ˆ r (2) ↑ (for E ↑ ), which takes place far from themeasuring apparatus, is evidently not a result of somenon-local physical e ff ect; it is merely a non-local infer-ence by the experimenter, based on his knowledge of theinitial intricate state ˆ r (0) and his possibility to select theruns of the sub-ensemble E ↑ by a retroactive use of in-formation gathered through the pointer. If experimentsinvolving the spin 2 are performed at an arbitrary time t > r (2) ↑ = | ↓i h↓| for the runs belonging to E ↑ andthe state ˆ r (2) ↓ = | ↑i h↑ | for the runs belonging to E ↓ .However, the very sorting of runs requires observationof the pointer and transfer of this information towardsthe processing point; the preparation of the spin 2 in ei-ther the state ˆ r (2) ↑ or the state ˆ r (2) ↓ through measurementof ˆ s (1) z can therefore be acknowledged only after the time t f . Altogether, nonlocality lies only in the q-correlationsbetween the two spins 1 and 2 that exist in their initialstate; these two parts do not communicate later on. Allphysical processes involved in the measurement are lo-cal.The uniqueness of the outcome of individual runs for an ideal measurement process also emerges theoret-ically from the identification of the weights q ( k ) i as fre-quencies, since we can characterise after the process asingle compound system S + A belonging to E i by thestate ˆ D i . A dynamical solution of the measurementproblem (Subsec. 1.1) has thus come out.In general, qualitatively new physical propertiesemerge in a change of scale, and their theoretical expla-nation goes through some interpretative principle whichcomplements the formalism. For instance, in statisticalmechanics, the principle 2 of Sec. 2.3 is used to explainhow macroscopic continuity of matter emerges froma discrete microscopic structure, or how irreversibilityemerges from reversible equations of motion. Here,the qualitative changes that result from the macroscopicsize of the apparatus concern not only phenomena (the measurement process is irreversible), but also, remark-ably, concepts : Classical features emerge from a merelyformal quantum approach supplemented by the interpre-tative principle 5 which concerns only the pointer of theapparatus. The principle 5 of Subsection 6.1 cannot be extendedcarelessly, as it is founded on several stringent require-ments.( i ) The e ff ective commutation of the projectors ˆ Π i with the full algebra relies on the macroscopic charac-ter of these projectors, since Eq. (12b) holds only for amacroscopic pointer ( G ≫ ( A i is then replaced by thevalue of the order parameter).( ii ) Moreover, this e ff ective commutativity of the pro-jectors ˆ Π i is ensured only at the final time , as Eqs. (12a-b) involve the final states ˆ D ( k )sub . During the process, thenon-Abelian nature of ˆ Π i cannot be neglected since thepointer must evolve from its initial metastable state toone of the stable states, and this time-dependence oftr ˆ D ( t ) ˆ Π i requires that [ ˆ Π i , ˆ H ] is e ff ective until equilib-rium is reached. Note that, whereas the projector ˆ Π i per-taining to A does not commute with ˆ H ( ˆ Π i is e ff ectivelyconserved only after t ′ f ), the projector ˆ π i pertaining to Scommutes with ˆ H , so that tr ˆ D ( t ) ˆ π i remains constant atall times. However, being microscopic, π i cannot sat-isfy relations such as (12b), and the principle 5 does notapply to it.( iii ) The consideration of sub-ensembles has alsobeen essential. If we wish individual runs to providethe outcomes ˆ D i , the necessary conditions (3) must befulfilled. Due to the existence of incompatible decom-positions of ˆ D ( t f ) (Subsec. 