A Review of Unitary Quantum Premeasurement Theory An Algebraic Study of Basic Kinds of Premeasurements
aa r X i v : . [ qu a n t - ph ] D ec A Review of Unitary Quantum Premeasurement TheoryAn Algebraic Study of Basic Kinds of Premeasurements
Fedor Herbut
Serbian Academy of Sciences and Arts, Knez Mihajlova 35, 11000 Belgrade, Serbia ∗ (Dated: February 23, 2018)A detailed theory of quantum premeasurement dynamics is presented in which a unitarycomposite-system operator that contains the relevant object-measuring-instrument interactionbrings about the final premeasurement state. It does not include collapse, and it does not con-sider the environment. It is assumed that a discrete degenerate or non-degenerate observable ismeasured. Premeasurement is defined by the calibration condition, which requires that every ini-tially statistically sharp value of the measured observable has to be detected with statistical certaintyby the measuring instrument. The entire theory is derived as a logical consequence of this definitionusing the standard quantum formalism. The study has a comprehensive coverage, hence the articleis actually a topical review. Connection is made with results of other authors, particularly withbasic works on premeasurement. The article is a conceptual review, not a historical one.General exact premeasurement is defined in 7 equivalent ways. A step is taken towards com-plete measurement (that includes collapse). Nondemolition premeasurement, defined by requiringpreservation of any sharp value of the measured observable, is characterized in 10 equivalent ways.Overmeasurement, i. e., a process in which the observable is measured on account of being afunction of a finer observable that is actually measured, is discussed. Disentangled premeasure-ment, in which, by definition, to each result corresponds only one pointer-observable state in thefinal composite-system state, is investigated. Ideal premeasurement, a special case of both non-demolition premeasurement and disentangled premeasurement, is defined, and its most importantproperties are discussed. Finally, disentangled and entangled premeasurements, in conjunction withnondemolition or demolition premeasurements, are used for classification of all premeasurements.In concluding remarks the omitted aspects of the intricate topic of quantum measurement theoryare shortly enumerated. PACS numbers: 03.65-w, 03.65 Ca, 03.65 Ta
CONTENTS
I. Introduction: Glance at History; Large InformationalGap; Balance and Dirac Notation; Basic Tools; BasicAim; ConventionsII. General PremeasurementIII. Towards Complete MeasurementIV. Nondemolition PremeasurementV. Functions of the Masured Observable; Minimal Pre-measurement and OvermeasurementVI. Disentangled PremeasurementsVII. Ideal PremeasurementsVIII. Subsystem Measurement and Distant MeasurementIX. ClassificationX. Summing Up the Equivalent DefinitionsXI. Concluding RemarksApp Proof of an auxiliary algebraic certainty claim
I. INTRODUCTION
To cast a quick glance at history , one can say thatthe quantum-mechanical theory of premeasurement be-gan in the last chapter of von Neumann’s celebrated book(von Neumann, 1955). Since then numerous authors gave ∗ Electronic address: [email protected] contributions, but none as many, so it seems to me, asPeka J. Lahti with various coauthors and by himself. (Seehis review Lahti, 1993. Some of his articles and those ofothers will be cited when they overlap with the claimsmade and proved in the present article.) The effortsof Lahti and coauthors seem to have culminated in thebook by Busch, Lahti, and Mittelstaedt, 1996, which, nodoubt, completed the mentioned work of von Neumannin a masterful fashion.Nevertheless, a large informational gap was left inthe wake of this awesome work. The gap is betweenits highbrow mathematical language and wide general-ity adopted in the book on the one hand, and the con-cepts and symbols that are standard in most quantum-mechanical textbooks and articles on the other.Besides, there is the complementarity between thewhole and the part well-known even in everyday experi-ence: viewing the wood, you don’t see the trees; watchingthe trees, the wood is out of sight. Comparing Busch,Lahti, and Mittelstaedt 1996 to a magnificent ’wood’,the ’trees’ would be the generalized observables (so-calledpositive-operator valued or POV measures; see conclud-ing remark IV in section X), and the ordinary observablescould be likened to ’bushes’.If I am allowed to follow this metaphor a step further,I think that most of us quantum physicists ’live’ in the’bushes’. I think that a balanced approach between thegeneral and the special is desirable, in which one confinesoneself to premeasurement of ordinary observableswith purely discrete spectra and with arbitrarily de-generate or non-degenerate eigenvalues. As to the formal-ism, it is good to stick to the widely used practical
Diracnotation . It follows von Neumann’s representation-independent (or abstract) treatment, which is best suitedfor fundamental quantum-mechanical investigations. Be-sides, unitary evolution operators are made use of. Theseare the confines of the present topical review, which iswritten mostly in the way of a research article becausealmost everything that is claimed is derived from the ba-sic simple definition of premeasurement.Measurement theory investigates composite systemsconsisting of the measured object, we will denote it assubsystem A, and the measuring instrument, subsystemB. The basic tool for the treatment of subsystems isthe use of partial traces . These have three very prac-tical accompanying rules (much utilized in the presentarticle). tr B (cid:16) Y A X AB (cid:17) = Y A tr B X AB ≡ Z (1) A , (1 a )tr B (cid:16) X AB Y A (cid:17) = (cid:16) tr B X AB (cid:17) Y A ≡ Z (2) A , (1 b )where Y A (cid:16) = Y A ⊗ I B (cid:17) ( I B being the identity op-erator for subsystem B) and X AB are arbitrary sub-system and composite-system operators respectively, andthe operators Z ( i ) A i = 1 , B (cid:16) Y B X AB (cid:17) = tr B (cid:16) X AB Y B (cid:17) ≡ Z (3) A . (1 c )We will refer to (1c) as to ’under-the-partial-tracecommutativity’ .These general rules are easily proved by straightfor-ward evaluation of both sides in an arbitrary pair of com-plete orthonormal bases {| k i A : ∀ k } , {| n i B : ∀ n } .The use of partial traces with the rules (1a-c) is widerthan measurement theory; it can be applied in anyquantum-mechanical treatment of composite systems. Literature
To my knowledge, the partial trace wasintroduced by von Neumann (1955, section 2 of ChapterVI).
The basic aim of this article is to present generalpremeasurement and its important special case nonde-molition premeasurement in as many details as possible.The above explained balance has enabled the presentauthor to derive 7 equivalent definitions for generalpremeasurement, and 10 equivalent definitions for non-demolition premeasurement. The rest is a conceptual framework.In premeasurement the separate results are notselected out, which would require theoretically somekind of collapse.
Literature
In Busch, Lahti, and Mittelstaedt 1996the term ”premeasurement” is used. Complete measure-ment is called ”premeasurement” with ”objectification”.The latter term is a synonym of ”collapse”.
Complete measurement , which by definition endsin one definite result, consists of two parts: In the first,the measured object interacts in a specific way with themeasuring instrument, and specific quantum correlationsare created. In the second part, the correlations are ’read’as information about the object on part of the measuringinstrument, which thus becomes a ’subject’ (it is here a’technical’ term). In the second part, as we know fromexperience, collapse to one definite result takes place. Inthe present article only the first part is investigated byapplying unitary evolution operators , and by treat-ing the measuring instruments as quantum-mechanicalsystems.One should keep in mind that unitary quantum me-chanics, opponents call it sometimes ”bare quantum me-chanics”, is actually the textbook quantum mechanics, inwhich all dynamical changes are expressed via the famousSchr¨odinger equation.Collapse has been for a long time paradoxical from thepoint of view of unitary dynamics, because it cannot bederived from the latter in a consistent and expected way(von Neumann, 1955, last section VI.3). It can be donein an unexpected way called relative-state or Everettiantheory Everett 1957, Everett 1973 (cf remark II in theConcluding remarks).By ”measurement” one usually means completemeasurement in the literature. This term will be usedin the present article when one has both or either ofpremeasurement and complete measurement in mind.The present article will use the following conven-tions . Some physical notions and their mathematicalrepresentatives will be used interchangeably throughout,like ’pure state’ and ’state vector’ (vector of norm one);’state’, ’density operator’, and ’state operator’; ’observ-able’ and ’Hermitian operator’; ’compatible’ observablesand ’commuting’ Hermitian operators’; event’ and ’pro-jector’. Complete orthonormal bases in a given statespace will be called simply ’bases’. Vectors of normthat is not necessarily one will be written overlined; non-overlined vectors will always be of norm one.Tensor products of vectors will mostly be writ-ten without a multiplication sign, but sometimes,for emphasis, such a sign will be utilized, e. g., | φ i i A | φ i i B = | φ i i A ⊗ | φ i i B (as will be used below).Whenever possible, strings of symbols will begin bynumbers that will be followed by the sign ” × ”. We usethe convention that if in a string of entities the firstis zero, the rest need not be defined; the string is bydefinition zero.It should be noted that a ”necessary and sufficientcondition” is another definition. Occasionally also theterms ”characterization”, ”characteristic properties”,”criterion”, and ”condition” will be used as furthersynonyms of ”definition”.In order to avoid overburdening the article with math-ematics, mathematical presentation in terms of lemmata,theorems etc. will be avoided and boldface ”claims”,strictly identified via the numbers of the correspondingrelations, will be used instead. This seems more appro-priate for a physical text. Besides, it is easier for thereader to look up a given relation than to find a lemmaor a theorem etc. because the former are all consecutivelyenumerated, whereas in the latter case the lemmata, the-orems etc are usually each separately consecutively enu-merated.Both the proofs and the remarks on literature arewritten in italics to enable the reader to skip them easilyin a first reading. II. GENERAL PREMEASUREMENT
Let subsystem A be the object of measurement, andlet O A = X k o k E kA , k = k ′ ⇒ o k = o k ′ (2 a )be the unique spectral form of the measured discreteobservable, which may have an infinite purely discretespectrum and each eigenvalue may have an arbitrary (fi-nite or infinite) degeneracy or lack of it. By ’uniqueness’is meant the non-repetition of the eigenvalues { o k : ∀ k } in (2a). Henceforth, we always mean by ’spectral form’the unique one unless otherwise stated. The symbol ” ⇒ ”denotes logical implication.Also the completeness relation P k E kA = I A is satis-fied.Let, further, subsystem B be the measuring instrumentequipped with a so-called pointer observable P B = X k p k F kB , (2 b )in its spectral form. Also (2b) is accompanied by thecorresponding completeness relation P k F kB = I B .Theeigen-projectors F kB of the pointer observable will becalled ’ pointer positions ’ in the present article . (Theterm usually applies rather to the eigenvalues p k inthe literature, but they will be seen to play an inferiorrole.) Literature
We make O A and P B the basic’players’ in the theory following von Neumann; cf von Neumann 1955, end of the third page of section 3. ofchapter VI. The premeasurement interaction establishes entangle-ment between object and measuring instrument in a spe-cific way in the final composite premeasurement state.It is investigated in what follows. The unitary oper-ator incorporating the premeasurement interaction and mapping the initial composite-system state vector | φ i i A ⊗ | φ i i B into the final state (at the end of premea-surement interaction) we denote by U AB : ∀ | φ i i A ∈ H A : | Φ i f AB ≡ U AB (cid:16) | φ i i A | φ i i B (cid:17) . (3)Here | φ i i A is an arbitrary initial state vector of themeasured system A , H A is the state space (complexseparable Hilbert space) of the object, and | φ i i B is the initial or ready-to-measure state vector of theinstrument. The superscripts refer to the initial and thefinal state respectively.The three-tuple ( | φ i i B , P B , U AB ) is sometimescalled the ” measuringinstrument ” in the formalismof unitary measurement theory.The degeneracies (or multiplicities) of the eigenvalues o k of the measured observable may be arbitrary. Alsothe ranges R ( F kB ) of the ’pointer positions’ F kB may be degenerate with any degeneracy including thepossibility of all being non-degenerate, when one canwrite P B = P k p k | k i B h k | B (cf (2b)). Literature
For the last mentioned entirely non-degenerate choice of the pointer positions, Busch, Lahti,and Mittelstaedt (1996) say that ”it is minimal in thesense that it is just sufficient to distinguish the eigen-values of” the measured observable (2a) (cf subsectionIII.2.3 in their book, where a different notation is used).
