Generalized Observables and the Measurement Process in the ESR Model Sandro Sozzo Dipartimento di Fisica Università del Salento INFN – Sezione di Lecce.

Generalized Observables and the Measurement Process in the ESR Model

Sandro Sozzo

Dipartimento di Fisica

Università del Salento

INFN – Sezione di Lecce

NONLINEAR PHYSICS. THEORY AND EXPERIMENT. V

Gallipoli, June 12-21, 2008

Nonobjectivity of properties in standard QM

After one century since its birth, the conceptual and mathematical foundations of nonrelativistic QM are still subject of research work.

The conceptual problems of QM are mainly connected with the standard, or Copenhagen, interpretation, i.e., the interpretation of QM that arose from the discussions of the founding fathers, and that is actually accepted by the majority of scientific community and expounded in all manuals of QM.

Nonobjectivity of physical properties. A property of a physical system cannot be thought of as possessed or not possessed by the individual samples of the system if a measurement of that property is not performed.

Logical end epistemological problems arising from nonobjectivity.

Nonobjectivity is incompatible with realism, causality, locality.

The quantum-classical transition.

A consistent quantum theory for gravity has not been worked out.

The measurement problem in standard QM

Nonobjectivity is the deep root of the so-called objectification problem, the main and unsolved problem of the quantum theory of measurement.

Laws of evolution for a physical system in standard QM.

(i) Deterministic, linear, unitary, reversible evolution described by the Schrödinger equation.

(ii) Stochastic, nonlinear, nonunitary, irreversible evolution described by the projection postulate (the Lüders postulate).

Attempts to conciliate the two descriptions within standard QM.

1. Unitary evolution of the compound system made up of the microscopic system that is measured plus the macroscopic apparatus.

2. Formal justification of the projection postulate when the subsystems are considered separately.

3. The properties of the apparatus are nonobjective, which contradicts the observative data (objectification problem).

4. Von Neumann’s chain and ensuing quantum paradoxes (Schrödinger’s cat, Wigner’s friend, etc.).

Quantum Mechanics

Minimal Interpretation

Relative frequency of measurement outcomes

Statistical Interpretation

Only measurement outcomes

Objectification problem excluded

Referent ?

Realistic Interpretation

Properties of individual systems

OBJECTIVITY VS COMPLETENESS

Completeness

NonobjectivityIncompleteness

Hidden variables with quantum probabilities as absolute

All properties objective

Objectification problem excluded

OBJECTIFICATION VS UNIVERSALITY

Universal Validity

(a) Without objectification

(b) Aiming at objectification

Limited Validity

Objectification

Incompleteness

Hidden variables with quantum probabilities as conditional

ESR model

All properties objective

Objectification problem excluded

No-go theorems

Objectivity and “no-go arguments” Should an objective interpretation of the mathematical formalism of standard QM be possible, the objectification problem would be avoided.

Every attempt at recovering objectivity must be based on a preliminary criticism of these arguments.

The idea of recovering objectivity by means of a reinterpretation of the mathematical apparatus of standard QM seems at first unsound because of the huge number of “no-go arguments” which seem to show that nonobjectivity is an inherent feature of the quantum formalism. In particular,

The standard interpretation of the two-slit and similar experiments.

The no-hidden variables theorems, e.g.,

(i) Bell-Kochen-Specker theorem. Standard QM is contextual, i.e., the value of an observable for an individual sample of a physical system in a given state is not assigned a priori but is actualized in a measurement and can be different if a different measurement context is considered.

(ii) Bell theorem. Standard QM is nonlocal, i.e., contextuality holds also at a distance, since a measurement on a particle 1 may instantaneously influence the properties of a particle 2, even if 1 and 2 are far away, if 1 and 2 have interacted in the past.