5.1), it is not justified to pos-tulate directly , as generally done, that the coe ffi cients p i in ˆ D ( t f ) = P i p i ˆ D i might be interpreted as frequenciesof the outcomes ˆ D i in the full ensemble E : this fallacyis the measurement problem that we addressed. We es-caped this loophole (Sec. 5.4), by eliminating the quan-tum ambiguity through a dynamical process, the poly-microcanonical relaxation [10], which provides the ex-pected structure for the states ˆ D ( k )sub . The relative frequency p i of occurrence of the macro-scopic value A i of the pointer has been found, accordingto the above principle, as p i = tr ˆ D ( t f ) ˆ Π i in terms of18he final state of S + A. However, such a proportion iscurrently expressed by Born’s rule tr S ˆ r (0) ˆ π i , which dis-regards the apparatus and involves only the initial state of the tested system . In the light of the restrictions aboutprinciple 5, we are not entitled in the present approachto directly interpret the q-probability tr S ˆ r (0) ˆ π i as a gen-uine probability and to admit blindly Born’s rule. Toderive it theoretically, we need to rely on the followingtwo properties.( i ) In the final state (2) of S + A, the marginal stateˆ r i of S is fully correlated with the macroscopic indi-cation A i . Using the identity tr A ˆ R i ˆ Π j = tr S ˆ r i ˆ π j = δ i j ,we can thus identify tr ˆ D ( t f ) ˆ π i with the true probability p i = tr ˆ D ( t f ) ˆ Π i . However, this feature was not granted apriori, as it results from the dynamics of the process. In-deed, if the coupling ˆ H SA is too weak , the “registration”process considered in Sec. 4 may be imperfect, driv-ing ˆ R ii ( t ) with some probability to a wrong equilibriumstate ˆ R j with j , i (see ref. [10], sec. 8). The resultingimperfection of the correlation between s i and A i thenproduces a violation of Born’s rule , with a q-probabilitytr ˆ D ( t f ) ˆ π i of s i in the final state di ff erent from the ob-served frequency tr ˆ D ( t f ) ˆ Π i of A i .( ii ) The conservation law [ ˆ H , ˆ s ] = D ( t f ) ˆ π i = tr ˆ D (0) ˆ π i = tr S ˆ r (0) ˆ π i between q-probabilities of s i at the initial and final times, againas a consequence of the dynamics of S + A during themeasurement process. With the above property p i ≡ tr ˆ D ( t f ) ˆ Π i = tr ˆ D ( t f ) ˆ π i , this finally leads to Born’s ex-pression p i = tr S ˆ r (0) ˆ π i .The complete correlation between s i and A i allowsus to extend the ordinary probabilistic interpretation of p i = tr ˆ D ( t f ) ˆ Π i , issued from the principle 5, to some mi-croscopic quantities. In the present approach, we mayfor instance determine from p i the ordinary expectationvalue of ˆ s and its variance, or write the standard con-ditional probability of ˆ s to be equal to s j if the pointertakes the value A i as tr ˆ D ( t f ) ˆ π j ˆ Π i / tr ˆ D ( t f ) ˆ Π i = δ i j . Infact, we have shown that such identifications are licit only at the final time , when ordinary probabilities haveemerged after interaction with an apparatus designed tomeasure the observable ˆ s . The occurrence of the initialstate in Born’s expression p i = tr S ˆ r (0) ˆ π i is somewhatmisleading. Although one can infer from the measure-ment of ˆ s some formal properties of ˆ r (0), one shouldnot interpret it as a probability of ˆ s to take the value s i in the initial state ˆ r (0) of S (Subsec. 7.3). We haveavoided such an over-interpretation of the quantum for-malism, which would lead to the logical contradictionsexemplified by Bell’s inequalities or the GHZ paradox.
7. Epilogue: Quantum mechanics as a half-blindtheory
The above reasoning appears as the complete oppo-site of a recent approach [41, 42] which introduces asa starting point some physical axioms pertaining to thesystem S placed in all imaginable contexts. There, anordinary probabilistic description applies for each con-text. Gleason’s theorem is then used to unify the con-texts [43, 44, 45], and thus to construct for S the stan-dard mathematical formalism of quantum mechanics.Here, we conversely start from this abstract formalism.We consider for S a single context which is materialisedby a macroscopic apparatus A, and analyse the dynam-ics of the compound quantum system S + A. The proper-ties of the measurement emerge at the end of this pro-cess owing to the introduction of a few physical prin-ciples. We comment below the main features of thepresent approach.