Premeasurement is defined by the so-called cali-bration condition : If the initial state of the objecthas a sharp (or definite) value of the measured observ-able, then the final composite-system state has the cor-responding sharp value of the pointer observable . ’Cor-responding’ we write as ’having the same index value’ k .It is firmly established that the quantum-mechanicalrelations have a statistical meaning and are tested on en-sembles of equally prepared systems. In particular, theprecise statistical form of the calibration condition is expressed in terms of the standard probability for-mulae : For any fixed value ¯ k of the index k , onehas h φ | i A E ¯ kA | φ i i A = 1 ⇒ h Φ | f AB F ¯ kB | Φ i f AB = 1 , (4)and the final state | Φ i f AB is given by (3). Equivalentdefinitions of the calibration condition that are more op-erational are derived below.The obvious physical meaning of the calibrationcondition (4) is that a (statistically) sharp value ofthe measured observable must be (statistically) sharplydetected by the measuring instrument. The calibrationcondition is obviously a necessary requirement for pre-measurement. Following Busch, Lahti, and Mittelstaedt1966 for our restricted choice of observable, we takethe calibration condition as the definition, i. e., as anecessary and sufficient condition, for general exactpremeasurement. Literature
The calibration condition is given inBusch, Lahti, and Mittelstaedt 1996, subsection III.2.3.
The recent ontic breakthrough (Pussey, Barrett, andRudolph 2012 etc.; cf Herbut 2014a and Leifer 2014)makes it plausible that the state vector is a property ofthe individual system. Then it is desirable to understandthe calibration condition as a primarily individual-system requirement. This means that it may go be-yond statistical (and ensemblewise) meaning (relations(4)) towards the individual systems in the following sense.If the calibration condition were a purely statistical re-quirement, then it would allow not to be true on someindividual systems in any finite ensemble , say on N’ ofthem. But the relative number N’/N of such exceptionsis required to tend to zero as N, the number of systems inthe ensemble, tends to infinity. If the calibration condi-tion is an individual-system requirement, then it is neverallowed to be violated by any individual system in pre-measurement .Nevertheless, the purely statistical notion (4) forpremeasurement is more suited in view of the factthat all quantum-mechanical formulae have, actually,a statistical meaning. One must also be aware thatthere exist strictly ensemble measurements that cannotbe given individual-system meaning. A well-knownelementary example is two-slit interference. (When theindividual photon hits the second screen, this does nottell much about it. Only an ensemble of photons canreveal interference.) Such measurements are outside thescope of this investigation.To derive an equivalent, more practical, form of (4),we need a useful, general, known, but perhaps not wellknown, auxiliary algebraic certainty claim .An event E is certain, i. e., has probability one, ina pure state | ψ i if and only if the latter is invariantunder the action of E : h ψ | E | ψ i = 1 ⇔ E | ψ i = | ψ i . (5)(The symbol ” ⇔ ” denotes logical implication in bothdirections.) Proof is given, for the reader’s convenience,in the Appendix.Equivalence (5) makes it obvious that the calibrationcondition can be equivalently defined by the more opera- tional invariance form of the calibration condition ,which is: | φ i i A = E ¯ kA | φ i i A ⇒ | Φ i f AB = F ¯ kB | Φ i f AB . (6) Literature
The invariance form (6) of the calibra-tion condition was utilized in Herbut 2014c.
As to the measuring instrument , the following claim is valid. The set of all states | φ i i B that satisfythe calibration condition in premeasurement span a sub-space S i B (cid:16) ≡ { all | φ i i B } (cid:17) in the state space H B ofthe measuring instrument in which every state vector isa ’ready-to-measure’ state, i. e. | φ i i B ∈ S i B , ⇒ | φ i i B ∈ { all | φ i i B } . (7) Proof is a straightforward consequence of the form (6) ofthe calibration condition and the linearity and the continuityof all operations involved. Let | φ i i A = E ¯ kA | φ i i A , and let | φ i i B and | χ i i B be two initial states of the measuringinstrument for which the calibration condition is valid. Then,for any complex numbers α, β satisfying | α | + | β | = 1 ,(6) implies U AB h | φ i i A ⊗ (cid:16) α | φ i i B + β | χ i i B (cid:17)i = αF ¯ kB U AB (cid:16) | φ i i A | φ i i B (cid:17) + βF ¯ kB U AB (cid:16) | φ i i A | χ i i B (cid:17) = F ¯ kB U AB h | φ i i A ⊗ (cid:16) α | φ i i B + β | χ i i B (cid:17)i . Thus, the calibration condition is valid also for ( α | φ i i B + β | χ i i B ) .Analogously, one proves preservation under the limitprocess: Let { ( | φ i i B ) n : n = 1 , , . . . , ∞} be a sequenceof states observing the calibration condition that convergeto | φ i i B , then the calibration condition is preserved underthe limes due to the continuity of both the unitary evolutionoperator and the projector F ¯ kB . The subspace S i B can be one-dimensional. As analternative, one can define each premeasurement witha fixed initial state | φ i i B disregarding the dimensionof S i B . This will be done throughout in this article inorder to avoid overburdening the presentation.Now we state and prove the claim of the dynamicalnecessary and sufficient condition for premeasure-ment.One has premeasurement if and only if ∀ | φ i i A , ∀ k : (cid:16) F kB U AB (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) = (cid:16) U AB E kA (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) (8)is valid. One proves necessity as follows. For each k valuethe completeness relation P k ′ E k ′ A = I A , repeated use of thecalibration condition in its invariance form (6), and finallyorthogonality and idempotency of the F kB projectors enableone to write F kB U AB (cid:16) | φ i i A | φ i i B (cid:17) = X k ′ || E k ′ A | φ i i A || × F kB U AB h(cid:16) E k ′ A | φ i i A . || E k ′ A | φ i i A || (cid:17) | φ i i B i = X k ′ || E k ′ A | φ i i A ||× F kB F k ′ B U AB h(cid:16) E k ′ A | φ i i A . || E k ′ A | φ i i A || (cid:17) | φ i i B i = || E kA | φ i i A || × F kB U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) | φ i i B i = || E kA | φ i i A || × U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) | φ i i B i . After cancellation of || E kA | φ i i A || , the claimed relation(8) follows.Note that the argument covers both the case || E kA | φ i i A || > and || E kA | φ i i A || = 0 due to the convention that, whenthe first factor in a term is zero, then, by definition, the entireterm is zero. Namely, in the latter case, it has just been shownthat if E kA | φ i i A = 0 , then also F kB | Φ i f AB = 0 , i. e., (8)is valid.To prove sufficiency , let (8) be valid, and let | φ i i A = E ¯ kA | φ i i A be satisfied. Then, one has (cid:16) U AB E ¯ kA (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) = (cid:16) F ¯ kB U AB (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) . One can here omit E ¯ kA due to the above assumption, andthus the calibration condition in its invariance form (6) | Φ i f AB = F ¯ kB | Φ i f AB is obtained. Hence, we do havepremeasurement. Literature
The dynamical definition of generalpremeasurement (8) is introduced and proved in Herbut2014c.
In an attempt to comprehend the meaning of thedynamical criterion (8) intuitively , one may realizethat complete measurement, which is beyond thisstudy, would collapse the final premeasurement state | Φ i f AB given by (3) into one of its sharp pointer-position projections F kB | Φ i f AB . || F kB | Φ i f AB || , and E kA | φ i i A . || E kA | φ i i A || is the corresponding collapsedform of the initial state. Then (8) says that it amountsto the same whether the collapse takes place at thebeginning and then evolution sets in or at the end afterthe evolution.Important consequences of the dynamical definition(8) of premeasurement apply to a connection with com-plete measurement. These consequences are discussed inthe next section. To express the claim of the basis-dynamical char-acterization of general premeasurement, we take anarbitrary basis {| k, q k i A : ∀ q k } in each eigen-subspace R ( E kA ) of the measured observable O A (cid:16) = P k o k E kA (cid:17) . Then, the final state (3) is that of a pre-measurement if and only if ∀ k, ∀ q k : U AB (cid:16) | k, q k i A | φ i i B (cid:17) ∈ R ( I A ⊗ F kB ) . (9 a ) Proof of necessity is obvious, and so is that of sufficiency if one has in mind that an initial state | φ i i A has a sharp value o ¯ k of O A if and only if it canbe expanded as | φ i i A = P q ¯ k (cid:16) h ¯ k, q ¯ k | A | φ i i A (cid:17) × | ¯ k, q ¯ k i A (cfthe calibration condition (6)). Since the power of the set {| k, q k i A | φ i i B : ∀ q k } equals the dimension of R ( E kA ) and since dim (cid:16) R ( E kA ) (cid:17) ≤ dim ( H A ) ≤ dim ( H A ) × dim (cid:16) R ( F kB ) (cid:17) ,a unitary operator U AB satisfying the basis-dynamicalcondition can always be constructed. Thus, all discreteobservables are, in principle, exactly measurable. (Atleast there is no algebraic obstacle for this; cf concludingremark VI in section X.) Literature
In the book Busch, Lahti, and Mittel-staedt 1996 this is proved in Theorem 2.3.1 with theassumption of a pointer observable that is discrete andnondegenerate ( ∀ k : dim (cid:16) R ( F kB ) (cid:17) = 1 ). It is obvious that the basis-dynamical condition has itsequivalent form in the subspace-dynamical condition(like the other face of the coin): The final state | Φ i f AB is that of an premeasurement if and only if each eigen-subspace R ( E kA ) of the measured observable is takeninto the corresponding eigen-subspace R ( I A ⊗ F kB ) bythe unitary evolution operator U AB . In terms of asystem of formulae the characterization has the preciseform ∀ k : U AB h R ( E kA ) ⊗R ( | φ i i B h φ | i B ) i ⊆ R ( I A ⊗ F kB ) , (9 b )where by a tensor (or direct) product of subspaces ismeant the span of the set of all tensor products of anelement from the first and an element from the secondfactor space.Our next claim of the probability reproducibilitycondition as a necessary and sufficient condition for premeasurement reads: The probability of a result E kA predicted by any initial state | φ i i A of the object equals the probability of the corresponding pointerposition F kB in the final composite-system state: ∀ | φ i i A , ∀ k : h φ | i A E kA | φ i i A = h Φ | f AB F kB | Φ i f AB . (10)One should note that one way to put the calibrationcondition is to claim that every Kronecker distributionof probability in the sense of LHS(10) equals the corre-sponding Kronecker probability distribution in the senseof RHS(10). Obviously, the probability reproducibilitycondition is valid only if so is the calibration condi-tion. Though the probability reproducibility conditionapparently requires much more than the calibrationcondition, perhaps surprisingly, the former is valid ifand only if so is the latter (both in the statistical sense). Literature
This is a known fact, cf Busch, Lahti,and Mittelstaedt 1996, subsection III.1.2). To prove that the calibration condition implies the prob-ability reproducibility condition, we argue as follows. Onaccount of (8), (3), and the idempotency of the projectors, RHS (10) equals h φ | A h φ | i B ( E kA U − AB )( U AB E kA ) | φ i A | φ i i B = LHS (10) . Intuitively , one may remark that the probabilityreproducibility condition (10) displays the kind of in-formation that is transmitted from object to measuringinstrument when measurement entanglement in the finalstate is established.Our next claim is another necessary and sufficientcondition for premeasurement.