The SR interpretation of QM

The research group of Lecce has inquired into the proofs of the impossibility of reconciling standard QM with objectivity. It has been shown that they can actually be criticized. In particular, the no-go theorems require accepting an epistemological principle (MCP) that does not fit in well with the operational philosophy of QM. If MCP is replaced by a weaker principle (MGP) the proofs cannot be given. On this basis a new interpretation of the mathematical formalism of standard QM has been provided which assumes objectivity of properties and preserves all quantum laws and predictions, but those following from MCP (Semantic Realism, or SR, interpretation). In this interpretation QM is semantically incomplete.

The SR interpretation is mainly based on epistemological arguments. In order to show its consistency, an extended SR (ESR) set-theoretical model has been worked out in which objectivity holds from the very beginning, and incorporates the mathematical formalism of standard QM together with its rules for calculating probabilities, but introduces a new interpretation of the latter instead of substituting MCP with MGP.

The ESR model: basic notions I

New theoretical entities.

• Set E of microscopic properties. E characterizes and is such that, for every

physical object x, every fE either is possessed or it is not possessed by x, independently of any measurement procedure (objectivity of microscopic properties).

• The set of microscopic properties possessed by x defines a microscopic state Si.

Standard primitive and derived notions.

• Physical system Ω.

• Set S of (pure) states of Ω. Each state is operationally defined as a class of physically equivalent preparing devices.

• Physical object. A physical object is an individual sample x of Ω; we say that “x is in the state S” if the device π preparing x belongs to the equivalence class S.

The ESR model: basic notions II

New observative entities.

• Generalized observables. Every generalized observable A0 is obtained by considering an observable A of standard QM with spectrum Ξ and adding a further outcome a0 (no-registration outcome of A0) that does not belong to Ξ, so that the spectrum Ξ0 of A0 is given by Ξ0= Ξ {a0}.

• Set F0 of all macroscopic properties of Ω.

F0 = { (A0, Δ), A0 generalized observable, Δ Borel set on R }.

• Set F F0 of all macroscopic properties associated with observables of standard QM.

F = { (A0, Δ), A0 generalized observable, a0 Δ }.

Ax. 1. A bijective mapping φ: E → F F0 exists.

F = φ(f),

E F

F0

φ

φ-1

fF=(A0,Δ)

F'=(A0,Δ{a0})

F0=(A0,{a0})

g G=(B0,Г)

a0 Δ

G = φ(g)

A measurement scheme I

Whenever a physical object x is prepared in a given state S and the generalized observable A0 is measured on it, the set of microscopic properties possessed by x induces a probability that the apparatus does not react, so that the outcome a0 may be obtained. In this case, x is not detected and we cannot get any explicit information about the microscopic properties possessed by x. If, on the contrary, the apparatus reacts, an outcome different from a0, say a, is obtained if and only if x possesses the microscopic property f=φ-1((A0 ,{a})) (hence if and only if x possesses all microscopic properties associated, via φ-1, with macroscopic properties of the form F=(A0, Δ), where a Δ, a0 Δ).

A measurement provides information on microscopic properties. Conversely, the microscopic properties determine the probability of a result, which therefore does not depend on features of the measuring apparatus nor is influenced by the environment. In this sense the measurements considered here are “perfectly efficient”, or idealized.

A measurement scheme II

Suppose that a set S of physical objects in the state S is prepared. S can be

partitioned into subsets S1, S2, …, Sn such that in each subset all objects

possess the same microscopic properties. We briefly say that the objects in Si

are in some microscopic state Si. Hence, in general, every state S can be associated with a (not necessarily finite) family of microscopic states S1, S2, …, where each Si is characterized by the set of microscopic properties that are possessed by any physical object in Si.

Let us consider a physical object x in the microstate Si, and suppose that an idealized measurement of a macroscopic property F=(A0,Δ), a0Δ, is performed on x. Whenever x is detected, it turns out to possess F iff it possesses the microscopic property f=φ-1(F) (i.e., f is one of the microscopic properties characterizing Si).