Measurement theory is often treated in close connec-tion with interpretation of quantum mechanics. Ourscope here has been more limited. We did not attemptto interpret the quantum formalism taken as granted, butonly proposed an interpretation of ideal quantum mea-surement processes. We followed two paths in parallel.On the formal side, we discussed which features of theHamiltonian are needed to ensure that the process hasall required properties, and we brought out the most de-tailed results that quantum dynamics (without interpre-tation) can provide. On the conceptual side, we lookedfor the least numerous and narrowest possible interpre-tative principles needed to establish, for ideal measure-ments, a bridge between formal quantum results andphysical reality. All other mathematical objects manip-ulated in quantum theory, the operator-valued observ-ables and states as well as their scalar products, the “q-expectation values”, have remained abstract.Thus, only the final indications of the macroscopicpointer were eventually described by means of ordinaryprobabilities for individual runs. Some microscopicphysical properties selected by the process could sub-sequently be grasped as the result of an inference. In-deed, most interpretative principles introduced above ina natural way refer to macroscopic properties. They lie astride macrophysics and microphysics and are consis-tent both with our macroscopic experience and with thequantum formalism.Some of these principles are minimalist, in the sensethat they are submitted to drastic conditions, which19owever are su ffi cient for our purpose. The princi-ples 1 and 5, which identify some q-probabilities withrelative frequencies, do not apply to arbitrary observ-ables and states, but only to particular macroscopic ob-servables and particular states satisfying stringent con-ditions (Subsec. 6.3). The principle 4 helped us toexplain through a sub-ensemble analysis the apparent“bifurcation” (or “multifurcation”) which leads fromthe single initial state ˆ D (0) to several final states ˆ D i .However, this principle introduced such sub-ensemblesonly by the end of the measurement process, after thetime t ′ f at which ˆ D ( t ) had already reached the form P i p i ˆ D i . In fact, the possibility of recognising physi-cal sub-ensembles within ˆ D exists only at the last stageof the measurement process and emerges from the dy-namics of A. The principles 2 (Subsec. 2.3) and 3 (Subsec. 3.2)are consistent with the conception of a quantum state asa catalogue of q-information referring to a statistical en-semble. In this prospect, dynamics produce transfer ofq-information within the set of observables; selectionof a sub-ensemble a ff ords updating of q-information.An ideal measurement appears as a processing of in-formation , which involves transformations of the q-information carried formally by quantum states, andconversion of q-information into ordinary informationaccessible to experiment . Such changes can be madequantitative by evaluation of the relevant von Neumannentropies. Let us review these informational aspects ofthe above treatment.The initial metastable density operator ˆ R (0) of A,defined by some macroscopic data, is provided by theprinciple 3 as the least informative one that accountsfor these data (Section 3), while ˆ r (0) encodes the q-information characterising initially the ensemble E inwhich S is embedded. The relaxation of the coupledsystem S + A (Section 4) consists in a transfer of q-information from some degrees of freedom to others, States being viewed as catalogues of knowledge (Sec. 2), q-information about S is updated in an ideal measurement by replacingthe initial state ˆ r (0) by ˆ r i if A i is selected, or by P i p i ˆ r i if the indica-tions of A are not selected. If the tested observable is not fully spec-ified, the least biased subsequent predictions should rely on a stateobtained by averaging over all possible interaction processes. If forinstance, one is aware that an ensemble of spins initially prepared inthe state ˆ r (0) have been measured in some direction, but if one knowsneither in which direction nor the results obtained, one should assignto the final state the density operator [ˆ1 + ˆ r (0)] as being the best (butimperfect) description. (To show this, write ˆ r (0) in its polar form andthen as the projected form after a measurement.) with