The strong invariance form of the calibrationcondition reads: The calibration condition is valid if andonly if the premeasurement entities { U AB , P B , | φ i i B } lead to a final state | Φ i f AB so that | φ i i A = E ¯ kA | φ i i A ⇔ | Φ i f AB = F ¯ kB | Φ i f AB (11)holds true. Claim (11) follows immediately from the probability re-producibility condition (10) because, according to the latter, h φ | i A E ¯ kA | φ i i A = 1 = h Φ | f AB F ¯ kB | Φ i f AB (cf (4), (5) and (6)). The strong invariance form of the calibration conditionimplies that the calibration condition (6) is essentiallysatisfied also in the time-reversed situation. Hence, thestrong invariance form (11) of the calibration conditionimplies a kind of time reversal symmetry in premea-surement : If we slightly generalize the premeasurementconcept allowing the initial object+instrument system tobe correlated (due to previous interaction that was com-pletely independent of the premeasurement), and we ap-ply time reversal to the premeasurement process, then | Ψ i f AB becomes the initial state, the instrument, sub-system B, is then the measured object, the former pointerobservable P B (cid:16) = P k p k F kB (cid:17) is the measured observ-able, and | φ i i A | φ i i B the final state. The formerly measured observable O A (cid:16) = P k o k E kA (cid:17) is now thepointer observable with its eigen-projectors { E kA : ∀ k } as the ’pointer positions’.Now | φ i i A reproduces the relevant information con-tained in | Ψ i f AB in terms of the probability repro-ducibility condition ∀ | Φ i f AB , ∀ k : h φ | i A E kA | φ i i A = h Ψ | f AB F kB | Ψ i f AB . (12) Literature
Peres gave a similar discussion inPeres 1974.
Finally, we turn to the canonical subsystem-basis expansion criterion for premeasurement. The claim reads: an object+(measuring instrument) state | Φ i AB (cid:16) ≡ U AB ( | φ i i A | φ i i B ) (cid:17) is the final state ofpremeasurement, i. e., | Φ i AB = | Φ i f AB , if , expandedin some eigen-basis {| k, s k i B : ∀ k, s k } of the pointerobservable P B (cid:16) = P k p k F kB (cid:17) , in | Φ i AB = X k X s k | k, s k i A | k, s k i B , (13 a )the ’expansion coefficients’ (elements of H A ) {| k, s k i A : ∀ k, s k } satisfy the following relations: ∀ k : X s k ||| k, s k i A || = h φ | i A E kA | φ i i A . (13 b )One is dealing with the final state of an premeasurement only if relations (13a,b) are valid for every eigenbasis ofthe pointer observable. Proof of the claim follows immediately from the fact that(13a) and (13b) together are an equivalent formulation of theprobability reproducibility condition. Namely, ∀ k : LHS (13 b ) = || F kB | Φ i f AB || = h Φ | f AB F kB | Φ i f AB = RHS (10) . Note that the ’expansion coefficients’ in (13a) are,apart from (13b), arbitrary vectors in H A . This iswhat makes the premeasurement that we investigate inthis section general. Kinds of premeasurement will bespecified in the special cases studied further below. Literature
A somewhat simplified form of theabove characterization of premeasurement in terms ofthe form of the final state was obtained by Lahti (1990,relation (13) there).
III. TOWARDS COMPLETE MEASUREMENT
To begin with, we specify the explicit expanded formof the final premeasurement state (3) in terms of finalcomplete-measurement states F kB | Φ i f AB . || F kB | Φ i AB || : | Φ i f AB = X k ( h φ | i A E kA | φ i i A ) / × (cid:16) F kB | Φ i f AB . || F kB | Φ i f AB || (cid:17) (14)valid for every | φ i i A (cf (3)). Naturally, it followsfrom || F kB | Φ i f AB || = (cid:16) h Φ | f AB F kB | Φ i f AB (cid:17) / and theprobability reproducibility condition (10).Expansion (14) displays a connection between pre-measurement and complete measurement .The concept of complete measurement utilized in (14)is without overmeasurement (cf section V). In otherwords, each value of the measured observable is measured minimally (more in section V). Literature : The notion of minimal measurementwas introduced and elaborated in previous work Herbut1969.
Incidentally, the probability, given by the square of theexpansion coefficient in (14), is seen to be independent of the kind of premeasurement performed. This is aknown (often tacit) textbook claim.To formulate a further important consequence of thedynamical condition (8), attention is called to conceptsrelevant for complete measurement .If h φ | i A E kA | φ i i A > E kA | φ i iA in the expansion | φ i i A = P k ′ E k ′ A | φ i i A as the k -th initial compo-nent (of the object state). Further, we refer to thecorresponding final vector F kB | Φ i fAB in the expnsion | Φ i f AB = P k ′ F k ′ B | Φ i f AB , due to P k ′ F k ′ B = I B ,as the k -th complete-measurement final com-ponent . (Note that both are not of norm one in general.)The initial and final components are closely connected.Namely, the dynamical definition (8) implies that onlythe k -th initial component contributes to the k -th complete-measurement final component in theunitary evolution of premeasurement : ∀ | φ i i A , ∀ k : F kB | Φ i f AB = U AB (cid:16) E kA | φ i i A ⊗ | φ i i B (cid:17) . (15)Relation (15) is actually (8) rewritten.Claim (15) is relevant for complete measurementof the observable O A in which the result o k is obtained because this process ends in the state F kB | Φ i f AB . || F kB | Φ i f AB || or in a normalized projectionof this state by a sub-projector of F kB in case ofovermeasurement.We refer to the initial and the final k -th components together as to the k -th branch (or channel) of pre-measurement. This is a concept that applies to the en-tire premeasurement process, not to any fixed moment t, t i ≤ t ≤ t f in it. What relation (15) ’tells us’ can be put in intuitive terms as follows: The measurement process takesplace within each branch separately , independentlyof the rest of the branches.
IV. NONDEMOLITION PREMEASUREMENTNondemolition (synonyms: predictive, repeatable,first kind) premeasurement is defined as a premea-surement satisfying the additional requirement that,if the measured observable O A (cid:16) = P k o k E kA (cid:17) has a sharp value in the initial state , then, in the final state,the sharp value is preserved (it is not demolished).In the statistical language of quantum-mechans the additional nondemolition requirement reads: h φ | i A E ¯ kA | φ i i A = 1 ⇒ h Φ | f AB E ¯ kA | Φ i f AB = 1 . (16 a )This together with the statistical form of the calibra-tion condition (4) is the extended statistical calibra-tion condition definition of nondemolition premea-surement.Using the more practical form (5) of certainty, the non-demolition additional condition (16a) can be given themore practical equivalent invariance form | φ i i A = E ¯ kA | φ i i A ⇒ | Φ i f AB = E ¯ kA | Φ i f AB . (16 b )If joined to the invariance form of the calibrationcondition (6) of general premeasurement, then we havethe extended invariance form of the definition ofnondemolition premeasurement.We have also the additional strong invariance re-quirement | φ i i A = E ¯ kA | φ i i A ⇔ | Φ i f AB = E ¯ kA | Φ i f AB , (17)which together with (11) is the extended strong in-variance definition of nondemolition premeasurement.The equivalent definitions of nondemolition premea-surement are presented, as far as possible, in an orderparallelling that of the definitions of general premeasure-ment. For this reason, (17) is given here without proof.Proof is supplied below; (17) is a consequence of (20).Since requirement (16b) of nondemolition of the mea-sured eigenvalue of O A , joined to the calibration con-dition, makes the premeasurement evolution operator U AB more specified, it is to be expected that it standsin some additional relation to the eigen-projectors E kA (with respect to that in the dynamical relation (8)). In-deed, so it is. The following additional system of dy-namical necessary and sufficient conditions for ageneral premeasurement to be a nondemolition one isvalid.The claim of the additional dynamical condition goes as follows. A premeasurement is a nondemolitionone if and only if the evolution operator U AB com-mutes with each eigen-projector E kA of the mea-sured observable O A when acting on (cid:16) | φ i i A | φ i i B (cid:17) . Interms of formulae, a premeasurement is a nondemolitionpremeasurement if and only if ∀ | φ i i A , ∀ k : (cid:16) U AB E kA (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) = (cid:16) E kA U AB (cid:17)(cid:16) | φ i i A | φ i i B (cid:17) (18)is satisfied. To prove necessity , let {| k, q k i A : ∀ k, ∀ q k } be aneigenbasis of the measured observable: ∀ k, k ′ , q k , q k ′ : E kA | k ′ , q k ′ i A = δ k,k ′ | k, q k i A , and let | φ i i B observe the calibration condition and the non-demolition requirements (16b) with all pure initial states ofthe object. Then the eigenvalue relations and the nondemoli-tion requirement (16b) imply ∀ k, k ′ , q k ′ : (cid:16) U AB E kA (cid:17)(cid:16) | k ′ , q k ′ i A | φ i i B (cid:17) = δ k,k ′ × U AB (cid:16) | k, q k i A | φ i i B (cid:17) = δ k,k ′ E kA h U AB (cid:16) | k, q k i A | φ i i B (cid:17)i . On the other hand, using (16b) again, one has (cid:16) E kA U AB (cid:17)(cid:16) | k ′ , q k ′ i A | φ i i B (cid:17) = E kA E k ′ A (cid:16) U AB | k ′ , q k ′ i A | φ i i B (cid:17) = δ k,k ′ E kA U AB (cid:16) | k ′ , q k ′ i A | φ i i B (cid:17) . Since the right-hand sides are equal (if k = k ′ , and both arezero otherwise), so are the left-hand sides. Sufficiency is proved in the following way. Let | φ i i A = E ¯ kA | φ i i A and let condition (18) be valid. Then U AB (cid:16) | φ i i A | φ i i B (cid:17) = E ¯ kA U AB (cid:16) | φ i i A | φ i i B (cid:17) . Hence, on account of definition (3), also | Φ i f AB = E ¯ kA | Φ i f AB , i. e., relation (16b) is satisfied. The dynamical condition (8) and (18) together givethe extended dynamical definition of nondemolitionpremeasurement.Next, the claim of the additional basis-dynamicalcondition can be put as follows. A given premeasure-ment is a nondemolition one if and only if, under theassumptions for claim (9a), ∀ k, ∀ q k : U AB (cid:16) | k, q k i A | φ i i B (cid:17) ∈ R ( E kA ⊗ I B ) (19 a )is satisfied. Proof is obvious (cf that of (9a)).