A measurement scheme III

)()()( ,, FpFpFp iS

diS

tiS

Joint probability that x in the state Si be detected and turn out to possess F.

Probability that x in the state Si be detected when F is measured on it.

Conditional probability (either 0 or 1) that x in the state Si turn out to possess F when detected.

i

tiS

itS FpSSpFp )()|()( ,

i

diS

idS FpSSpFp )()|()( ,

Probability that x in the state S be detected.

Joint probability that x in the state S be detected and turn out to possess F.

i

diS

ii

tiS

i

S FpSSp

FpSSpFp

)()|(

)()|()(

,

,

Conditional probability that x in the macroscopic state S be in the microscopic state Si.

The fundamental equation of the ESR model

)()()( FpFpFp SdS

tS

Ax. 2. The probability pS(F) can be evaluated by using the same rules that yield the probability of F in the state S in standard QM.

The detection probability pSd(F). Since we deal with idealized measurements, the

occurrence of the outcome a0 is attributed to the set of microscopic properties possessed by x, which determines the probability pS

i,d(F). Furthemore, the conditional probability p(Si|S) depends only on S. Hence also pS

d(F) depends only on the microscopic properties of x. The ESR model does not provide a general mathematical treatment of pS

d(F), but some predictions on it can be obtained.

The probability pS(F). It can be interpreted as the conditional probability that a physical object x in the state S turn out to possess the macroscopic property F when it is detected.

The main assumption of the ESR model.

Physical implications of Ax. 2

Ax. 2 implies a new interpretation of the probabilities following from standard quantum rules, which are indeed interpreted as conditional instead of absolute.

All standard quantum rules for evaluating probabilities are preserved, hence the ESR model provides a general framework in which the mathematical formalism of standard QM is embodied. In particular, states and macroscopic properties in F are still represented by trace class operators and orthogonal projection operators, respectively. Mathematical representations of microscopic states and properties, generalized observables and macroscopic properties of the form (A0, Δ), a0 Δ, are still lacking.

Microscopic states can be seen as possible values of a hidden variable, and the probability pS

i,t (F) is completely determined by the value Si of the hidden variable. The ESR model can be classified as a hidden variables theory with reinterpretation of quantum probabilities. Moreover, the ESR model preserves a stochastic form of local realism.

Because of Ax. 3, a parameter μ exists which determines, together with Si, whether the physical object x is detected whenever the property F is measured on it, i.e., whether the outcome a0 occurs or not. Since Si determines all macroscopic properties of x whenever x is detected, all macroscopic properties are determined by the pair (μ, Si), hence they are objective.

Objectivity of macroscopic properties

Macroscopic objectivity, hence local realism, is thus recovered in the ESR model.

Ax. 3. For every microscopic state Si, the probability pSi,d (F) admits an epistemic

interpretation in terms of further unknown features of the physical objects in the state Si.

Objectivity of macroscopic properties cannot be deduced from Axs. 1 and 2, but we can introduce a further assumption which implies it.

The above result is highly unconventional, since it shows that the mathematical formalism of standard QM can be embodied in an objective framework, at variance with well established beliefs (of course, this is possible because of the reinterpretation of quantum probabilities). Besides implying local realism, it entails that the objectification problem disappears together with a number of paradoxes in the ESR model.

The predictions of the ESR model

Predictions concerning the subensemble of physical objects that are detected by the measurements. They are obtained by using the quantum formalism (Ax. 2), hence formally coincide with the predictions of standard QM, but standard QM would interpret them as referring to the ensemble of all objects that are produced. One expects that these predictions are matched by experimental data whenever idealized measurements are performed and only detected physical objects are considered.

Predictions concerning the ensemble of all objects in S. Here the detection probability pS

d(F) plays an essential role, hence these predictions are mostly qualitative because of the lack of a general theory for pS

d(F). Notwithstanding this, one can also obtain some quantitative predictions by imposing some consistency conditions for the validity of the ESR model.