possible loss towards inaccessible ones (principle2). Thus, in truncation, q-information leaks towardsinaccessible q-correlations between S and an increas-ingly large number of degrees of freedom of A (witha huge recurrence time). In registration, the loss of q-information towards the bath is partly compensated forby the creation of complete q-correlations between s i and A i . Some downgrading is also produced for eachsub-ensemble E ( k )sub by the poly-microcanonical mecha-nism of relaxation (Sec. 5).The principle 5 (Sec. 6.1) finally expresses thatthe q-information about the pointer variable, embeddedwithin the state ˆ D ( t f ), can be converted into ordinary,readable information , and disclosed after the final timein the form of relative frequencies p i of occurrence of A i . As usual, selection of the sub-ensembles E i , taggedby the value A i of the pointer, increases q-information.As regards the system S itself, some ordinary infor-mation pertaining to the tested observable ˆ s has thusbeen extracted owing to the correlations between s i and A i built up dynamically by the coupling with the appa-ratus. The dynamical process undergone by S + A haspulled out from the latent q-information contained in ˆ r (0) the part associated with ˆ s , and has converted itinto true information . The initial q-informations aboutobservables that commute with ˆ s have been preserved;they are encoded within the reduced states ˆ r i , which maysubsequently be used for further experiments.However, gaining information on S through a quan-tum measurement requires an irreversibility of the phys-ical process of interaction between S and A, hence aloss of q-information. Not only does this loss take placewithin A, but von Neumann’s reduction expresses thevanishing after ideal measurement of the q-expectationvalues tr S ˆ r i ˆ O o ff of all o ff -diagonal observables ˆ O o ff ofS such that ˆ π i ˆ O o ff ˆ π i = i . Remarkably, gainingfull information about ˆ s requires perturbing S so as todestroy the whole q-information about the observablesthat do not commute with ˆ s . This unavoidable loss ofq-information about the observables of S incompatiblewith ˆ s is a price to pay for testing the quantum observ-able ˆ s . The apparatus plays a major role, not only experi-mentally but also in the theory of ideal measurements.As usual in statistical mechanics, it is owing to themacroscopic size of the apparatus that the irreversibilityof the measurement process emerges from the reversiblemicroscopic dynamics (Secs. 4 and 5). It is also thismacroscopic size which produces at our scale other re-20arkable types of emergence of features qualitativelydi ff erent from those of quantum theory (Sec. 6).Technically, we have seen that the dynamical equa-tions (6) which govern the relaxation of S + A for thefull ensemble of runs are expressed only in terms of theapparatus. The tested system only appears through thefactors ˆ h i of the coupling ˆ H SA which trigger the evo-lution of A towards one or another of its stable states.The system S does not even intervene at all in the poly-microcanonical relaxation that takes place for the sub-ensembles (Sec. 5).As regards the interpretation of the measurement out-comes, most principles that we have been led to intro-duce also concern only the macroscopic apparatus . Wehave stressed that the probability p i refers to the pointerobservable, and is associated only indirectly with theeigenvalues of ˆ s through the full correlation between Sand A. Due to the omnipresence of the apparatus in the anal-ysis of ideal quantum measurements, their outcomesshould not be viewed as intrinsic properties of the sys-tem S irrespective of A, but as joint properties of S andA. In particular, the relative frequencies of occurrenceof the pointer indications A i came out theoretically as p i = tr ˆ D ( t f ) ˆ Π i . This expression has pre-eminence overBorn’s formula p i = tr S ˆ r (0) ˆ π i , which although impor-tant is only a by-product of the dynamics of S + A (Sub-sec. 6.4) and has no fundamental character, as is obvi-ous when the measurement is imperfect.We also stressed that one should not be misled bythe occurrence of the initial state ˆ r (0) in Born’s rule.Retrodiction from the outcomes of measurements to-wards properties of S at the initial time is legitimate onlyfor abstract q-probabilities, and p i should not be inter-preted