Conditions (9a) and (19a) can be given the form of one condition : ∀ k, ∀ q k : U AB (cid:16) | k, q k i A | φ i i B (cid:17) ∈ R ( E kA ⊗ F kB ) . (19 b )It is the extended basis-dynamical definition of non-demolition premeasurement.Since dim (cid:16) R ( E kA ) (cid:17) ≤ dim (cid:16) R ( E kA ) ⊗ R ( F kB ) (cid:17) , theconstruction of U AB is always possible. Hence, non-demolition premeasurement of any discrete observableis, in principle, possible (as far as the algebra of unitarypremeasurement theory goes, cf concluding remark VIin section X).The ’other face of the coin’ of the one relation express-ing the basis-dynamical condition (19b), is the extendedsubspace-dynamical condition of nondemolition pre-measurement. It is: ∀ k : U AB h R ( E kA ) ⊗ R ( | φ i i B h φ | i B ) i ⊆ R ( E kA ⊗ F kB ) . (19 c )There is another additional system of necessaryand sufficient conditions for a premeasurement tobe a nondemolition one. It is the twin-observablescondition .A premeasurement is a nondemolition one if and onlyif its final state satisfies the conditions ∀ | φ i i A , ∀ k : E kA | Φ i f AB = F kB | Φ i f AB (20)(cf (3)), i. e., if and only if all pairs of events E kA and F kB are so-called twin projectors in it, and hence themeasured observable O A and the pointer observable P B are twin observables in it. Literature
Twin observables were introduced inHerbut and Vujiˇci´c 1976, and elaborated in Herbut 2002.(They are presented in detail in the unpublished reviewHerbut 2014b.)
Proof of Necessity follows from P k E kA = I A ,(3), the calibration condition (6), and the nondemolition con-dition (16b): | Φ i f AB = X k || E kA | φ i i A || × U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) ⊗ | φ i i B i = X k || E kA | φ i i A ||× E kA F kB U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) ⊗ | φ i i B i . The rest is obvious.To prove sufficiency , let | φ i i A = E ¯ kA | φ i i A . Onaccont of (6), one has | Φ i f AB = F ¯ kB | Φ i f AB . Further, (20)implies | Φ i f AB = E ¯ kA | Φ i f AB , i. e., the sharp value of themeasured observable is not demolished ((16b) is valid). One should note that there is no condition for gen-eral premeasurement that would be parallel to the ’twin-observables condition’ (20).Condition (20) has important consequences . Firstof all, it implies the additional strong invariance form ofthe calibration condition (17) in an obvious way (if onehas in mind the strong invariance form of the calibrationcondition (11)). But there is more. A) In the final subsystem states (reduced density op-erators) ρ f A ≡ tr B (cid:16) | Φ i f AB h Φ | f AB (cid:17) and ρ f B ≡ tr A (cid:16) | Φ i f AB h Φ | f AB (cid:17) there is no coherence with respect to the events E kA and F kB respectively: ∀ | φ i i A ∈ H A : ρ f A = X k E kA ρ f A E kA , (21 a ) ∀ | φ i i A : ρ f B = X k F kB ρ f B F kB . (21 b ) B) The following commutations are valid: ∀ | φ i i A , ∀ k :[ E kA , ρ f A ] = 0 , [ F kB , ρ f B ] = 0 . (22 a, b ) Proof of claim A) : Since P k F kB = I B , one has ρ f A = X k,k ′ tr B (cid:16) F kB | Φ i f AB h Φ | f AB F k ′ B (cid:17) . Utilizing under-the-partial-trace commutativity (1c) twice,and orthogonality of the projectors, one further obtains ρ f A = X k,k ′ tr B h(cid:16) F k ′ B F kB (cid:17) | Φ i f AB h Φ | f AB i = X k tr B (cid:16) F kB | Φ i f AB h Φ | f AB F kB (cid:17) . The twin relations (20) and the fact that opposite-subsystemoperators can be taken outside the partial trace (cf (1a,b))enable one to write further ρ f A = X k tr B (cid:16) E kA | Φ i f AB h Φ | f AB E kA (cid:17) = X k E kA h tr B (cid:16) | Φ i f AB h Φ | f AB (cid:17)i E kA = X k E kA ρ f A E kA . One proves symmetrically, exchanging the roles of the twosubsystems and of F kB and E kA , that ρ f B = X k F kB ρ f B F kB . Proof of claim B) . It is straightforward to see that claimsA and B are equivalent. . One should note that the relation ρ f A = P k,k ′ E kA ρ f A E k ′ A is an identity on account of thecompleteness relation P k E kA = I A . Due to the factthat none of the coherence terms E kA ρ f A E k ′ A k = k ′ is non-zero (cf (22a)), one says that there is no co-herence in the final subsystem state ρ f A as far asthe measured observable O A is concerned. So itis symmetrically in ρ B . This means that one isdealing with improper mixtures (D’Espagnat, 1976) ρ f A = P k [tr( ρ f A E kA )] × (cid:16) E kA ρ f A E kA . [tr( ρ f A E kA )] (cid:17) , etc. Literature
The twin relations (20) are charac-terized in Busch and Lahti 1996 , sections 4 and 5,as ”strong correlations”, which are an extremal caseof ”correlations”. The latter are applicable also togeneral premeasurements. They are described in detailin subsections III.3.1-III.3.4 .
Another characterization of nondemolition premea-surement that has no parrallel claim for general pre-measurements is the claim of repeatability . It assertsthat nondemolition complete measurement can be equiv-alently defined by requiring that, for every initial stateof the measured object, immediate repetition of the samecomplete measurement necessarily (with statistical neces-sity) gives the same result: ∀ | φ i i A , ∀ k, h φ | i A E kA | φ i i A > (cid:16) h Φ | f AB F kB (cid:17) E kA (cid:16) F kB | Φ i f AB (cid:17). || F kB | Φ i f AB || = 1 . (23 a )Making use of (5), (23a) can be equivalently written as ∀ | φ i i A , ∀ k, h φ | i A E kA | φ i i A > E kA F kB | Φ i f AB = F kB | Φ i f AB . (23 b ) Proof of necessity
If the premeasurement is a nonde-molition one, then the twin observables condition (20) is valid.Hence ∀ | φ i i A , ∀ k, h φ | i A E kA | φ i i A > E kA F kB | Φ i f AB = E kA E kA | Φ i f AB = F kB | Φ i f AB , i. e. (23b) is satisfied.Proof of sufficiency Let | φ i i A = E ¯ kA | φ i i A be valid.Then, utilizing the calibration condition (6) twice, and makinguse of (23b), we have | Φ i f AB = E ¯ kA F ¯ kB | Φ i f AB = E ¯ kA | Φ i f AB .Thus, the additional nondemolition condition (16b) fornondemolition premeasurement is satisfied. Literature
The ’repeatability condition’ is one ofthe ways how Pauli defined nondemolition measurementin Pauli 1933. additional necessary and suf-ficient condition for nondemolition premeasurement.We shall call it the extended probability repro-ducibility condition . It reads ∀ | φ i i A , ∀ k : h φ | i A E kA | φ i i A = h Φ | f AB E kA | Φ i f AB . (24) Proof of necessity follows from the probability repro-ducibility condition (10): ∀ | φ i i A , ∀ k : h φ | i A E kA | φ i i A = h Φ | f AB F kB | Φ i f AB . Taking into account the validity of the twin-observables con-dition (20), this immediately becomes (24).To prove sufficiency , we take | φ i i A = E ¯ kA | φ i i A .Then (24) implies h Φ | f AB E ¯ kA | Φ i f AB = 1 , or, equivalentlyon account of (5), E ¯ kA | Φ i f AB = | Φ i f AB . Thus, the sharpvalue is preserved, and we have nondemolition premeasure-ment. Literature
Pauli (1933) gave the condition at issue thefollowing catchy physical meaning : ”The initial state andthe final state of premeasurement predict the same probabili-ties for the measurement in question.In the book by Busch, Lahti, and Mittelstaedt (1996),which treats a wider class of measurement processes, the’repeatability condition’ and that of the ’extended probabilityreproducibility condition’ are not equivalent; the formeris a special case of the latter (see subsections III.3.5 andIII.3.6 there). Both in Pauli 1933 and in Busch, Lahti, andMittelstaedt 1996 the latter measurement is called ”of thefirst kind”. And so it is in most of the literature. (The termis abandoned in this review because ”first kind” does notsuggest the essence of the condition.)