Let a set of idealized measurements be performed on an ensemble of physical objects in a state S. It follows from the general features of the ESR model (in particular, from Ax. 2) that the predictions of this model can be partitioned in two classes.

A mathematical representation of generalized observables

Let A0 be a generalized observable, obtained from the observable A of standard QM represented by the self-adjoint operator Â, and assume that A0

has a discrete spectrum Ξ0= {a0} {a1, a2,…}. Let us study the special case in which the detection probability depends on A0 but not on the outcome an.

Let P1Â , P2

Â, … be the (orthogonal) projection operators associated by the spectral decomposition of  to a1, a2, …, respectively.

Let Ω be a physical system associated with the (separable) Hilbert space H , and let x be a sample of it.

)(1)(

)()()(

00

0

Apap

apApapdS

tS

nSdSn

tS

Generalized measurement operators

We introduce, for every generalized observable A0 and unit vector |ψH, a set

MψA0={Mψk}kN0 of generalized measurement

operators. ...,)ˆ(,...

)ˆ(,)ˆ(

)ˆ(1

ˆ

ˆ

22

ˆ

11

0

An

dn

AdAd

d

PApM

PApMPApM

IApM

pψd(Â) denotes the detection

probability of the generalized observable A0 in the state S.

All the operators in MψA0 are linear, bounded, self-adjoint and positive. Moreover,

MψA0 is complete and commutative, i.e.,

and, for every n, mN,

Nn

nnkNk

k IMMMMMM )()()()()( †0

†0

†

0

0],[],[ 0 nmn MMMM

Because of the completeness relation, the probability of getting the outcome ak, kN0, when measuring A0 on a physical object x in the state S represented by the unit vector |ψ is given by

Generalized observables as families of POV measures

The properties of the generalized measurement operators suggest one to represent the generalized observable A0 by means of the family of positive operator valued (POV) measures

kkk

NkkkNkk

MMaE

MMaE

†

†

)()(

}}){(}{:{00

)(M)(M)(ap k†

kkA

ktS ap

|ψH, ψ|ψ=1

Generalized projection postulate

GPP. Let an idealized measurement of the generalized observable A0 on a physical object x in the state S, represented by the unit vector |ψ, yield outcome ak, kN0. Then, the final state Sk of x immediately after the measurement is represented by the unit vector

k†

k

kM)(M

kM

Whenever an outcome different from a0, say an, is obtained, the final state of x coincides with the state predicted by using the standard projection postulate which thus continues to hold whenever only detected physical objects are taken into account.

Whenever the outcome a0 is obtained, the state of x is not changed.

Nonselective measurements

Let an idealized measurement of the generalized observable A0 on a physical object x in the state S, represented by the unit vector |ψ, yield outcome ak, kN0, but assume that we do not know the actual outcome (nonselective measurement). Then, the final state S' of x immediately after the measurement is a mixed state represented by the density operator

Nn

nnkNk

k MMMMMMW 00

0

'

Nn

nnndd aacApApW

2)ˆ()]ˆ(1['

Nn

nn ac

If the spectrum of the observable A of standard QM from which A0 is obtained is nondegenerate, and |a1, |a2, …, are the eigenvectors of the self-adjoint operator  that represents A corresponding to the eigenvalues a1, a2, …, respectively, then

Classification of measurements in the ESR model

The representation of generalized observables by means of sets of POV measures in the ESR model defines a subclass of idealized measurements.

Suppose that a first measurement of the generalized observable A0 is performed on a physical object x in the state S and then repeated on x in the final state. If the first measurement yields outcome an a0, the second could yield either an or a0, but it can never yield am, 0 m n (generalized measurement of the first kind).

The outcome of the measurement determines the final state of the physical object x (generalized ideal measurement).

The idealized measurements of the ESR model are generalized ideal measurements of the first kind.

The unsharp extension of standard QM

In the unsharp extension of standard QM (unsharp) observables are represented by POV measures which reduce to projection valued (or PV) measures in the case of sharp observables.