as a true probability for ˆ s to take the value s i inthe state ˆ r (0), as is often taught.For instance, in experiments testing Bell’s inequali-ties, spin pairs are all prepared similarly in a given initialstate ˆ r (0) of S, and several series of measurements areperformed, each one using a pair of detectors oriented intwo given directions. Each single run provides a value + each such setting, one getsfrom the ensemble of runs a correlation between the twospins, which it is legitimate to interpret as a true corre-lation , but only in the final state . However, by retrodic-tion towards the initial state ˆ r (0), one may interpret thisquantity only as an abstract q-correlation , not as a truecorrelation. Indeed, putting together such q-correlationsissued from di ff erent series of measurements violates Bell’s inequalities, which should be satisfied by truephysical correlations. As a quantum measurement is ajoint property of S and A, we are not allowed to interpret simultaneously as real properties of the initial state of Sthe results of experiments obtained with di ff erent appa-ratuses (here with di ff erent directions of the detectors).This deep property of quantum measurements is in linewith the absence, for quantum states, of a sample spaceas in ordinary probability theory [46, 47, 48, 49, 50].The situation is even worse with the GHZ paradox,experimentally verified. There, complete q-correlationsbetween several observables are exhibited by measure-ments, performed on identically prepared systems S andinvolving di ff erent apparatuses (hence di ff erent ensem-bles of joint systems S + A). If one then combines someidentities implied by these q-correlations, using elemen-tary algebraic rules that would hold for ordinary corre-lations, one stumbles on a logical contradiction [51]:one would find a result + − mathematically expressed in terms of S alone , they ac-quire a consistent physical meaning only in the presenceof a dedicated measurement apparatus . Probabilistic orlogical paradoxes occur when results obtained in di ff er-ent experimental contexts are interpreted as ordinary ex-pectation values or correlations and are put together. The 3-spin example of the GHZ paradox involves several ob-servables ˆ a ( j ) , ˆ b ( j ) ( j = , ,
3) with ˆ b (3) ≡ ˆ b (1) ˆ b (2) , defined byˆ a (1) = ˆ σ (1) x , ˆ b (1) = ˆ σ (2) z ˆ σ (3) z , etc. All these operators have eigen-values a ( j ) = ± b ( j ) = ± a ( j ) and ˆ b ( k ) which anticommute when j , k . The sys-tem is prepared in a pure state, the common eigenstate of ˆ a ( j ) ˆ b ( j ) ( j = , ,
3) with eigenvalues 1. At the formal level, this yields the q-expectation values h ˆ a ( j ) i = h ˆ b ( j ) i = h ˆ a ( j ) ˆ b ( j ) i = j = , , h ˆ a (1) ˆ a (2) ˆ a (3) i = −
1. Physically, the prop-erty h ˆ a (1) ˆ b (1) i = a (1) and ˆ b (1) ; each run provides a fullycorrelated outcome a (1) , b (1) such that a (1) = b (1) , in agreement with h ˆ a (1) ˆ b (1) i =
1. Likewise, other sets of measurements provide out-comes satisfying a (2) = b (2) and a (3) = b (3) , while simultaneous mea-surement of ˆ a (1) , ˆ a (2) , ˆ a (3) yields a (3) = − a (1) a (2) for each run. How-ever, accounting for ˆ b (3) = ˆ b (1) ˆ b (2) and naively combining the identi-ties a (1) = b (1) , a (2) = b (2) and a (3) = b (3) would yield a (3) = + a (1) a (2) ,in flagrant contradiction with the quantum result h ˆ a (1) ˆ a (2) ˆ a (3) i = − a (1) = b (1) , a (2) = b (2) , a (3) = b (3) = b (1) b (2) and a (3) = − a (1) a (2) , each of which is satisfied by a measurement per-formed with a specific apparatus, holds even if a single run is consid-ered for each of the four measurements. O , one should discrimi-nate when teaching quantum mechanics q-expectationvalues tr ˆ D ˆ O , which are latent mathematical objects,from true expectation values tr ˆ D ˆ s , which emerge atthe issue of the measurement of an observable ˆ s . Dis-tinguishing formal q-probabilities, which characterisequantum “states”, from ordinary probabilities, whichgovern data issued from measurements, would help tounderstand the status of Born’s rule, and to circumventapparent contradictions that arise when one combinesq-correlations which cannot be measured with a singleexperimental measurement setting. Acknowledgment
It is a great pleasure to acknowledge extensive, pas-sionate and thorough discussions with Franck Laloë.