Finally we give two canonical-expansion criteria fornondemolition premeasurement. Let us first specify the subsystem-basis canonical expansion one . A) A sufficient additional condition for nondemolition pe-measurement reads: A final state | Φ i f AB of premeasurement(cf (3)) is that of a nondemolition one if there exists an eigen-basis {| k, s k i B : ∀ k, s k } of the pointer observable P B (cf (2b)) such that, when | Φ i f AB is expanded in it, eachnonzero ’expansion coefficient’ (vector in H A ) is an eigen-vector of the measured observable O A (cid:16) = P k o k E kA (cid:17) withthe same k -value: ∀ | φ i i A : | Φ i f AB = X k,s k | k, s k i A | k, s k i B (25 a )implies ∀ k, s k : E kA | k, s k i A = | k, s k i A . (25 b ) B) The final state | Φ i f AB is that of nondemolitionpremeasurement only if (25a,b) is valid for its expansion in every eigen-basis of the pointer observable. To prove necessity , we point out that (25a) is equiv-alent to the partial scalar product ∀ k : | k, s k i A = h k, s k | B | Φ i f AB . Applying E kA to both sides, and utilizing the twin-observables condition (20), we obtain ∀ k : E kA | k, s k i A = (cid:16) h k, s k | B F kB (cid:17) | Φ i f AB = | k, s k i A . (Since ”partial scalar product” is not used often in the liter-ature, the reader may be unfamiliar with it. Perhaps he (orshe) should read Appendix A in Herbut 2014b.)To prove sufficiency , we assume the validity of(25a,b), and we apply E ¯ kA and alternatively F ¯ kB to(25a) for an arbitrary fixed value k = ¯ k . Then thetwin-observables definition (20) follows. Another criterion for a premeasurement to be a nonde-molition one is, what we shall call, the twin-correlatedcanonical Schmidt decomposition criterion. It is in termsof a special kind of a canonical Schmidt or a bi-orthonormaldecomposition with positive numerical expansion coefficientsof the final state | Φ i f AB . A) A premeasurement is a nondemolition one if thereexists a canonical Schmidt decomposition ∀ | φ i i A : | Φ i f AB = X k r / k X s k | k, s k i A | k, s k i B , (26)where {| k, s k i B : ∀ k, ∀ s k } are simultaneous eigen-vectorsof the pointer observable P B (cid:16) = P k p k F kB (cid:17) , i. e., ∀ k : F kB = X s k | k, s k i B h k, s k | B , and the final measuring-instrument state operator (reduceddensity operator) ρ f B h ≡ tr A (cid:16) | Φ i f AB h Φ | f AB (cid:17)i spanningthe range R ( ρ f B ) .Note that symmetrically {| k, s k i A : ∀ k, ∀ s k } are simul-taneous eigen-vectors of the measured observable O A (cid:16) = P k o k E kA (cid:17) and the final object state operator (reduced den-sity operator) ρ f A h ≡ tr B (cid:16) | Φ i f AB h Φ | f AB (cid:17)i spanning therange R ( ρ f A ) .Actually, (26) is a subsystem-basis expansion with respectto the instrument-subsystem sub-basis {| k, s k i B : ∀ k, ∀ s k } (spanning R ( ρ f B ) ). Hence, the above common eigen-sub-basis {| k, s k i A : ∀ k, ∀ s k } for O A and ρ f A of the objectsubsystem is uniquely determined by the state | Φ i f AB .(The former is determined via the antiunitary so-called”correlation operator” U a inherent in the compositestate. Decomposition (26) is called twin-correlated canonicalSchmidt decomposition due to the twin-observables condition (20). More about this special kind of Schmidt canonicalexpansion see in section 5 of Herbut 2014b.) B) A premeasurement is a nondemolition one only if the twin-correlated canonical Schmidt decomposition (26) isvalid for every common eigen-bases of P B and ρ f B thatspan the range of ρ f B . Proof of sufficiency is immediately obtained bynoticing that (26) is a special case of (25a,b). Proof of necessity also follows from this remark and the propertiesof any twin-correlated canonical Schmidt decomposition. (Ifin doubt, consult Herbut 2014b.)
Literature
Twin-correlated canonical Schmidt decom-positions were introduced into quantum mechanics by vonNeumann (1955, see section 2 in chapter VI, in particularpp. 434-436). Von Neumann did not utilize our term.The antiunitary correlation operator was introduced in Her-but and Vujiˇci´c 1976. The canonical Schmidt (or biorthogo-nal) decomposition of a bipartite state vector (with and with-out the explicit correlation operator U a , but without twincorrelation) was reviewed in subsection 2.1 of Herbut 2007a.The twin-correlated canonical Schmidt decompositioncriterion for a kind of premeasurement was investigated alsoby Beltrametti, Cassinelli, and Lahti (1990). They referredto it as to ”strong correlations premeasurement”. Some founders of quantum mechanics and many founda-tionally oriented physicists consider only nondemolition pre-measurement for individual quantum systems. The reasonis, of course, the fact that unless one can check the result ob-tained (repeatability), the individual-system physical mean-ing of the result of premeasurement is doubtful.Nevertheless, nowadays we are witnessing an ever-increasing permeation of physics by information theory (seee. g. Vedral, 2006). General premeasurement, treatedin section II, which includes besides nondemolition alsodemolition premeasurement, transmits information fromsystem to measuring apparatus. In premeasurement this isapparent for individual systems if the initial state has a sharpvalue of the measured observable; otherwise only probabilityis transmitted (the probability reproducibility condition),and it is an ensemble phenomenon. But premeasurementunderlies, in some way, complete measurement, and thelatter transmits individual-system information. Besides,premeasurement theory primarily concerns ensembles. Forthese reasons thorough investigation of premeasurementmore general than the nondemolition one in section II ismade the basis of this review.
V. FUNCTIONS OF THE MEASURED OBSERV-ABLE; MINIMAL PREMEASUREMENT ANDOVERMEASUREMENT
We denote by f ( . . . ) any single-valued real function, i.e., any map of the real axis into itself. It determines a Hermi-tian operator function ¯ O of any given Hermitian operator O A (cid:16) = P k o k E kA (cid:17) of the object subsystem in spectral formas follows:¯ O A ≡ f ( O A ) ≡ X k f ( o k ) E kA = X l o l E lA , (27 a ) where the second spectral form is, by definition, unique, i.e., it is the same as the first spectral decomposition , butrewritten in the unique form. It satisfies l = l ′ ⇒ o l = o l ′ .In general, the first spectral form is not unique.The inverse function f − ( . . . ) is, in general, multi-valued,i. e., its images are sets: ∀ l : f − ( o l ) = { o k : f ( o k ) = o l } .We can omit the eigenvalues and keep only their indices be-cause they are in a one-to-one relation (in the unique spectralform). Hence, one can write ∀ l : l = f ( k ) , l being theindex of o l = f ( o k ) , and ∀ l : f − ( l ) = { k : f ( k ) = l } .Parallelly, one defines a corresponding pointer observ-able ¯ P B ≡ X l p l F lB , (27 b )where ∀ l : F lB ≡ X k ∈ f − ( l ) F kB , (27 c )and { p l : ∀ l } are arbitrary distinct real numbers.Whenever the function f ( . . . ) in (27a-c) is not one-to-one, i. e., whenever this function is singular, the measurementof ¯ O A that is obtained due to a measurement of O A , iscalled overmeasurement of ¯ O A . If O A is a completeobservable, i. e., if all its eigenvalues are non-degenerate,then ¯ O A is said to be overmeasured maximally (or just”measured maximally” cf Herbut, 1969). Literature
In von Neumann 1955 it is mostly assumedthat all observables are overmeasured maximally (cf p. 348not far from the beginning of section 1 of Chapter V).
If an observable is measured so that it is not overmea-sured, then one says that it is measured minimally (a termintroduced in previous work Herbut, 1969).
Literature
It was, actually, L¨uders (1951) who intro-duced minimal premeasurement via ideal premeasurement,though he did not call it so. The way I see it, it was animportant development in unitary measurement theory aftervon Neumann. (Unfortunately, some physicists still laborunder the illusion that,if there is any measurement otherthan maximal, then it is only ideal measurement as L¨udersintroduced it.)
The opposite concept of overmeasurement is undermea-surement . If, in the notation used above, the observable¯ O A is measured minimally, then O A (cf (27a)) is under-measured. The best known example of undermeasurementis that of values in a continuous spectrum. The latter is, asa rule, an interval. One breaks it up into a set-theoreticalsum of smaller (non-overlapping) intervals. One evaluatesthe spectral measures of these subintervals and, utilizingthem, one defines a discrete observable. Its measurementundermeasures the continuous spectrum. As it is well known,the values of a continuous spectrum cannot be exactly mea-sured. (There are no eigen-projectors corresponding to them.) Literature
Von Neumann has claimed that a con-tinuous spectrum is normally measured undermeasuring itby a discrete observable as just described (cf von Neumann1955, chapter III, section 3. p. 220). His term for under-measurement is ”measurement with only limited accuracy”. Regarding the continuous spectrum, see also Ozawa (1984).
Now we state and prove the basic claims on overmea-surement :For every function f ( . . . ) the following claims are valid. A) If | φ i i B is calibration-condition-satisfying for the pre-measurement of O A , then so it is also for that of ¯ O B ≡ f ( O A ) in terms of the pointer observable ¯ P B (cf (27a-c)).Remembering the dynamical definition of general measure-ment, this claim can read: ∀ | φ i i A , ∀ k : F kB U AB (cid:16) | φ i i A | φ i i B (cid:17) = U AB E kA (cid:16) | φ i i A | φ i i B (cid:17) ⇒∀ l (cid:16) ≡ f ( k ) (cid:17) : F lB U AB (cid:16) | φ i i A | φ i i B (cid:17) = U AB E lA (cid:16) | φ i i A | φ i i B (cid:17) . (28 a ) B) If the premeasurement of O A is a nondemolition one, then so is that of its function ¯ O A ≡ f ( O A ) . Havingin mind the definition of nondemolition premeasurement interms of twin observables (20), this additional claim can beput as follows: ∀ | φ i i A , ∀ k : F kB | Φ i f AB = E kA | Φ i f AB ⇒∀ l (cid:16) ≡ f ( k ) (cid:17) , F lB | Φ i f AB = E lA | Φ i f AB . (28 b ) To prove claim A , we argue as follows. On accountof the assumption that the calibration condition (6) is validfor the premeasurement of O A , one has ∀ | φ i i A : | Φ i f AB ≡ U AB ( | φ i i A | φ i i B ) = X k || E kA | φ i i A || × F kB U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) ⊗ | φ i i B i . Hence, ∀ | φ i i A : | Φ i f AB = X k F kB U AB (cid:16) E kA | φ i i A | φ i i B (cid:17) . ( aux ) Further, we rewrite this as two sums, and we take | φ i i A = E ¯ lA | φ i i A . Then, we replace | φ i i A by E ¯lA | φ i i A in thesecond sum, obtaining | Φ i f AB = X k ∈ f − (¯ l ) F kB U AB (cid:16) E kA | φ i i A | φ i i B (cid:17) + X k/ ∈ f − (¯ l ) F kB U AB (cid:16) E kA E ¯lA | φ i i A | φ i i B (cid:17) . Next, we take into account the facts that E ¯ lA E kA = E kA if k ∈ f − (¯ l ) , E kA E ¯ lA = 0 if k / ∈ f − (¯ l ) and cor-respondingly (cf (27a-c)) F ¯ lB F kB = F kB if k ∈ f − (¯ l ) , F ¯ lB F kB = 0 if k / ∈ f − (¯ l ) . We thus obtain | Φ i f AB = F ¯lB X k ∈ f − (¯ l ) F kB U AB (cid:16) E kA | φ i i A | φ i i B (cid:17) = F ¯ lB X k F kB U AB (cid:16) E kA | φ i i A | φ i i B (cid:17) . (We were able to include the terms for k / ∈ f − (¯ l ) becausethey are zero.) Finally, on account of the above general aux-iliary relation (aux), we have in our special case | Φ i f AB = F ¯ lB | Φ i f AB . Therefore, the calibration condition for ¯ O A is valid (cf(6)), and we are dealing with a premeasurement of thisobservable. Proof of claim B
We assume that we have nonde-molition premeasurement of O A , and we utilize the twin-relations definition (20) for this.. Then, on account of (27a-c), ∀ l : F lB | Φ i f AB = (cid:16) X k ∈ f − ( l ) F kB (cid:17) | Φ i f AB = (cid:16) X k ∈ f − ( l ) E kA (cid:17) | Φ i f AB = E lA | Φ i f AB follows for an arbitrary initial state of the object. We seethat the premeasurement is nondemolition also for that of ¯ O A because the twin relations (20) are valid also for thisobservable. VI. DISENTANGLED PREMEASUREMENTS