Unsharp observables are usually introduced in order to take into account:

what actually occurs in concrete measurements where the inefficiencies of measuring apparatuses make measurements nonideal. In this case, unsharp observables can be interpreted as smeared versions of sharp observables (such an interpretation is consistent only for commutative POV measures);

the intrinsic “quantum noise” inherent in the measurement outcomes and revealing an underlying “unsharpness”, or indeterminacy, of nature.

ESR model versus unsharp QM

The ESR model cannot be embodied into the framework of unsharp QM because of both conceptual and theoretical reasons.

1. The occurrence of the no-registration outcome depends on intrinsic features of the physical object that is considered. These features can be seen as a consequence of the objectivity of properties and neither depend on the measuring apparatus nor have an unsharp source.

2. The mathematical representation of generalized observables introduces a dependence on the state of the physical object that is considered which does not appear in unsharp QM, hence some predictions of the ESR model cannot be obtained in the framework of unsharp QM.

A unified treatment of the measurement process

Let Ωm be a microscopic physical system associated with the Hilbert space Hm, x an individual sample of Ωm prepared in the pure state S represented by the unit vector |ψ, and let A0 be a generalized observable that is measured on x with discrete and nondegenerate spectrum Ξ0= {a0} {a1, a2,…}. Assume that A0 is associated in S with

the set MψA0 of generalized measurement operators.

(i) Schematize A0 by means of a macroscopic apparatus, sample of the physical system ΩM

associated with the Hilbert space HM.

(ii) |1, |2,…: unit vectors of HM representing the macroscopic states of the apparatus and corresponding to the eigenvectors |a1, |a2,…, respectively.

(iii) Unit vector |0. It represents the macroscopic state of the apparatus when it is ready to perform a measurement or when x is not detected.

(iv) Suppose that {|0, |1, |2, …} is an orthonormal basis in HM.

(v) Consider the compound system made up of Ωm plus ΩM.

Unitary evolution of the compound system

,)ˆ(1

,)ˆ(

0 id

id

eAp

eAp

0)(0 *0*0 macnacW

mmm

nnnf

000

0)(

20**

*0

,

2

mac

nacmnaac

mm

m

nn

nmn

mnn

Assume that the time evolution of the compound system is unitary, as follows.

000 0),( 0

n

nnttUn

nn nacac

1202 The density operator associated

with the state of the compound system after the interaction is

Recovering the generalized projection postulate

The final state of the measured object x can be represented by the density operator obtained by performing the partial trace of Wf with respect to HM, as follows.

nnn

ndd

NkffMm aacApApkWkWTrW

2' )ˆ()]ˆ(1[0

The generalized projection postulate has been justified for the measured object on the basis of the unitary evolution of the compound system made up of the measured object plus the measuring apparatus.

The justification is complete since the two descriptions coincide also from an interpretative viewpoint. Indeed, because of objectivity of macroscopic properties, all probabilities of both Ωm and ΩM are epistemic.

The objectification problem disappears and the above description opens the way towards a consistent quantum theory of measurement.

Conclusions

The ESR model recovers in an objective framework all probability values predicted by standard QM, but interprets them as referring only to samples of the physical system which are detected (conditional probabilities). When referring to the set of all samples of the physical system (absolute probabilities) the ESR model provides predictions that are different from those of standard QM.

The generalized observables of the model can be represented by sets of POV measures labeled by the pure states of the physical system that is considered. Moreover, the standard projection postulate must be replaced by a generalized projection postulate.

The measurement process can be described in terms of an interaction between the measured system and the measuring apparatus. By assuming a unitary evolution of the compound system, one can derive the generalized projection postulate for the measured system. The standard projection postulate is an approximate evolution law valid only for detected objects, hence FAPP.

The ESR model brings into every measurement a no-registration outcome which is interpreted as providing information about the measured system.