Appendix A. Super-systems, q-expectation values vsaverage values, and maximum entropy
Inspired by the equivalence, in ordinary probabil-ity theory, between expectation values and average val-ues, we wish here to compare q-expectation values withquantum averages over a large number of samples. Tothis aim, we introduce a large set of systems S [ n ] ( n = , , · · · , N ) similar to the (small or large) system S ofinterest [29]. We regard the merger of S ≡ S [1] withall its siblings S [2] , · · · , S [ N ] as a large compound super-system S = { S [1] , S [2] , · · · , S [ N ] } . One should not con-fuse the super-system S , which is a single compoundGedanken system, with the ensemble E to which S be-longs. (In fact, the quantum description of S involves a“super-ensemble” E of copies of S .)With each observable ˆ O ≡ ˆ O [1] in the Hilbert space H [1] of S ≡ S [1] , we associate the average observable O ˆ = N − P n ˆ O [ n ] in the Hilbert space Π n ⊗H [ n ] of S . (Eachterm of this sum is meant as the tensor product of ˆ O [ n ] byall the unit operators associated with the other systems S [ n ′ ] with n ′ , n .) The various average observables,pertaining to the macroscopic super-system S , nearlycommute with one another, as the commutation relation[ ˆ O , ˆ O ] = i ˆ O for S implies [ O ˆ 1 , O ˆ 2 ] = i O ˆ3 / N for S ,with N ≫
1. (The set of average observables constitutea Lie algebra, but not a full algebra since their productslie outside their set.) Accordingly, these average ob-servables behave quasi-classically in the large N limitand may be assigned simultaneously rather well definedvalues.Let us consider a state D ˆ of S , invariant under permu-tations of the subsystems S [ n ] . These subsystems S [ n ] alllie in the same marginal state ˆ D , and the q-expectationvalues tr ˆ D ˆ O for S and Tr D ˆ O ˆ for S are equal. If thecorrelations between subsystems are weak, of order lessthan N − , while the q-variance of ˆ O in the state ˆ D isfinite, the q-variance of O ˆ for the super-system S , i.e.,Tr D ˆ O ˆ 2 − (Tr D ˆ O ˆ ) ∼ [tr ˆ D ˆ O − (tr ˆ D ˆ O ) ] / N , is negli-gible in relative size. We can thus apply the principle 1of Subsec. 2.2 to the average observable O ˆ of the largesupersystem S , and interpret the q-expectation value Tr D ˆ O ˆ as an ordinary value. This leads us to identify a formal q-expectation value h ˆ O i for the (possibly small)system S with the macroscopic value of the correspond-ing average observable O ˆ for the super-system S .Though somewhat artificial due to the virtual natureof the supersystem S , the latter identification was a keypoint in a general proof [29] of the maximum von Neu-mann entropy criterion based upon Laplace’s indi ff er-ence or equiprobability principle . The derivation ex-tended a classical argument by Gibbs, who had shownthat, for given h ˆ H i , a microcanonical equilibrium statefor the super-system S entails for the (possibly small)system S a canonical state. A similar purpose in quan-tum mechanics (Subsec. 3.2) is to assign the least biasedstate to a quantum system S when the sole q-expectationvalues h ˆ O p i of some observables ˆ O p of S are given(1 ≤ p ≤ p max ). The quantities h ˆ O p i are identifiedwith the values h O ˆ p i of the average observables O ˆ p ofthe associated super-system S , which nearly commuteand present small fluctuations. As the macroscopic data h O ˆ p i for the supersystem S are defined within a smallmargin (like the energy for the microcanonical state),many kets are compatible with them for large N , andunitary invariance in the Hilbert space of S sets thesekets on the same footing. Laplace’s indi ff erence prin-ciple then leads to assign to S a density operator con-centrated over these kets, which generalises the micro-canonical state. The corresponding density operator ˆ D of S then results by tracing out from S its subsystemsS [ n ] with 2 ≤ n ≤ N . Such a program, which presentsdi ffi culties when the observables ˆ O p do not commute,22as been achieved in Ref. [29] . It provides for ˆ D theexponential of a weighted sum of the observables ˆ O p ,the same result as the outcome of the maximisation ofvon Neumann’s entropy under constraints on h ˆ O p i . Theprinciple 3 of subsection 3.2 may therefore be replacedby Laplace’s indi ff erence principle, used in connectionwith the equivalence between q-expectation values andensemble averages (principle 1 of Subsec. 2.2), andwith unitary invariance. References [1] W. Heisenberg, Über quantentheoretische Umdeutung kinema-tischer und mechanischer Beziehungen, Zeitschrift für Physik33 (1925) 879–893.[2] J. A. Wheeler, W. H. Zurek, Quantum theory and measurement,Princeton University Press, 2014.[3] O. Alter, Y. Yamamoto, Quantum measurement of a single sys-tem, Wiley, New York, 2001.[4] V. B. Braginsky and F. Y. Khalili, Quantum measurement, Cam-bridge University Press, 1995.[5] D. Home, M. A. 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