Now the question of inverting the described functionalrelation ¯ O A ≡ f ( O A ) (cf (27a-c)) arises. For any answerthere is a need for a more precise terminology. We haveseen that the eigenvalues do not play an essential role inpremeasurement; only the eigen-projectors and their indicesdo.Having in mind the relation between the indices k and l (cf (27a-c)), one says, by definition , that ¯ O A is coarser than O A , or that the former is a coarsening of the latter.The same relation can be expressed also by the terms: O A is finer than ¯ O A , or the former is a refinement of thelatter.If a coarsening ¯ O A ≡ f ( O A ) (cf (27a-c)) is mathe-matically given, and one measures the coarser observable¯ O A minimally, one may wonder if this means anything forthe finer observable O A . Clearly, the answer is: one hasan undermeasurement of O A . This would belong toapproximate measurement theory, which is outside the scopeof this article.It may happen that a given premeasurement of anobservable ¯ O A is actually a premeasurement of one ofits refinements, i. e., that ¯ O A = f ( O A ) is valid, andthat ¯ O A is overmeasured. If this is the case, then, asit was explained, there must exist also a correspondingrefinement of the pointer observable P B with co-indexedeigen-projectors, and a subspace of the subspace spannedby all the calibration-condition-satisfying states | φ i i B ,the unit vectors in which are calibration-condition-satisfyingfor the refinement. Evidently, it is not always easy to findout if a given observable ¯ O A (cid:16) = f ( O A ) (cid:17) is actuallyovermeasured.We now turn to a class of cases in which, though O A itselfmay have refinements, its given measurement is certainly notan overmeasurement, i. e., it is a minimal measurement .By definition , one has a disentangled or uncorrelatedpremeasurement of an observable O A (cid:16) = P k o k E kA (cid:17) ifthere exists an eigen-sub-basis {| φ i kB : ∀ k, | φ i kB = F kB | φ i kB } of the pointer observable P B (cf (2b)), and for each k value only one eigenvector | φ i kB from this sub-basis ap-pears in the canonical subsystem-basis expansion form (13a)of the final premeasurement state. Equivalently, for everyinitial state | φ i i A of the object, the final state | Φ i f AB = U AB (cid:16) | φ i i A | φ i i B (cid:17) can be expanded in this sub-basis: ∀ | φ i i A : | Φ i f AB = X k | φ i f ,kA | φ i kB , (29 a )where | φ i f ,kA are vectors in H A . Then, equivalently ∀ k : | φ i f ,kA = h φ | kB | Φ i f AB , (29 b )the right-hand sides being partial scalar products (cf possiblyAppendix A in Herbut 2014b).The term ’disentangled’ (introduced by Schr¨odinger,1935) applies to the uncorrelated (disentangled) vectors | φ i f ,kA | φ i kB in the final state (29a).Disentangled premeasurement can be of the nondemolitionkind, when the result of measurement is preserved throughoutthe branches ∀ k : | φ i f ,kA = E kA | φ i f ,kA , or of the demolitionkind if the preservation fails for some k value. Among thelatter, there is an interesting possibility: the set of ’expansioncoefficients’ {∀ k : | φ i f ,kA } in (28a) may turn out orthog-onal ∀ | φ i i A . Then they determine another observableof subsystem A with sharp values in the selective final states F kB | Φ i f AB . || F kB | Φ i f AB || possibly independently of the initialstate.It is worth mentioning that, whereas most measurementproperties are valid (or not valid) for each branch separately;the property at issue is a premeasurement property, valid forthe entirety of the final state. Literature
This case is referred to as ”strong statecorrelation” in Busch, Lahti, and Mittelstaedt 1996 (subsec-tion III.3.2). A sufficient condition for disentangled premea-surement appears in nondemolition premeasurement of a complete observable O A : O A = P k o k | k i A h k | A , k = k ′ ⇒ o k = o k ′ . Then the preservation of E kA = | k i A h k | A requirement implies that in each complete-measurementbranch F kB | Φ i f AB only | k i A can appear (cf (20)): | Φ i f AB = P k | k i A ⊗ | φ i f ,kB . (The state vector | k i A , asany state vector, cannot be entangled.)In many discussions of the paradox of the emergence ofcomplete measurement in quantum mechanics, one confinesoneself to the simplified case of disentangled measurement.(Most likely, one keeps unitary dynamics as simple as possiblein order not to cloud the issue of the paradox.)Another sufficient condition for disentangled pre-measurement is non-degeneracy of all ’pointerpositions’ for obvious reasons (cf in section 2 the passagenext to the one in which relation (3) is). Literature
In Busch, Lahti, and Mittelstaedt 1996 thiscase is referred to as ”the pointer observable is minimal inthe sense that it is just sufficient to distinguish between theeigenvalues” of the measured observable (in passage next to(7) in III.2.3).
Let us next state and prove the basic property of disen-tangled measurements. It goes as follows.If a given premeasurement of an observable O A is dis-entangled , then the evolution operator U AB can be re-placed by a set of partial isomorphisms { U kA : ∀ k } eachmapping the range R ( E kA ) of the corresponding eigen-projector E kA (cf (1a)) into H A isometrically, i. e., lin-early and preserving scalar products. The replacement is per-formed so that : ∀ | φ i i A : | Φ i f AB ≡ U AB ( | φ i i A | φ i i B ) = X k ( U kA E kA | φ i i A ) ⊗ | φ i kB (30)is valid.One should note that the operators { U kA : ∀ k } , beingpartial isometries can be extended into a unitaryoperator in H A , and they map orthonormal vectors intoorthonormal ones. To prove the claim, let us take a complete orthonormaleigen-basis {| k, q k i A : ∀ k, q k } of O A , and expand | φ i i A = P k,q k h k, q k | A | φ i i A × | k, q k i A . Next, we evaluate the finalpremeasurement state using this expansion. U AB ( | φ i i A | φ i i B ) = X k,q k h k, q k | A | φ i i A × U AB ( | k, q k i A | φ i i B ) . According to the calibration condition (6), each final state U AB ( | k, q k i A | φ i i B ) must be invariant under the action of F kB on the one hand, and, according to the definition of dis-entangled premeasurement, only one of the given eigen-vectors | φ i kB , independent of the q k values, can appear in the ex-pansion. Hence, U AB , being unitary, maps, for each valueof k , the orthonormal sub-basis {| k, q k i A | φ i i B : ∀ q k } intosome other orthonormal sub-basis {| k, q k i ′ A | φ i kB : ∀ q k } in H A ⊗ H B . (Note that the state vectors | φ i kB , as anystate vectors, cannot be entangled.) It follows from the or-thonormality of {| k, q k i ′ A | φ i kB : ∀ q k } that also the set {| k, q k i ′ A : ∀ q k } is orthonormal.Each pair of orthonormal sub-bases determines a partialisometry ∀ k, U kA : ∀ q k : U kA | k, q k i A = | k, q k i ′ A . Hence,we can write U AB ( | φ i i A | φ i i B ) = X k,q k h k, q k | A | φ i i A × ( U kA | k, q k i A ) | φ i kB = X k [( U kA E kA | φ i i A ) ⊗ | φ i kB ] . The characterization of disentangled premeasurement by(30) is the essence of this kind of premeasurement. Theoperators { U kA E kA : ∀ k } are called state transformers because, in disentangled complete measurement with theresult o k , they transform the initial state | φ i i A into U kA E kA | φ i i A . || U kA E kA | φ i i A || (cf Herbut, 2004). Literature
The state transformers in relation (30) arealso called Kraus operators because they have been introduced(as far as I know) by Kraus (1983). The above claim is statedand proved as Theorem 3.2 in Lahti 1990. In Busch, Lahti,and Mittelstaedt 1996 claim (30) is presented in subsectionIII.2.3 .
As it is easily seen, any disentangled premeasurement is a nondemolition one if and only if ∀ k : U kA E kA = E kA U kA E kA ; (31)otherwise, it is a demolition premeasurement (cf (16b)). VII. IDEAL PREMEASUREMENTS
The simplest among disentangled premeasurements arethe ideal premeasurements . They can be defined inseveral equivalent ways. The ones that are most used are thefollowing. (I) ∀ k : U kA = I k , where I k is the identity operator in H A restricted to the range R ( E kA ) of E kA (cf (30)): ∀ | φ i i A : | Φ i f AB = X k ( E kA | φ i i A ) ⊗ | φ i kB . (32)Expansion (32) is obviously a twin-correlated canonicalSchmidt (or bi-orthogonal) decomposition (having themeasured observable and the pointer observable as twinobservables in mind, cf (20) and (26)). We may call itshortly the canonical-final-state definition of idealpremeasurement. (II) The definition that ensues may be called the
L¨uderschange-of -state one . It says that that the general finalstate ρ f A of the object in ideal premeasurement of an ob-servable O A (cid:16) = P k o k E kA (cid:17) is given by: ∀ | φ i i A : ρ f A ≡ tr B (cid:16) | Φ i f AB h Φ | f AB (cid:17) = X k E kA | φ i i A h φ | i A E kA . (33 a )One can rewrite (33a) as ∀ | φ i i A : ρ f A = X k h φ | i A E kA | φ i i A (cid:16) E kA | φ i i A h φ | i A E kA (cid:17). tr (cid:16) E kA | φ i i A h φ | i A E kA (cid:17) , (33 b )as seen if one applies, in the denominator, under-the-tracecommutativity, idempotency, and if one evaluates the tracein a basis containing | φ i i A .In this way it is seen that the final complete-measurement states are the pure states ∀ k, h φ | i A E kA | φ i i A > E kA | φ i i A . || E kA | φ i i A || , (33 c )where h φ | i A E kA | φ i i A are, of course, the probabilities in theimproper mixture (33b) (cf D’Espagnat 1976). (III) The third definition may be called the strongly ex-tended calibration condition . It reads: Every initial statethat has a sharp value of the measured observable does notchange at all in ideal premeasurement: | φ i i A = E ¯ kA | φ i i A ⇒ ρ f A = | φ i i A h φ | i A . (34 a )This definition of ideal premeasurement is in line with ourfirst definitions of general premeasurement (6) and nondemo-lition premeasurement (16b). Comparing (34a) with (16b), itis obvious that ideal premeasurement is a special caseof nondemolition premeasurement .One should note that the first definition of ideal premea-surement is not additional to that of general premeasurement;it defines the premeasurement in question by itself com-pletely. The second and third definitions, on the contrary,are additional requirements because premeasurement cannotbe defined only by changes in the object subsystem. We give an ’in-circle’ proof (to be distinguishedfrom a circular one) of the equivalence of the three definitions.The first implies the second : Substituting thefinal premeasurement state from (32) both in its ket and braforms, one obtains ρ f A ≡ tr B (cid:16) | Φ i f AB h Φ | f AB (cid:17) = X k,k ′ E kA | φ i i A h φ | i A E k ′ A (cid:16) tr B ( | φ i kB h φ | k ′ B ) (cid:17) = X k E kA | φ i i A h φ | i A E kA = RHS (33 a ) . The second implies the third : Substitut-ing | φ i i A = E ¯ kA | φ i i A in (33a) immediately lads to ρ f A = | φ i i A h φ | i A .The third implies the first . The third definition(34a) can be completed into | φ i i A = E ¯ kA | φ i i A ⇒ | Φ i f AB = | φ i i A | φ i ¯ kB (34 b ) with | φ i ¯ kB = F ¯ kB | φ i ¯ kB . (34 c ) Relations (34b,c) imply that each complete-measurementbranch E kA | φ i i A . || E kA | φ i i A || in the decomposition | φ i i A = P k E kA | φ i i A gives a disentangled component inthe final state. Hence, the final premeasurement state is dis-entangled. Having in mind I A = P k E kA , and (34b) with(34c), one obtains: | Φ i f AB = X k U AB (cid:16) ( E kA | φ i i A ) | φ B i i (cid:17) = X k || E kA | φ i i A || × U AB h(cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) | φ B i i i = X k || E kA | φ i i A || × (cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) | φ B i k = X k ( E kA | φ i i A ) | φ B i k (cf (32)). Literature
The change of state (33a-c) was postulatedby L¨uders (1951) (cf Messiah, 1961, or Cohen-Tannoudji,Diu, and Laloe, 1977). For this reason ideal measurementis sometimes called L¨uders measurement. If the measuredobservable is complete (no degenerate eigenvlues), then theL¨uders change of state coincides with that postulated by vonNeumann (1955) as his process 1. Therefore ideal measure-ment is sometimes called also von Neumann-L¨uders measure-ment.Among others, also the present author studied some deriva-tions of the L¨uders change of state Herbut (1969), Herbut(1974), Herbut (2007b). Also Khrennikov (2009a, 2008, and2009b) has paid great attention to ideal measurement.Incidentally, the postulate of L¨uders (33a-c) was called’minimal measurement’ in Herbut, 1969 to stress the fact thatit has introduced a difference with respect to von Neumann(1955), who restricted his discussion to maximal overmea-surement of degenerate observables, i. e., measurement ofcomplete observables whose function the initial observable is.
Ideal measurement is not very important in practicebecause it cannot be achieved in direct measurement. Thecharacteristic property (34a) contradicts the empirical factthat in every direct interaction at least one quantum ofaction has to be exchanged, i. e., some change in the statehas to be brought about. But this measurement is of greattheoretical importance. This will be elaborated elsewhere.
VIII. SUBSYSTEM MEASUREMENT AND DIS-TANT MEASUREMENT
In this section the main results of a recent article (Herbut2014c) are shortly reviewed to complete the picture on basickinds of measurement.Let us envisage a composite quantum system consist-ing of subsystems A and A and being initially inan, in general, entangled state | φ i i A A . Further, letus performs an arbitrary exact subsystem premeasurement,i. e.,,premeasurement of a subsystem observable, e. g., O A (cid:16) = P k o k E kA (cid:17) , on subsystem A . We call sub-system A the nearby subsystem, and subsystem A thedistant one. (The spatial connotation is here metaphorical;the terms are actually dynamical.)Let the premeasurement end up in the final premeasure-ment state | Φ i f A A B ≡ U A U A B (cid:16) | φ i i A A | φ i i B (cid:17) , (35 a ) U A being the unitary evolution operator of the dynami-cally isolated distant subsystem. Then the subsystem pre-measurement exerts no influence on the distant subsystem A that is ’untouched’ by the measurement interaction.Actually, even more is true. It is a known fact that anyunitary change to the nearby subsystem A , with or with-out an ancilla A , does not have any influence on the stateof a dynamically isolated subsystem A . (For details, seesection 4 in Herbut 2014c.)If, on the other hand, one considers the final state F kB | Φ i f A A B . || F kB | Φ i f A A B || , (35 b ) of a complete measurement, then there is, in general, a change in the state of the ’interactionally untouched’ dis-tant subsystem A , which is due to the entanglement in the initial state | φ i i A A .Then, the claim is that the final distant-subsystem statein question has the form: ρ f ,kA = U A (cid:16) ρ A ( E kA ) (cid:17) U † A , (36 a )where by ρ A ( E kA ) ≡ tr A (cid:16) | φ i i A A h φ | i A A E kA (cid:17). tr (cid:16) | φ i i A A h φ | i A A E kA (cid:17) (36 b )is denoted the conditional state of the distant subsystem A under the condition of the ideal occurrence of the event E kA in the composite-system state | φ i i A A instanta-neously at the initial moment. This interpretation of (36b) isobvious if one rewrites the relation (utilizing idempotency andunder-the-partial-trace commutativity (1c)) in the equivalentform ρ A ( E kA ) =tr A (cid:16) E kA | φ i i A A h φ | i A A E kA (cid:17). || E kA | φ i i A A || , (36 c )utilizing the trivial equalitytr (cid:16) | φ i i A A h φ | i A A E kA (cid:17) = || E kA | φ i i A A || (37)and having in mind relation (33c) for ideal measurement.(The lengthy proof of claim (36a,b) is given in the Appendixof Herbut 2014c.)On account of the fact that the unitary interaction opera-tor U A B has dropped out of the final expression (36a,b),it is an important corollary of the result that, whateverthe kind of complete measurement performed on A , thechange caused to A is the same as if the subsystemmeasurement were ideal .The change of state due to ideal complete measure-ment is easily evaluated on account of relation (32) andthe probability reproducibility condition || F kB | Φ i f A A B || = || E kA | φ i i A A || (cf (10)): F kB | Φ i f A A B . || F kB | Φ i f A A B || = U A E kA | φ i i A A | φ i kB . || E kA | φ i i A A || . (38)As to the state of the distant subsystem A , one has: ρ f ,kA ≡ tr A B h(cid:16) F kB | Φ i f A A B . || F kB | Φ i f A A B || (cid:17)(cid:16) h Φ | f A A B F kB . || F kB | Φ i f A A B || (cid:17)i . Further, substituting here (38) and taking into account (37),one finally obtains ρ f ,kA = U A h tr A (cid:16) E kA | φ i i A A h φ | i A A E kA (cid:17). || E kA | φ i i A A || i U − A . (39) Now we come to distant measurement . We assume thatinitially we have a twin-observables relation ∀ k : E kA | φ i A A = E kA | φ i A A , (40)where E kA are the eigen-projectors of a distant-subsystemobservable O A (cid:16) = P k ¯ o k E kA (cid:17) . (The twin-observablescriterion for nondemolition observables (20) is an example for(40)).)Making use of (40), relation (39) implies ρ f ,kA = U A h tr A (cid:16) E kA | φ i i A A h φ | i A A E kA (cid:17). || E kA | φ i i A A || i U − A . (41)In view of (33c) for ideal measurement, relation (41) givesthe final distant-subsystem state due to the instantaneousideal occurrence of the (twin) distant subsystem event E kA in the state | φ i i A A at the initial moment.Comparing (39) and (41), one can see that both the in-stantaneous ideal occurrence of the event E kA on the nearbysubsystem and that of the event E kA on the distant subsys-tem lead to the same final state ρ f ,kA of the distant subsys-tem A . In other words, one can say that the instantaneousideal occurrence of E kA in the state | φ i i A A at the ini-tial moment gives rise to the instantaneous ideal occurrenceof the distant twin event E kA in the same state at the samemoment.This was introduced and called distant measurement along time ago (Herbut and Vujiˇci´c 1976). But now we havethe recent (above mentioned) result that any exact nearby-subsystem complete measurement leads to the same final stateof the distant subsystem as nearby-subsystem ideal completemeasurement. Hence, one can say that, if twin observables areinvolved (cf (40)), any exact subsystem measurement givesrise to a change of state in the opposite subsystem which isthe same as caused by instantaneous ideal measurement of thecorresponding distant twin event at the initial moment in thegiven composite-system state | φ i i A A with entanglement.We can keep the term ”distant measurement” for this moregeneral measurement.One should note that as long as the nearby-subsystemcomplete measurement is ideal, it gives the same change ofthe global (composite) state as the analogous ideal completemeasurement on the distant subsystem. If the formercomplete measurement is more general than ideal, then itseffect coincides with that of ideal complete measurement onthe distant subsystem only locally on the distant subsystem.Distant measurement plays a natural role in the paradoxi-cal so-called Einstein-Podolsky-Rosen (EPR) phenomenon. Ifa bipartite state vector | φ i i A A allows distant measurementof two mutually incompoatible observables (non-commutingoperators) O A and ¯ O A , [ O A , ¯ O A ] = 0 , then wesay that we are dealing with an EPR state (following theseminal Einstein-Podolsky-Rosen 1935 article).A very simple example of an EPR state is the well known singlet two-particle spin state | φ i i A A ≡ (1 / / (cid:16) ( | + i A | −i A − | −i A | + i A ) (cid:17) , (42) where + and − denote spin-up and spin-down respec-tively along any fixed axis . One can see, in obvious no-tation, that the oppositely oriented spin-projection operatorsare twin observables: s ~k ,A | φ i i A A = s ~ ( − k ) ,A | φ i i A A , (43)where the unit vector ~k is arbitrary. One can suitablychoose ~k either along the positive z-axis or along the pos-itive x-axis.If one performs an ideal complete measurement of the spinprojection along the z-axis in a subsystem measurement onthe nearby subsystem A , it is easily seen that the compos-ite final state is, e. g., | z, + i A | z, −i A , and the (this timepure) state of the distant subsystem A is | z, + i A . Anal-ogously, one can obtain by distant complete measurement, e.g., | x, + i A .Einstein et al. pointed out that in transition from (42) tothe final state of distant complete measurement , the men-tioned result of opposite spin projection along the same axiswas brought about in a distant action without interaction (a ”spooky” action), which could not be reconciled with ba-sic physical ideas that reigned outside quantum mechanics.(More in section 6.2 of Herbit 2014b or in Bohm 1952, chap-ter 22, section 15. and further.)One can find articles in the literature in which all entangledbipartite states are called EPR states. Perhaps because anyentanglement allows distant complete measurement (withintuitively ”spooky” action). VIII. CLASSIFICATION
In the classification that follows we disregard overmea-surements because in them the measured observable is afunction of a finer observable, and the measurement is just aconsequence of the minimal measurement of the latter.
Five kinds of final complete-measurement compo-nents of minimal measurements can be distinguished,and they are displayed in the ARRAY below, and denotedby M x , y . They are (in reading order on the Diagram):1) the ideal ones, M , ; a , characterized by preserving the sharp-value component states (cf (34a));2) the nondemolition non-ideal disentangled ones,M , ; b , which do not preserve the sharp-value componentstates, but they preserve the sharp values themselves (cf(16b)), and they map the initial pure states | φ i i A intopure states U kA (cid:16) E kA | φ i i A . || E kA | φ i i A || (cid:17) , U kA = I A ; (44)3) the nondemolition entangled ones, M , , whichalso preserve the mentioned sharp values , but they map the initial pure states into mixtures (improper or second-kind ones cf D’Espagnat, 1976) ρ f A ≡ tr B h(cid:16) U AB ( | φ i i A | φ i i B ) (cid:17)(cid:16) ( h φ | i A h φ | i B ) U † AB (cid:17)i . (45)In other words, redundant entanglement is created be-tween the subsystems A and B (’redundant’ regardingthe premeasurement).
4) the demolition disentangled ones, M , , which donot preserve the sharp values , but they do map the intialstates | φ i i A into pure states given by (44); and finally5) the demolition entangled ones, M , , which neither preserve the sharp values , nor do they mapthe pure states into pure states. They map the formerinto improper mixtures given by (45), i. e., they create redundant entanglement . ARRAY (with rows and columns) disentangled : entangled : nondemolition : M , ; a , b M , demolition : M , M , To understand the
ARRAY , one must take into accountthat a kind of measurement M x , y is defined by the row x and the column y . In the special case of M , , onehas two kinds: M , ; a , which is ideal measurement , and M , ; b , which is non-ideal .One can utilize the classification into M x , y completemeasurements also for premeasurements if they are homoge-neous in the sense that all the final complete-measurementcomponents of the premeasurement belong to one and thesame kind M x , y of the 5 complete measurements. Thosethat lack such homogeneity can be classified by the worst finalcomplete-measurement component, ”worst” meaning thatit is farthest from the beginning in reading order on the array. IX. SUMMING UP THE EQUIVALENT DEFINI-TIONS
Let us sum up that we have obtained defini-tions of general premeasurement in section 2 : the ’statistical form’ of the calibration condition (4), its ’invariance form’ (6), its ’strong invariance’ form (11); the ’probability reproducibility condition’ (10), the ’basic dynamical’ characterization (8), the ’basis-dynamical’ characterization (9a), and, as theother side of the coin, the ’subspace-dynamical’ criterion (9b),and finally the ’canonical subsystem-basis expansion’ criterion (13a)with (13b).To the author’s knowledge, new are 3), 5) and 7).Let us sum up the results of section (4) . We have defined nondemolition premeasurement by additional require-ments in 10 equivalent ways . We have extended the 7 equiva-lent definitions of general premeasurement, and thus we haveobtained: the ’extended statistical calibration condition’ definition(4) + (16a), the ’extended invariance calibration condition’ criterion(6) + (16b), the ’extended strong invariance’ characterization (11) +(17), the ’extended probability reproducibility condition’ (10)+ (24), the ’extended dynamical’ definition (8) + (18), the ’extended basis-dynamical’ characterization(9)+(19a) or equivalently (19b) by itself; further, the’extended subspace-dynamical’ condition (19c); the ’canonical subsystem-basis expansion’ criterion (25a)with (25b), and the ’twin-corre;ated canonical Schmidt decomposition’definition (26).The ’canonical subsystem-basis expansion’ characterizationof general premeasurement (13a) with (13b) was extended intwo ways (items 7) and 8)).Two additional conditions without a counterpart in generalpremeasurement were given: The Pauli ’definition of repeatability’ (23a) or (23b), and the ’twin-observables relation’ (20).As far as the author can tell, new are 3) and 8).Characterization in item 10) had the consequences of lackof coherence (21a) in the final object state with respectto the measured observable O A and analogously (21b)regarding the final measuring-instrument state and thepointer observable P B .Finally, let us sum up the three equivalent definitions ofideal premeasurement: The canonical-final-state definition (32), the L¨uders-change-of-state definition (33a-c), and the strongly extended calibration condition definition(34a-c). X. CONCLUDING REMARKS
Let us conclude this review by a few remarks on someaspects that have been omitted . What has been covered isoutlined at the very beginning of the article. (I)
Complete measurement was viewed only as a con-stituent of premeasurement.
Literature
A quote from the first passage of the lastchapter of the book on quantum mechanics by Peres (2002)reads:”In order to observe a physical system, we make it interactwith an apparatus. The latter must be described by quantummechanics, because there is no consistent dynamical schemein which a quantum system interacts with a classical one. Onthe other hand, the result of the observation is recorded by theapparatus in a classical form ...”Transition from quantum-mechanical description of themeasuring instrument to that of classical physics Peres calls”dequantization”. It seems to be way out of the scope ofunitary quantum measurement theory. See also Hay, andPeres, 1998. ng other things, the expounded theory allows general-ization (extension) in a number of its basic concepts thathave not been covered. (II)
Both the initial state of the object | φ i i A and thatof the measuring instrument | φ i i B allow generalization to general states (density operators) ρ i A and ρ i B respec-tively.It is to be expected that, as it is usually the case withdensity operators, the relations valid for general states caneasily be obtained from the corresponding relations valid forpure states when one decomposes the general states into pure ones. Besides, there is so-called purification: Every densityoperator can be viewed as the reduced density operator of abipartite pure state. Literature
As to general states, von Neumann hasproved in his famous no-go theorem (von Neumann, 1955,first part of section 3. in chapter VI) that unitary measure-ment theory cannot explain complete measurement, i. e., thefact that the individual systems end up in one branch F kB | Φ i f AB . || F kB | Φ i f AB || (46) of the final premeasurement state | Φ i f AB (if the completemeasurement is a minimal one, cf section V). We outline theclaim of this theorem.In a purely pure-state measurement theory, as in the presentreview, it is clear that the coherence | φ i i A = P k E kA | φ i i A with respect to the eigenvalues o k of the measured observable O A (cid:16) = P k o k E kA (cid:17) in the initial state | φ i i A of the objectis not destroyed; it is only transformed into coherence | Φ i fAB = X k F kB | Φ i fAB (47 a ) with respect to the pointer observable P B (cid:16) = P k p k F kB (cid:17) in the final premeasurement state | Φ i fAB (cf (15) for theseparate evolution of each branch). Though von Neumann didnot expound a detailed theory of premeasurement, he did notconsider the measurement paradox in the pure state case be-cause the unitary evolution operator takes the pure compositestate | φ i i A | φ i i B into a pure state | Φ i fAB , and there is noway how coherence could disappear.What von Neumann did was to assume that the initial stateof the measuring instrument is in a mixed state ρ i B ; a propermixture due to incomplete knowledge about the state. Thenone might conjecture that this mixture would lead to the mix-ture ρ f AB = X k h φ | i A E kA | φ i i A × (cid:16) F kB | Φ i f AB . || F kB | Φ i f AB || (cid:17)(cid:16) h Φ | f AB F kB . || F kB | Φ i f AB || (cid:17) (47 b ) (cf (14)), which is observed in the laboratory. Von Neumanndisproves in detail this conjecture.Von Neumann’s no-go theorem has been often consideredas a proof of the measurement paradox, i.e., of the puzzlewhy unitary quantum mechanics does not furnish separatebranches (terms in (47b)) for the individual measured objects.In particular, it is puzzling how a deterministic theory, as theone reviewed in this article, can lead to the indeterministicbranches for the individual objects, which becomes determin-istic (described by (47b)) on the ensemble level.No other conceivable way (than the one treated in vonNeumann’s no-go theorem) how one could obtain (47b)within unitary dynamics of measurement was seen till Everett1957 and 1973 and De Witt 1973 shocked the world byhypothesizing that the separate branches of premeasurementmight become parallel worlds and that we, and everythingthat we know, somehow become one of these worlds. (III) The discrete observables O A = P k o k E kA to whichthis review has been confined can be generalized to include also observables that contain a continuous part in theirspectrum, in particular purely continuous observables asposition and linear momentum. Literature
Busch and Lahti (1987), and also Busch,Grabowski, and Lahti (1995a) have investigated this subject. (IV)
The concept of observables that are measured canbe extended from ordinary ones to ones that are expressedas POV (positive-operator valued) measures (cf Busch,Lahti, and Mittelstaedt, 1996). The PV (projector-valued)measures corresponding to ordinary observables are specialcases.
Literature
One should read section 3.6 in Part I ofVedral (2006). Generalized observables (POV measures) aredescribed in detail in the book by Busch, Grabowski, andLahti (1995b). Also the study in Busch, Kiukas, and Lahti(2008, section 3.), on connection between POV and PVmeasures based on the Neumark (1940) dilation theorem isrecommended. Also the article Peres (1990) is relevant andinteresting. (V)
The eigen-projectors E kA of an ordinary observablecan be generalized by positive operators, the physical meaningof which is ’effects’. The projectors, with the meaning ofevents or properties or quantum statements, are special cases.In the most general case, which is studied in Busch, Lahti,and Mittelstaedt 1996, premeasurement is defined by theprobability reproducibility condition, and generally this isnot equivalent to the calibration condition. The presentinvestigation was undertaken in the hope that restrictionto the physically most important case of ordinary discreteobservables will help to delve deeper into the subject, obtainmore results, and see them with more clarity and simplicity. (VI) The theory presented in the present review is purelyalgebraic. The question of feasibility of the particularpremeasurement procedures is not discussed.
Literature
Concerning this important aspect of mea-surement, one should read Cassinelli and Lahti (1990) andsection 6. in chapter III of Busch, Lahti, and Mittelstaedt1996. One may also learn about the Wigner-Yanase-Arakitheorem, which claims to set serious limitations on what canbe measured exactly. One may read the critical short articleOhira and Pearle (1988), and the references therein. (VII)
Measurements cannot be performed without preparation , a procedure that brings about the initialstates | φ i i A . Literature
One can read about preparation in section8. of chapter III in Busch, Lahti, and Mittelstaedt 1996 orin Herbut 2001. (VIII) Information-theoretical aspects of measure-ment theory have not been discussed in the present revieweither.
Literature
Information gain in various kinds of mea-surement is investigated in Lahti, Busch, and Mittelstaedt1991. One can read about this aspect in section 4. ofchapter III of Busch, Lahti, and Mittelstaedt 1996. Entropic, information-theoretical and coherence aspects of measurementwere studied in a number of articles by the present authorHerbut (2002, 2003a, 2003b, and 2005). I am grateful to my onetime associates: the late MilanVujiˇci´c, further Milan Damnjanovi´c, Igor Ivanovi´c, and MajaBuri´c for helpful and inspiring discussions on measurementtheory.
Appendix Proof of an auxiliary algebraic certaintyclaim We prove now the general claim that the following equiva-lence is valid for a pure state | ψ i and an event E : h ψ | E | ψ i = 1 ⇔ | ψ i = E | ψ i . Evidently h ψ | E | ψ i = 1 ⇒ h ψ | E c | ψ i = 0 , where E c ≡ I − E is the ortho-complementary projec-tor and I is the identity operator. Further, one has || E c | ψ i|| = 0 , E c | ψ i = 0 , and E | ψ i = | ψ i as claimed.The inverse implication is obvious. REFERENCES
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